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Elementary Theory of the Optical Properties of Solids
Elementary Theory of the Optical Properties of Solids FRANKSTERN* United States Naval Ordnance Laboratory, White Oak, Silver Spring, Maryland
...... . . 300 . . . . . . . . . . . . . . 300 1. Introduction. ......................... 2. Maxwell’s Equations. ....................................... 301 3. Conservation Equations. ...................... ..... . . . . . . . 304 305 4. Plane Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Inhomogeneous Case ........................ 307 6. Energy Density.. . . ........................ 308 7. Sources, Potentials, and the Wavelength-Dependeht Dielectric Constant. . 310 8. Alternative Treatments of the Magnetization. . . . . . . . . 11. Reflection and Transmission. . . . . . . . . 9. Boundary Conditions. . . . . . . . . . . . . . . . . . . . 10. Amplitude Reflection Coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 . . . . . . . . . . . . . . . . . 317 11. Snell’s Law and Generalizations. . . 12. Energy Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 13. Reflectivity. .............................. 14. Total Reflection.. ......................... 15. Transmission of Isotropic Radiation. . . . . . . . . . 16. Thin Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 111. Dispersion Relations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 328 17. Dispersion Relations for the Dielectric Constant.. . . . . . . . . . . . . . . . . . . . . . 18. Dispersion Relations for the Index of Refraction.. . . . . 19. Qualitative Consequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 20. Phase Shift Dispersion Relation. . . . . . . . . . 21. Numerical Procedures for Calculating the P 22. Extrapolation Procedures. . . . .................................... 338 23. Derivation of Sum Rules. . . . .................................... 340 IV. Optical Properties of Simple Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 24. The Effective Field. . . . . . ..... ... . 342 A. The Free-Electron Gas.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 25. General Considerations. . ................................ 344 26. Conductivity and Mobilit ................................ 346 27. Optical Properties in Various Frequency Ranges.. ..................... 347 28. Reflectivity Minimum. . . . . . . . . . . . . . . . ...... 350
I. Electromagnetic Preliminaries. . . . . . . . . . . . . .
.
* Present address: IBM Thomas J. Watson Research Center, Yorktown Heights, New York. 299
300 29. Energy Density.
FRANK STERN
...........
............................
B. Optical Modes in Ionic Crystals. ..........................
30. Phenomenological Discussion. . . . . . . . . . . . . . . . . . . . 31. Atomic Theory of Optical Modes. ...................... 32. Longitudinal and Transverse Modes; Effective Charge.. . . . . . . . . . . . . . . . 33. Reflectivity. .................................. .............. 34. Free-Carrier Effects.. .......................... .............. V. Quantum Theory of Absorption and Emission.. ..........................
350
356 359 361 364
.......... 37. Optical Absorption for a Simple Model. 38. Free-Carrier Absorption. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39. Oscillator Strengths .............................. VI. Wavelength-Dependent Dielectric Constant of a Free-Electron Gas 40. Introduction. ...... ........................ ................... 41. Calculation of the Co 42. Verification of the Su ........................................
370 374 375
45. Surface Impedance and Anomalous Skin Effect.. ...................... 46. Real Crystals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII. Interactions of Charged Particles.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47. Longitudinal Dielectric Constant of a Free-Electron Gas .......... 48. Screening of a Static Point Charge.. . . . . . . . . . . . . . . . . . 49. Energy Loss of a Moving Charged Particle. .......... 50. Forces between Charged Particles.. .................................. Acknowledgments............ .....................................
389
383
394 394 406 408
1. Electromagnetic Preliminaries
1. INTRODUCTION One of the most powerful tools for studying the properties of solids is the measurement and analysis of their optical properties. In this work some of the results required for such an analysis are presented, with emphasis on the detailed development of simple models. We shall express many of the results in numerical form, and this in part has dictated the use of mks units throughout. The treatment is elementary in the sense that no physics beyond Maxwell’s equations and simple quantum mechanics is used. Our detailed considerations are limited to materials for which the usual optical constants are scalars, i.e., to isotropic substances and to cubic crystals, and to the case of zero magnetic field other than that of the electromagnetic radiation itself. We neglect the distinction between
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
301
energy and free energy, a procedure which is strictly correct only a t absolute zero.’ Most of the material in the first two parts is a review of standard results of electromagnetic theory, and serves mainly to establish the notation used here, and to list results for later reference. Sections 7 and 8, which give a formal introduction to the wavelength-dependent dielectric constant, may be somewhat less familiar. Dispersion relations as applied to the analysis of optical properties are discussed in Part 111. We make frequent use of dispersion relations in later chapters. I n Part IV we summarize some of the classical results for two very important physical systems, the free-electron gas and the optical lattice vibrations in ionic crystals. These systems are sufficiently simple that detailed results can be obtained Yery easily, yet realistic enough that the results give quite a good representation of a t least some of the properties of real solids. Some of the more basic results of the quantum theory of absorption and emission of radiation are given in Part V. N o attempt is made to derive specific results for real solids, since this requires detailed calculations beyond the scope of this work. In the last two parts we consider the transverse and longitudinal wavelength-dependent dielectric constants for a free electron gas, for which detailed results can be obtained relatively easily. These results are applied in a number of examples, including the anomalous skin effect, the screening of a static point charge, and the energy loss of moving charged particles. 2. MAXWELL’S EQUATIONS
The theoretical framework for optical properties is given by Maxwell’s equations, which we give in mks units2:
V - B = 0,
(2.la)
V - D = p,
(2.lb)
V x E
=
-dB/dt,
(2.lc)
V x H
=
J
(2.ld)
+ aD/at,
where p is the externally introduced charge density and J is the current density including both the conduction current Jcond and the externally 1
*
H. Frohlich, “Theory of Dielectrics,” p. 9. Oxford Univ. Press, London and New York, 1949. J. A. Stratton, “Electromagnetic Theory.” McGraw-Hill, New York, 1941.
302
FRANK STERN
TABLE I. QUANTITIES ENTERINQ IN MAXWELL’S EQUATIONS Quantity
mks Units
E (electric field) D (electric displacement) p (charge density) J (current density) B (magnetic induction) H (magnetic field)
1 volt/m 1 coul/m* 1 coul/ms 1 amp/m2 1 weber/m* 1 amp-turn/m
F~
Practical units = 10-2 volt/cm = lo-‘ coul/cmz = 10-6 coul/cma = 10-4 amp/cma = 104gauss = 47 X 10-0 oersted
= 4u X 10-7 henry/m = 1.25664 X 10- henry/m
eo = ( M O C * ) - ~ = (go/eo)i
=
farad/m
8.8542 X
FOC =
( c o c ) - ~= 376.73 ohms
introduced current Jmt. The mks units for the quantities appearing above are given in Table I. In the presence of matter Maxwell’s equations can be written: (2.2a)
V*B = 0, e0V.E = p
v
x
E
+ aB/at
=
V x B - c-*dE/dt
We used the relations
- v.P,
0,
= po(Jex.t
+
Jcond
(2.2b)
+ aP/dt + V
(2.2c) x
M). (2.2d)
,
+ P,
H pO-lB - M, (2.3) where P is the polarization (the electric dipole moment per unit volume), and M is the magnetization (the magnetic moment per unit volume) of the matter present; po and e0 have the values given in Table I. The charge density can be thought of as the sum of an externally introduced charge density p and an effective charge density -7.P.Similarly, the current density contains the external current, the conduction current, and V x M. The absence of source terms in the other additional terms dP/dt two equations indicates that the electromagnetic field can be thought to originate entirely in changes and currents, since there are no free magnetic charges in nature.* When the fields vary with time there will generally be a phase shift
D
=
+
W. I(.H. Panofsky and hi. Phillips, “Classical Electricity and Magnetism,” p. 128. Addison-Wesley, Reading, hiassachusetts, 1955; see also P. -4.M. Dirac, Phys. Rev. 74, 817 (1948); H. A. Wilson, ibid. 76, 309 (1949); E. M. Purcell, G. B. Collins, T. Fujii, J. Hornbostel, and F. Turkot, ibid. 129, 2326 (1963).
ELEMENTARY OPTICAL PROPERTIES O F SOLIDS
303
between the polarization and the electric field. It is simplest in such cases to represent the fields as the real parts of complex quantities. For example, we represent a plane wave b ~ 3 ~ :
E
=
Re
e = Re [6:0exp (&or - i w t ) ] =
Re [go exp (zlrl.r
- id)]exp ( -kz*r),
(2.4)
where w = 21v is the angular frequency of the field and is the polarization vector. The symbol Re means that the real part is to be taken: Re A = +(A A*). I n general we will have a complex propagation vector f = kl zk~,where klis perpendicular to the planes of constant phase, while kg is perpendicular to the planes of constant amplitude. We can now easily include the possibility of a phase shift between D and E by defining
+ +
D
=
Re
m0exp (&-r - i w t ) ] = Re 6,
fi = &,
(2.5)
where fi, fi, and i may all be complex. Note that D and E, which are real quantities, are no longer proportional to each other a t arbitrary times. One often omits the symbol Re in writing the expressions for the fields. This does not lead to difficulties as long as relations linear in the fields are considered, since Maxwell’s equations are themselves linear in the fields. The current Joond which appears as a source term in Maxwell’s equation (2.2d) includes the current which flows in a conductor in the presence of an electric field, as given in the linear range by Ohm’s law Joond = uE. Furthermore, the displacement in phase with the field is given by D = EE, where both u and e are real. We can combine these terms if we note that for fields whose time variation is given by Eq. (2.4), Jcond
+ dD/dt = uE + =
e
dE/dt = Re [(u
Re { - i w [ c
- iwe)g]
+ i(u/w)]g].
(2.6)
In everything which follows we shall, unless otherwise indicated, combine the effects of the conduction and displacement currents in a complex dielectric constant X defined by X ( w ) = i(.)/€O, i(0) =
E(O)
+ i[u(w)/w]
=
€1
+ ies.
(2.7a) (2.7b)
Where it is desirable to distinguish complex quantities from real quantities, we indicate complex quantities with a tilde, and their real and imaginary parts by subscripts 1 and 2; for example, 6 = a1 ia~.
+
304
FRANK STERN
Alternatively, we can represent Eq. (2.6) as the real part of a complex current Z (a)8, where ".w)
- iwe(w).
= .(Lo)
(2.8)
Only at low frequencies will the conductivity equal the dc value uo. We uniformly use the time variation exp (-id). If we had used then i would be replaced everywhere by -i. In applying exexp (id), pressions taken from various sources, one should be careful to check which time variation was used. For the magnetic fields we can write by analogy:
H
= Re H =
Re [Hoexp
B
=
Reg,
P
=
fir,
p
=
- id)],
(&or
+ i/& =
(2.9)
where p is the permeability and Z,,, is the relative permeability of the medium. 3. CONSERVATION EQUATIONS
The fact that electric charge can be neither created nor destroyed is expressed by the conservation equation:
+ ap/at = 0.
V -J
(3.1) This equation is easily obtained from Maxwell's equations (2.lb) and (2.ld). A second conservation equation can be deduced from Maxwell's equations if we use the vector identity:
V*(E x H) = H.(V
x
E) - E*(V x H).
(3.2)
) (2.ld) we find: Substituting from Eqs. ( 2 . 1 ~ and
V. (E x H)
+ E-jext+ E-(jcond + aD/at) + H-(aB/at) = 0.
(3.3)
When we deal with fields varying sinusoidally with time we can make use of the easily proved relation#: Re A*Re B = Re A x Re B
=
4 R d - 8 * = 4 Re A**B
3 Re
(A x 8*) = + Re (A* x 8 )
(3.4a) (3.4b)
where the bar over the quantities on the left means that their time average 4
See p. 136 in Stratton?
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
305
over one period of the field is to be taken. Thus we find that for sinusoidally varying fields :
+
E*(Joond dD/at)
+ H-(dB/dt) = +(wezfi*g* + o p $ H * H * ) .
(3.5)
We have already identified WE^ with the electrical conductivity in Eq. (2.7). Thus $ ~ & - f i * gives the power dissipation due to Joule heating and, by analogy, 3wp2H*H* gives the magnetic power dissipation. I n both cases the power dissipation arises from the phase difference between the field and the polarization or magnetization, since it is the imaginary part of I or p which appears. This connection between phase shifts and power dissipation is quite general.6 The term E. Jext in Eq. (3.3) can be associated with the power dissipated by externally introduced currents. Note that since we are dealing here with sinusoidally varying fields, the time-averaged energy density in the field is constant. The power dissipated in the material is supplied by the external sources which maintain the fields. Conservation of energy requires that the first term in Eq. (3.3) give the net outflow of energy per unit volume. By analogy with Eq. (3.1), we see that S = E x H (3.6) is the power flow across a unit area normal to S. S is called the Poynting vector. 4. PLANE WAVES
For the plane waves defined in Eqs. (2.4) and (2.9), we find that in a homogeneous material containing no externally introduced charges or currents Maxwell’s equations can be written:
pf.H
=
0,
(4.la)
af.B
=
0,
(4.lb)
f xE
=
WpI3,
(4.1~)
ii x H
= -w&,
(4.ld)
If the real and imaginary parts of f are parallel to each other (a special case of this is a vanishing imaginary part) these equaiions show that for electromagnetic waves in isotropic media g, E, and H form a mutually orthogonal set of vectors. 5
C. Zener, “Elasticity and Anelaaticity of Metals.” Univ. of Chicago Press, Chicago, Illinois, 1948.
306
FRANK STERN
From Eq. (4.1) we find:
i;
x
(f
x
0) = w p f
x H = -dp&
=
(f.f)f- (f.f)B
=
-(f*f)B. (4.2)
Thus
f - f = pi02 =
[ kl
= "K"Kn'w2/cz = (n
l2 - I k2 l2
+ ik)W/C2
+ 22kak2.
(4.3)
We have introduced the index of refraction n, and the extinction coefficient k. The real and imaginary parts of Eq. (4.3) are sufficient to determine I kl I and I ka from n, k, and the angle between kl and k2.If that angle is zero, we have
I
1 kl I
If, in addition,
K~
= ~u/c;
1 k2 I
= ~u/c.
(4.4)
is real, we have: ?z
k
(1 2 I + Ki)]"', = [(~m/2)(1 2 I - K1)I1/'. = [(%/2)
(4.5)
The time-averaged energy flow is given by:
8
=
4 Re (g* x
H)
=
Re
[B*
x
(ff
x B)/2wp]
[(g**fi)ff- (g**ff)&] ( ~ C C ~ / ~ K(f*.E)k, ~ ) if COB (kl, kz) = 1.
= ( 2 ~ ~ w p o )Re -~
(4.6~~)
=
(4.6b)
We can express this nlimerically as:
9 (watts/cm2)
= 1.3272 X 1 0 - * ( n / ~ [I~ )B
I
( v o l t / ~ r n ) ] ~ ~( ~4 . 6 ~ )
Here 2, is a unit vector in the direction of kl_and kz, which are assumed to be parallel to each other. Substituting for E from Eq. (2.4) we find in this case :
8=
(nce0/2rc,)(80*-8~)k exp (-2k2-r).
(kl I / k2).
(4.7)
Thus the flux is attenuated by a factor exp ( - 2 J k2 I d ) on traveling a distance d through the material. The coefficient of -d in the exponent is called the absorption coefficient, and is given by .(w)
= 2
1 k2 I
= 2kw/c = 4nk/X,
(4.8)
where h is the wavelength in vacuum of electromagnetic radiation of angular frequency w, and k is the extinction coefficient for that frequency. When the real part of the dielectric constant is negative, k and a! can be quite large, corresponding to strong attenuation of the light beam, even
307
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
though K Z , which measures the power dissipation in the medium, is small. The attenuation in this case arises because the incident power is strongly reflected. I n metals the distanre k2 1-l = c/kw is called the skin depth 6. From Eq. (4.5) we see that when 1 c1 , << € 2 , i.e., wkl << U / E ” , we have: 6
(4.9 1
= [tL/(puw)]1’2.
Numerical expressions for a and 6 are given in Part IV. 5 . ISHOMOGESEOUS CASE Let us now consider the energy flow in the more general case in which
kland k2 are not parallel. Such waves are sometimes called inhomogeneous
waves. A thorough discussion of the inhomogeneous case is given in the article by Konip in the old Handbuch der Physilc. I n order to discuss Eq. (4.6a) in the inhomogeneous case, we must introduce a more explicit representation for the polarization. Let us define Such a choice can always be made for a field of the form (2.4). El and Ez are the directions of the major and minor axes of the ellipse traced by the end point of E during a complete period. For a linearly polarized wave, Ez will be zero. The phase factor exp (i&) will drop out in any expression in which E is multiplied by E*. We can cause it to vanish by a n appropriate choice of origin on our time scale, and will therefore ignore it henceforth. From the real and imaginary parts of Eq. (4.lb) we deduce that if # 0:
Elski - E2.kz
0;
=
E1*k2
+ Ez*k,
=
0.
(5.2)
Thus we can write: -
w
E*.k
= =
+ (E* - E)].k 2E2.kz + 2iE1*k2. [E
-
4
-
=
-2iE2.G
=
2Ez.kz - 2iEz.kl (5.3)
In the homogeneous case (kl , I kz) we have E * k * = 0, E*.k write : Re [- (E*.G)E]
5
=
0. We now
+ iEl.k2)(El + 2X2)]
=
- 2 Re [(E2*k2
=
-2[(Ez*h)Ei - (Ei.kz)Ez]
=
2k2 x (Ep x El).
(5.4)
W . Konig, in “Handbuch der Physik” (H. Geiger and K . Scheel, eds.), 5’01. 20, p . 194ff.Springer, Berlin, 1928.
308
FRANK STERN
Substituting in Eq. (4.6a) we find for the time-averaged energy flow in the inhomogeneous case :
S=
(21c,wpo)-’[(E*~E)k1
+ 2k2 x
(Ez x El)].
(5.5)
6. ENERGY DENSITY The conservation equation [Eq. (3.3)] for the energy in the electromagnetic field is the basis for the conventional electrostatic result that the energy density is +E.D 4H.B. This result cannot be carried over directly to the electromagnetic case if e or p are functions of frequency. The extension to the dispersive case has been given in a number of places708and is given below. At a fixed point in space, let the electric field vary with time as
+
E(t)
=
Re [EoG(t) exp (-iwot)]
=
Rea(t),
(6.1)
where the envelope function G ( t ) is a slowly varying function of time which we can write in the form ~ ( t= )
/
m
g”(w)
-m
g”(w) =
(2~)-’
m
J-m
exp (-iwt) dw,
(6.2a)
G(t) exp ( i d ) dt.
(6.2b)
The Fourier integral for E(t) is ~ ( t= )
Im ~ ( w exp )
-m
(6.3)
( - i w t > dw,
where B ( w ) = &(w - uo). The Fourier expansion for B ( t ) is similar to Eq. (6.3), with f i ( w ) = Z ( w ) B ( w ) . Thus, on taking the time derivative, we obtain
1
m
dD((t)/dt
=
-m
uZ(w)g”(w
If we introduce a new variable w’ c i i j ( t )/dt
=
-iEo
exp ( --coot)
= w m
J-m
(wo
-
-
UO) exp
wo,
we find
(-id)dw.
+ a’)~ ( w o+ w ’ ) ~ ( w ’ ) X exp (-iw’t)
7
(6.4)
dw’.
(6.5)
L. Brillouin, “Wave Propagation and Group Velocity.” Academic Press, New York, 1960.
* L. D.
Landau and E. M. Lifshita, “Electrodynamics of Continuous Media,” p. 253. Pergamon Press, New York, 1960.
309
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
Since G ( t ) is slowly varying, the range of values of o’ will be small compared with 00, and we can expand Eq. (6.5) in a power series, using (WO
+ o’)Z(WO + w’)
=
wOZ(WO)
+
u’[Z(UO)
+
+
UOZ’(WO)]
* * a ,
(6.6)
where Z’(w) = d a ( w ) / d w , and the series has been terminated with terms linear in w’. On substituting Eq. (6.6) into Eq. (6.5) we obtain: ~ ( t ) / d= t
goexp (-iwot) { -iooi(wo)G(t) + [Z(wo)
+ woZ’(wo)]G’(t))
(6.7) where we have omitted terms involving the second and higher derivatives of G ( t ) . We can now calculate the value of the term E ( t ) -D( t ) from Eq. (3.3). To suppress the rapid sinusoidal oscillations of angular frequency wo we use Eq. (3.4a), which is still valid when the sinusoidal functions are multiplied by a slowly varying function of time. We find:
+ $[rl(wo) + woei’(wo)](d)dt) [B*(t) * B ( t ) ] . (6.8) We have dropped terms which depend on higher derivatives of G ( t ) . The first term on the right in Eq. (6.8) has already been identified in Eq. (3.5) as the power dissipation. The second term represents the rate at which the energy density is being changed. We may therefore give the time-averaged energy density associated with an electromagnetic field in a dispersive medium, including magnetic field terms, in the form:
V(t>
= *[(q
+ wel’)B*(t)
*B(t) + (p1 + w p l ’ ) H * ( t ) - H ( t ) ] .
(6.0)
The relation between H and E is given by Eq. (4.lc), from which we can deduce that, for the case of homogeneous fields:
H*.H
=
(ii
=
[(Mi*) (E.E*) - (&.B*)(E*.E)]/(pp*w2)
=
I c I E*.B// p I.
x E).(ii* x E*)/(pp*w2)
(6.10)
We will consider the energy density for some simple examples in Part IV. If the material has a nondispersive real permeability and a real dielectric constant, then n2 = KK,,,, and we can write:
U
=
nro[n
+ w(dn/dw)]E*.E/(2~,)).
(6.11)
310
FRANK STERN
It follows from Eqs. (4.6b) and (4.4) that the velocity of energy transport in this case is 8 / 0 = c/[n 4-w(dn/dw)] = dw/dkl (6.12) i.e., the group velocity? 7. SOURCES, POTENTIALS, AND THE WAVELENGTH-DEPENDENT DIELECTRIC CONSTANT In this section we shall give formal expressions for the electromagnetic fields produced by given external sources p(r, t ) and Jext(r,t ) . We do this by Fourier-analyzing the sources, finding the response of the medium to the component waves, and integrating over these responses. For example we write:
Jext(r,t )
=
/
W
jext(k,w ) exp (zk-r - i w t ) dak dw,
(7.la)
J-W
for which the inverse Fourier transform is
jext(k, w )
=
J
m
( 2 ~ ) - ~ Jext(r,t ) exp (iwt - 2k.r) d3r dt. -m
(7.lb)
The externally introduced charge density p, and the fields E, B, D, and H can be similarly represented. In these expansions k and w are taken to be real quantities. Fourier components such as jext(k, w ) will generally be complex, but they must satisfy a condition of the form:
Bext(k, w ) ] *
=
j=t(-k,
(7.2)
-w)
to assure the reality of physical quantities such as Jext(r, t ) . We introduce the scalar potential C$ and the vector potential A by noting that Maxwell's equations (2.la) and (2.lc) are identically satisfied if we take:
B(r, t )
=
V x A(r, t ) ,
E(r, t )
=
-vC$(r, t ) - A(r, t ) ,
(7.3a)
which implies that the Fourier-transformed quantities satisfy :
g(k, w )
=
zk x A(k, w ) ,
B(k, w )
=
-zkd,(k, a)
+ iwA(k, w ) . (7.3b)
If we now generalize the complex dielectric constant and permeability of Eqs. (2.4) and (2.9) to permit them to depend on the wave vector k as well as on the angular frequency w, we can write: fi(k, w ) = Z(k, w)B(k, w ) ,
(7.4a)
Wk,
(7.4b)
w) =
w,w)/P(k,
w),
31 1
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
where the conductivity is included in Z(k, w ) , as in Eq. (2.7b). These definitions and the reality condition (7.2) imply:
[i(k, u)]* = i (-k,
(7.4c)
-u),
(7.4d) CP(k, .)I* = P(-k, - w ) . Maxwell's equations (2.lb) and (2.ld), which relate the fields to the external sources, can now be written:
k.k$ - wk-A
+ [wPZ$ - k*A]k
p * k - w2pi]A
= ?/Z,
(7.5a)
=
(7.5b)
PJextI
where each quantity with a tilde is a function of k and Lorentz condition
w.
If we use the (7.6a)
k.A(k, a) = wp(k, w)Z(k, w)$(k, a) to reduce the arbitrariness in the potentials, we find:
- w2p(k, w)Z(k, ~ ) ] $ ( k , a) [k*k - w2p(k, w)Z(k, w)]A(k, U ) [k*k
=
P(k, w)/i(k, o),
(7.6b)
=
p(k, W)Je.t(k,
(7.6~)
w).
Another way to simplify Eq. (7.5) is to separate the vector potential A and the current Jextinto longitudinal and transverse parts:
k
X
A(k, W )
=
L ( k , m ) 4-
Jext(k, w )
=
JL(k a)
+ JT(k,
w),
= 0,
k*&
k*JT = 0.
AL = k
X JL
=
w),
(7.7)
Then Eq. (7.5) reduces to:
k*k$ - wk*AL = z/Z,
[knk
(7.8a)
- o2pZ]A~ = PJT,
(7.8b)
provided the sources satisfy the conservation equation
k*J,,t(k, W ) = w?@, w ) , (7.9) as in Eq. (3.1). Thus the transverse part of the vector potential is determined by the transverse current sources. For the longitudinal part of the field we may choose either the gauge $ = 0,
or the gauge
AL = 0,
AL(k, U ) $(k,
=
W)
-kF(k, w)/[wk*ki(k, =
"pk, w)/[k*kZ(k,
w)],
w)].
(7.10) (7.11)
312
FRANK STERN
It is clear from Eq. (7.3) that Eqs. (7.10) and (7.11) lead to the same electric field, and that both of them lead to a vanishing magnetic field. The wavelength-dependent dielectric constant arises in quite a natural way in this formal treatment. The solution of Maxwell’s equations to find the fields produced by given sources in an homogeneous isotropic medium is reduced to the evaluation of Fourier integrals, provided i ( k , w ) and p(k, 0)are known. Of course the integrals will generally be quite formidable. For most ordinary optical problems one can neglect the dependence of the dielectric constant on k, but for some phenomena, such as the anomalous skin effect, the more complete expressions are required. In Parts III-V we shall assume that the dielectric constant is independent of k, but in the last two parts we return to the wavelength-dependent formalism for the particular case of a free-electron gas, with applications to the anomalous skin effect and to the screening of a static point charge. Some simple consequences of the relations given in this section can be deduced for waves propagating in the absence of externally introduced charges or currents. Then Eq. (7.8b) shows that a transverse wave of wave vector k and angular frequency w can exist only if: k-k
=
(7.12)
p(k, w ) i ( k , w ) w z .
This equation will not have a solution for real k and w if, as is generally the case, pZ is complex. But if we somewhat cavalierly disregard the restriction of k and w earlier in this section to real values, Eq. (7.12) is equivalent to the resul,t found for a single plane wave in Eq. (4.2). A longitudinal wave can exist in the absence of external sources only if
Z(k, w )
=
(7.13)
0
as is easily seen from Eqs. (7.10) and (7.11). Although Z(k, w ) cannot vanish in a real solid in equilibrium, since there will always be a t least a small positive imaginary part of Z, those values of k and w for which Z(k, w ) is close to zero characterize waves for which a fixed external source a(k, w ) will produce a large longitudinal response in the medium. Such waves are called plasma oscillations, and a frequency for which Eq. (7.13) is approximately satisfied is called a plasma frequency. We shall return to this subject later. 8. ALTERNATIVE TREATMENTS OF THE MAGNETIZATION
I n the expressions given so far we have described the induced magnetization through the permeability p&,, i.e. M(k, a)
=
[1
- Z(k, o)]B(k,
PO,
(8.h)
313
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
where we have introduced the reciprocal relative permeability (8.lb) because many equations take a somewhat simpler form in terms of (R than in terms of "K. Equation (&la) is not the only way in which magnetic effects can be described within the framework of Maxwell's equations. The reason for this is that the magnetization enters Maxwell's equations (2.2) only through the term V x M in (2.2d). We can consider this. term to be simply another current term, just as the conduction current and the displacement current are so considered. If we do this, we find by an argument similar to the one in Eq. (2.6) that the magnetic effects can be included in the complex conductivity or in the complex dielectric constant. Thus we can write: CCO-'~
+
+ CTE+ (aD/at) + V
M, (8.2s)
B
=
Jext
Jind
=
Re ( i w ( a -
=
Re [iwutr(k, w)i(k, w ) ] ,
(8.2~)
=
Re [wzeoP(k, w)i(k, w ) ] ,
(8.2d)
X
Jind
=
Jext
h e )
X
+ k * k ( l - %)po-l}L(k, w ) ,
(8.2b)
if no external charges are present and we choose the gauge t j = 0. In Eq. (8.2) the magnetic effects have been absorbed in Ztr or P, and we must then write : f8.2e) R(k, u) = &B(k, 0). The notation for the modified constants P and Fr follows that used by Lindhardsa in his paper on the properties of a gas of charged particles. We use Ztr or Ztr only for transverse fields, since B, H,and M vanish for longitudinal fields, It is possible to absorb only part of the magnetic effects into the conductivity. Thus we could absorb the imaginary part of % in the complex conductivity, and would then write = &p0. If we use Eq. (8.2), the condition for the propagation of an eleclromagnetic wave in the absence of sources is:
k*k = (wa/ca)Pr(k,w ) . But from Eq. (8.2) we see that P(k, w)
=
X(k, 0)
+ C % * k d ( l- %),
(8.3) (8.4)
for which Eq. (8.3) reduces to (7.12), as it should. A treatment based on a wavelength-dependent dielectric constant 88
J. Lindherd, Kgl. Dam& Videnskab. Selskab, Mat.=&s. Medd. 20, No. 8 (1954).
314
F K A S K STERK
must be used with some caution if a boundary between different media is presents*b,soOne example is given in the discussion of the anomaIous skin effect in Part VI. If one wanted to use the formalism of Eq. (8.2) for substances which are magnetic even in the absence of an external field, one would include the current associated with the spontaneous magnetization in Jext. The considerations of the present work, however, assume that there is no spontaneous magnetization or polarization in the media we deal with. II. Reflection and Transmission
9. BOUNDARY COSDITIOSS When electromagnetic waves pass from one medium into another, certain boundary conditions must be satisfied, giving rise to the phenomena of reflection and refraction. These boundary conditions, which may be obtained from Maxwell’s equations, are9:
n.(DA - DB)
= 7,
n x (EA - EB) = 0, n-(BA - BB)
(9.la) (9.lb)
0,
(9.lc)
n x (HA - HB)= j,
(9.ld)
=
where n is a unit vector, directed from medium B toward medium A at right angles to the boundary between them, and j and r represent the surface densities of current and of externally introduced charge, respectively. Equations (9.lb) and ( 9 . 1 ~ )express the continuity across the boundary of the tangential component of t,he electric field and the normal component of the magnetic indue tion. We shall be concerned only with radiation incident upon a boundary containing no currents or charges of external origin. Consequently, we may write : ,. n. (ZAEA - ZBEB) = 0, (9.2a)
-
nx(L -
EB)
=
0,
n.(,izAHA - E ~ H =~ 0,)
nx
-
HB)
=
0,
(9.2~) (9.2d)
I am indebted t o Dr. John C. Sloncxewski for an interesting discussion of this point. Boundary conditions for the case of excitons have been discussed by V. M. Agronovich and V. L. Ginzburg, Usp. Fiz. Nauk 77, 663 (1962) [Soviet Phys.-Usp. (English Transl.) 6, 675 (1963)], and by J. J. Hopfield (unpublished). See p. 185 in Panofsky and Phillips.3
8b 80
9
(HA
(9.2b)
315
ELEMENTARY OPTICAL PROPERTIES O F SOLIDS
These relations follow from the fact that we may set EA = F ~ E A and HA = Re HA as in Eqs. (2.4) and (2.9). Let radiation with the electric vector E = Re [Eoexp (2g.r - id)] be incident upon the plane boundary between two media. In order to satisfy the boundary conditions, we assume the presence of a reflected wave in the first medium and a transmitted wave in the second medium having, respectively, the electric vectors E” = Re [Eo” exp (z%”*r- id)] and E’ = Re exp (&.r - id)]. Similar equations may be written for the vectors representing the magnetic fields. We shall assume that the real and imaginary paits of k (k, and k2) and the normal n, which we shall take as directed into the first medium, are coplanar, referring to their common plane as the plane of incidence.
mot
Y
t
, .
I K2’ I
FIG.1. Wave vectors and angles in the plane of incidence.
The similar time dependence of all three waves follows from the boundary conditions. Furthermore, it is necessary that a t each point ro on the boundary, tern = E’era = E”-ro. (9.3) Therefore the components of k, k’,and k” in the boundary plane must be equal :
Cxn
=
ii”xn,
Cxn
=
ii’xn.
(9.4)
Equation (9.4) shows that fl and g’lie in the plane of incidence. If we take this plane to be the x-y plane, with the normal n in the y direction, as shown in Fig. 1, then E, = E,’ = k,”. (9.5)
-
From Eq. (4.3) we obtain:
k.E Pel;’
=
p.p’=
= (n’
(n
+ ik)2(02/c2)
= &,2
+ ik’)~(w~/cz)= &;z + fE,?
+ &,2 = &,”2 +
fl.,”Z,
(9.6)
316
FRANK STERN
+
Here n ilc is the complex index of refraction of the medium from which the wave is incident, and n‘ f ik’ is the complex index of refraction of the second medium. We find from Eqs. (9.5), (9.6), and Fig. 1 that:
li,“ 6,’
=
-kv,
= - [ ( w z / c 2 ) (n’
+ ik’)2 - Lz2]1’2.
(9.7)
10. AMPLITUDE REFLECTION COEFFICIENTS We are now in a position to determine the components of the reflected and transmitted waves in terms of those of the incident, wave. Equations (9.2) may be written n.[Eo E:’ - ( ~ ’ / ~ ) E P=, , I0, (1O.la)
+
nx
n.[Ho
(Eo +
- EP,,)= 0,
+ Hd’ - (Z/X,,JHP,,I = 0, n x (Ho+ Hi’ - Hot)= 0,
(1O.lb)
(1O.lc) (10.ld)
where we used i = “KO, p = Zmp0. To evaluate the amplitude of the reflected and transmitted waves, it is convenient to consider separately cases with the electric vector polarized perpendicular to and parallel to the plane of incidence. There is no loss of generality in doing so, since any electric or magnetic vector may be resolved into components normal to and in the plane. We shall employ the subscript s to refer to the normal component of a vector and p to refer to the parallel component. The s component will be considered positive in the +z direction. Equation (10.lb) then gives:
Eoa + Ell;‘
=
Eo;.
(10.2a)
Combining Eq. ( 4 . 1 ~ )with (1O.ld) we have
n x [(E xEo.)
+ (k” xEos”) - (XJk’)
(k xEo8’)] = 0.
(10.2h)
Expanding the triple vector product and noting that n9EoS = n.EO8’= n.Eol’ = 0, we find: &(Bob, -
EOa”)
=
(~~2n’)li:Eoa’.
(10.2c)
We define the amplitude reflection coefficient P, and the amplitude transmission coefficient t8by :
El8‘’= P*Eo,,
(10.3a)
Elh‘ = f # E 0 8 ,
(10.3b)
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
317
and find from Eq. (10.2) that: (10.4a)
la =
fEu
+
2fEU (zm/;rn’>G’
= 1
+ P,.
(10.4b)
We will usually assume that Zm = Zm’ = 1. That will be closely true for any material a t optical frequencies,’O and will hold quite well also for many materials a t lower frequencies. When the electric vector lies in the plane of incidence, it is convenient to carry out the calculations in terms of the magnetic vector, which is perpendicular to the plane. Proceeding as in Eq. (10.2) we find for this case : Roa” = P p R O a , (10.5a) R06’ =
Z S o I ,
where
P,
=
fE,
- (“K/“K’)JEy’
iu + (“.?)
JEu”
(10.5b) (10.6a) (10.6b)
11. SNELL’S LAWASD GESERALIZATIONS The various components of the propagation vectors can also be expressed in terms of the angles they make with the normal n. Let kl make an angle 0 with the normal, as in Fig. 1; we may call this the angle of incidence. Similarly, we may let kl”and kl’make angles of 0” and 0’ with the normal, referring to them as the angles of reflection and refraction respectively. The angles that kz, kz’, and k2” make with the normal will be referred to as $, $’, and $”, In terms of these angles we find, from Eq. (9.5), that:
I kl 1 sin 0 1 ke 1 sin $
= =
1 kl’ sin 0’, 1 4’1 sin $’,
which form a generalized form of Snell’s law of refraction. lo
See p. 252 in Landau and Lifshitr.8
(1l.la) (1l.lb)
318
FRANK STERN
Equation (4.3) allows us to solve for the magnitudes of kl and kz (or I 4’’I and 1 ki’ I). We find:
Here x = 0 - 6 (see Fig. l ) , and is assumed to be less than go”, and K,,, is assumed to be real in the second line of each equation. Equation (11.2) is the extension of Eqs. (4.4) and (4.5) to the inhomogeneous case. To find the wave vector in the second medium, we assume for simplicity that the first medium is a lossless dielectric with index of refraction n. Then, if K,,’ = 1, Eqs. (9.7) and (11.2) yield:
I 4I k,’
=
na/c,
= (nw/c) sin 0,
J kz I k’,
=
0,
= -(w/c)
[X’ - n2 sin20]1/2.
(11.3)
It follows that for the case we are considering, with a real wave vector in the first medium, ki is always normal to the boundary plane. This can also be seen from Eq. (11.1b) . The phase velocity V’ = w / /
k1’ 1
(11.4)
in the second medium is no longer a function only of the properties of the medium, but depends also on the angle of incidence. This is sometimes6 taken into account by defining an effective index of refraction d e f t = c/v’. The effective index equals the index defined in Eq. (4.3) for normal incidence, or for arbitrary incidence when the second medium is also lossless. 12. ENERGY FLOW
We now turn our attention to the energy flow in the two media, which is given by the Poynting vector [see Eq. (4.6a)l. We note that the field in the first medium is the sum of the contributions of the incident and reflected waves. Thus, when the first medium is a dielectric, and kz =
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
kit
=
319
0, the average energy flow is
8 (medium 1)
4 Re (8* + 8"*)x (H+ H = Re (fo*X H , + &'* xiid' =
*
)
- exp [-i(k1 - k:') ar] + go*xHd' + 80"*x% exp [i(k1 - kl") or]} (12.1) = s + s" + sin*. In evaluating the energy flow we take the general case of elliptic + f o p , and HO= Ho,+ Hop. We choose polarization, with 80= 208
and go, as the independent components, with the components in the plane of incidence determined from these by Eq. (4.1). For simplicity we assume that the first medium is a lossless dielectric, so that the incident and reflected beams are both homogeneous, and that Km = Km' = 1. &a and Ho.can both be chosen to be real (by an appropriate choice of zero on the time scale) if the incident wave is linearly polarized, but not otherwise. Straightforward substitution in Eq. (12.1) 'gives for the energy flow in the incident beam:
S
=
+ K-'[~Oa*~:,/(2€0w)]k1.
[Eoa*E0a/(2p0~)]kl
(12.2)
The energy flow in the reflected beam is
B"
=
~ a * ~ * [ ~ 0 , * ~ 0 a / ( 2 / 1 0 . ) (]?kp ~ * ?" p / ~ ) [g08*fl0a/(2WJ)]k1", (12.3)
where we used Eqs. (10.3a) and (10.5a). A somewhat lengthier calculation gives for the interference terms: sint
=
[&a*&/
+ 4 sin 28 b
( P O W ) lkiz
(Hoa*&[Pp*
Re
[Pa
exp ( -2iykiy) I
exp (2iyk1,)
-
?8
exp (-2dgk~]).
(12.4)
The interference terms give energy flow only parallel to the boundary plane. These terms arise from the overlap of the incident and reflected beams, and can be ignored in the typical reflectance experiment, in which collimated beams are used. The interference terms must, however, be considered near the boundary where the beams overlap. Since the incident beam normally contains a range of wave vectors, say between k and k Ak, phase cancellation will occur and we can expect Eq. (12.4) to go to zero at distances 2 (Ak)-' from the surface.
+
320
FRANK STERN
The energy flow in the second medium is
s’ = 3 Re ( E ’ * x H ) = i e x p (-2ki.r)
Re (E0’*xH0’),
(12.5)
which gives
s’ = { [ E o ~ * ~ o(2poW) : o . / ]f8*@’1
+
2’
[~oII*~o‘/(2EoW)](~p*tp/(
12)
(Kl’k,’
+ Kz’ki)
+ ( ~ ~ k z y ) ’ k l ~Re/ ~ (~-)i t , t , * E ~ , ~ ~ , * / i ?1* )exp ( -22ykzy).
(12.6)
Conservation of energy requires that the component of energy flow normal to the surface must be continuous across the boundary. This is seen to be the case if we compare Eq. (12.6) with (12.2) and (12.3), and use (10.4b) and (10.6b). The presence in Eq. (12.6) of a component of energy flow normal to the plane of incidence seems paradoxical until we consider that the overlap of the incident and reflected waves in the first medium also gives such a component of energy flow. 13. REFLECTIVITY The reflectivity of the material is defined for the general case by R = and can be calculated from Eqs. (12.2) and (12.3). When the electric field is polarized perpendicular to the plane of incidence, or in that plane, we have, respectively:
-S”.n/S.n,
R,
=
P,*P,,
R,
=
P,*Pp.
(13.1)
For unpolarized light, fbr light polarized at 45” to the plane of incidence, and for circularly polarized light, R = +(R, R p ) . Introducing 20’ = n’ ilc‘ = (2’)1/2, we can show from Eqs. (9.7), (10.4), and (10.6) that when the first medium is a lossless dielectric the equations for the reflection and transmission coefficients are :
+
+
(13.2a) (13.2b)
32 1
ELEMENTARY OPTICAL PROPERTIES O F SOLIDS
At normal incidence we find from Eqs. (13.1) and (13.2) that the reflectivity is: (13.2e)
If the first medium is also lossy, we must replace kI2 by (k - k ' ) 2 in the numerator and by ( k k') in the denominator. To understand better the phenomena which occur during reflection and refraction, it is helpful to discuss the simple case where the second, as well as the first medium, is a lossless dielectric. Then we can simply by n' in Eq. (13.2). If n' 1 72, we see that the square roots in replace Eqs. (13.2) and in (11.3) are real for all values of 8. Thus in this case Eq. (1l.la) can be written:
+
m'
n sin e
=
n' sin 8'
(13.3)
the familiar form of Snell's law of refraction. TABLE 11.AMPLITUDEREFLECTION COEFFICIEKTS Fa AN D P, FOR LIGHTWHOSEELECTRIC PERPENDICULAR TO AKD IN THE PLANE OF INCIDENCE, RESPECTIVELY FIELDIs POLARIZED
The coefficients are defined in Eqs. (10.3) and (10.5)) and explicit expressions are given in Eq. (13.2). The special values of the angle of incidence 0 are Brewster's angle Be = arctan (n'ln) and the angle for = arcsin (n'ln),where 72 and n' are the real onset of total reflection, indices of refraction of the medium from which the beam is incident, and of the second medium, respectively. The and - signs indicate real coefficients with magnitudes between 0 and 1, and correspond to phase shifts of Q" and 180" respectively in the reflected beam. The real phase n). shifts 6, and 6, are given in Eq. (14.1) ; we use ro = I n' - n I/(n'
+
+
n'
>n
n'
e
7,
i.,
0
- To -
+
o
+
-
-1
-1
eB
+
o
To
0
e
0
es
eT
eT
< e < 90" 90"
.Fa
ro
+
1 exp (is,) -1
FP
-ro
0
+1
exp (isp) -1
322
FRANK STERN
It is instructive to consider the phase changes which light undergoes on reflection. These are summarized in Table 11. It is easily verified from Eq. (13.2) that for a beam incident a t angle 8’ from the second medium the amplitude reflection coefficient is the negative of the coefficient for incidence a t angle 0 from the first medium, where 0 and 0’ are real angles related by Eq. (13.3). This opposition of signs is shown in the first five rows of Table 11. The table also shows that a t normal incidence the electric field has a 180” change of phase on reflection for waves incident from the medium of lower index, while it is the magnetic field which changes phase when the wave is incident from the medium of higher index. 14. TOTAL REFLECTION When n’ < n, the angle of refraction 0’ given in Eq. (13.3) becomes imaginary when the angle of incidence 0 exceeds OT = arcsin (n’/n). For greater angles of incidence the beam is totally reflected, but with a phase shift whose magnitude increases from 0, reaching 180’ a t glancing incidence. Thus a t glancing incidence the phase change on reflection is 180” in all cases; this is verified experimentally by the dark fringe observed a t the surface of Lloyd’s mirror.“ The phase shifts for total reflection, as determined from Eq. (13.2))are: f, =
exp (i6,),
=
exp (is,),
f p
6,
=
-2 arctan [(sin2 e - [n’/n12)1/2/cosel,
6, = - 2 arctan [(sin2e - [n’/n12) (n’/n)-2/cos el.
(14.1) I n the case of total reflection it is seen from Eqs. (9.5) and (9.7) that
k,’ lies in the boundary plane while kz’is normal to it. Thus the plane of
constant phase is perpendicular to the plane of constant amplitude, and the field in the second medium is an extreme example of an inhomogeneous field. There is no energy flow normal to the boundary plane, as is implied by the term total reflection. When the second medium becomes conducting, i.e., ~ 2 ’ # 0, it follows from Eqs. (11.3) and (12.6) that there mill be a component of energy flow normal to the boundary plane in the second medium, and total reflection is no longer possible. 1s. TRASSMISSION
O F ISOTROPIC
RADIATIOX
It is sometimes useful to know the value of the reflectivity averaged over all directions of incidence. To calculate the average, note that the 11
F. A. Jenkins and H. E. White, “Fundamentals of Physical Optics,’’ p. 66. McGrawHill, New York, 1937.
ELEMENTARY OPTICAL PHOPEltTIES OF SOLIDS
323
power striking an element of area d A a t an angle of incidence B and in a cone of directions covering solid angle ds2 is I cos 0 dAds2, where I measures the intensity of the radiation, as discussed for blackbodies in Part V. Thus the appropriate average for the reflectivity R is
R (0) cos B sin -9 &d+ R A= ~
1cos
(15.1) B sin 8 dBd+
where we introduced E = sin2&It is more convenient to consider the radiation transmitted through the surface, rather than that which is reflected, so we introduce T = l - R . (15.2) The average transmissivity will be TAv= J T ( 5 ) d& For this section we will restrict ourselves to materials with real indices of refraction. We will use n = n>/n<, (15.3) where n> and n< are, respectively, the larger and smaller of the indices of the media on opposite sides of the boundary plane. We will indicate as unprimed the reflectivity and transmission for beams incident from the medium with the lower index, and will label with a prime the corresponding quantities when the beam is incident from the material with a larger index. We will consider the radiation to be unpolarized. Thus T = $ ( T , T p ) .Using Eqs. ( E l ) , (13.2), and (15.2) we find:
+
(15.4a)
The primed quantities are obtained by replacing ‘n by n-1 in Ey. (15.4); By a straightforward transformation of both T,’ and Tpl vanish for > r2. integrals we can show that
T i , & = n-2T,,h,
=
n-2Tp,Av.
(15.5)
This old12 and important result is used in Part V in discussing thermal equilibrium in an isothermal enclosure containing two different media. 1*
P. Drude, “The Theory of Optics,” Part 111, Chapter 11, Article 6. Longmans, Green, New York, 1907;Dover Publications, New York, 1959.
324
FRANK STERN
Explicit evaluation of the integrals is somewhat more troublesome, but involves no unusual steps. We find128:
Ta.Av
=
Tp,Av=
+ 1) (n + I)-', 4n3(n2+ 2n - 1) (n' + 1)-2(n2 - l)-I
(15.6~3)
Q(2n
+ + 2n2(n2- 1)2(n2+ 1)-3In [n(n + 1) ( n - 2n2(n2 1) (ne- 1)-2In [n]
l)-I].
(l5.6b)
The transmissivity goes to 1 for all polarizations and angles of incidence when n = 1, since the two media are then identical and the boundary does not affect the propagation of radiation. This is easily verified for Eq. (15.6a), but cannot be checked directly for (15.6b), which is undefined for n = 1. If we put n = 1 u we find for u << 1 :
+
For all values of n the average transmission is higher for light polarized in the plane of incidence, as one expects from the behavior of the reflectivity. 16. THISFILMS An interesting application of the theory already discussed is to the phenomena of reflection from and transmission through thin solid films. In this case there are'multiple reflections a t both surfaces of the film, and interference effects are present in the reflected and transmitted light. In our discussion, we shall limit ourselves to light that is plane polarized in, or normal to, the plane of incidence. Assume that the film is bounded on one side by a dielectric, in which the incident beam is located, and o n the other side by any other medium. Let f l and tl be the reflection and transmission coefficients for light going from the first medium toward the second, 72 and tz the coefficients for light going from the second into the third, and fl' and 2,' the coefficients for light going from the second into the first. The light will undergo reflection and refraction a t the first surface, pass to the second where it will again be reflected and refracted, and so on. Thus there will be an infinite number of beams reflected back into the first medium by the film and an infinite number transmitted into the third. It is easy to see from Eqs. (9.5) and (9.7) that the propagation vectors are identical for each of the reflected beams in the first medium 12*
Some further details are given in F. Stern, A p p l . O pt ., t o be published.
325
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
and that they are also identical for each of the transmitted beams in the third. If there were no absorption in the film and no phase difference between reflected waves, the ratios of the amplitudes of the fields of the reflected waves would be given1* by the sequence: PI, Pz&B’, P1‘7~~&&’, 71’28~s&&’ * . However, upon passage one way through the film, the field of each beam is attenuated by a factor exp ( - I kz,’ 1 d ) where d is the thickness of the film. Furthermore a t a given value of z there is a phase difference of - 2kl,‘d between successive reflected beams, and between successive transmitted ones. We can write, using Eq. (11.3) :
6
= -&,’d
=
(2nd/X) (2’
-
sin2 0)1/2,
116.1)
where refers t o the first medium, i? to the second, and X is the wavelength in vacuum. The sum of the reflected fields will then give for the reflection coefficient F of the film:
P
=
fl
+ P2&f11 exp (22%) + P1’P~2flfl’ exp ( 4 4 + P1’2P~811&’exp (6i5) +
0
a .
(16.2) With the help of Eqs. (10.4b) and (10.6b) and of the fact that F1’ = we obtain tlB’ = 1 - TIz. Then Eq. (16.2) becomes
- pl,
(16.3)
Using the same approach, it is easily shown that the transmission coefficient t of the film is: (16.4) Let us first discuss the case in which there is air on both sides of the film. Then we may set 72 = -Pl in Eq. (16.3). We obtain for the reflection coefficient of a film when the electric vector is polarized normal to the plane of incidence :
P, =
la
{ [cos { [cos .e/(i?
e/(? - sin2 e)1/2] - [(Z’ - sin2 e)1/2/cos e l ) sin 5
- sin2 e p ]
+
[(;I
- sin2 e)1’2/cos e l ) sin 5
(1 - “K) sin5
(1
+ i’ - 2 sin2 e) sin 5 + 2i cos e cos
- sin2 e]l/z’
+ 2i COB 5 (1G.h)
0. S. Heavens, “Optical Properties of Thin Solid Films,” p. 56. Academic Press, New York, 1955.
326
FRANK STERN
and when the electric vector is polarized in the plane of incidence:
rp
=
-
-.
1 [itcos e/ (i’ - sin2 e) 1 / 2 1 - [(x‘- sin2 e) 1 / 2 / i ’ cos e l } sin i { [itcos e/ (x’ - sin2 e) 1/21 + [(2’ - sin2 e) 1 / 2 / z f cos e l 1 sin 5 + 2i cos 5 (P e - i’ + sin2 e) sin I + X’ - sin2 e) sin i + 2izt cos e cos i(xt- sin2 e>l/z’ COS~
(P cos2 e
(16.5b) The square of the magnitude of Eqs. (16.5a) and (16.5b) will give the expressions for the reflectivity of a thin film in air. Ferrell and Stern14note that there will be a sharp rise in the reflectivity of a thin film near the plasma frequency for light polarized in the plane of incidence when the imaginary part of the dielectric constant is small. Inspection of Eq. (16.5b) shows that FP + -1 when 2’ -+ 0, except for normal incidence. Numerical calculations by G ~ e r t i n ’show ~ that in the case of silver one may expect a small rise in the reflectivity for films of thickness 200 A or less; increasing the angle of incidence increases the effect. There is never any rise in the reflectivity of light polarized normal to the plane of incidence; therefore the effect disappears for normal incidence. Even when the theory of reflection at a single surface suggests that most of the incident energy.should be reflected at the first surface, there can be interference effects that considerably reduce the strength of the reflected beam when one has a thin film. This is due to the previous observation that, even at total reflection, the field does not vanish within the second medium, but decays exponentially with distance. Therefore, as the numerical results15 demonstrate, the thinner the film, the greater the interference effects one should expect. One could, however, experience difficulties because of the failure of the bulk optical constants to hold for thin films.16 Since it appears that for most metals films no more than a few hundred angstroms thick would be required in order to observe any rise in reflectivity at the plasma frequency, a backing will be required. When the incident beam is in air and the film has a backing, the expressions for the reflection coefficients of light polarized perpendicular to and in the plane of incidence, R. A. Ferrell and E. A. Stern, Am. J . Phys. 30, 810 (1962). See also the note added in proof on page 408. 16 R. F. Guertin, unpublished work, 1961. 16 H. Mayer, “Physik dunner Schichten,” Part I. Wiss. Verlagsges., Stuttgart, 1950. 14
327
ELEMESlARY OPTICAL PROPERTIES OF SOLIDS
P,
=
] cosb-i[ i’(iB-sin28)1/2 (;’Isin2,9)1/2
(XB - sin28)112
25 (Xl-sin28) 1 / 2 1
+
XI
case
sin8J
(16.6) where we have introduced XB for the dielectric constant of the backing. The expressions for the reflectivity obtained from the above two expressions are extremely complicated. Numerical calculations have been carried out for several metals with a fused quartz backing.ls For silver, which was the only metal investigated both with ,and without a backing, the calculations predict that for angles of incidence less than Brewster’s angle for the backing, the maxima in the reflectivity will be less pronounced than with no backing. At the Brewster’s angle the results are approximately the same, and for larger angles of incidence the maxima are more pronounced. 111. Dispersion Relations
In this chapter we discuss some properties of the optical constants of materials which can be derived without reference to specific models. We shall give a brief account of dispersion relations, often called KramersKronig relations, and shall show how they lead to several sum rules. The reader interested in further details should consult the books of Bode,” Frohlich,l* Landau and Lifshitz,lg MOSS,^^^ and Solodovnikov,lgb or the papers by TollJ20and Wu.)Oa H. W. Bode, “Network Analysis and Feedback Amplifier Design,” pp. 278, 303ff. Van Nostrand, Princeton, New Jersey, 1946. See p. 4 in Fr0hlich.l 10 See p. 256 in Landau and Lifshitz.8 lB. T. S. Moss, “Optical Properties of Semiconductors,” Chapter 2. Butterworth, London, 1959. I O b V. V. Solodovnikov, LLIntroduction to the Statistical Dynamics of Automatic Control Systems.” Dover, New York, 1960. 2O J. 9. Toll, Phys. Rev. 104, 1760 (1956). ma T. T. Wu, J. Math. Phys. 3, 262 (1962), 1’
328
FRANK STERN
17. DISPERSION RELATIONS FOR
THE
DIELECTRIC CONSTANT
We have already seen in Part I that the existence of a phase shift between the displacement D and the electric field E can be characterized by a complex dielectric constant 2 = Z/eO. The relation between the frequency components of fi and E is: fi(w)
=
Z(w)E(w).
(17.1)
For arbitrary time variations at a fixed point we takezob
J-lw
The physically required reality of D ( t ) and E ( t ) is achieved by requiring that f i ( - w ) = f i * ( u ) and E ( - w ) = E*(w). From Eq. (17.1) it therefore follows that i(-w) = i*(w). (17.3) These relations show that the real parts of fi (a), fi (a),and 0 (a)are even functions of w , and that the imaginary parts are odd functions of w. The response of the dielectric to an electric field is given by the polarisation P(t),where P(t) = D(t) - e O E ( t ) . Thus
J--0,
If E(t) is a square-integrable function, corresponding to a wave train of finite energy, we can invert Eq. (17.2b) and obtain ~ ( t= )
/
m
-m
J
-m
[~(w)
- EO]
m
dw
i,
~ ( t ’ exp )
[iw(t’
- t ) ] dt’/2n,
-m
(17.4b) Thus G ( t 2Ob
-
t‘) gives the polarization response a t time
t to a unit delta-
We use the same symbol both for D ( t ) and for its Fourier transform. The argument will usually be explicitly given.
ELEMENTARY OPTICAL PROPERTIES O F SOLIDS
329
function electric field pulse a t time t’, and is seen to be the Fourier transform of i ( w ) - e0. If the Fourier integral defining G can be inverted, then we can write : i(w)
-
m
e0 =
i,
G ( T ) exp
(iwT)
dT.
(17.5)
The causality requirement, that there can be no response before the field is on, means that G( T ) = 0 for T < 0. The integration in Eq. (17.;) can therefore be limited to positive values of T . Equation (17.5) then defines a function which is analytic for complex values of w having a positive imaginary part r. To show this we note that the derivative of Eq. (17.5) with respect to w will be independent of the path in the complex plane, and will converge in the upper half-plane because of the convergence factor exp (-I’T). Thus : ( w ) - e0, and therefore i ( w ) , itself, are analytic functions in the upper half of the complex w plane. It is well to point out
w
w’
FIG.2. Contour of integration in the complex a’plane.
that we consider functions whose time variation is given by exp ( - i d ) . Had we used exp ( i w t ) in Eq. (17.2), the roles of upper and lower halfplanes would be interchanged. To deduce the dispersion relations from the analyticity of ; ( w ) - €0, we make use of the result that the integral of an analytic function over a contour that encloses no singularities is zero. Thus we can write $[i(w’) - E O ] ( W ’ - w ) - I dw’ = 0 when the integral is evaluated along the closed contour shown in Fig. 2, where the pole a t w on the real w‘ axis is excluded by a small semicircle of radius 6. The integral along the real axis, from - ~0 to w - 6 and from w 6 to a, in the limit 6 + 0 is, by definition, the Cauchy principal value of the integral along the real axis. The large circle which closes the contour contributes nothing, since we assume that Z(w’) - eo goes to zero there. Finally, the small semicircle around the pole contributes - i ~ [ i ( w ) - eo]. Thus we find that
+
;(w)
-
e0 =
(in)-’P J
m
[;(w’) -m
- E ~ ] ( w ’- w ) - I dw’,
(17.6a)
330
FRANK STERN
where P stands for the Cauchy principal value. If we separate real and imaginary parts, we find that el(o)
- €0 ea(w)
=
(I/T)P
=
(l/r)P [m
Q(o’)[w’
J
[el(w’)
-m
- ~1-l dw’,
- e O ] ( w - w’)-l
(17.6b) dw’.
(17.6~) We can transform to integrals over positive frequencies only by noting that;
(17.7) Inserting this result in Eq. (17.6), and making use of (17.3), we find the following form for the dispersion relations between the real and imaginary parts of i : ~ ( u)
€0
=
l
(2/r)P
m
u’e2(w’)
(d2- w2)-l du’,
(17.811)
The constant co has been dropped in the integrand of Eq. (17.8b1, since it is easily verified that
P
lm -
( u ’ ~ wO2)-l dw’ = 0
(17.9)
for wo # 0. The conditions underlying the derivation of Eq. (17.8) have not been stated here with precision. In Toll’s paper,20 boundedness of i ( w ) was assumed. This led to the presence of the constant €0 in Eq. (17.8a), which we know to be the limiting value of i ( w ) for very high frequency. Toll points out that singularities at finite frequencies or divergences at infinite frequency can be treated in modified dispersion relations which contain additional arbitrary constants. One such case occurs for conductors, since, as we showed in Eq. (2.8), the imaginary part of i ( w ) is (u/w), and is not bounded as w + 0 if the material has a nonvanishing dc conductivity uo.
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
331
This is most easily dealt with by subtracting ao/o from ep(w) throughout the derivation of the dispersion relations. No change in Eq. (17.8a) is required, since this additional term contributes nothing to the integral. Thus the dispersion relations for the dielectric constant, in a form in which they are readily applied, are :
-1=
K~(w)
m
(2/n)P
(d2- 09)-~dw’,
O’K~(O’)
(17.10a)
where we have divided by eo to make the expressions dimensionless. The derivation of the dispersion relations (17.10) contains only three major assumptions : boundedness, linearity, and causality. Thus these dispersion relations will be valid for many other physical quantities which express a linear relation between an input and an output if the output cannot precede the input. Our interest here is in the optical properties of solids, but dispersion relations like those discussed here are of importance in many other parts of physics, as well as in electrical engineering.” 18. DISPERSION RELATIONS FOR
THE
INDEXOF REFRACTION
Tol120has defined B dispersion relation as “a simple integral formula relating a dispersive process to an absorption process,” and that is the sense in which we use the term here. Dispersion relations are often called Kramers-Kronig relations because of the work by H. A. KramersZ1and R. de L. Kronig,22*22a who gave dispersion relations for the dielectric constant and for the index of refraction, with application originally to the dispersion of X-rays. We have given the dispersion relations for the dielectric constant in Eq. (17.10). The analog to Eq. (17.10a) for the complex index of refraction n(w) a ( w ) is:
+
n(.)
-1=
1
0
(2/n)P
w’k(o’) (w’2
- w2)--1 dw’.
(18.1)
The derivation of this result proceeds along somewhat different lines from the derivation leading to Eq. (17.6), since the index of refraction H. A. Kramers, Atti del Congress0 Znternazionale dei Fisici, 11-30 Settembre 1967, Como-Pauia-Row (Published by Nicola Zanichelli, Bologna), 2, 545 (1928). R. de L. Kronig, J . Opt. SOC.Am. 12, 547 (1926). *I* R. de L. Kronig, Ned. Tijdschr. Natuurk. 9, 402 (1942). s1
332
FRANK STERN
does not express a linear relation between two physical amplitudes. A detailed derivation of Eq. (18.1) from the causality requirement has been given by Toll,” who also points out that the analog to Eq. (17.10b) can be used if u[n(w) - 11 is square-integrable. An alternative form for Eq. (18.1) is obtained if we substitute $ca(u’) for u’k(w’), where a is the absorption coefficient. 19. QUALITATIVE CONSEQUENCES Some interesting consequences of the dispersion relations (17.10) are best displayed by carrying out an integration by parts which puts them in the form: xI(u)
- 1 = (1/r) Im [dK4(w’)/dw’] 0
KZ(w)
=
In
- (1/r)/m[dK1(w’)/~w’~In 0
(1
u’2
([a’
- w2 1-1)
+
w’
(19.la)
dw’,
- w 1)
dw‘. ( 19.1b)
For simplicity we have assumed that we are dealing with a material whose dc conductivity vanishes. The logarithms in both Eqs. (19.la) and (19.lb) will have strong peaks in the neighborhood of w’ = w. Thus K ~ ( w )will tend ) large and positive, to peak at frequencies for which the derivative of K ~ ( w is and will have minima near frequencies for which the derivative of K Z ( W ) is large and negative. The behavior of K Z ( W ) depends in the opposite way on the derivatives of K ~ ( w ) ,maxima of K ~ ( w )occurring where the slope of K~ ( w ) is large and negative. An important application of Eq. (19.h) makes use of the result, derived in Part V, that K ~ ( w )is proportional to w - ~I m 12p(w), where p ( w ) is the density of final states for the transitions near energy hw, and 3n is the matrix element for the transition as given in (35.9). Our interest here centers on p (a), the density of final states. In a periodic crystal the energy difference Ruj, between initial and final states can be represented by a periodic function in the reciprocal lattice, and therefore must have certain critical points24 where VkWj, will vanish. At the critical points the density of states in three-dimensional crystals remains finite and continuous, but Thus maxima and minima its derivative has a square-root ~ingularity.~4 in K ~ ( w )may be associated with critical points in the energy difference between initial and final states. These critical points need not occur a t 88
J. S. Toll, Ph.D. Thesis, Princeton University, 1952 (unpublished), Chapter 1. I am indebted t o Professor Toll for several valuable discussions of this work. L. Van Hove, Phy.9. Rev. 89, 1189 (1953).
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
333
the same points in reciprocal space as the critical points for either the initial or the final bandlXabut they will often coincide with one or both. A particularly simple example in the dielectric constant of a semiconductor is the behavior near an allowed direct energy gap, where the density of states vanes as ( w - wu)1/2?4b If I 312 l2 is slowly varying, then drcz/dw varies as ( w - wu)-1’2 when w is greater than wg. Equation (19.la) shows that this singularity will result in a peak in the dielectric constant near wu. Furthermore, since the absorption is still weak near the edge, the reflectivity, (13.2e), is controlled mainly by the index of refraction, and will also peak near wg. In some materials, like germanium and the III-V semiconductors, the density of states near the lowest direct energy gap is very small, and the effect is not easily observed. The first pronounced reflectivity peak may occur near the energy gap for transitions at another point in the Brillouin zone. Beyond the first major energy gap the situation is less clear-cut, but it should still be possible, once a rough energy band structure is known, to correlate maxima and minima in K~ ( w ) with critical points in the energy difference between Our arguments to this point have been based’on a matrix element 312 which varies slowly near a critical point. For forbidden transitions, or for indirect transitions, for which the absorption coefficient increases as a higher power than ( w - wu)1/2, there is no singularity in dKa/dw, and no pronounced structure is expected near a band edge. On the other hand, if exciton2592Sa or magnetic field26 effects are important, the absorption coefficient near a band edge may rise more rapidly than for allowed direct transitions, and K ~ ( w )may tend toward a singularity near the band edge. Singular behavior may also occur if one or more of the second derivatives of the energy difference between bands goes to zero at a critical point.24
SHIFT DISPERSION RELATION 20. PHASE The dispersion relations between the real and imaginary parts of the dielectric constant are very powerful tools in the analysis of optical properties. One important use, the derivation of sum rules, is discussed later in D. Brust;,’J.C. Phillips, and F. Bassani, Phys. Rev. Letters 9, 94 (1962) for an example. *4b See Section 37 for a more detailed discussion. 240 B. Velick9, Czech. J. Phys. B11, 787 (1961). 25 R. J. Elliott, Phys. Rev. 108, 1384 (1957). 268 T. P. McLean, in “Progress in Semiconductors” (A. F. Gibson, ed.), Vol. 5, p. 53. Heywood, London, 1960. 26 See, for example, B.[Lax and S. Zwerdling, in “Progress in Semiconductors” (A. F. Gibson, ed.), Vol. 5, p. 221. Heywood, London, 1960. U* See
334
FRANK STERN
this Part. For direct application to experiments on optical properties, Equations (17.10a) and (17.10b) have only limited usefulness, since we usually know the real part or the imaginary part of the dielectric constant only in limited frequency ranges. Another useful relation is one involving the reflectivity a t normal incidence, since this is a quantity that can be directly measured. The reflectivity R is the ratio of reflected to incident intensities. It can be expressed in terms of 7, the complex ratio of outgoing to incoming electric or magnetic fields, by:
R(w)
= P(w)[P(w)]* = [ ~ ( w ) ] ~ .
(20.1)
For normal incidence the amplitude reflection coefficient P ( w ) is given, ilc,by in terms of the complex index of refraction n
+
~ ( w )= r ( w )
exp [ i e ( w ) ]
(n
=
+ ik - l ) / ( n + ilc + 1).
(20.2)
We have chosen the amplitude reflection coefficient for the magnetic field Fp from Eq. (13.2~).The amplitude reflection coefficient for the electric field at normal incidence P, differs from Eq. (20.2) only by a factor - 1. Since P ( w ) expresses a linear relation between two amplitudes, it will be analytic in the upper half of the complex w plane, and relations analogous to Eq. (17.8) exist between its real and imaginary parts. But what we really need in order to analyze reflectivity data is a relation between the phase e ( w ) and the magnitude r ( w ) of the amplitude reflection coefficient, since it is r ( w ) = R1I2.whichis given by the measurements. The optical constants are determined from e ( w ) and r ( w ) by the relations
+ r2 - 2r cos e), + r2 - 2r cos O ) ,
n
=
(1 - rZ)>(l
(20.3a)
k
=
2r sin e / ( l
(20.3b)
which are direct consequences of Eq. (20.2). Thus e ( w ) and r ( w ) specify the optical constants completely, and this shows the great value of a dispersion relation for e ( w ) . To show that the derivation of a phase shift dispersion relation cannot be as straightforward as the derivation leading to Eq. (17.8), we note that P(ij) could be multiplied by any function of ij which has absolute value 1 for real values of 6 without changing the reflectivity. Such a function, called the Blaschke product,20is B(S) =
IT [(S - Pn)/(Pn* n
- 41,
(20.4)
where the pn are complex constants. They must have non-negative imagi-
336
ELEMENTARY OPTICAL PHOPERTIES OF SOLIDS
nary parts, since, as we mentioned above, P ( i j ) is analytic in the upper half of the ij plane. Furthermore, any jZn whose real part is nonzero must be paired with another whose real part is of opposite sign, since f ( w ) , and therefore B ( w ) , must satisfy Eq. (17.3). TollZOhas shown that the dispersion relation for the phase shift e ( w ) is O(w) = ( 2 w / r ) P
Lrn
In
[r(w’)3(w2
- ~ ’ ~ ) - ~ d+ w ’[(a),
(20.5)
where two other terms given by Toll, which arise from pathological behavior at finite or infinite frequencies, have been dropped. The last term in Eq. (20.5) is the phase shift of the Blaschke product: [(w) =
-iln
[B(w)]
=2
+
C tan-l
[(u
n
- ~nr)//~ns],
(20.6)
where jZn = bnr ipni, and the tan-’ takes values between -3. and .3. We find that [ ( w ) is an increasing function of frequency, and that27
- €(O>
€(a>
= (2P
+ d.,
(20.7)
where p is the number of factors in the Blaschke ptoduct in which Pnr = 0, and p is the number of pairs in which llnr # 0. While the Blaschke product cannot be ruled out mathematically, it is possible to show on simple physical grounds that the Blaschke product cannot contribute to the phase of the amplitude reflection coefficient of a solid which is in thermal equilibrium or in its ground state. Under these conditions the extinction coefficient k is nonnegative, and one can then easily show that, both for conductors and for insulators, the phase of P(w) in Eq. (20.2) is zero when w = 0, and is restricted to the range 0 I 6 I ?r. When w passes through an absorption region, the phase shift increases, reaches a maximum less than or equal to ?r, and then decreases again. When the last absorption peak has been passed (this corresponds to excitation of the Is shell of the heaviest atom) we oan approximate the absorption by expressions for the photoelectric effect, for which the cross section (and therefore the absorption coefficient a) decreases approximately as w-a.28 Therefore the extinction coefficient k = c a / ( 2 w ) varies as w-4. On the other hand, at high frequencies K (1 - wp2w-2) , i.e., n 1+ ~ , ~ w - ~as , will be shown in Eq. (23.2). These asymptotic relations for n and k, together with Eq. (20.2), imply that O ( w ) -+ ?r as w -+ a. The
-
-
n N. G. Van Kempen, Phys. Rev. 89, 1072 (1953). 18 A. H. Compton and S.K. Allison, “X-rays in Theory and Experiment,” Chapter VII. Van Nostrand, Princeton, New Jersey, 1935.
336
FRANK STERN
amplitude reflection coefficient r ( w ) approaches awpaw-2 a t these high frequencies, and from this behavior it is not hard to show that the integral in Eq. (20.5) gives a canonical phase shift which approaches ir asymptotically. Thus the Blaschke product, which always adds to the phase shift, as we showed in Eq. (20.7) , cannot be present for the reflectivity of solids a t normal incidence. Our proof that the Blaschke product cannot be present must be modified in one respect. The amplitude reflection coefficient for the electric field a t normal incidence P g ( w ) is just the negative of Eq. (20.2)-This factor of - 1 is provided by a single factor in the Blaschke product (20.4), with p,, equal to zero. Except for the restriction just mentioned, the phase shift dispersion relation which applies to the reflectivity of solids a t normal incidence is 6 ( w ) = (w/ir)
la
[ln R(w’)
- In R ( w ) ] ( w * - d2)--1 dw’,
(20.8b)
where e ( w ) is called the canonical phase shift by Toll. We have subtracted In R(w) from In R(w’) in the integrand of Eq. (20.8b). Because of Eq. (17.9), the value of the integral is not affected, but the singularity for w’ = w is removed, and the principal value is no longer required. Equation (20.8b) is a dispersion relation giving the imaginary part of the function In [ P ( w ) ] as an integral over the real part, and is therefore the analog of (17.8b). Because of the different properties of In [ P ( w ) ] , the proof used to derive Eq. (17.8) will not be valid here, and, in particular, the analog of (17.8a)’ cannot be written down directly. Teutsch and Gottlieb%have given dispersion relations for the real and imaginary parts of In [ P ( w ) ] based on the vanishing of its derivative as w + a . They find an expression equivalent to (20.8b) , and give for the other dispersion relation a result which can be written: In [ r ( w ) / r ( v ) ] = (2/n)
/
m
w’e(W’)[(w’2
0
- U21-1 -
(U’2
- v2)-11
dw’.
(20.8a) The proof given by Teutsch and Gottlieb assumes that the derivative is analytic in the upper half-plane, and therefore ignores the possibility, represented by the Blaschke product, that P(&) may have zeros in the upper 2 plane. A proof of Eq. (20.8b) was first published by VelickQ,sO * 8 M. Gottlieb, Ph.D. Thesis, University of Pennsylvania, 1959 (unpublished) , Appendix 11. I am indebted to Dr. P. Miller and to Dr. W. Teutsch for information about this
so
work. B. VelickQ, Czech J. Phye. B11, 541 (1961),
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
337
who eliminates the Blaschke product in essentially the same way as above, by using Eq. (20.2) and the requirement that k > 0. 21. NUMERICAL PROCEDURES FOR CALCULATING THE PHASE Numerical evaluation of Eq. (20.8b) is quite easy if the reflectivity is known over a wide enough range of frequencies. The simplest case is one in which the reflectivity is known over a frequency range w1 5 w 5 WZ, and is constant outside this range. Examples of this are the reflectivity associated with lattice vibrations in LiF,29*81 and the reflectivity associated with free carriers in semiconductors, for example PbTe,82where one may generally take w1 = 0. In these simple cases the phase is easily found from Eq. (20.8b) to be (21.la) = ez(w) ~ 4 ,
e m+
e,(~)
=
+ ( 2 ~ ) - 'In [ R I / R ( u ) ]In [ ( w +
&(w) = (w/T) &(w)
=
C'
[ln R(o')
w>/l
~1
- 0 11,
- In R(w)](w2- d2)--1do',
(2?r)-'ln [Rs/R(w)] In
[I
w2
-
l/(wz
-I- w ) l ,
(21.1b) (21.1~) (21.Id)
where R1and R2are the constant values of the reflectivity for w 5 w1 and for o 2 02,respectively. If o becomes equal to w1, B l ( w ) is undefined. But we can expand Rl/R(w) about w = wl, and find that we must take el(w1) = 0. Similarly we find that 8 4 4 = 0. Note that the dimensions of w do not affect the values of the expressions in Eq. (21.1). Thus it is possibIe to express o in any convenient units, such as wave numbers or electron volts. The integration in Eq. (21.1~)is easily carried out numerically, particularly if the reflectivity is known for equally spaced values of d. The only special case arises for o' = w, in which case one takes for the integrand the average of the values at the grid point just preceding and just following. Thus provision must be made for evaluating the integrand one grid interval outside the range of integration in either direction. If w1 = 0, we put = 0, and the procedure is slightly simpler. One possible source of error in the phase & ( w ) arises if the minimum reflectivity is very small, since the reflectivity minimum will make a large contribution to the integral. Experimental errors can well increase near a reflectivity minimum, particularly if it occurs in a narrow frequency interval. A low reflectivity, however, implies that the extinction coefficient 81 8
M. Gottlieb, J . Opt. SOC.Am. SO, 343 (1960),[LiF]. J. R.Dixon, Bull. Am. Phys. Soc. 6,312 (1961); and private communications, 1961.
338
FHANK STEltN
is small. Therefore the absorption coefficient a = 41k/X may be small enough to be measured directly in transmission experiments. It may also be possible to determine the index of refraction in the neighborhood of the reflectivity minimum. Even if the index cannot be measured, it will generally be useful to know the extinction coefficient, since the minimum reflectivity is k2/4. This technique was used successfully in LiF,a8where the and would have been very minimum reflectivity was about 3 X difficult to measure directly. The phase e(w) given by Eq. (21.1) equals zero when w = 0, and approaches zero as w + 0 0 . It represents the phase associated with a single absorption band which is well separated in frequency from other absorption bands, since we have assumed that the reflectivity is constant both above w2 and below WI. Thus we completely neglect the contribution of other absorption bands to the phase e ( w ) in the region being studied. The validity of this procedure can be established in general, in much the same way that it was established for the free-carrier case.84 A numerical test of the phase shift dispersion relation has been made for the lattice vibrations of quartz by Spitzer and Kleinman,aSwho found that there were relatively large errors for small values of the phase or the extinction coefficient k. 22. EXTRAPOLATION PROCEDURES In many cases of interest the criteria for application of Eq. (21.1) are not met, since the reflectivity has not approached a constant limit at the highest frequencies for which experimental results are available. Reflectivity measurements become increasingly difficult with increasing photon energy. Above about 6 ev vacuum ultraviolet techniques must be used, and above about 18 ev the transmission of windows is sufficiently poor that the light source must be placed inside the vacuum system with the sample. No really satisfactory way to proceed is available when the reflectivity is still varying at the highest frequency attainable. However, several extrapolation methods have been used, often with very good results.a6a One common way86-asto extrapolate the reflectivity beyond the cutoff See Figures 4-7 in Gottlieb.20 F. Stern, J . A p p l . Phys. 32, 2166 (1961),Eq. (10). 86 W.G. Spitzer and D. A. Kleinman, Phys. Rev. 121, 1314 (1961). as. A power series expansion of O(w) has been used for extrapolation by Velickf.Plc 80 H. R. Philipp and E. A. Taft, Phys. Rev. 113, 1002 (1959),[Gel. 8' M. Aven, n. T. F. Marple, and B. Segall, J . A p p l . Phys. 32, 2261 (1961), [ZnSe]. 88 R.E. Morrison, Phys. Rev. 124, 1314 (1961),[ I n k , InSb, GaAs]. 88
M
339
ELEMENTARY OPTICAL PROPERTIES O F SOLIDS
of the experimental data is to assume that R
=
Rz(wz/w)P,
w
2
wz
(22.1)
where Rz is the measured reflectivity at w2, and p is an empirical parameter chosen to give best agreement with the experimental results for small W . A sensitive criterion is that the extinction coefficient, and therefore, because of Eq. (20,3b), the phase associated with interband transitions must vanish for energies less than the energy gap. For extrapolations based on Eq. (22.1) we must replace (21.ld) by:
(22.2)
Another extrapolation pro~edure3~~40 is to assume that R = Rzexp [B(wz - w ) ] ,
w
2
WZ.
(22.3)
The constant B is usually chosen to make the derivative of the extrapolated reflectivity equal the measured derivative a t w 2 , I n this case the integration for e3(u) must be truncated a t a n upper limit w3, or it will diverge. The cutoff frequency w 3 is determined by optimizing the agreement with experiment at low frequencies. In this case we find: &(w)
=
[In
~R2! w+) Bwz] In
+
( ~ 3
W)
1
~
-2 1
(w2+w);w3-w1
BW +-In 2n
1 ~ 3 '
- 0' 1
/uz'-wZ/'
(22.4) Neither of the two extrapolation procedures just described is fundamentally sound, although both give good results in many cases. An alternative approach would be to recognize that on the high-frequency side of an absorption band the extinction coefficient falls off approximately as w-4.z8 Thus, far above an absorption band the reflectivity is determined by n = K ~ ' ~where , K = K, - sW-2. (22.5a) Here 40
40.
K,
is the contribution to the dielectric constant of all absorption
M. P. Rimmer and D. L. Dexter, J . A p p l . Phys. 31, 775 (1960), [Gel. M. G. Doane, M.S. Thesis, University of Rochester, 1961, (unpublished), [KCl, KBr, NaCl]. I am indebted to Dr. K. Teegarden for making a copy of this thesis available. Perhaps the first application of the phase shift dispersion relation to optical properties of solids was made by T.s. Robinson, Proc. Phys. SOC.(London) B66,QlO (1952), [polythene].
340
FRASK STERS
processes a t energies above those of the band we are considering. From the dispersion relation (17.10a) we deduce that
8
= (2/T)
1
W’Kz(W’)
(22.5b)
dw’,
the integral being taken over the band being studied and all lower bands. Unfortunately, experimental measurements have not been made to high enough frequencies to make extrapolation procedures based on Eq. (22.5) applicable. In the event that data which included the highest absorption bands were available, we would have K , = 1, and the reflectivity would go to zero asymptotically as wW4. The procedures we have described are not hard to carry out. Equations (20.3) and (20.8b) have been used by many workers to determine the optical constants of ~ o l i d s . ~ ~ JThe ~ J ~advantage -~* of this method over other methods of determining optical constants is that measurements may be made near normal incidence, and do not require polarization of the light beam. There are disadvantages also. One is that measurements must be made over a very wide frequency range, and must often be extrapolated beyond the upper limit of experimental data, usually in the ultraviolet. Another disadvantage is that errors in reflectivity in a small wavelength range will affect the calculated optical constants over a rather wide range. For the experiments cited above, these disadvantages have not been so serious as to offset the advantages offered by use of the dispersion relations. 23. DERIVATION OF S U M RULES
At frequencies higher than those of the highest absorption band we can write, using Eq. (17.10a), Kl(W)
= 1
-
(2/T)U-’
[
aJ
W’Kz(W’)
dw’.
(23.1)
This expression becomes more accurate as w increases. (We restrict ourselves, however, to photon energies low enough that photoionization is the dominant process.) At very high frequencies we can neglect the periodic potential in the crystal, and even the binding of the inner atomic electrons, and can treat all the electrons as free. A very simple derivation, to be given in Part IV, shows that in this case the dielectric constant is K ~ ( w )= (1
1
- Ne2/(meow2)= 1 - wp2w-2,
(23.2)
E. A. Taft and H. R. Philipp, Phys. Rev. 121, 1100 (1961), [Ag]; H. Ehrenreich and H.R. Philipp, ibid. 128, 1622 (1962), [Ag, Cu].
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
341
where N is the total number of electrons per unit volume, and m is the free-electron mass. Identifying the coefficients of in the two equations above allows us to write the sum rule O ’ K (~w ’ )
dw’ = (7r/2)Ne2/ (meo).
(23.3)
Another sum rule which can be deduced in the same way applies to the imaginary part of [ ; ( w ) ] - ’ . The reciprocal of the dielectric constant obeys dispersion relations of the same form as does the dielectric constant. If we again apply Eq. ( I 7.1Oa), and take the reciprocal of (2:3.2),we find that
/om
w’
Im ([-Z((W)]-~I
dw’ =
(7r/2)Ne2/(meo).
(23.4)
+
The imaginary part of -[;(a)]-* equals K 2 / ( K 1 2 K?), and is of considerable importance in describing the interaction of charged particles with matter, to be discussed further in Part VII. Finally, we may write a sum rule, derived from the Kramers-Kronig relation (18.1), in the form:
/om
o ’ k ( w ’ ) dw’ = (7r/4)Ne2/(meo).
(23.5)
This result, in a slightly different form, was given by Kronig.22 The sum rules (23.3)-(23.5) are formally incomplete, in that we neglected the motion of the nuclei, which will also move as free particles at the high frequencies being considered. For an atom with atomic number Z and atomic weight A , the nuclear contribution to the dielectric constant is approximately Z/ ( 1 8 3 6 4 ) as big as the electronic contribution, and can be neglected. Less trivial is the use of the free-electron mass, instead of an effective mass, in Eq. (23.2). We justify this qualitatively by noting that for a very high-frequency disturbance the effect of the periodic potential is negligible compared with the kinetic energy of the electron. The binding of the inner electrons is ignored on the same basis. The sum rule (23.3) is formally equivalent to the Thomas-ReicheKuhn f-sum rule for atom^.^**^^ We defer a discussion of this equivalence until Part V, in which oscillator strengths will be defined. An interesting consequence of Eqs. (23.3) and (23.5) is that %Av = 1 if the average is computed with o k ( w ) as a weighting factor. 42
H. A. Bethe and E. E. Salpeter, in “Handbuch der Physik” (S.Fliigge, ed.), Vol. 35, Sect. 61. Springer, Berlin, 1957.
342
FRANK STERN
We note in conclusion that the derivations leading to the sum rules do not assume the system to be in its ground state, or in thermal equilibrium. If there is emission in certain frequency ranges, as in masers, it is associated with negative values of K ~ ( w or ) k ( w ) . The sum rule shows that this emission must be compensated by additional absorption in other parts of the spectrum. IV. Optical Properties of Simple Systems
I n this Part we shall give in some detail the optical properties for two simple physical systems, the free-electron gas and the lattice vibrations of ionic crystals. These models involve phenomenological constants whose exact significance requires a much more detailed analysis than that given here. The models do, however, describe the behavior of optical properties of real solids quite well in many cases, and provide a good basis for understanding these properties. 24. THEEFFECTIVE FIELD
One of the simplest systems whose optical properties can be calculated in detail consists of particles, each of charge q and mass m*, bound to a fixed center by a force -m*wo2x,where x is the displacement. If an effective field F = Re POexp (--id) is applied, and x = Re f, the equation of motion for g is: m*(d28/dP)
+ m*y
id%/dl)
+ m*oo28 = qpoexp (-id)= qP.
(24.1)
The middle term on the left in Eq. (24.1) is a damping term proportional to the velocity of the particle, and y is a phenomenological damping constant. Such a damping constant is generally introduced in simple models to represent the processes that dissipate energy or momentum in a system being driven by external forces, and return the system to thermal equilibrium when the external forces are removed. In the electronic case, y represents the combined effect of collisions with lattice vibrations, with impurities or lattice defects, with dislocations or boundaries, or with any other departure from the perfectly periodic potential of an infinite single crystal. I n very few cases can a single phenomenological damping constant adequately describe all these scattering processes for all the carriers. We use it nevertheless, because it gives simple results and because the results have a t least qualitative, and . sometimes quantitative, validity in describing the optical properties of free-carrier systems. We will refer in Part V to some quantum-mechanical treatments in which scattering processes are taken into account more accurately.
ELEMESTARY OPTICAL PROPERTIES O F SOLIDS
343
The solution of Ey. (24.1) is: f = (*P/rn*)/(wo2
-
- iyw).
w2
(24.2)
If there are N particles per unit volume, the polarization can be written:
P
=
Nqf
=
(Nq2P/m*)/ (uo2
- w*
- i y w )*
(24.3)
The effective field P is different for different types of carriers. If the carriers are free, as in metals or in the ionosphere, Darwin43found that the effective field is the electric field E. His criteria suggest that this will also be the effective field acting on conduction electrons and holes in a semiconductor. Thus we can write
E, P, = &,E,
(24.4a)
$, =
(24.4b)
where 2, is the electric susceptibility of the free charges. For bound charges, the effective field can be deduced by drawing an imaginary sphere around the point a t which the field is required, and summing the contributions to the field from the bound charges inside the outside the The material outside the sphere is treated as a continuum, and gives a field equal to P b / ( 3 ~ 0 ) inside. For crystals of cubic symmetry, or for amorphous materials, the field at the origin due to all the other particles in the sphere vanishes.44-d6 DarwinAghas shown that the free charges do not contribute to the field seen by the bound charges. Thus the effective field seen by bound charges is
8,
=
E + Pb/(363)
(24.5a)
If we write the relation between the polarization of bound charges and the effective field they see in the form
P,
(24.5b)
= eOxb8b,
then it follows from Eq. (24.5a) that: jib Ja
=
Pb/(&)
= zb/(l
- gxb),
(24.5~)
C. G. Darwin, Proc. Roy. SOC.A146, 17 (1934); A182, 152 (1943). Experimental
evidence concerning the effective field acting on electrons in the ionosphere is discussed by J. A. Ratcliffe, “The Magneto-Ionic Theory and its Applications to the Ionosphere,” p , 154. Cambridge Univ. Press, London and New York, 1959. 44 See p. 31ff in Panofsky and Phillips.8 46 M. Born and K. Huang, “Dynamical Theory of Crystal Lattices,” p. 103. Oxford Univ. Press, London and New York, 1956. 4 6 Other calculations on effective fields in dielectrics are described by W. F. Brown, Jr., in “Handbuch der Physik” (S. Fliigge, ed.), Vol. 17, p. 1. Springer, Berlin, 1956.
344
FRANK STERN
where Sib is the susceptibility of the bound charges. Equation (24.5~)gives the Clausius-Mossotti-L~rentz-Lorenz~~ expression for the dielectric constant of liquids and cubic crystals if we replace 2 b by Z - 1. When both free and bound charges are present, it follows from Eqs. (24.4) and (24.5) that their susceptibilities add. Thus the dielectric constant is : Z
=
1
+ %a + 2,.
(24.6)
A . The Free-Electron Gas 25. GENERAL CONSIDERATIONS When w is small compared with the frequencies wo that characterize the bound electrons, the contribution of the bound electrons to the susceptibility becomes independent of frequency. Thus for frequencies from the infrared down to dc, except for that part of the infrared where the contribution of the ions is important, we can restrict our attention to the free carriers, and replace the effect of the bound charges by a frequencyindependent dielectric constant which we shall call K,. For the general case in which there are several groups of free carriers, for which wo = 0, the dielectric constant deduced from Eqs. (24.3) and (24.6) can be written: i(w> =
K,{
1-
C [wii2/ j
.
(w2
+ yjz)I + ( i / w > C
Cwlj2Yj/ (wz
j
+ ~i")II , (25.1)
where y j is the damping constant for carriers of type j , and wlj =
[Nje2/(mjcoK,)]1/2.
(25.2)
Here Nj is the concentration of free carriers in the j t h group, and mj is their effective mass. We have replaced q2 by e2, since the carriers have charge f e . Some useful numerical expressions are: wlj
[sec-'1
=
5.641 X lo4 [Nj [~rn-~]rn/(rnj~,)]~/~,
vlj
[sec-'1
=
wlj/2~ = 8.978 X
Elj [ev]
=
fiwlj = 3.713 X lO-"[Nj
Xlj [microns]
=
c/vlj
=
lo3 [Nj [~ m -~ ] m/(mj~ ,)]~ /~ , [~m-~]m/(mj~,)]~'~,
3.339 X lOlo[Nj [~rn-~]m/(mj~,)]-'/~.
(25.3)
When the damping is sufficiently weak we can drop the y's in Eq.
345
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
(25.1), and obtain the simple result:
where
The plasma frequency w , / ~ x ,defined in general as a frequency for which the real part of the dielectric constant vanishes while the imaginary part is TABLE111. OPTICALPROPERTIES OF ALKALIMETALSNEARTHE PLASMA FREQUENCY
We give the approximate wavelength A, at which the real part of the dielectric constant equals zero, and the corresponding photon energy &., The damping is measured by K ~ ( W , ) , the imaginary part of the dielectric constant a t the plasma frequency. Also given are the lattice constant a, the carrier concentration N (assuming 1 electron per atom), and the value of m*K,/m deduced from Eq. (25.2) using the observed plasma frequency and carrier concentration. XP"
Metal [microns] Li Na K Rb CS
0.21 0.34 0.395 0.435
~ ~ [ e v ] KZ (w,) 5.9 3.65 3.15 2.85
0.012 0.1 0.23 0.35
a[A]* 3.509 4.291 5.32 5.70 6.16
N[cm-a] 4.63 2.53 1.33 1.08 8.56
X X X X X
m*k/ni
m*/mc
1.00 1.38 1.51 1.45
1.45 0.98 0.93 0.89 0.83
lozz 10"
loz2
1022 10"
Taken from Ives and Brigg~.~8
* The values are for 20"C,and are taken from Pear~on.~g The value for Cs is extrapolated. c
Taken from Pines.m
small, is one of the most striking features of the optical properties of a solid. Below the plasma frequency the dielectric constant is negative, and the imaginary part of the index of refraction will be greater than the real part. Above the plasma frequency the imaginary part of the index drops sharply, and the real part becomes dominant. This behavior is demonstrated by the experiments of Wood4' on transmission through thin films of alkali metals. He found a sharp onset of transmission as the wavelength decreased. More detailed studies of the optical constants of the alkali metals were made by 47
R. W. Wood, Phys. Rev. 44, 353 (1933); R. W. Wood and C. Lukens, ibid. 64, 332 (1938).
346
FRANK STEHN
Ives and B r i g g ~ and , ~ ~ qualitatively confirm the behavior predicted by Eq. (25.1), Some r e ~ u l t s ~on~ properties -~~ of alkali metals are summarized in Table 111. 26, COSDUCTIVITY ASD MOBILITY The conductivity of a free-carrier. system containing several groups of carriers is: U = QWK2 = Uoj/(l W2T?), (26.1)
+
j
where
~j
is the relaxation or scattering time: (26.2)
r j = yj-1.
The dc conductivity uoj is usually written in the form uoj = Njepj,
pj =
erj/mj,
(26.3)
where pj is the mobility of carriers in the j t h group, and is conventionally defined to be positive for both electrons and holes. (Kote that in our notation the charge of the electron is - e . ) Some examples of mobilities a t room temperature are : Cu, 40 cm2/volt-sec ; n-type InSb, 70,000 cm2/volt-sec; PbS, PbSe, PbTe, Ge, lo3to 4 X los cm2/volt-sec. Some useful numerical relations are : pj
[cm2/volt-sec]
=
1.7589 X 10'6(m/mj)7j [sec],
rJ [sec] = yJ-' = 5.085 X
(m,/m)pj [cm2/volt-sec],
u3 [ohm-' cm-l] = 1.6020 X 10-'9NJ [ ~ m - ~ ] p [cm2/volt-sec]. j
(26.4)
A few words are necessary to explain the significance of the effective mass m* and the scattering time r which we have used as phenomenological constants in our discussion of the optical properties. The effective mass has a unique meaning only for bands whose energy can be represented as a quadratic, isotropic function of the wave vector k. For cubic crystals whose energy bands are ellipsoids with principal effective masses ml, (8
40
60
H. E. Ives and H. B. Briggs, J. O p t . SOC.A m . 28, 238 (1936), [K]; 27, 181 (1937), [Na]; 27, 395 (1937), [Rb, Cs]. A relation between the plasma frequency and the photoelectric effect was found by H. E. Ives and H. B. Briggs, J. Opt. SOC.Am. 28, 330 (1938); see also the note added in proof on page 408. W. €3. Pearson, "A Handbook of Lattice Spacings and Structures of Metals and Alloys." Pergamon Press, New York, 1958. D. Pines, Solid State Phys. 1, 414 (1955).
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
347
m2, and ma, the effective mass to be used in describing the free-carrier optical effects is6‘ m* = 3(rnl-I mz-l msl)-l, (26.5)
+
+
while for a spherical but nonparabolic band in a degenerate sample with Fermi energy E p S 1 : m* = fi2k I d k / d E I E - E ~ . (26.6) The scattering time T which enters in the theory of transport properties is generally a function of carrier energy and in many cases cannot be defined at a11.62 Thus in characterizing a whole group of carriers by a single scattering time T we make a grossly oversimplified assumption. One can make a rough correction for this by multiplying expression (27.5) for the absorption constant by T A ~ T - ~ A ~ . ~ * However, for reliable results one must use a quantum-mechanical treatment. We will refer to such treatments in Part V, and have restricted our attention here to simple models which give the over-all features of the optical behavior. IN VARIOUS FREQ U E N C ~RANGES 27. OPTICALPROPERTIES
Simple expressions for the optical properties of a free-carrier system can be obtained from Eq. (25.1) in certain wavelength ranges. For this purpose we will suppose that there is only one type of carrier present, with effective mass m*.Then we can put A, = XI, using Eqs. (25.3) and (25.5). The results can be generalized to a many-carrier case without difficulty. The two characteristic wavelengths for the system are A,, already given by Eq. (25.3)) and A, = 2ac7, or A, [microns] = 1.0709 (.t*/m) p [cm2/volt-sec].
(27.1)
At room temperature A, M 40 microns for Cu, while A, M 1000 microns for pure InSb. Most semiconductor measurements are made at wavelengths short compared with both X, and A,. In that range we find 7i M
Km[l
-
-
(wp2/w2)
+
i(ywp2/w3)].
.(
>> UP, 7 )
(27.2)
The imaginary part of the dielectric constant is much smaller than the real (~~)1/2, and IC = ~ ~ / 2 nThe . optical properties for which part. Thus n W. G. Spitzer and H. Y. Fan, Phys. Rev. 106, 882 (1957). See, for example, C. Herring, Bell System Tech. J . 34, 237 (1955), Appendix A; C. Herring and E. Vogt, Phys. Rev. 101, 944 (1956). 6) R. A. Smith, “Semiconductors,” p. 219. Cambridge Univ. Press, London and New York, 1959. 61
61
348
FRANK STERN
numerical expressions are must useful are the absorption coefficient a, and the reflectivity at normal incidence R. From Parts I and I1 we find that: (27.3a) a = 4&/A = 2kw/c = O K ~ K , , , / ( ~ C ) = U K ~ / ( ~ C E ~ ) , a [cm-l]
=
1.014 X 106k&[ev],
(27.3b)
k
=
0.9866 X 10-sa [~m-~]/&[ev],
(27.3~)
R
=
[(n
- 1)' + k']/[(n
+ l ) z + k'].
(27.4)
In Eqs. (27.3b) and (27.3c), & is the photon energy, and K~ = 1. For short wavelengths we find, making use of the numerical expressions given earlier, that Eq. (26.2) leads to a [cm-I]
=
5.262 X 1~-17n-l[Nm2/(m*2~)](A[microns])2.
>> up,y)
(w
(27.5)
We will express N in cm-8 and p in cm2/volt-sec in all numerical expressions. The reflectivity is
R
- Ra{l-
1.794 X 10-21[Nm/(m*n(~, - l))](A micron^])^], (a
+
>> u p , 7)
(27.6)
where R, = (K,*/' - l)z(~,1/2 1)-' is the asymptotic value of the reflectivity at short wavelengths. Equations (27.2), (27.5), and (27.6) have ignored interband transitions, and thus are no longer valid if these transitions are present. For semiconductors, A, and A, are big enough that Eqs. (27.5) and (27.6) should hold for a fairly wide range of wavelengths, with the wavelength of the absorption edge as a lower limit and the upper limit determined by lattice vibration effects, which have also been ignored in (27.2). The region of validity of Eqs. (27.5) and (27.6) may be small or nonexistent for metals, in which interband transitions generally play a role at wavelengths comparable with A,. The second simple case, that of very long wavelengths, can be reached without difficulty for metals, and also for impure semiconductors with low mobility, in which A, and A, lie in the not-too-distant infrared. Thus when w << w,, y, Eq. (25.1) can be approximated by M M
In this case
K~
>> I ~1 1,
+ ~ , ( 1- w p 2 ~ . "+ ) iuo/(eow). K,[(1 - Wp2'?)
iOp'T/W]
(O
<< wp, y)
(27.7)
and we have
n M k M (K2/2)'/' M 5.4753 X 10-2(A [microns]
U O ) ~ / ~ .
(a <
ELEhfENTARY OPTICAL P R OP E R T I ES O F SOLIDS
849
We give the dc conductivity u0 in ohm-' cm-I in all numerical expressions. From Eq. (27.7) we find: a
<< u p , y) 1 - 36.327(A [ r n i c r o n ~ ] u ~ ) - ~( w~ < ~<. up,y )
[em-l] w 6.8803 X 103(u0/A [mi~rons])"~,
Rw 1
-
(2/n)
-
(w
(27.9) (27.10)
The second of these is the famous Hagen-Rubens formula,54which gives the deviation of the reflectivity from 1 with an error of 20: or less for a great variety of metals a t wavelengths beyond about 4 microns. An alternative to Eq. (27.9) is an expression for the skin depth, 6 = 2 / a , which equals : 6 [microns] = 2.9068 ( A [microns]/ao)
8[cm]
=
l/*,
5.0330( Y [bIc/sec]~~)-'/~.
(w
<< w p , r'r
(27.11)
For metals i t will generally be possible to find a region of wavelengths short in comparison with A,, but long in comparison with A,. That will be true because WpT
=
3.207
Up/?
x
lo-11[NWZ*/(WZK,)]1/2/L
(27.12)
is generally very large for metals. When y << w << w, the complex dielectric constant is given by Eq. (%7.2),with ~2 << ' K~ ,. Now K~ is negative, and therefore li >> n. \Te find that both the absorption coefficient and the reflectivity are approximately independent of wavelength in this case, with the values [Cm-l]
R
=
2WpKm1"/C
=
3.763
x
10-'(Nm/?7Z*)'/*, (y << 0 << U p ) (27.1:3)
W 1
- ( 4 n / k 2 )X 1 -
X 1
- 6.236 x 1010(m*N/m)-"*p-'.
2/(OprK,112)
(y
<< w << w,)
(27.14)
The only remaining case, which applies if wPr << 1, can be realized for very pure germanium, and perhaps in a few other semiconductors. Then for w p << w << y the complex dielectric constant will be given by Eq. (27.7), but with ~2 << K', and k << 1. We find: a
R
FZ
R,[l -
[em-'] w :376.75n-1~o, ~K,W~*T'/(K,
<< w << y)
(up
(27.13)
- I)]
M Rcl{l- 2.037 X l o - * ' [ ( ~ ~ * / m ) . V ~ ' ( ~ , l)-']], (w,
where R, 54
= (K~'/*-
1)"~ m
<< w << y)
+ l)-', as for Eq. (27.G).
E. Hagen and H. Rubens, Ann. Physzk [4] 11, 873 (1903)
(27.16)
350
FRANK STERE
28. REFLECTIVITY MINIMUM The reflectivity of a free-carrier system a t normal incidence has been shown in Eq. (27.6) to decrease with increasing wavelength a t short wavelengths and, in (27.10), to approach 1 asymptotically a t long wavelengths. Thus there will be a minimum in the reflectivity, whose spectral position gives a good measure of (N/m*). I n the simplest case of negligible damping the dielectric constant is given by Eq. (25.4) and it is clear from (27.4) that the reflectivity a t normal incidence will have a minimum value of 0 when n = 1, which occurs at the angular frequency w = w~[K,/(K, - 1)] l i z . From Eq. (27.4) it can be shown without difficulty that the condition for a reflectivity extremum is (K1
- 1) (dn/dW)
+ Ki(dk/dW)
=
0,
(28.1)
where we have considered the material to be nonmagnetic. From Eq. (28.1) it is possible to obtain an implicit equation for the frequency a t which the reflectivity is a minimum. The equation has a simple solution if we assume that only one type of carrier is present and if wPr >> K,. I n that case we find ;
where w p r is given by Eq. (27.12), and X, by (25.3). The position of the minimum is given fairly accurately by Eq. (28.2a) if wpr >> K,, but the magnitude of the minimum reflectivity given by (28.2b) should be considered only a crude estimate. When w P r >> K,, the predicted value of Rminis of the order of 15; or less, and experimental difficulties may prevent an accurate measurement from being made. I n the case of metals the plasma frequency is in the ultraviolet or the visible, where interband transitions can occur, so the damping constant y = 7-l deduced from the dc conductivity no longer characterizes the actual losses. 29. EKERGY DENSITY The energy density in a material medium can be considered as the sum of the energy in the field and the energy in the particles of the medium. We illustrate this for a free-carrier system, ignoring damping for simplicity. Then the dielectric constant is given by Eqs. (24.10) and (24.11),
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
35 1
and the time-averaged energy density given by (6.9) is:
+
T J = r~ & * * E [ K I w ( d K l / d u ) which gives:
O 0
+1
11,
K
. . I
(29.1)
=
$c~E**EK,,
w
=
iE$*'EK,(Wp2/W2),
w
1 up; 5 up.
(29.2a) (29.2b)
The kinetic energy of the free-carrier motion is found from Eqs. (24.2) and (24.11) to be:
0 (carriers)
=
$
c Nlml(dxj/dt)2 4 c NjmjW29,*'fj =
j
I
= $ & * - E K , ( w ~ ~ / w ~(29.3) ).
The remaining energy density can be written in the form
0 (field)
= =
+ tp$t**H t€&*'E(K, + I 1). +&**EK,
(29.4)
K
The sum of Eqs. (29.3) and (29.4) is equal to 'the total energy density (29.2). We find that for frequencies below the plasma frequency the kinetic energy of the carriers accounts for half of the total energy. In this frequency range & is imaginary, and the fields decay exponentially. At higher frequencies the kinetic energy of the carriers is a fraction uP2/(2u2)of the total energy, dropping off as their contribution to the dielectric constant becomes less important. When dissipation is introduced, the separation of the energy density into two parts like Eqs. (29.3) and (29.4) can no longer be made. In the extreme case for which w << y, the kinetic energy of the carriers becomes unimportant, and the energy density is mainly that of the magnetic field. The time-averaged energy density in this limiting case is 0 = ( u / u ) B * * R . The power dissipation is &uB*.E,and if we define the Q of the system to be w 0 divided by the power dissipation, we find that Q 3 in the limiting case of high conductivity.
-
B. Optical Modes in Ionic Crystals 30. PHENOMESOLOGICAL DISCUSSION
The lattice vibration spectrum of a crystal with 2 atoms per primitive cell contains 6 branches, 3 of which correspond a t long wavelengths to sound waves in the crystal, and are therefore called the acoustical branches. At long wavelengths the remaining 3 branches, with frequencies of the
352
FRAPiK S T E R S
order of 1013 sec-l in most crystals, correspond to motion of the 2 atoms in opposite directions. I n ionic crystals these modes produce a strong polarization, and interact strongly with electromagnetic radiation (transverse modes) and with charged particles (longitudinal modes). They are called the optical modes. Interaction between photons and phonons requires that both energy and momentum be conserved. This condition can be met for single-phonon processes only for the long-wavelength optical modes in a crystal, since the wave vector of a photon with frequency sec-l is 2 X lo3 cm-l, and is very small compared with the range of wave vectors (=los em-l) allowed for lattice vibrations. There are also weaker interactions involving 2 or more phonons. These have been extensively studied recently in highresolution absorption measurements, and can give detailed information about key points in the lattice vibration spectrum.55We will not consider the multiphonon processes here. In this section we shall make use of the result, to be established in a later section, that the absorption of infrared light by an ionic crystal is strongly peaked at the angular frequency W T of the long-wavelength transverse optical modes. From this fact alone we can predict the frequency dependence of the dielectric constant, making use of the dispersion relation (17.lOa), which we rewrite here in the form: K ~ ( w )=
K,
1
$. ( ~ / R ) P o ’ K ~ ( u ’ )
-
w2)-l
dw’,
(30.1)
where the integral is extended only over the absorption associated with the lattice vibrations. The effect of all absorption mechanisms a t higher frequencies is included in K,, which is assumed to be real and frequencyindependent in the frequency range being considered. If we approximate the absorption peak a t wT by a delta function, W’KZ(W’) = ( n C / 2 ) 6 ( w ’ - C O T ) , then Eq. (30.1) leads to: Ki(U)
-
K,
=
~ ( W T ’
-
d-’.
(30.2)
The constant of proportionality C is easily found in terms of K ~ ,the lowfrequency limit for the dielectric constant in the lattice vibration range. Then we find: K ~ ( C O ) = K,
+
-
( K ~
K,)uT~(wT~
-
w2)-l.
(30.3)
This result will be derived again in the following from a microscopic treatO5
See, for example, the paper of W. G. Spitzer, J . A p p l . Phys. 34, 792 (1963), which discusses phonon assignments for GaAs and CdS and refers to experimental results for a number of other semiconductors.
ELEMENTARY OPTICAL PROPERTIES O F SOLIDS
353
ment of lattice vibrations. Comparison of Eq. (30.3) with (30.1) shows that
1 w'K~(u')
dw' = ( n / 2 )( K ~
K,)wT',
(30.4)
where the integration is limited to the absorption peak near COT. Since ~2 is nonnegative for a system in its ground state or in thermal equilibrium, it follows that K~ 2. K., We have already seen that the frequency for which the dielectric constant vanishes has a special significance. I n particular, a longitudinal electric field can exist at that frequency even in the absence of external sources. This was shown in the discussion leading to Eq. (7.13), and also follows from Eq. (4.1), which shows that H = 0 if the dielectric constant vanishes, and that E is parallel to the wave vector k if H = 0. It is therefore natural to suppose that the long-wavelength longitudinal optical mode will occur a t the frequency for which K = 0. This result is confirmed by the detailed theory to be given below. We then find from Eq. (30.3) that (30.5) where w l ; / 2 a is the longitudinal optical frequency. Equation (30.5) is often called the Lyddane-Sachs-Teller relation.66 In crystals having two identical particles in the primitive cell, such as crystals with the diamond structure, the long-wavelength optical modes produce no polarization, and cannot give the one-phonon absorption at COT which ordinarily dominates the optical absorption. Other absorption mechanisms, such as multiphonon processes, will be present, but they will be relatively weak. Thus Eq. (30.4) shows that K~ - K, will be small in crystals with the diamond structure. It follows from Eq. (30.5) that the longitudinal and transverse optical modes of long wavelength are degenerate in these crystals. The experimental value for the microwave dielectric constant of germanium is 16.0 f 0.3 at 4.2°K,5' while the optical value at 16 microns is 16.01 at room temperatureu and 3%, to 4% smaller at very low temperature^.^^ I n gallium arsenide the static and optical dielectric constants differ by a considerably larger amount, the room temperature values being K~ = 12.5, and K, = 10.9.60 R. H. Lyddane, R. G. Sachs, and E. Teller, Phys. Rev. 69, 673 (1941); see also p. 155 in Frohlich,' and p. 86 in Born and Huang.46 KT F. A. D'Altroy and H. Y. Fan, Phys. Rev. 103, 1671 (1956); see also W. C. Dunlap, Jr. and R. L. Watters, ibid. 92, 1396 (1953). 68 C. D. Salzberg and J. J. Villa, J . O p t . Soc. Am. 47, 244 (1957); 48, 579 (1958). M. Cardona, W. Paul, and H. Brooks, Phys. Chem. Solids 8, 204 (1959). 6o K. G. Hambleton, C. Hilsum, and B. R. Holeman, Proc. Phys. Soc. (London) 77, 1147 (1961).
354
FRANK STERN
For angular frequencies between W T and W L the dielectric constant (30.3) is negative. If damping is confined to a very narrow frequency range near wT1 we can ignore the imaginary part of the dielectric constant, and find that between W T and W L the real part of the index of refraction vanishes. Thus the crystal will be totally reflecting throughout this range. In real crystals there will be some damping, and the reflectivity will be less than 100%:. There will be a peak in the reflectivity between W T and W L ; several reflections from the surface of an ionic crystal will give a fairly narrow range of infrared frequencies. The reflection peak is at the Reststrahl frequency, important both because of its origin in the vibration spectrum of the crystal and as a practical source of relatively monochromatic infrared radiation. We have seen that the main features of the optical properties of ionic crystals in the lattice vibration frequency range can be found very simply if we assume that absorption is strongly peak near LOT. I n the more detailed considerations below we show how the parameters in Eq. (30.3) are related to the atomic properties of the medium. 31. ATOMICTHEORY OF OPTICALMODES The polarization in an ionic crystal with one ion pair per primitive cell is given by
P
=
NQI
+ be$,
(31.1)
where Q is an effective charge, s' = (5, - ii-) is the relative displacement of the positive ions with respect to the negative ions, N is the number of ion pairs per unit volume, p is the effective field, and b is a constant describing the combined polarizability of the two ions. If we substitute p = E P'/(3to) from Eq. (24.5a), as appropriate for the motion of bound charges, we find:
+
P
=
(NQI
+ b&)/(l
- Qb).
(31.2)
For high-frequency oscillations the displacement of the ions is small, and the polarization is given in terms of the high-frequency dielectric constant K, by P = ( K , - l)&. Thus b / ( l - +b) = K , - 1, and Eq. (31.2) can be rewritten :
-
P
=
NQ's'
+
where
Q'
=
(K,
(K,
- 1)~&
+ 2)Q/3.
(31.3) (31.4)
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
355
The equations of motion of the ions can be written in the form
+ QP,
M+(d%+/dt2)
=
-K(fi+ - ii-)
M-(d2ii-/dt2)
=
K(ii+ - ii-) -
QP,
(31.5a) (31.5b)
where M+ and M- are the masses of the two ions, and K+ and Ei_ are their displacements. Born and Huang61 show that for crystals with tetrahedral or higher cubic symmetry the force constant K is a scalar. If we multiply Eq. (31.5a) by M - and (31.5b) by M+, subtract, and divide by (M+ M - ) , we obtain:
+
d4,(d2P/dt2)
+
=
+ QP,
-KI
(31.6)
M-) . Replacing P by E where M , = M+M-/(M+ and using Eq. (31.3) to eliminate p, we find:
+ p/(3e0) as before,
M,(d21/dt2) = - ( K - [NQQ'/(3eo)])I
(31.7a)
+ Q'E.
If we define M*T2 to be the quantity in braces in Eq. (31.7a), and introduce a phenomenological damping constant y, we can rewrite the equation of motion in the form :
+ y(ds"/dt) +
(d2S/dt2)
WT%
=
(Q'/Mr)E.
(31.7b)
Equations (31.3) and (31.7), together with Maxwell's equations, are the fundamental relations describing the long-wavelength optical modes. They apply to waves whose wavelength is large compared with the lattice constant (so that the force constant K in Ey. (31.5) is independent of wavelength) but small compared with sample dimensions. To proceed further, let us consider a plane wave I
(31.8)
I. exp (iq-r - i d ) ,
=
with similar expressions for p and k. Equations (31.3) and (31.7) show t,hat s', E, and are parallel. Substituting (31.8) into (31.7b) leads to: =
(&'/M,)
(UT2
-
fJJ2
- i'yu)-'E.
(31.9)
If Eq. (31.9) is substituted into (31.3) we find:
-
P/Q =
[K,
-1
+ (NQ"/(M,eo))
(COT'
-
u2 - iyu)-']E. (31.10)
K , ) w T ~ ( w T ~-
w2
- iyw)-l,
The dielectric constant is: Z((w) =
where
K,
+
K8 61
-
( K ~
-
K,
See p. 105 in Born and Huang.46
=
NQ'2/(MreoW~2).
(31.11) (31.12)
356
FRANK STERN
We have thus verified the dielectric constant found in the previous section, and can rewrite Eq. (30.4) in the form:
/
W ’ K ~( w ’ )
dw’ = (7r/2) NQ’2/ ( M,Q) .
(31.13)
A numerical expression for evaluating Eq. (31.12) for crystals with rocksalt or zincblende structure is: K~
-
K,
=
1.968(&’/e)2(X~ [rnicroi~~])~A,-~(a[A])-3, (31.14)
where e is the magnitude of the charge of the electron, a is the lattice constant, A , is the reduced atomic weight of the ions on the C’* scale, and A T = 27rc/wT is the wavelength in vacuum corresponding to angular frequency W T . For a CsCl structure the numerical coefficient in Eq. (31.14) is 0.984. The force constant K in Eq. (31.5) is related to the elastic properties of the crystal, as well as to the optical mode frequencies. This leads to the relation6*Ia B = ~ N L ~ IK,, T ~2)~ ((K , ~)-‘wT*, (31.15a)
+
+
where B is the bulk modulus (the reciprocal of the compressibility), and T O is the nearest neighbor distance. I n numerical form this gives:
B [dyne/cm2] = 5.891
x
1 0 1 4 C A r (f ~ 82) ( K ,
+ 2)-’(a[Al)-’(X~
[microns])-*
a
(3I. l5b)
where C is 4 for the rocksalt structure and for the zincblende and CsCl structures, and the other quantities have the same significance as in Eq. (31.14). For many alkali halides the calculated and observed bulk moduli agree within about lot;, 32. LOSGITUDISAL .4ND TRASSVERSE J I O D E S ; EFFECTIVE CHaRGE
A simple physical picture of the significance of the transverse optical mode angular frequency W T is best obtained from Eq. (31.9), which gives the relative displacement s produced by an electric field of angular frequency w. We see that in the absence of damping a displacement can be present without a driving field only a t the angular frequency w T . Furthermore, an external field with w near w T will produce large displacements 62 63
B. Szigeti, Proc. Roy. Soc. A204, 51 (1960). Equations (9.29) and (9.30) on p . 111 in Born and Huang46 are incorrect; the factor d l should be deleted.
ELEMESTARY OPTICAL PROPERTIES O F SOLIDS
357
of the ions, limited by the damping constant y. Thus a transverse field with w near W T will excite large-amplitude oscillations. The energy transfer from these modes to other modes, caused by anharmonicity of the effective potential, by other phoiion scattering mechanisms, and by the nonlinear dependence of the electric moment of the crystal on the lattice displacements, is responsible for the energy absorption peak near W T . Calculations for several damping mechanisms have been made for the combination bands of GaP by Kleinman and Spitzer,6*who conclude that the anharmonic potential is the major source of damping in that case. The physical discussion for longitudinal modes is similar to the one we have just given for transverse modes, except that the driving force arises from a lionstatic charge distribution. If we assume the presence of a source p =
Re
[po
exp ( i q - r - i w t ) ]
=
Re F,
(32.la)
we have a n electric displacement D given by
D = Re [Do exp (iq-r - id)]= Re D, D
qgq2,
=
(32.lb) (32.lc)
where Eq. ( 3 2 . 1 ~ follows ) from blaxwell’s Eq. (2 .lb ). If we now take the divergence of Eq. (31.7b), and use the auxiliary results
E WL*
=
(D - NQ’S)/(QK,),
-
WT*
=
NQ’*/(i1fr~o~,),
(32.2a) (32.2b)
which follow from (31.3) and (31.12), we find: v . [ ( d 2 6 / d t 2 ) f y(ds’/dt)
+
WL2S]
=
[&’/K,]V’D,/(hl,Eo)
For the transverse part of E, V ~ S ‘ T= 0. Thus Eq. (32.3) shows that the Fourier components of an external charge are sources for longitudinal lattice vibrations. This is consistent with Eq. (7.10), which shows that an external charge component F(q, w ) is the source for the corresponding component of the longitudinal vector potential. Equation (32.3) also shows that in the absence of a source F, we can have a longitudinal lattice vibration only at the angular frequency wL, and only if there is no damping. In a real crystal, for which y is not zero, a fixed source j(q,W) with angular frequency near W Lwill give rise to large longitudinal lattice displacements, and these will lead to large pou-er dissipation in the medium. The discussion in the preceding paragraphs gives the physical basis for 64
1). A. Meinman and W. G. Spitzer, Phys. Rev. 118, 118 (1960).
358
FRAEK STERN
the identification of W T and W L as transverse and longitudinal optical angular frequencies, and also for the assumption that the absorption of electromagnetic radiation is strongly peaked near w T , One interesting feature must be added to our discussion of transverse waves. It follows from Eq. (4.3) that the relation between the wave vector and the frequency of a plane wave is
where we have assumed the material to be nonmagnetic. For a given w this equation uniquely determines the real and imaginary parts of the propagation vector q (but not its direction) if we assume that the wave is homogeneous. On the other hand if we fix q, there will be two solutions for w . When 1 q I >> W T K , ~ / ~ / C ,the solutions are w N W T and w N c 1 q 1 corresponding respectively to a pure lattice vibration of angular frequency W T and to a pure electromagnetic wave traveling a t the velocity K,-% On the other hand, for longer wavelengths there is substantial coupling of electromagnetic and mechanical energy in both frequency branches. This is discussed in some detail by Huang.E6 When the wave vector becomes much smaller than WTK,'/'/C the two solutions of Eq. ( 3 2 . 4 ) approach 0 and W L . Thus a t these long wavelengths the 3 lattice vibration modes approach the same frequency. The longwavelength optical lattice vibrations have been discussed recently by several authors.66nfJ a . Efective C h a r g e
The effective charge Q which was introduced in Eq. (31.1) is called the SzigetP charge; it is a measure of the polarization associated with the relative displacement of the positive and negative ions. I n ionic crystals it is generally smaller than the static charge per ion that one would use in a calculation of the cohesive energy of the crystal. Only if the ions are small and rigid will the two effective charges be the same. A material in which the Szigeti charge is surprisingly large is Sic, for which Q = 0 , 9 4 e . @ One might characterize S i c as a covalent crystal, and expect it to have zero static charge. The large value of Q in S i c and its deviation from the nominal value in alkali halides arise from the distortion a K. Huang, Proc. Roy. Soc. A208, 352 (1951); see also pp. 90-98 in Born and Huang.dE (6 H. B. Rosenstock, Phys. Rev. 121, 416 (1961). 87 A. A. Maradudin and G. H. Weiss, Phys. Rev. 123, 1968 (1961); T. H. K. Barron, ibid. 125, 1995 (1961). (8 W. G.Spitzer, D. Kleinman, and D. Walsh, Phys. Reo. 113, 127 (1959) ; W. G. Spitzer, D. A. Kleinman, and C. J. Frosch, ibid. 113, 133 (1959).
359
ELEMENTARY OPTICAL PHOPEHTIES OF SOLIDS
of the atoms when they are displaced. Szigeti,B2Matossi,6Band Cochran’o have discussed the meaning of effective charge in some detail. The electric field accompanying longitudinal lattice vibrations can also be described by an effective charge. To show this we note that if y = 0, the electric displacement D vanishes at the longitudinal mode angular frequency W L , and P = -eOE. If we substitute this relation in (31.3), we find EL
= -N[&’/K,]SL/EO
= -N[(K,
+ 2)&/(3K,)]s~/Eo
(32.5)
to be the electric field associated with a longitudinal wave in which the relative displacement of the positive and negative ions from their equilibrium positions is SL. The expression in brackets in Eq. (32.5), which also appears in (32.3), is called the Callen7’charge. It is the effective charge used in transport calculation^,^^ since the electric field of a longitudinal optical phonon, as given in Eq. (32.5), will scatter charge carriers. A numerical expression for the Callen charge, &’/K,, in units of the electronic charge is: &’/ex,
=
0.7128{A,(a[A])8(~,-*-
) 1 / 2 ( X ~C.mi~rons])-~,
(32.6)
K~-I)
where X L = 2rc/uL, and the other quantities are the same as in Eq. (31.14). For some materials XL?* and K, may be known even though AT and K* are not known. I n such cases Eq. (32.6) gives an upper limit for the Callen charge if we let K#-’ + 0. The Szigeti charge Q is easily obtained from Eq. (32.6) if we remember that Q’ was defined in (31.4) to be ( K , 2)&/3. Thus in materials with a high optical dielectric constant the Callen charge is about one-third of the Szigeti charge.
+
33. REFLECTIVITY
The most commonly used experimental method for studying the optical properties of ionic crystals in the lattice-vibration region of the spectrum is through measurement of reflectivity. Absorption measurements can also be used if thin samples are available. The reflectivity maximum is of considerable interest, since this gives the position of the Reststrahlen, the band of wavelengths obtained after several reflections F. Matossi, 2.Nuturforsch. 14a, 791 (1959). W. Cochran, Nature 191, 60 (1961). H. B. Callen, Phys. Rev. 76, 1394 (1949). T S See, for example, H. Ehrenreich, Phys. Chem. Solids 2, 131 (1957). 78 The energy of longitudinal optical phonons in several semiconductors has been deduced from tunnel diode characteristics by R. N. Hall and J. H. Racette, J . Appl. Phys. 32, 2078 (1961). (0
70
360
FRANK STERN
from an ionic crystal. The maximum is not very sharp, and thus cannot be used for reliable determination of optical parameters of the crystal. A maximum in the reflectivity R, given in Eq. (27.4), occurs where there is a maximum in (1 - R)-I, or in (n2 IC’ l ) / n , which, for K:, << I ~1 1, implies a maximum in ( 1 ~1 ,312 I ~1 i1”)/K2. If we define
+ +
+
-
2 = (W2
and assume that y
WT2)/(WL2
-
<< w T , we find from the dielectric K1
K2
(1
+
(33.2a) (33.2b)
UZ) ‘ I 2 / (UWTZ‘),
where a =
constant (31.11) :
- 1)/2,
= K,(Z
= YK,
(33.1)
W?),
b
( K ~- K , ) / K , ,
=
(
K
-~ l ) / ~ , .
(33.3)
The position of the Reststrahl peak, urnax,is given by ~ m a x= W T (
where zm is a root of 3abz3
1
+ azm)
(33.4)
l”,
+ (46 - u - 2ab)z2 - (3b f 2 ) +~ 1 = O
(33.5)
lying between 0 and 1, since this is the range of 100% reflectivity when y = 0. The solutions of Eq. (33.5) can be given in simple form around the edges of the range of values taken by a and b : a > 0, 0 < b < 1. These solutions are :
+ 2) - ((3b + 2 ) ’ - 16b)”2]/(8b), = [a(2b + 1)]--1’*, = [(l + - l]/u, zm = [(4 + 3a)”2 - 2]/(3a), Z,
=
[(3b
2,
2 ,
a)1’2
0;
(33.6a)
m;
(33.6b)
a
=
a
-+
b
=
0;
(33.6~)
b
=
1.
(33.6d)
To facilitate the determination of z,, the solution of Eq. (33.5) for a rather coarse grid of values of a and b is given in Table IV. Rough interpolation in the table can be used to give a good trial root to substitute in Eq. (33.5) for determination of an accurate value. It is somewhat disconcerting that this procedure, when applied to a number of alkali halides, gives slightly poorer agreement with experiment than the approximation zm = (4b 2)-l given by Havelock.74 These considerations concerning the reflectivity maximum are of some formal interest, but do not always describe the observed reflectivity
+
74
T. H. Havelock, Proc. Roy. Soc. A106, 488 (1924) ; see also p . 123 in Born and H ~ a n g . ~ 6
361
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
TABLEIV. REFLECTIVITY MAXIMUM IN IONIC CRYSTALS
We tabulate the solution of Eq. (33.5) lying between 0 and 1 for a range of values of a = ( K , - K , ) / K , and b = ( K , - 1)/~,. The angular frequency of the reflectivity maximum is given in terms of the tabulated value z, by wmsx = o T ( 1 az,)”2, where W T is the transverse optical mode angular frequency.
+
a
b: 0 . 0
0.25
0.5
0.75
1 .o
0.0 0.5
0.500 0.449 0.414 0.366 0.232
0.431 0.385 0.353 0.310 0.193
0.360 0.323 0.297 0.262 0.165
0.298 0.269 0.251 0.224 0.144
0.250 0.230 0.215 0.194 0.128
.o
2.0 10.0
accurately. This is because the reflectivity near the peak is rather slowly varying, and can easily be changed by fairly small deviations from the simple model we have used, based on the dielectric constant (31.11) with negligible damping. Reflectivity curves showing’ one or more subsidiary maxima between W T and W L are not uncommon. The determination of the reflectivity minimum is more straightforward. When the imaginary part of the dielectric constant is small, the minimum is close to the point K~ = 1. If we define w0 by K ~ ( w O ) = 1, then the reflectivity minimum occurs for Wmin
M
WO
+ [(
KZ/K~’)
(4.1’
- (K ~ ’ / K ’ )
(33.7a)
}]w=wg,
where the prime indicates the derivative with respect to the reflectivity at the minimum is
w.
The value of (33.7b)
Rmin N [ K Z ( W O ) ] ’ / ~ ~ .
If we use Eq. (31.11) for the dielectric constant, we find:
Rmin
(Ks
- 1) ( K , - l)ay2/[16(K~-
K,)’WT2].
(33.813)
EFFECTS 34. FREE-CARRIER In most ionic crystals the concentration of free electrons and holes is negligibly small. But in semiconductors the free carrier concentration can be appreciable. We shall see that this modifies our conclusions about longitudinal lattice modes, but not about the transverse modes.
362
FRASK S T E H S
In a system that does not contain externally introduced charges, Maxwell’s equation [Eq. ( 4 . l b ) l can be written K(w)k.E = 0, for which the solutions are E = 0, K ( W ) = 0, or k - E = 0. The first of these is uninteresting, since it follows from Eq. (4.1) that all the fields will be zero. In the second case, Eq. (4.1) shows that H = 0, and k /I El which is a longitudinal mode. The third case is a transverse mode. If we neglect damping, the equation of motion for the lattice vibrations, (31.7b), shows that the modes at the angular frequency W T will not produce an electric field. Thus the transverse optical modes do not excite the free carriers, and are unaffected by their presence. The longitudinal modes can exist without a driving force at frequencies for which K ( O ) = 0, a condition which can be exactly satisfied only if there is no damping. Making use of the discussion of effective fields a t the beginning of this chapter, in particular of Eq. (24.6), we find that the polarization can be expressed as the sum of the contributions of the free carriers] the ions, and the bound electrons:
P
=
NeqX
+ N&’s +
(K,
-
l)~cE,
(34.1)
where N , and N I are the concentrations of free carriers and of ions, respectively, and the other symbols are defined in Eqs. (24.3) and (31.3). If we assume that there is only one group of free carriers] characterized by effective mass m* and damping constant ye, we find from Eqs. (24.6), (25.1), and (31.11) that ;(a)
- w * ~ ( w ?+ iysw)-’
= K,[1
+ ( w 2 - w$) (w$
- w2 - i7rw)4], (34.2)
where 71 is the damping constant for the lattice motion. If both damping constants vanish we find that the dielectric constant will have 2 zeros, given by : w2
= +(w,~
+
WL.”)
* [t(wp2 +
WL*)2
- WP*WT2-J1’2.
(34.3)
These two longitudinal modes of the system will in general consist both of ionic and of free-carrier motion. Equation (34.3) was derived by Yokota’j directly from the equations of motion. The longitudinal mode frequencies are easily understood in the limiting cases when up << w L or w,, >> w L , If the free-carrier effects are minor, then the high-fiequency mode will be close to W L , while the low-frequency mode will be a t w p ( w T / w L ) = w ~ ( K , / K ~ ) ~ / ~This , reduction in the free-carrier plasma frequency arises because the low-frequency dielectric constant has , of the value K, which it would have in the absence of the value K ~ instead 75
I. Yokota, J. Phys. SOC.Japan 16, 2075
(1961).
ELEMENTARY
owrcfi
363
PROPERTIES OF SOLIDS
the ions. If wP >> WL, the free-carrier plasma frequency is affected very little by the lattice, as we would expect. But the lattice mode now occurs at WT, instead of at WL. This happens because the free-carrier dielectric constant is large and negative at low frequencies. Thus the dielectric constant can vanish only if the ionic contribution is large and positive, which occurs only in the region of maximum dispersion near up. This zero in the dielectric constant will probably not occur when damping effects are considered. The effect of the lattice vibrations on the dielectric constant must be taken into account if optical properties are used to determine the freecarrier effective mass. The necessary corrections for a general case are best obtained from Eq. (34.2), but a simpler expression is possible at short wavelengths. We find :
- cA2, R ( w ) = R,[1 - 2Km-'12(K, - 1)-'CA2], K(W)
=
>> up,W L ) (W > > wP, W L )
K,
C [P-~] = 8.97 X 10-22N [~rn-~](rn/rn*)
(34.4a)
(W
+ (K#
K,)
(34.4b)
(AT [P])-~,
(34.4c) where R, is the short wavelength reflectivity as in Eq. (26.6), and wavelengths are expressed in microns. Thus the effective mass calculated without considering the lattice vibrations will be smaller than the actual effective mass. The effect of the lattice vibrations on the position of the reflectivity minimum is easily calculated if up>> W L and damping is neglected. We find: Wmin
=
WpK,I/'(
K,
- 1)-'"[1
+$
(Ka
-
K,)
Km-1W~2Wp-2]l
(Up
>> W L ) (34.5)
which is to be compared with Eq. (28.2a). When the carriers cause only a small perturbation of the longitudinal lattice mode, we find that in the absence of damping
which is to be compared with Eq. (33.8a). The effect of screening on the interaction of free carriers and lattice vibrations has been discussed by Ehrenrei~h.7~ 70
H. Ehrenreich, Phys. Chem. Solids 8, 130 (1959).
364
FRANK STERN
V. Quantum Theory of Absorption and Emission
35. ABSORPTION The interaction of a system of charged particles with electromagnetic radiation is described in nonrelativistic quantum mechanics by the Hamiltonian”: i-1
a-1
where 4 and A are the scalar and vector potentials of the applied field, p i and mi are the charge and mass of the ith particle and the ( p , . - q , A , ) /mi are its velocity components in rectangular coordinates. ‘Uois the potential energy of the system when the external fields are removed. The relation between E and B and the potentials is given in Eq. (7.3a). We shall consider here a single plane wave with 4 = 0 and with vector potential
A = Re
[A, exp (z1.r - i d ) ] = Re A,
(35.2)
and shall not restrict ourselves to the usual case of transverse fields, because we want to apply the results to the longitudinal case in Part VII. We saw in Eqs. (7.8b) and (7.10) that we can always choose the gauge 4 = 0 even if the field is not purely transverse. From Eq. ( 3 5 . 2 ) and the relation p = -ifiV, we find: , Asp
+ p * A = 2A. (p + ihk) = 2A.p+.
(35.3)
The additional term 6Ak in p+ will have no effect for transverse fields, for which k e i = 0, but must be included when we discuss longitudinal fields. The interaction between the field (35.2) and the charges is thus XI=
0=
0 exp
- iwt)
+ Q* exp ( i w t ),
- C (q;/2mt)A0 exp (z1.r) -p,+.
(35.4a) (35.4b)
t=I
To obtain &* from &, replace io by i o * , k by -k, and p,+ by pi - ink. One of these terms will induce upward transitions, and the other will induce downward transitions at the same rate?* Considering only one of these terms, we find that if the system was initially in the state Qjl the 77
78
J. H. Van Vleck, “The Theory of Electric and Magnetic Susceptibilities,” p. 20. Ox-
ford Univ. Press, London and Sew York, 1932. L. I. Schiff, “Quantum Mechanics,” 2nd ed., p, 252. McGraw-Hill, New York, 1955.
365
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
probability of finding it in state 9, a time t later is given by78a
1 Cj 1 qj) i2 sin2 [ + ( u m j - ~ ) t ] / [ f i ( ~ m j - ~ ) ] 2 , (35.5) where fiu,j = Em - G j is the energy difference between the upper and lower states. If the number of states for which Em - G j lies between liw 1
(qm
+
and fiu dG is transition rate
p(&)
d&, an integration of (35.5) over these states gives a = (21/fi)
1
(qm
i 0 i qj) 1 2 p ( & ) -
(35.6)
If the initial and final states are discrete, then a constant transition rate will be obtained only if the perturbation X I contains a band of frequencies which includes u m j , 7 8 The rate at which the interaction Hamiltonian (35.4)induces transitions is thus given by TO
= [ire2
1 Lo j2/(2m2h)]1 ( q m I iio exp
N
(zk-r)
C pi+ I \kj)
(35.7)
12p(E),
i-1
where we have taken the electronic case (qi/mi) = -e/m, and have introduced a unit polarization vector &, Induced transitions will take place both from j to m and from m to j . The net rate of transitions from state m to state j is given by rind
=
To{ f ( E m )
- f ( E j ) 3 - f ( E j ) C1 - f ( E m ) 11 - f(M1,
[1
(35.8) roc f ( E m ) where f(&) is the probability that the state of energy & is occupied. If the lower state, with energy E j is more likely to be occupied, as in thermal equilibrium, then our sign convention for r i n d gives a negative net rate of induced transitions, and corresponds to absorption. If the upper state can be populated more than the lower state, induced emission will result, as in masers, and r i n d will be positive. To this point we have talked about transitions of the entire system. In most cases, however, a one-electron approximation is adequate, and we can describe the transition as a change in the state of a single electron. If the total wave function \k can be taken to be a determinant of oneelectron wave functions $ i , then it is easily shown7gthat the induced (or stimulated) transition rate is given by =
rind
m ‘*a 79
= Lire2
I
=
I Po exp
($m
1 2 / / ( 2 m 2 f1 im )~( ~ P ( E ) C J F ( E~ )f
(zknr) ‘p+ I
$j)j
(35.9a)
( ~ j ) ~ ,
(35,gb)
See Eq. 29.17 in Schiff.78 E. U. Condon and G. H. Shortley, “The Theory of Atomic Spectra,” p , 171. Cambridge Univ. Press, London and New York, 1953.
366
FRASK STERN
where p ( & ) now refers to the density of one-electron states per unit energy, and &j and &, are now one-electron energies. Our considerations here assume that k is real, although we shall later apply the results occasionally to cases in which k is complex. When the imaginary part of f is small we can factor out exp ( -2kz.r) in Eq. (35.9a), which shows that the transition rate is proportional to the local field intensity. I n such cases the matrix element (35.9b) should involve only k,, the real part of G. The absorption coefficient CY is the power absorbed per unit volume divided by the incident flux. The power absorption per unit volume is given by fiurind provided p ( & ) in Eq. (33.9) is referred to unit volume. The incident flux is
s=
3 Re [(-id*) x (& x i ) / p ] = ( w / 2 p ) Re [ ( i * . i ) f- (6 * .G)i] = [nwz/(2cp)] I I2k1, (35.10) where n is the index of refraction, f - i *is assumed to be zero (i.e., we now Re (E* x H )
=
restrict our attention to homogeneous transverse waves), and Eq. (4.4) was used. fl is a unit vector in the direction of propagation. The absorption coefficient for light polarized along Po is therefore given by a(w) =
[ncpe2/(nwm2)] 1311 j2p(&)[f(&j)
- f(Em)].
(35.11a)
We can also express the power dissipation per unit volume, AwT,,~, as
$wezE.fi*= 1* w3 E ~ K z ~ - ~using * , Eqs. (3.5) and (35.2), and noting that 9 = 0. Then, from Eq, (35.9), we find: KZ
=
[ne2/(m2eow2)1I3n I2p(E)[f(&,) - f(Em)].
(35.llb)
One-electron wave functions in a periodic lattice have the Bloch form u,(r) exp (zk,-r) where k, is the wave vector of the state, and u,(r) has the periodicity of the lattice. The matrix element 311, defined in Eq. (35.9b), will therefore vanish unless k, = k, k. For visible or infrared radiation the wave vector k is negligibly small compared to the size of the Brillouin zone (the ratio is of the order of the lattice constant divided by the wavelength), and can be ignored. Thus direct radiative transitions in a solid are described by vertical lines on an & versus k diagram. For cubic crystals the absorption coefficient will not depend on the direction of the polarization vector lo, and can be written J/)
=
+
a(u) =
Cnwez//3nwmZ>11 (hI P
I
+j)
I2p(&)Cf(E3)
- f(Em)l.
(35.12)
This equation can be transformed from mks to Gaussian units by replacing p by ~TK,/C*; a t optical frequencies K, = 1. If several processes contribute to the absorption, we must sum Eq. (35.12) over them.
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
3 87
We have assumed throughout this section that the field acting at a point is just the macroscopic field at that point. We saw in Section 24 that this is valid for free or nearly free electrons, but not for localized charges. For the localixed case we must include corrections to Eq. (35.1), the socalled local field corrections.80We shall not attempt to incorporate these in the results given here, but shall discuss them in Section 39 in connection with the oscillator strengths of F centers in ionic crystals. 36. BLACKBODY RADIATION
AND SPONTANEOUS EMISSION
A medium in thermal equilibrium at temperature T contains blackbody radiation which will induce transitions between states of the system, Thermal equilibrium is maintained by spontaneous emission from the excited states of the system into lower states. We can use this fact to calculate the rate of spontaneous emission.80*z80b In a sample of volume V the number of plane waves modes having wave vectors in an element dk,dk,dk, of k space is (2?r)-*Vdk&k,dk.,81 and does not depend on the shape of the sample or on the boundary conditions82 provided the dimensions are large compared to the wavelength. Thus for electromagnetic radiation, if we take k = 1 k I = nw/c, from Eq. (4.4), and note that two independent directions of polarization are possible for each wave vector, we find the density of modes per unit volume between w and w dw to be G ( w ) dw, where
+
G(w)
=
(27r)-*(47rk2)(dlc/dw) (2)
=
n2w2/(?r2c2~,).
(36.1)
We have introduced the group velocity v, = dw/dk, and consider only frequency regions where there are no strong absorption peaks with their resulting anomalous dispersion. In a system in thermal equilibrium at temperature T, the average See, for example, R. L. Adler, Phys. Rev. 126, 413 (1962); N. Wiser, ibid. 129, 62 (1963). 80s A. Einstein, Physik. 2.18, 121 (1917). gob Our treatment assumes that the electronic states involved in the optical transitions interact only with the radiation field. In real solids there will be interactions with lattice vibrations which can change the state of the system between absorption and emission, and which can substantially change the relation of emission and absorption rates from the one described here. This is discussed by W. B. Fowler and D. L. Dexter, Phys. Rev. 128, 2154 (1962). I am indebted to Dr. Dexter for sending me a preprint of this pa.per. 8 1 N. F. Mott and H. Jones, “The Theory of the Properties of Metals and Alloys,” p. 56. Oxford Univ. Press, London and New York, 1936. 82 See p. 45 and Appendix IV in Born and Huang.46 80
368
FRANK STERN
energy of a mode with angular frequency w ism
-
E(w) = fiw [exp (fiw/KT)
- l]-I,
(36.2)
where K is Boltzmann’s constant. Thus the energy density of blackbody dw is u(w)dw, with radiation in a dielectric in the range w to w
+
u (w ) = [n%w8/ (T ~ C ~ U ,][exp )
(fiw/K T )
- 134.
(36.3)
The velocity with which energy flows in a dielectric is the group velocity, as was shown in Eq. (6.12). We assume that the radiation is isotropic. Then the time-averaged flux of radiation with wave vector k lying in an element of solid angle dn, with polarization vector do lying in an angular interval dc9 in a plane perpendicular to k, and with angular frequency in a range dw, is
ISI
= u(w)v,(dn/4r) (dO/2r) dw = [n2fiws/(&2)][exp (fiw/KT)
- 1]-1(dn/4r)
(d0/2?r) dw.
(36.4)
We can use Eqs. (36.4) and (35.10) to eliminate the vector potential from (35.9). We find that the transition rate induced by blackbody radiation is: rM
=
ne2pw
-1 m 1 2 p ( ~ ) [ f ( ~ m ) Tm2c
- f(~j)][exp (fiw/KT) - I]-’ X (dn/4r) (dt9/27r) dw,
(36.5)
where 3n is the matrix element which appears in Eq. (35.9). For a system in thermal equilibrium a t temperature T the probability that a one-electron state of energy E i is occupied is the familiar expression for Fermi-Dirac statisticsaan: fo(Ei) = {exp [(Ei
where Ej
&F
- &F)/KT] + l}-l,
(36.6a)
is the Fermi energy. If we consider two states such that Em =
+ fiw, we find fo(Ej)
- fo(&m)
=
fo(L>[1
- fo(Ej)][exp
(fiw/KT)
- 11.
(36.613)
In thermal equilibrium the induced rate rbb is negative, corresponding to absorption, and the induced transitions from E j to Em must be balanced by spontaneous transitions from E m to Ej. From Eqs. (36.5) and (36.6b) we See, for example, C. Kittel, “Introduction to Solid State Physics,” 2nd ed., p. 56. Wiley, New York, 1956. 888 T his expression must be modified for impurity levels in semiconductors. See p. 87 in Smith.“ 88
369
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
find that the rate of spontaneous emission from states is rspont= [ne2w/(.rrm2c)] I 3n2 I p ( G j ) f ( G m ) [ l
$m
to
and adjacent
- f(E,)](dfl/4~)(dO/2~)dw. (36.7)
This result was derived for a system in thermal equilibrium, for which we should put fofor f. But the result (36.7) is valid even when the system is not in thermal equilibrium, since f(&,) [l - f(&,)]is the probability that the initial state is occupied and the final state empty, and the remainder of the expression depends only on the properties of the states themselves. I n a system not in thermal equilibrium, the radiation density may exceed the blackbody value, and the rate of stimulated emission will then exceed the rate given by Eq. (36..5), since we know from Eq. (35.7) that this rate is proportional to the energy density. If we let X stand for the number of photons in a radiation mode, we can combine the induced and spontaneous emission rates, and find:
(36.8) where f m stands for f(Gm).The spontaneous part of Eq. (36.8) is equal to Eq. (36.7)) and the stimulated part reduces to Eq. (36.5) if X has its thermal equilibrium value, [exp (fiw/KT) - 13-'. The spontaneous emission rate can be expressed in terms of the absorption coefficient. For a cubic crystal we can average Eq. (36.5) over all directions, and can eliminate the matrix element by using (35.12). We find that in thermal equilibrium: TBpont =
[nzwz~(w)/(n2c2)][exp (RwlKT) - 11-l dw.
(36.9)
If we integrate over all values of w we find that the total rate of spontaneous emission per unit volume is given by
where we introduced x = Rw/K T . The coefficient in brackets in Eq. (36.10) has the numerical value 2.53 X 1011T 3~ m sec-I - ~ = 6.83 X lo1* ( T/300)8 cm-3 sec-l if LY is given in cm-I. Equation (36.10) is equivalent to the expression given by Van Roosbroeck and Shockley in their calculation of the radiative recombination of electron-hole pairs in semiconductors.& 84
W. van Roosbroeck and W. Shockley, Phys. Rev. 94, 1558 (1954).
370
FRANK S T E R N
If the absorption coefficient near the absorption edge is given by a(.) = A ( A w - & G ) P , and the energy gap &G is >>K T , then the total emission rate is given approximately by (Ro=
[ K aT3/(7r2Aac2)]n2A( KT )p [ (&G/KT )
+ p I 2 exp ( -&a/KT )I? ( p + l), (36.11)
and equals the rate of radiative recombination of electron-hole pairs in thermal equilibrium. Our results for spontaneous emission can easily be extended to the more familiar atomic case. The density of states can be written p ( & ) = k l p ( w ) = NA-'G(w - w m j ) , where N is the number of atoms per unit volume and 6 is the Dirac delta function. The matrix element can be written in more familiar form if we use the dipole approximation, exp (zkar) = 1, and the resultss
( h n I P I $i)
=
imUmj(#m
I r 1 rClj)
1
(36.12)
where #j and $m are the one-electron wave functions of the initial and final states. Then the spontaneous emission rate per atom between states m and j is given by substituting these expressions for the density of states and for the matrix element in Eq. (36.7) and integrating over solid angle d0, over polarization directions do, and over a range of frequencies near w j . We find TRpont
1 (h1 r
= CneZwa/(3nch)]
$j)
;2.ff(Sm)[1- f(&j)].
(36.13)
This agrees with the usual result86 if we put n = 1 and replace 1 by 47r/c2. An interesting incidental point related to blackbody radiation inside a dielectric is that the energy flux given in Eq. (36.4) is greater by a factor n2 than the corresponding flux in a vacuum. One might a t first suspect that thermal equilibrium could not be maintained near an interface between a dielectric and a vacuum in an isothermal enclosure. But the necessary balance in the energy flow across the interface is provided by the ordinary laws of reflection,12which lead to Eq. ( 1 5 . 5 ) . 37. OPTICALABSORPTION FOR
A
SIMPLEMODEL
The optical absorption is easily found for direct transitions from Eq.
(36.11), We consider transitions between conduction and valence bands
whose band edges occur a t k = 0, snd assume that &, and & j are functions only of k = I k 1. Then the density of states per unit volume and per ed 88
See the equation preceding (35.20) in Schiff .7* See Eq. (36.22) in Sohiff
ELEMESTARY OPTICAL PROPERTIES OF SOLIDS
371
unit energy whose wave vectors have magnitudes near k is given by p(&) =
(k?/T*) 1 (d€,/dF;) - (d&,/dk)
1-1.
(37.1)
The derivation of Eq. (37.1) is closely analogous to that of (36.1), except that the factor 2 here refers to spin, and the wave vector k in (37.1) is determined by the relation &,(k) = &,(k) Aw. If we introduce conduction band and valence band effective masses mc and m,,and write :
+
&,(k) = &,a
+ (fiW/2mc),
(37.2)
we find that the density of states is:
(37.3b) Iiote that if m, >> m,,m, FZ m,. The model Tve shall use to calculate the band structure and matrix elements needed to calculate the optical absorption is based on the assumption that the conduction band is s-like, and the valence band is p-like. ITe shall use the k - p perturbation methods7to find the energy dependence of these bands near k = 0, and shall neglect interactions with all other bands. The Schrodinger equation for a Bloch wave function #k(r) = uk(r) exp (zk-r) can be written:
+
[- ( f i 2 / 2 m ) v 2 ih/m)k-p
+ (fi2k2/2m)+ V - Ek]uk(r) = 0,
(37.h)
n-here p is the momentum operator - i f i V ) and Ti is the periodic potential energy in the crystal. When k = 0 we can take the solutions to be
( H o - &)u8(r) = 0,
(Ho- &)up(r) = 0,
(37.4b)
where u p(r) is one of the three functions uz(r), u,(r), u.(r), whose symmetry properties are much like those of atomic p functions with angular dependence ( z / T ), (y/r) and ( z / T ) , respectively. If we use these functions 8’
E. 0 . Kane, Phys. C h e m Solids 1, 249 (1957).
372
FRANK STERN
as a basis and neglect the free electron energy h2k2/2m,the matrix elements of the bracketed expression in (37.4a) are:
-
E,
k,P
E
k,P
-
&,
&
kuP
k,P
0
0 0
p
0
&,-E
kJ’
0
0
I
0 = (% I Pu
kU
(37.5a)
E, -
where
P
=
(fi/m) (7-b I P z
I Uz)
I Ps I
(37.5b)
=
0)
(37.6a)
+
k2P2]1’2,
= (Us
The secular equation for the eigenvalues is:
-
(G,
- C)(EP -
&)2[(&,
whose solutions are: & = E,,
E,,
$(E8
W &,,
&,,
&,
-
+ Ep)
E)
- k2Pq
f {[&(C8 -
k2P2/&G,
&,
&,)I2
fk2P2/&~.
(37.6b)
The four bands in (37.613) are the two “heavy hole” bands, the “light hole” band, and the conduction band. If we compare these with (37.2), we find that the effective masses for the last two are I
m,
=
m,
=
h2&G/2P2.
(37.6~)
The results are quite simple, because we have neglected the free electron contribution to the energy, and the interactions with other bands which lead to the warped energy surfaces found in real and which determine the effective mass of the heavy hole band. Within the accuracy of our approximation, ( 3 7 . 6 ~ is ) consistent with the general relation=: (m/mj)zz =
+ (2/m) C
1
z
(&i
-
&*I-’ 1
1
(+i ~z
/+j)
1.’
(37.6d)
We now have all the information necessary to calculate the absorption using Eq. (35.11). Near k = 0 the matrix element between the conduction band and each of the three valence bands, when averaged over all polarizations of the incident light, is given by
j 3n
la
=
m2P2/3fi2= m2&~/6m,.
(37.7)
To take spin-orbit coupling into account very crudely, we assume that 88
A. H. Wilson, “The Theory of Metals,” 2nd ed., p. 47. Cambridge Univ. Press, London and New York, 1953.
373
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
one of the valence bands is well below the other two, and that near the absorption edge only transitions between the conduction band and the light hole band and one heavy hole band enter. We then find:
+ 2a12)~~~2(12rrnli2)-1m,112(liw -
(37.8a)
3.38 X lo5 n-1(m,/m)1~2(&~/~w) (Aw -
(37.8b)
a(w) = (1
a[crn-l]
=
&G)~/~(&G/~~u),
where n is the index of refraction. The energy in Eq. (37.8b) is expressed in electron volts. We have assumed that f(&i)- f(Ern) = 1, i.e., that the valence band states are occupied and the conduction band states are empty, which applies to pure materials a t low temperature. We have taken the material to be non-magnetic; to write Eq. (37.8a) in Gaussian units, replace P Oby 47r/c2. Equation (37.8) gives a fairly good fit to the optical absorption of the 111-V intermetallic semiconductors InAslSa InSb18' and room-temperature GaAsssbfor photon energies near their absorption edges, even though a very crude model was used to derive it. A detailed theory of the band structure and optical absorption of the 111-V semiconductors has been given by Kane.87 I n many semiconductors the bottom of the conduction band and the top of the valence band are not a t the same point in the Brillouin zone, and the treatment given here, based on Eq. (35.11), is not applicable. Another interaction mechanism must be present to provide the crystal momentum required for a transition from one of these states to the other. Lattice vibrations can provide this momentum, but other scattering mechanisms may sometimes be involved. The theory of this type of transition has been reviewed by M ~ L e a n For . ~ ~a~participating phonon group with energy near KO the absorption coefficient contains terms proportional to (Aw - &a f KO)2,where &G is the indirect energy gap. The positive sign indicates phonon absorption, which has a temperature dependence given by A( T) = [exp (el?") - l]-l, while the negative sign indicates phonon 1 = emission, which has a temperature dependence given by A( T) [l - exp (-O/T)]-l. An important effect, neglected so far, is the interaction between the electron and the hole which are produced when optical absorption causes a transition from a valence band to a conduction band. This interaction can lead to bound states of the electron-hole pair, called excitons, which give characteristic absorption spectra at low temperature^.^^ They also
+
asa 88b
J. R. Dixon, Proc. Intern. Conf. Semicond. Phys., Prague, 1960, p. 366 (1961).
I. Kudman and T. Seidel, J . A p p l . Phys. 33, 771 (1962).
See, for example, E. F. Gross, Usp. Fiz. Nuuk 78, 433 (1962);see Soviet Phys.-Usp. (English Trunsl.) 6 , 195 (1962).
374
F R A S K STERN
affect the absorption coefficient near the absorption edge, causing an abrupt increase a t the edge, followed by a gradual rise which merges a t energies far from the edge into the absorption coefficient that would be calculated without the interaction, using Eq. (35.11). The theory of this effect was given by Elliottz5and has been reviewed by & I ~ L e a n . ~ ~ ~
38. FREE-CARRIER ABSORPTION The classical theory of free-carrier absorption, for frequencies such that (27.5) and shows that the absorption is proportional to the square of the wavelength and to the reciprocal of the mobility. Physically one can think of the scattering processes as removing energy from the electron motion and transferring it to other modes of the system. This energy must be supplied by the electromagnetic wave that is maintaining the electronic motion, and the wave is attenuated. A quantum-mechanical description of free-carrier absorption requires that we consider not only the interaction (35.4) between the radiation and the carriers, but also the interaction between the charge carriers and the lattice vibrations or scattering centers. The transition from the initial state to the final state proceeds by a two-step process through a virtual intermediate state. A good description of the quantum theory of freecarrier absorption has recently been given by Dumke.go He finds that for photon energies less than K T the quantum-mechanical result reduces to the classical value, and that the leading correction term is of second order in (&w/KT).For degenerate systems K T is replaced by the Fermi energy E F . Dumke shows that intermediate states in other bands can be neglected provided the photon energy is small compared with the energy separation of the interacting states. The most detailed quantum-mechanical free-carrier absorption calculations have been made for n-type g e r m a n i ~ m , ~ and ~ J ' ~have taken into account the band structure and deformation potentials of this material. The results are in good agreement with experiment, though not for all wavelength and temperature ranges. Expressions for free-carrier absorption when the principal scattering mechanisms are optical mode scattering or impurity scattering have been given,93 and predict a wavelength dependence stronger than X2 a t short wavelengths. Free-carrier processes need not always involve scattering within a WT
>> 1, is given by Eq.
W. P. Dumke, Phys. Rev. 124, 1813 (1961). H. J. G. Meyer, Phys. Rev. 112, 298 (1958). 92 R. Rosenberg and M. Lax, Phys. Rev. 112, 843 (1958). 93 S. Visvanathan, Phys. Rev. 120, 376, 379 (1960). 90
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
375
single band. A calculation of the absorption that arises because of scattering between nonequivalent valleys in a conduction or valence band has been made by Risken and Meyerg4for the case of nondegenerate statistics. One of the early problems of free-carrier absorption was the unusual absorption spectrum of p-type g e r m a n i ~ m The . ~ ~ absorption was found to be proportional to the carrier concentration, but had considerable structure in no way resembling a X2 behavior. The explanation of the effect was given by Kahnlg6and in more detail by Kane,g7in terms of direct interband transitions between different valence bands, and gives good agreement with the observed absorption. 39. OSCILLATOR STREXGTHS
It is often useful to discuss the strength of the transition between two atomic levels in terms of the oscillator strength, defined as fji
=
2
1
I P,
(*j
~
qi)
12/(mfiuji),
(39.1)
where p , is the 5 component of the total momentum of the electrons, and qi are the wave functions of two atomic states, and fiw,i is the difference of their energies. It can be shown on quite general that C fji = 2, (39.2)
\kj
j
where 2 is the total number of electrons on the atom (or ion). The sum is over all possible final states j , and is independent of the initial state i. Because of the presence of coji in the denominator of Eq. (39.1), the oscillator strength is positive for upward transitions, in which the final state j has higher energy, and is negative for downward transitions. The rigorous sum rule (39.2) , often called the Thomas-Reiche-Kuhn sum rule, applies only to transitions of the atom as a whole. It applies exactly to one-electron transitions only for hydrogen or for ions with only one electron. I n solids, however, it is often a good approximation to think of the transitions between states of the whole system as involving only a change in a single electronic stat,e. The oscillator strength can then be approximated by fji
where $i and O4 O5 O6 O7
#j
=
2
I
(#j
I pz I
#i)
12/(mfiuji),
(39.3)
are one-electron wave functions, and p, is the x component
H. Risken and H. J. G. Meyer, Phys. Rev. 123, 416 (1961). W. Kaiser, R. J. Collins, and H. Y. Fan. Phys. Rev. 91, 1380 (1953). A. H. Kahn, Phys. Rev. 97, 1647 (1955). E. 0. Kane, Phys. Chem. Solids 1, 83 (1956).
376
FRANK STERN
of the momentum of a single electron, Thus Eq. (39.3) bears the same relation to (39.1) that (35.9) does to (35.7). The simplified expressions are somewhat easier to deal with, but one must bear in mind that they are approximations to the more exact relations (39.1) and (35.7). One important difference between the matrix element in Eq. (39.3) and the matrix element in (35.9) is the absence in (39.3) of the factor exp (zk-r). Neglect of this factor is called the dipole approximation in atomic transitions, since the neglected terms in the expansion of exp (zlr-r) lead to quadrupole and higher multipole transitions. For solids, taking exp (zk-r) m 1 corresponds to the long wavelength limit. When the states of the system lie in a continuum, we define fji(w) dw to be the sum of all the oscillator strengths (per unit volume of the crystal) dw. The initial condition of the for which wj; lies in the range w to w system is specified by the function f(&), which gives the probability that a one-electron state of energy & is occupied. The sum of all the oscillator strengths in a given angular frequency range is then found by multiplying the oscillator strength for a single transition by p ( w ) , the density of final states per unit angular frequency interval, and by f(&J - f ( & j ) , which is the probability that the initial state is occupied and the final state is empty, as in Eq. (35.8). If we note that p(w) = A p ( & ) , we find that
+
fii(w)
=
[2/(mwji)l
I (+i I P. I +J 12~(&)Cf(&i) - f(&i)I-
(39.4)
The oscillator strength is positive, corresponding to absorption, if the state j has higher energy than the state i and is less populated. If we specify that the state j in Eq: (39.4) shall always be the state of higher energy, we can remove the subscripts on wji, and can deal only with positive angular frequencies. For solids the subscripts on fji(w) can be thought to label the various bands that contribute to transitions of angular frequency w. Where more than one pair of bands contributes, the total oscillator strength, ftot ( w ) , is the sum over all contributing f j i ( w ) . Thus the imaginary part of the dielectric constant is given in terms of ftot by: KZ ( w )
= [?re2/
( 2 m w d I ftot (a) ,
(39.5)
which is obtained from Eqs. (35.11b) and (39.4) if we make the approximationexp (zk-r) m 1. The sum rule (23.3) for the imaginary part of the dielectric constant takes a very simple form if we use Eq. (39.5) to express tcZ(w) in terms of the oscillator strength. We find: (39.6)
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
377
where N is the total number of electrons per unit volume. This is the natural analog of the sum rule (39.2), since our oscillator strength was defined for a unit volume. The integrated oscillator strength is the total number of electrons, as in the atomic case. Thus we have established that the sum rule (23.2) and the oscillator strength sum rule (39.2) are equivalent for a simple case, using the long-wavelength approximation. The long-wavelength restriction is imposed so that the amplitude of the field should be roughly constant over the volume which contributes to the matrix element for the transition. A strict interpretation would therefore require that the wavelength be comparable to atomic dimensions, or larger, which limits the validity of the dipole approximation to photon energies of the order of lo3-lo4ev or less. An example will show that this is in an unnecessarily severe restriction. Consider a photon whose energy is just below 1.2 X lo6ev, the threshold energy for removing an electron from the 1s shell of lead.98 The corresponding wavelength is 0.11 A, which is much smaller than the size of a lead atom or ion. But such a high-energy photon will not interact strongly with the outer electrons, since the photoelectric absorption cross section Thus the main interfa.llsoff approximately as u3beyond the action of the photon in our example will be with the L shell, the shell with principal quantum number n = 2. As a rough approximation for the radius of this shell we take 2ao/(Z - s), where a. is the Bohr radius and s is a screening parameter. For the L shell of lead we have 2 = 92 and s M 20g9 and the radius of the L shell is m0.015 A, which is much smaller than the wavelength of the photon in our example. Thus the long-wavelength approximation holds for the dominant interaction with the atom. The example we have chosen is actually a conservative one, since a photon of slightly higher energy would interact mainly with the 1s shell, whose radius is smaller, while a photon of lower energy would have a longer wavelength. The argument we have given, which has also been given by Jarnesl1Oo is qualitative, but suggests that the dipole approximation is a good approximation up to quite high photon energies, at least insofar as it concerns the wavelength dependence of the dominant matrix elements for the interaction of photons with atoms and crystals. Before concluding this section we will consider the absorption associated with F centers in alkali halides, since this is conveniently discussed in terms of oscillator strengths. This absorption is associated with excitation of an electron bound to a negative ion vacancy, and leads to a well-defined See Appendix VI in Compton and Allison.% See p. 626 in Compton and Allison.*s 100 R. W. James, “The Optical Principles of the Diffraction of X-Rays,” p. 141. Bell, London, 1948.
9s
99
378
FRANK STERN
absorption band in the visible portion of the spectrum. We find, on using Eqs. (35.1la) and (39.4), that the integrated absorption can be represented by :
J a (w)
l~fii,
dw = ~re2//2nmeoc)
(39.7)
where n is the index of refraction of the host crystal at the frequency of the absorption peak, N is the number of contributing defects per unit volume, and f j i is the integrated oscillator strength per defect for the transition that leads to the absorption peak. We have assumed that K,,, = 1. In deriving Eq. (39.7) we assumed that the electric field acting at the defect site is the average field in the medium, since this is essentially what was done in deriving (35.11). But the discussion at the beginning of Part IV showed that for the case of bound charges the effective field at a lattice site in a cubic material is given by
F
=
E
+ P/(3eo) = i(n2 + 2 ) E ,
(39.8)
where we have assumed that K M na. Thus, since the field enters quadratically into the absorption, the integrated absorption of the F-center line can be expressed, on considering the local field correction, in the form:
which, if CY is expressed in em-', N in cm-', and fiw in ev, gives:
/
a(w)
+ 2 ) 2 N [cm-*]f&~
d(fio)[cm-lev] = 1.22 X 10-17(n2
(39.913)
Equation (39.9) can be used to determine the concentration of F centers if the oscillator strength is known, or to find the oscillator strength if the defect concentration is known. The oscillator strength is found to be of the order of unity for F centers in the alkali halides that have been studied. The paper by Doyle1'' should be consulted for experimental details, and for reference to earlier experimental work. A number of authors have studied the validity of Eq. (39.9) as applied to defects in ionic crystals and have given corrections to this equation. References to several theoretical treatments can be found in Doyle's paper.'O1 101
W. T.Doyle, Phys. Rev. 111, 1072 (1958).
379
ELEMESTARY OPTICAL PROPERTIES OF SOLIDS
VI. Wavelength-Dependent Dielectric Constant of a Free-Electron Gas
40. INTRODUCTIOX The wavelength-dependent dielectric constant introduced in Section 7 provides a very powerful tool for discussing nonlocal effects in solids. An example of such an effect is the response of carriers to an external field which varies rapidly in a distance comparable to the mean free path I of the carriers. This mean free path is related to the electronic damping constant y of Part IV by: 1 = B/y, where 0 is a mean velocity whose exact value depends on the degree of degeneracy of the carriers and on the scattering mechanism. A carrier a t a given point in the crystal will have a velocity determined by the electric fields a t points it has passed on its trajectory since the time of its last randomizing collision. If the field is varying rapidly in a mean free path, the effective field along the trajectory may not correspond a t all closely to the local field. It can be shown that undei these circumstances the current flowing is related to the electric field byIo2:
J(r, 1)
=
1
( 3 N e 2 / [ 1 6 n 2 m e ~ v ~ ] )t[t.E(r’, 1’)]t-4exp
(-t/O
d 3 t , (40.1)
for degenerate statistics, where V F is the velocity of carriers a t the Fermi surface, t = r - r’, 4 = F 1, and t’ = f - ( E / v F ) is a retarded time. We see that Eq. (40.1) describes a nonlocal responsf: to the applied field. We shall derive below and in Part VII the wavelength-dependent dielectric constant of a really free-electron gas, by which we mean a system in which the electronic damping constant y, associated with collisions with lattice defects or phonons, goes to zero, and the mean free path for such collisions becomes very long. Such a system is sometimes called a n ideal conductor. It provides the extreme case for nonlocal effects, since there is no scattering to help confine the “memory” of the electron to a small region in space. In this treatment we shall neglect the spin magnetic moment of the electron. I n this Part we restrict ourselves to transverse fields, with a vector potential ~
=
i ( k , u ) exp (zk-r - id),
k-i
=
0.
(40.2)
We defer discussion of the longitudinal case until Part VII. lo2
See, for example, D. C. Mattis and J. Bardeen, Phys. Rev. 111,412 (1958) ; J. L. Warren and R. A. Ferrell, ibid. 117, 1252 (1960).
380
FRkUK STERN
The results of this Part are most easily obtained if we work with the transverse conductivity, defined in Eq. (8.2) :
-
qtr
= Z
+ ik2(%
- l)/(pow),
(40.3)
where 8l = [%,(k, w ) ] - I is the reciprocal relative permeability, and Z = u - iwe is the ordinary electrical contribution to the conductivity. The additional term (40.4) AZ = i k 2 ( ( R - l ) / ( p o w ) represents the effect of the current V xM associated with the magnetization. We shall assume here that (R is real. Thus, for real k and w this term ir, i.e., we consider a wave is purely imaginary. If we replace w by w growing slowly with time with amplitude proportional to exp (-id) exp pi), and then let r + 0, there will also be a contribution to the real part of A;, This follows from the result that in an integration
+
+
ir)-1 = PW-1- i r S ( w ) , (40.5) lim ( W r+o where 6 ( w ) is the Dirac delta function, and P denotes the principal value. This result is easily proved by noting that the imaginary part of Eq. (40.5) is - r / ( w 2 r2),which vanishes for all w # 0 as F + 0, and whose is - ir. Thus we find that the integral over all values of w from - co to magnetic current V x M contributes
+
+
AUI =
[irk2((R
-
(40.6)
l)/p0]8(0)
to the real part of the cbnductivity. Note that Eq. (40.6) is an even function of w . We shall later use the integral of Ztr over positive values of w , and must therefore take only one-half of the integral of Eq. (40.6) over all w . This delta function contribution to the conductivity will be important when we check the sum rule. 41. CALCULATION OF
THE
CONDUCTIVITY
It is a straightforward matter to calculate the conductivity for a freeelectron gas in a transverse electromagnetic field if we use the relation gi(k, W )
= C T e 2 / / m 2 w )1
I (hI go ~
X P(5k-r)'P
I
+j)
- f ( L )1,
I2p(G) Cf(&j)
(41.1) which is based on Eq. (35.11). Note that for the transverse vector potential (40.2) used here, Zo*k= 0, so that & * p + = &-p. We consider a light wave of angular frequency w and wave vector k, propagating in the z direction and polarized in the z direction. The wave function of the free-electron state of wave vector kj, normalized in volume V , is V-1/2 exp (ikj.r), Thus
381
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
the matrix element in Eq. (41.1) is h2kjxz,We must conserve momentum and energy, which, for a transition from kj to k, means:
+ k, = k j 2 + 2mwh-‘.
k, k,?
=
(41.2)
kj
(41.3)
Squaring Eq. (41.2) and subtracting (41.3) gives (41.4)
where we have used the fact that k is in the z direction. The density of states per unit energy is easily found from Eq. (41.4) by noting that changing w by A-’ dE changes k,, by (m/fi*k)d&. Thus the density of states per unit volume and unit energy for which the final states are in a range dk,’dk,’ is found, by proceeding as in Eq. (36.1), to be: p (G) =
[m/ ( 4a3A2k)] dk,’dk,’.
(41.3)
The conductivity is then found from Eq. (41.1) to be:
al(k,w)
=
[e2/(4a2mwk)]
where E,
=
G, =
[ v z w ~ / ( ~~ ~QAw )] Gj
+ Aw,
/Im -m
(41.6)
k , ’ 2 [ f ~ ( G l) f o ( E m ) ] dk,’dk,’,
+ [Azk2/(8m)] + [(fiz/(2m)3(kz’* + ky’2)1 (41.7)
and fo is the occupation probability for thermal equilibrium, as given in Eq. (36.6). The integral in Eq. (41.6) can be simplified if we integrate over angles in the k,‘k,‘ plane, which allows us to replace the factor k,‘2dk,’dk,‘ by ahJ3dk’. Kext, following LindhardlEame introduce the dimensionless parameters 2 = k/(2ko), u = w/(kz!o), (41.8) where we define : vo = fiko/m, Go = fi2k02//(2m). (41.9)
For the time being we will consider ko and the associated energy arbitrary. Then we find that the conductivity becomes:
where Gj
=
+
&[(u - z ) ~ 4.P],
&,
=
+ z ) +~ 4d2].
&o[(u
&o
to be
(41.11)
382
FRANK STERN
To proceed further we must make simplifying assumptions about the statistical distribution lo, since the integration in Eq. (41.10) cannot be carried out simply for the case of general statistics. b7e first consider degenerate statistics, for which Go in Eq. (6.15) becomes the Fermi energy &F, and ko is taken to be k ~ the , wave vector a t the Fermi surface. For degenerate statistics we can use the relation kpS =
3a2N,
(41.12)
where N is the number of electrons per unit volume. The coefficient of the integral in Eq. (41.10) then becomes 6m0wp2/(w~), where up2=
Ne2/(meo).
(41.13)
For a nonzero contribution to Eq. (41.10) we must have the initial state inside the Fermi sphere, and the final state outside, i.e., (U
- 2)’
+4
~ <’ 1, ~
(U
+
2)’
+ 42” > 1.
(41.14)
From the integration limits which follow from Eq. (41.14) we find: a,(k, u) = [Ne2/(mu)](3au/4)(1 =
-
u2 - z 2 ) ,
[Ne2/(mw)][3n/(32~)][1 -
(U
-
u
2)*12,
+z
< 1,
I U - 2 1
=o
1u
- z
< I
I>
1.
< U + Z ,
(41.15)
To this we must add the delta-function contribution in Eq. (40.6). We have found the conductivity for the degenerate case by straightforward application of Eq. (35.11) to the free-electron gas. The nondegenerate case is even easier. In that case we have f t
where
kT
(&) = ~ T ~ ‘ ~ exp N ~( T &/K - ~ T ),
(41.16)
is defined by fi2k~’/(2m) = K T .
(41.17)
For the nondegenerate case we shall let the parameter ko in Eq. (41.8) equal k r , which means that so = K T . Thus, using the relationfo(&j) fo(&,) = f o ( E j ) [l - exp (-fiw/KT)], we find that Eq. (41.10) becomes:
~,(k, W)
= [Ne2/(mw)][x1’2/(4z>] exp
[- (u- ~ ) ~ ] [-1exp (-fiw/KT)], (41.IS)
to which we must again add Eq. (40.6).
383
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
42. VERIFICATION OF THE SUMRULE
We are now in a position to check the sum rule for the conductivity, which is easily deduced from Eq. (23.3):
lm
(42.1)
al(k, u) du = (n/2)Ne2/m.
Here the wave vector k is considered to be a fixed parameter. If we treat the degenerate case first we find that for both z z > 1 integration of Eq. (41.15) gives:
<
1 and
(42.2)
to which we must add the contribution from Eq. (40.6), which can be written
Lm
Acr(k, u) dw
= nk2(6i
-
Let us first consider the case of long wavelengths, since in the long wavelength limit the contribution (42.3) to the sum rule will vanish. We find, on expanding t,he logarithm in Eq. (42.2) in powers of z, that
I, q ( k ,
u) dw = (n/2) (Ne2/m)[1
- z2 + (z4/5)],
z
<< 1.
(42.4)
If we assume lhat Eq. (42.3) exactly makes up the difference between (42.1) and (42.4), we can deduce, using (41,12), that for long wavelengths 1
-
R(k, 0) E - (rokp/37r) [I TO
-
(k2/20kp2)],
k
<< kF,
= poez/(4nm) = e2/(4aeomc2) = 2.818 X 10-13 cm.
(42.5) (42.6)
We can compare Eq. (42.5) with the conventional result for the diamagnetism of a free-electron gas’m: Km
where P 10)
=
-1
=
- (NP2r~/3KT)[F1/2(&E’/KT)/F1/2’(&P/KT)],(42.7)
efi/2m is the Bohr magneton, F1n(x) is one of the standard
See p. 167 in Wilson.**
384
FRANK STERN
Fermi-Dirac integrals defined by104 (42.8a) and the prime denotes differentiation with respect to the argument of F . We note for later reference the asymptotic results:
= 1,
F l ~ ( z M) +d/2e2,
F1,2’(x)/F1,2(2)
FI,z(z) M (2/3)9/’,
F l p ’ ( x ) / F l , z ( z ) NN
<< -1; 2 >> 1.
2
3/(2~),
(42.8b) (42.8~)
Equation (42.8b) applies to a nondegenerate carrier distribution, and Eq. (42.8~)to a degenerate distribution. In the latter case the bracketed factor in Eq. (42.6) equals 3KT/(2€p), Thus the Landau diamagnetism of a degenerate free-electron gas is: K~
- 1 = - NP2po/(2&~)=
-poe2A2N/(8m2&p)= -poe2kp/(12a2m) =
-?‘okp/(3a),
(42.9)
which agrees with the long-wavelength limit of Eq. (42.5) if we neglect the difference between K,,, - 1 and 1 - 6%. That difference is small if 1 - 6% itself is small, as will be the case for the systems considered here. This gives us some confidence in the assertion that Eq. (42.3) represents the deficiency in the sum rule for all values of z. With this assumption we find, from Eqs. (42.2) and (42.3)) that: 1 - R(k, 0) = [?‘ok~/(16rZ~)][ (1 - z*)21n[(l +%)/I
1-z
I] - 22 - 2z8}. (42.10)
This agrees with the corresponding expression obtained by Bardeenlo5if his result is evaluated for the case of an ideal conductor (Lee,one with no superconducting energy gap), For large z, Eq. (42.10) leads to the following expression for the diamagnetism of a free-electron gas: 1 - R(k, 0) S
-[?‘o~F/(~HZ~)][~
-
(k2)-’], z
>> 1.
(42.11)
When z = 1, Lea,k = 2kp, the approximate expressions (42.5) and (42.11) both give a diamagnetic susceptibility which is 80% of the long wavelength limit, while (42.10) gives 7’5% of the limit. J. MoDougall and E. C. Stoner, Phil. Trans. Roy. SOC.London A237, 67 (1938); A. C. Beer, M. N. Chase, and P. F. Choquard, Helv. Phus. Acta 28, 529 (1955). 106 J. Bardeen, in I‘Handbuch der Physik” (S. Fliigge, ed.), Vol. 15, p, 303. Springer, Berlin, 1956; see also 0. Klein, Arkiv Mat., Astron. Fysik 31A, No.12 (1945) I
385
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
We can carry out the same analysis for the case of the nondegenerate free-electron gas, using Eq. (41.18). We find in that case also that Eq. (40.6) correctly supplies the deficiency in the sum rule for long wavelengths. The general expression for the diamagnetism in the nondegenerate case is: 1 - R(k, 0)
=
- exp ( - z 2 )
-[p&e2/(mk2)][1
m
+
( ~ ~ ~ " / ( 2 nl)n!]]] (42.12)
where z2 = A21C2/(8mKT).For small z this becomes 1
- &(k, 0) NN
-[N@2po/(3KT)][1
- (2z2/5)],
z
<< 1,
(42.13)
which agrees with Eq. (42.7) in the long-wavelength limit, since, in the nondegenerate case, FlI2(z)= Fl,z'(x) = *d/zez. For large values of z we 2) in the expression in braces in Eq. can replace (2% 1) by (2n (42.12), and find very roughly that
+
1
- R(k, 0) M
+
-[Nb2po/(2KT~2)](1 - (22')-'),
z
>> 1.
(42.14)
For z = 1 the complete expression (42.12) gives 0.693 of the long-wavelength limit, while the approximate results (42.13) and (42.14) give 0.6 and 0.75 of the limiting value, respectively. The qualitative behavior of the diamagnetic susceptibility deduced from the foregoing results is similar for both the degenerate and nondegenerate cases: a deviation proportional to z2 from the classical result, dependence. followed at large values of z by a r2 43. CALCULATION OF THE DIELECTRIC CONSTANT
We can now use the dispersion relation to calculate the real part of the dielectric constant from the conductivity. The appropriate dispersion relation, corresponding to Eq. (17.lOa), is: Klt'(k, a) - 1
=
[2/(~eo)]P
q(k, w')[o'~ -
w2]-'
dw'.
(43.1)
For the case of degenerate statistics we take al(k, w) from Eq. (41.15) for nonzero frequencies, and add the delta-function contribution given by (40.6) and (42.10). The integration of Eq. (43.1) is tedious but straightforward. It must be carried out separately for z < 1 and for z 2 1, but
386
FRANK STERN
leads in both cases to the result:
Kltr(k,W )
=
+ [l
1
+ [ 3 / ( 3 2 ~ ) ] ( ~ , ~ / d )[1 -
(U
-
Z)2]21n
u-2-1 u-z+l
+ z ) ~ In] ~
- (u
which agrees with the value obtained by Lindhard,sa who calculated the dielectric constant by finding the total current induced by a vector potential like that in Eq. (40.2). For nondegenerate statistics we find:
Kltr(k,W )
=
1-
+
( ~ p 2 / ~ 2 )
up2exp
(-9)
/om
O?ZG
u' sinh (2u'z) exp ( - d 2 ) du', u'2 - 7.42
(13.3)
where u and z have the values given in Eqs. (41.8) and (41.17). While Eqs. (43.2) and (43.3) are rather formidable, they can be simplified for several cases of physical interest. One of these is the case u z >> 1, for which we obtain, for degenerate statistics:
+
Kltr(k,W )
M 1
+
- ( w ~ ~ / w ~ ) [ (1/5)(U2 ~ -
Z2)-1].
(43.4%)
This differs slightly from the corresponding result given in Lindhard'ssa Eq. (3.19). For nondegenerate statistics we find
Kltr(k,W )
W
+
1 - ( W ~ ~ / / . ~ ) [ (2u2)-' C~
Another case of interest is the case u,x degenerate statistics, the result
P(k, W) M 1 -
+ (3
2z2)/(4u4)]. (43.4b)
<< 1. In that
case we find, for
+ 3u2 - (3niu/4) (1 - u2 - z 2 ) ] ,
(W,~//W~)[Z~
(43.5a)
which agrees with Lindhard's Eq. (3.18). We have included the imaginary part of the dielectric constant in Eq. (43.5a), using (41.15). I n Eq. (43.4) there is no corresponding contribution to the imaginary part. For nondegenerate statistics we find from Eqs. (41.18) and (43.3) that for small u and z
"Kr(k,W )
FZ
+
1 - ( w , ~ / w ~ ) [ ( ~ z ~2u2 / ~ )- i t h ( 1 - u2 -
44. DIELECTRIC CONSTANT WITH
NO
2')).
(43.5b)
EXTERNAL CURRENTS
The expressions given in the previous section give the response of a free-electron gas to an electromagnetic wave whose wavelength dependence
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
387
and frequency dependence are both specified. But in many cases we are interested primarily in the simpler case of a freely propagating wave of fixed frequency. Then we know from Eqs. (4.3) and (7.12) that when no external currents are present, hIaxwell’s equations lead, for transverse waves, to the result:
&*f=
(44.1)
(W2/C2)KXrn.
As we have shown above, it is possible to include the magnetic effects in the dielectric constant “Kr(&, w ) . Then we find, from Eq. (8.3):
f .f
P(&, w)*
= (w’/c’)
(44.2)
This is a n implicit equation which must be solved for f . f . When the resulting value for &.& is substituted in P ( f , w ) , we obtain the wavelengthindependent dielectric constant that best describes the behavior of freely propagating waves. Note that this will lead to complex wave vectors in most cases. Thus the term “propagating” as used here implies only that these are solutions of Maxwell’s equations in the absence of externally introduced currents, and does not imply absence of damping. From Eq. (43.4) we see that as long as 1 u2 1 >> I, the dielectric constant is accurately given by the classical value Ktr
= 1-
(
(44.3)
J
as in Eq. (25.4). But if we make use of both Eqs. (44.2) and (44.3), we find that
u2
= w’/(uo’k2)
x (c’/uo2) Thus the condition 1 u21
=
C2/(2)02Kt’)
[w’/(w2
>> 1 is satisfied
- w,’)].
provided
(44.4) w
>> uowp/c.
I n metals,
vo is the Fermi velocity, and vo/c is less than 0.01 ; in nondegenerate systems
a t ordinary temperatures vo/c will be even smaller. There is therefore a considerable frequency range in which vowp/c << w << wpJ for which the dielectric constant is given by Eq. (44.3), and is negative. I n that range, Eq. (44.2) leads to the relation
L x i ( w p 2 - w2)1’2/c x iw,/c,
vowp/c
<< w << u p .
(44.5)
In this case the field decays by a factor e in a distance k2-1
= c/wp,
(44.6)
called the London penetration depth. We now consider the limiting case of very low frequency, and will assume that both u and x are much less than 1, so that Eq. (43.5) can be
388
FRANK STERN
used. For degenerate statistics Eq. (44.2) then takes the form: c2IE2
=
4C2kpV m w,2[(37riu/4)
- 221,
(44.7)
where only the lowest order terms in u and 2 have been retained. This equation implies that IE, and therefore u and %, are complex. For real metals ckp >> wp, so that 9 on the right in Eq. (44.7) can be neglected. We then solve for 2, and find that a t low frequency
k M [37riWp2W/ (4c2vF)]'/'.
(44.8a)
For nondegenerate statistics we use Eq. (43.5b) and obtain:
IE M
[T'~?iWp2W/
(c2vo)1118,
(44.8b)
where vo is given by Eqs. (41.9) and (41,17), and equals (2KT/m)1'2. The results can be put into another, more commonly used form if we note that for degenerate statistics the mean free path 1 is given by UFT, where r is the relaxation time. Then Eq. (44.8a) can be written
IE M
[3TiuOo/ (4eoc21)]'/8,
(44.9)
where (TO is the dc conductivity, Ne2r/m, We can put Eq. (44.813) into a similar form, although for nondegenerate statistics we must consider the energy dependence of the relaxation time. For the particular case of an energy-independent mean free path, which applies to acoustic mode lattice scattering in semiconductors, we findl06 that 1 = ( 9 ~ K T / 8 r n ) ' / ~=r (37r1/2/4)v0r,so that Eq. (44.813) can also be written in the form (44.9). For other energy dependences a factor of the order of unity is required in Eq. (44.9) for the nondegenerate case. Equation (44.9) is misleading, however, since the procedure we have used, based on (44.2), is valid only in an infinite homogeneous medium without external sources. When P ( k , 0) depends on wave vector very slightly or not at all, the effect of a boundary will not affect the validity of the results. But in the low-frequency region the dominant term in the dielectric constant Eq. (43.5) varies as it-', and a procedure based on (44.2) will not give correct results. A more exact treatment for the particular case of specular reflection is given in the next section, and shows that Eqs. (44.8) and (44.9), while qualitatively correct in their dependence on carrier concentration, mean carrier velocity, and frequency, are in error by a numerical factor of order unity, and by a phase factor. 108
W. Shockley, "Electrons and Holes in Semiconductors," p. 277. Van 'Nostrand, Princeton, New Jersey, 1950.
ELEMENTARY OPTICAL PROPERTIES O F SOLIDS
389
IMPEDASCE AND AXOMALOUS SKINEFFECT 45. SURFACE The misleading results obtained by the method of the preceding section arise when the dielectric constant is a strong function of wave vector. In such cases the current a t a point will depend on the field at other points, and the effects of a boundary can be felt at substantial depths. This leads to a nonexponential decay of the field as it penetrates into the medium, and to other deviations from simple classical behavior. The phenomenon is called the anomalous skin effect, and has led to important information about the properties of metals.lo7 An experimental quantity which characterizes the properties of a medium is the surface impedance for normal incidence, 2, given bylo8 E=
Z(H xn),
(45.1)
where E and H are the values of the electric and magnetic fields a t the surface of the medium, and n is a unit vector along the direction of propagation of the wave. For simplicity we will consider a wave propagating in the positive z direction, in vacuum in the lower half-space z < 0, and normally incident on the medium in the upper half-space z 1 0. We assume that the electric field is in the x direction. With these conventions we find:
z=
k ( 2 = O)/Hu(2
= 0).
(45.2)
The reflectivity is easily derived from the surface impedance. If we assume that the wave is incident from vacuum, and assume a common time dependence given by exp (-id)throughout, we can write the magnetic field for z < 0 as the sum of an incident wave and a reflected wave, or R,(z) = Ri exp ( i w z l c ) R, exp (-iwz/c). (453a)
+
Then, using Eq. (4.lc), we find that &(z)
=
Zo[Ri exp ( i w z l c ) - 8, exp (-iuz/c)],
(45.3b)
where Zo =
(w)-l
= poc =
376.73 ohm
(45.4)
is a constant, sometimes called the impedance of the vacuum or the impedance of free space. If we define the amplitude reflection coefficient i: by i: = 101
108
a,/s,,
(45.5)
A , B. Pippard, Advan. Electron. Electron Phys. 6 , 1 (1964);Rept. Progr. Phys. 23, 176 (1960). See p. 284 in StrattonQ2
390
FRANK STERN
and solve Eqs. (45.2) and (45.3) for P in terms of i: =
(20- Z)/(ZO
z",we find that
+ 2).
(45.6)
If the medium from which the wave is incident is not a vacuum, we need only replace Zo by the corresponding value for that medium. If the electric and magnetic fields of the transmitted wave are exponentially damped with wave vector &, then Maxwell's equation ( 4 . 1 ~ ) ~ H = f xE/(pow), combined with (45.2) leads to the result
z = pow/fE, I
(45.7a)
which, if we use Eq. (44.2), is equivalent to =
Zo(p-1/2.
(45.7b)
We see on comparing Eqs. (45.6) and (45.7) with (10.6a), that the present results are consistent with those of the conventional treatment of reflectivity given in Part 11. The reflectivity for a wave normally incident from a vacuum is given by
R
=
I f 1 2
=
1 zo - 2 \'/I zo + z" 12.
(45.8)
We now proceed to calculate the surface impedance for a solid with a wavelength-dependent dielectric constant, for which Eq. (45.7) does not apply. To calculate the surface impedance for this case we first find the spatial Fourier components of the vector potential for z 1 0, which we write &(z)
=
exp ( - i w t )
/
m
A,(k) exp (ikz) dk.
(45.9)
J-ca
We have assumed that the amplitude is independent of x and y, and therefore integrate only over the z component of the wave vector, dropping the subscript z. The relation between the vector potential and an externally introduced transverse current Jext is [k2 -
(W2/C2)"Kr(k,
w)]i(k,
w) = poJext(k, w),
(45.10)
which is analogous to Eq. (7.8b). I n the problem considered here, a wave normally incident on a plane surface, we would not need any external current sources if we used the conventionaI boundary condition treatment of Part 11. With the present formalism, however, we must introduce an effective surface current at the boundary to simulate the proper conditions there. We shall treat the simplest case, that in which the carriers of the medium are assumed to be scattered specularly when they strike the boundary.
391
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
For specular reflection we imagine that any particle incident on the plane z = 0 from above will have its z component of velocity reversed, while its component of velocity in the x - y plane remains unchanged. We can replace this model by another, equivalent to it, by assuming that particles incident from above pass through the plane z = 0, and that the reflected particles are represented by particles which cross the plane from below. I n order that this alternative model should be equivalent to the actual specular reflection, we must assume that the medium in the lower half-space has the same properties as the other medium, and that the fields in the lower half-space are the mirror images of those above. This means that & ( - z ) = & ( z ) . For the magnetic field the relation is flu(-2) = - f l u ( z ) , since the magnetic field is a pseudovector which changes sign on reflection through a plane containing H.Thus we have a discontinuity in the y component of the magnetic field, which implies a surface current density, according to Eq. (9.ld)) which we can write in the form J,(z) = -2flu(0)6(z), (45.11) where Thus
R,(O)is the
J,(k)
= (27r)-1
limiting value of
Jm
-m
flu as z
s , ( z ) exp ( - i k z ) dz
approaches 0 from above,
=
-T-~R~(O). (45.12)
If we introduce this current as an external current source into Eq. (45.10), and use the relation B(k, u) = iwi(k, o),we find that &(z)
Jm
= [~oo/(i?r>]~~(O [k2) -W
- (o2/cZ)>tr(k,
u)]-1
exp (ikz) dk. (45.13)
Note that in our example k is in the z direction, and k = k,. To make our model self-consistent we must assume that P (-k, o) = P ( k , u), since this will assure that &( -2) = BZ(2). To find the surface impedance we evaluate Eq. (45.13) at z = 0, and use (45.2), with the result:
Z
= [2poo/(&)]
/= 0
[ka - ( w z / c z ) P ( k ,o ) ] - ’ d k .
(45.14)
When i t r does not depend on k, this integral can be evaluated very easily, and gives just the result (45.7b) which we deduced from more elementary considerations, and which includes the normal skin effect. The nontrivial application of Eq. (45.14) is to the anomalous skin
392
FRANK STERN
effect. We shall apply it here only to the case of an ideal conductor, for which we find, from Eqs. (43.5) and (44.9), that ( ~ ~ / c ~ ) > ~u) ~ (= ( k[37ricr~/(4~&7k)] , =
i@/k,
(45.15)
where p has been introduced as a temporary abbreviation for the parameters in Eq. (45.15). We find, then, that
= [2pOw/(~i)]
lrn 0
k d k / ( k 3 - ip).
(45.16)
We can simplify this expression by substituting for k the value Ic = ( - i p ) 1 / 3 s , and find that the surface impedance is
2
= (2pow/~i)( -ip)-’l3
(45.17)
The value of the integral in Eq. (45.17) is 27r/3a‘2.109Thus, substituting for p from Eq. (45.15) we find that,
2 = 4 ~ 0 3 - 3 1 2 [ ~ o ~ u 2 / ( 6 n ~(1~ )-] ’ i ~ 8G).
(45.18)
This is the surface impedance for the extreme anomalous skin effect for specular reflection. TO obtain the usually quoted value in Gaussian 1 - iG), simply replace namely (8/9) 31~6~*~3[ZwZ/(c4cr~)]1~3( po by 4?r/c2, and €0 by (47r)-’. Note that we use a time dependence exp ( -id))while both Jones1l0and Pippard’O’ use exp (iut). There is an ambiguity in taking the cube root in Eq. (45.17), but this is easily resolved if we note that the wave must be propagating in the +x direction and damped, which shows, by analogy with (45.7a), that the physically acceptable solution for 2 in our case must have a positive real part and a negative imaginary part. For diffuse scattering at the boundary, which better describes real solids, there appears to be no simple device like the one that led to Eq. (45.12). The calculations”‘ are rather formidable, and lead to a surface impedance bigger than Eq. (45.18) by a factor 9/8. We note in passing that our simple D. Bierens de Haan, “Nouvelles Tables d’Int6grales D6finies,” Table 17. Stechert, New York, 1939. 110 H. Jones, in “Handbuch der Physik” (S. Flugge, ed.), Vol. 19, p. 308. Springer, Berlin, 1056. 111 G. E. H. Reuter and E. H. Sondheimer, Proc. Roy. SOC.A196, 336 (1948); R. B. Dingle, Physica 19, 311 (1953). 109
ELEMESTARY OPTICAL PROPERTIES O F SOLIDS
393
arguments which led to Eq. (44.9) would have given, via (45.7), a surface impedance equal to (45.18) times a factor (3v3/4)i1'*, i.e., a result too big in magnitude by 30%, and in phase by 30'. Let us summarize our discussion of the surface impedance of a freeelectron gas by giving the values for the simpler limiting cases. At low frequencies we must take the conductivity into account. If we use Eqs. (45.7) and (27.7), and note that in the free electron case K, = 1 and up2= Ne2/(meo),we obtain the classical skin effect result: (45.19a) (2 U P T ) -'I2(1 - i) . 2/20 = ( . / U p ) As the frequency increases the classical skin depth (4.9) decreases. When it becomes comparable to the mean free path of the carriers we reach the nonlocal regime described a t the beginning of this chapter. When the classical skin depth is much smaller than the carrier mean free path we have the extreme anomalous skin effect, given in Eq. (45.18), which we rewrite here for degenerate statistics in the form:
Z/2o
=
2'3"''
(V F / ~ T C I/* ) ( 1 - ifl)
(u/w,) '1'
I
At still higher frequencies we reach the range we can use Eqs. (44.5) and (45.7) to find:
qzo
= -iw/w,.
VOWJC
(45.1913)
<< w << wp, for which (45.19~)
Finally, when w >> w p , the surface impedance of a free-electron gas approaches Zo. At sufficiently low values of w , we can take the bound electrons into account and generalize these results somewhat if we replace ZOby ZOK,-~, and let up2 = Ne2/(m*K,eo) as in Section 25. More detailed consideration of the anomalous skin effect leads to information about the transition between the classical skin effect and the extreme anomalous region,lll and to information about details of the shape of the Fermi surface.lW 46. REALCRYSTALS The results obtained in this part apply only to a freeelectron gas, and neglect both the short-range interactions between the electrons, and the periodic potential which is present in a real crystal. Few detailed calculations of the wavelength-dependent dielectric constant have been carried out for specific solids,112but there are many papers which take electronelectron interaction into account, and relate the dielectric constant to the matrix elements or oscillator strengths that would describe any particular crystaLsO 112
D.R. Penn, Phys. Rev. 128, 2093 (1962)
I
394
FRANK STERN
If the wavelength dependence of the dielectric constant is weak, then one can expand the dielectric constant in a power series in the rectangular coordinates of the wave vector k. To second order we would have: where repeated indices are to be summed. For crystals with tetrahedral or higher cubic symmetry the fourth-rank tensor in Eq. (46.1) has only 3 independent components. Furthermore, if the crystal has inversion symmetry all coefficients of odd powers of k will vanish in the expansion,'l8 and the crystal will be optically inactive.114 Thus for cubic crystals with inversion symmetry Eq. (46.1) takes the form115:
+
+
+
D(k, U ) = ~ o ( X ( 0u)B , 6(k*k)B E(k*E)k dT*EI1 (46.2) where T is a second-rank tensor whose diagonal components, measured with respect to the cubic axes of the crystal, are ka2, kU2,and kS2,and whose off-diagonal components vanish. The constant d describes the anisotropy of the dielectric constant. If d vanishes, then 6 gives the dependence of the E dielectric constant on k to second order for transverse fields, while 6 gives this dependence for longitudinal fields. A review of the electrodynamics of media with spatial dispersion which describes the consequences of the wavelength dependence of the dielectric constant in some detail has been given by Rukhadze and Silir~.~l'
+
VII. Interactions of Charged Particles
When charged particles are present as sources, the electromagnetic field is largely or entirely longitudinal, We shall first derive the dielectric constant of a free-electron gas for the longitudinal case, and shall then give applications of the longitudinal dielectric constant to screening, to energy loss, and to the forces acting on charged particles.
47. LOSGITUDINAL DIELECTRIC CONSTANT OF
A
FREE-ELECTROS GAS
To calculate the longitudinal dielectric constant for an ideal free-electron gas we proceed as in Part VI. We first find the conductivity, cr1 (k,u) , and This differs from the formally similar expansion of the conductivity or resistivity in powers of the components of the magnetic field where the term linear in H, which gives the Hall effect, does not vanish because H is a pseudovector. 114 A. A. Rukhadze and V. P. Silin, Usp. Piz. Nauk 74, 223 (1961); see Soviet Phys.Usp. (English Transl.) 4, 459 (1961). 111 The linear term also vanishes for the cubic point group T d (83m) [W. A. Wooster, "A Text-Book on Crystal Physics," p. 160. Cambridge Univ. Press, London and New York, 19381 which has six reflection planes along the face diagonals of the cube, but no inversion center. Thus the linear term vanishes for the sphalerite (zincblende) structure, which has Td symmetry. 113
395
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
then use the dispersion relation (43.1) to calculate the real part of the dielectric constant. If we retain the gauge $C = 0, the relation between the Fourier components of the longitudinal vector potential and the external charge density is given by Eq. (7.10) :
A(k, w ) where k
=
=
-kp(k, w)/[eok2wK(k, w)],
k
=
0,
(47.1)
I k 1.
With a longitudinal vector potential we must use: p+
=
P
+ 4%
(47.2)
as in Eq. (35.3). Thus the matrix element (35.9b) is: 312
= (#j
I exp (2k.r) [go* (p + %k)l 1 h).
(47.3)
Except for this change, we can follow the procedures and notation of Part VI, and find without difficulty that the conductivity in the longitudinal case is:
For degenerate statistics we find, using Eqs. (41.11)-(41.14), that u1
(k, w ) = [3Ne2w/ (mk2v02 j ] (7ru/2) , =
u+z
[ 3 N e 2 w / ( m k 2 u ~ 2 ) ] [ ~ / ( 8 z) ] ([u 1 - z]~),
ju-z~
1 u - z I > 1.
0,
(47.5)
In the transverse case we included a delta-function term in the conductivity to account for the current V xM,and chose the coefficient of that term to be of such magnitude that the sum rule (42.1) was satisfied. No such term is needed in the longitudinal case, since a longitudinal vector potential does not give rise to a magnetic induction B, as is easily seen from Eq. (7.3b) , and therefore does not induce any magnetization. Furthermore, the conductivity given in Eq. (47.5) satisfies the sum rule (42.1) exactly. For nondegenerate statistics we find that the expression corresponding to Eq. (41.18) is ul(k, w )
=
[ ~ " ~ N e ~ 1 i ~ w / ( 3 2 m z ~ exp [ K T[) ~ )(u ] -z)~] X [l - exp (-fiw/KT)],
which also satisfies the sum rule (42.1).
(47.6)
396
FRANK STERN
To find the real part of the dielectric constant from the conductivity, we need only integrate Eq. (43.1).For degenerate statistics the integration leads to the result :
This agrees with the result obtained by Lindhard.118He showed that one
can take the scattering of the electrons into account phenomenologically by replacing u by (u i r ) / ( k v F ) in Eq. (47.7)and eliminating the absolute value signs. The imaginary part of the resulting expression reduces to Eq. (47.5)in the limit r + 0. The phenomenological introduction of a relaxation time into quantum-mechanical expressions for the dielectric constant is discussed by Ehrenreich and Philli~p.4~ For nondegenerate statistics we find :
+
(47.8) The integral in Eq. (47.8)is exactly the same as the one in (43.3).This allows us to use our eaqlier results for K~~ (k, u) and to express the dielectric constant for longitudinal polarization and nondegenerate statistics in the form: Kl(k, W ) = 1 2U2[~lt'(k,W) - 1 (W~~/W')]. (47.9)
+
+
We can expand the expressions for the real part of the dielectric constant for various ranges of interest, and have summarized in Table V the results of such expansions for both degenerate and nondegenerate statistics for 1, z << u,u,z << 1, and u, 1 << z. For the first 2 of these ranges we have also listed the corresponding values for the transverse dielectric constant, given in Eqs. (43.4)and (43.5). The frequencies for which 1 ?(k, u) 1 % 0 have particular significance, since for these a longitudinal mode can most easily be excited in the medium by a source p(k, u). The damping of the mode will be small, and we refer to it as a plasma oscillation. From the first line of Table V we can show that for both degenerate and nondegenerate statistics the angular frequency of 116
Equation (3.7)in Lindhard?'
397
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
TABLE V. APPROXIMATE EXPRESSIONS FOR THE LONGITUDINAL AND TRANSVERSE DIELECTRIC COXSTANTS
We tabdate in each case the value of xl(krw ) , using the parameters k = 1 k 1, vo = liko/m, and vo is given by ( 2 & ~ / m ) for degenerate statistics and by (2KTlm) for nondegenerate statistics. EF is the Fermi energy, and up2= Ne2/meo. z = k/2k0, and u = w/kv0, where
Approximation
Nondegenerate
Degenerate
Longitudinal dielectric constant [see Eqs. (47.7)-(47.9)] 1U)z<
1
+ 3wv2 [l - (u2 + k2v02
I +
+22)]
2 4
- [l -
k2v02
(2242
+ Q.81
Transverse dielectric constant [see Eqs. (43.4) and (43.511
u, z<
0
UP2
1 - - [3u2 0 2
+
221
This entry is based on an expansion containing one more term than in Eq. (43.4a).
plasma oscillations in a free-electron gas is given by:
[w(k)l2 =
up2
+ k2[v2]~v)
(47.10)
where wv2, as before, is Ne2/(meo). We have used the fact that for degenerate statistics (47.1l a ) [v2]Ay = 6Ep/5m = #vo2, while for nondegenerate atatistics
[vZlAy = 3KT/m
=
tv2.
(47.11b)
398
FRANK STERN
Interactions between electrons will result in a longitudinal dielectric constant rather different from the one we have found for a free-electron gas in Eqs. (47.7) and (47.8). The reader is referred to the work of Glick1’7 for further details. 48. SCREESIKG OF
A
STATICPOISTCHARGE
The longitudinal dielectric constant for static fields (a = 0, u = 0) describes the effect of the electrons in screening the Coulomb field of an external charge introduced into the medium. Before discussing this effect in terms of the wavelength-dependent longitudinal dielectric constant, let us give the conventional theory of the screening.ll* Consider a free-electron gas with N electrons per unit volume, to which we add a point charge Ze a t the origin and, to keep the system neutral, Z free electrons. I n the neighborhood of the charge there will be an electrostatic potential 4 ( r ) , which will add -e$(r) to the energy of each electron. Since in thermal equilibrium the Fermi level must be constant throughout the sample, the electronic charge density near the impurity will be given by
+ ed (r)] / K T ),
-en (r) = [- e/ ( 2a2)](2mKT/fi2) a ’ z F( [GF ~~
(48.1)
where F,(z) is the Fermi-Dirac function defined in Eq. (42.8), and GF is the Fermi energy measured with respect to the bottom of the electron energy band far from the impurity. The total charge density also includes a term Ne for the uniform background of positive charge and a term Ze6(r) for the added charge at the origin, where 6(r) = 6(z) 6(y) 6 ( z ) is a threedimensional delta function. Thus Poisson’s equation takes the form:
+
V24(r) = [ - e / ( ~ ~ ~ ) ] [ Z 6 ( r ) N
-
(27~~)-~(2mkT/fi*)a’2
+
X F I I ~ ( C E F e9(r)l/KT)l, (48.2)
where K~ is the static dielectric constant of the medium in which the electrons move. We must solve Eq. (48.2) subject to the boundary condition r$ (r) + 0 as r CQ , since the system is neutral. At large distances from the impurity n(r) + N , and the right-hand side of Eq. (48.2) goes to zero. Equation (48.2) is easily solved if we expand the right-hand side in powers of +(r), and drop second and higher powers. This approximation will be valid if we are not too close to the impurity. Under these conditions it is easy to show, using V2(l/r) = -4d(r),1l9 that --f
9 (r)
3 exp ( -r / L ),
= [Ze/ (47rK&oT)
(48.3)
A. J. Glick, Phys. Rev. 120, 1399 (1963). G.5008, “Theoretical Physics.” Hafner, New York, 1934; see also p. 87 in Mott and Jones.8l 118 See p. 3 in Panofsky and Phillips.? 117 118
ELEMENTARY OPTICAL P R OP E R T I E S OF SOLIDS
399
where the screening length L, sometimes called the Debye length or the Debye-Huckel120length, is given by:
L-2
=
CNe2/ WOK T )] [ F I I z ’ ( x lFllz ) ( x )] z = E p l K T +
(48.4)
For a strong electrolyte*1st120 in which there are several types of ions, with charges Z,e and concentrations N , , contributing to the screening, the factor Ne2 in Eq. (48.4) must be replaced by x , N , Z , 2 e 2 . We can use nondegenerate statistics, for which FlI2’/F1I2 = 1, in this case. A simple extension of the theory to the case of semiconductors121with many conduction and valence bands leads again to the potential given by Eq. (48.3), with L given by
where n; and p j are the concentrations of electrons and holes in various bands, &,; and &,,j are the corresponding band edges, and x i = ( I F - &,;)/KT,z j = ( & , j - & F ) / K T .Our use of the function F I p ( x ) is based on the assumption that the density of states in each band varies as the square root of the energy from the band edge. It is easy to find the spatial dependence of the charge density around the added point charge directly from Eq. (48.3), using
p(r)
=
-eEoKaV2qb(r) = Zes(r) - [ Z e / ( 4 x r L 2 ) ]exp ( - r / L ) .
(48.6)
The first term represents the added charge at the origin, and the second term represents the screening electrons. The electrons are attracted toward an added positive charge, and are repelled from a negative charge. The integral of Eq. (48.6) over all space gives zero, which confirms that the screening charge just cancels the added charge. We have so far considered only the terms in Eq. (48.2) which are linear in +(r). For small enough values of T , however, the potential will become large in comparison with K T/e, or with &/e, and can no longer be treated as a small quantity. Solutions of Eq. (48.2) which do not make the linear approximation have been carried out numerically for a number of cases.122 The solutions for positive Z and those for negative Z differ markedly, while our linearized solution (48.3) only changes sign. The screening of a static charge distributed on a plane is an important aspect of the description of surface properties of semiconductors and metals. A solution of the equation corresponding to (48.2) for planar P. Debye and E. Huckel, Physik. 2.24, 305 (1923). R. B. Dingle, Phil. Mag. [i46, ] 831 (1955). 112 L. C. R. Alfred and N.H. March, Phil. Mag. [8] 2, 985 (1957). 120
IZ1
400
FRANK STERN
geometry has been given for nondegenerate statistics by Garrett and Brattain.12a Let us now consider the screening of an external charge using the wavelength-dependent dielectric constant of a free-electron gas. If we choose a gauge in which the vector potential vanishes, the relation between the Fourier components of the scalar potential and the charge density is given by Eq. (7.11) : $(k, w) = F(k, w)/[eok2z(k, .)lo For the particular case of a static point charge, p(r, 1 ) = zes(r),
we have
1
- z1.r)
p(k, 0) = ( 2 ~ ) - 4 p(r, t ) exp (iwt =
(48.7a)
dar dt
[Ze/(8r*)]6(u).
(48.7b)
Thus, if we use the spherical symmetry of this case, we find that the scalar potential is given formally by:
$(r, t )
=
1$(k,
w)
exp (2ls.r
- iwt)
d*kdw
Only for degenerate statistics is a simple closed form available for K(k, 0). In that case we have from Eq. (47.7) : Kl(k,
0) = 1
+ [3wp2/(4k5p2~)][ (1 -
2')
In
z + l
It-11
+
2zl,
(48.9)
but integration of Eq. (48.8) is still formidable. The situation is much simpler if we ask only about the behavior of the potential where it is slowly varying, that is, if we restrict ourselves to small values of k. Then we find from the second line of Table V that
K(k, 0)
= 1
- 7 + L-2k-2,
~<
(48.10)
where, for degenerate statistics,
L-2 12s
=
3ive2/(2EOEF),
C. G. B. Garrett and W.
9 =
W,2/(4kFSF2),
H.Brattain, Phys. Rev. 99, 376 (1955)
I
(48.1la)
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
401
and for nondegenerate statistics
L-2
= Ne2/(e&T),
q = wp2/(3ko2v02).
(48.11b)
If we neglect the term q which appears in Eq. (48.10), we find, using methods to be described below, that substitution of Eq. (48.10) into Eq. (48.8) leads to exactly the result (48.3) given by the conventional treatment. Furthermore, the screening length L given in Eq. (48.11) agrees with the value given in (48.4) if we take the appropriate limits for degenerate and nondegenerate statistics, and take x8 = 1. Thus in first approximation the wavelength-dependent dielectric constant of a freeelectron gas leads to the same description of the screening as that given by the conventional treatment. Let us now look more closely a t the quantities in Eq. (48.10). This is most easily done for the degenerate case. We find, using Eqs. (41.12), (41.13))and (48.11a),that L-’
=
4 k ~ (/~ a o ) ,
7 = 1/ ( ~ T u o ~ F ) ,
(48.12)
0.2, where a. is the Bohr radius, 4neoh2/(mez). For a typical metal q and should not be neglected outright in the integration for the potential using Eq. (49.10). If we insert Eq. (49.10) into (48.8) and rearrange terms we find:
4(r)
=
r
[ Z e L z / ( 2 ~ z e o r ) ] k sin kr(Czkz
+ 1)-* dk,
(48.13a)
where 2 2 = (1 - q ) Lz.This integral can be evaluated using standard methods of contour integrati~n,’~~ and gives:
4(r) = [Ze/(4se0[1
- q l r ) ] exp
(-r/d:).
(48.13b)
This differs from the conventional result (48.3) through the appearance of the factor ( 1 - q ) in the denominator, and through the presence of d: = (1 - q)l/zL,instead of L, in the exponent. We shall establish below that Eq. (48.13b) is valid only for high carrier density, where q is small, so that the difference between (48.1313) and (48.3) is not great. Furthermore, Eq. (48.13b) cannot be correct near r = 0, where r4(r) must equal Ze/(4aeo) for all electron densities. If we use the formal expression for the static dielectric constant in the (4w,2m2)/(fi2k4),we find that limit of large x, given in Table V to be 1 Eq. (48.8) yields a potential:
+
4 (r)
=
[Ze/ (4reor)] cos (rid:') exp ( -r/d:’),
(48.14)
See, for example, E. T. Whittaker and G . N. Watson, “A Course of Modern Analysis,” 4th ed., Sect. 6.222. Cambridge Univ. Press, London and New York, 1950.
402
FRAKK STERN
where 2’ = [ f i / (mwp)]1‘2for both degenerate and nondegenerate statistics. This agrees with the result given by Bonch-Bruevich and Koga111?4~for this case. To estimate the validity of the potentials (48.14) and (48.13) or (48.3), it is convenient to introduce the Kigner-Seitz radius rs, the radius of a sphere containing 1 electron in an electron gas of density N :
rs
=
[3/(4aN)]lj3
=
(9a/4)l13kp-l
=
1.9192kp-l.
(48.15a)
We take the unit of distance for the free-electron gas to be the Bohr radius ao. We can give the results somewhat wider validity by using an effective Bohr radius a* = ( 4 7 T € 0 K , 6 ~ )/ ( m*ez) = (Ksm/m *) Uo, 148.15b) where m* is the effective mass and K~ is the static dielectric constant (apart from the contribution of free carriers) of the medium. The theory given here will apply to such a medium if we replace m by m* and e2 by e 2 / K , everywhere, so that a* represents a scaling factor. For some purposes, therefore, we can characterize a free or nearly free-electron gas by the dimensionless parameters ?-,/a* or a * k ~ . We can now write the following expressions for the quantities which enter in Eqs. (48.3), (48.13), and (48.14) :
L/a*
=
7 =
(7r/12)1~2~r,/a*)1~2 = 0 . 6 4 0 ( r , / ~ * ) ~=’ ~ 0 . 8 8 6 ( a * k ~ ) - ” ~(48.16~~) 2L’33-513a-4’3(rs/a*) = 0.0553(r8/a*)= O.lOG(a*kF)-l, (48.16b)
C’/a* = 3 - 1 / 4 ( ~ s / a * ) = 3/4 0 . 7 G O ( r , / ~ * ) ~=’ ~1 . 2 3 9 ( a * k ~ ) - ~ / ~ ,
(48.16~)
where degenerate statistics have been assumed. Equation (48.13) was derived using the expansion of the static dielectric constant valid for x << 1 or X: << 2X.p. Thus a condition for the validity of Eq. (48.13) is that a typical wave vector in its Fourier expansion, say 2-l, is < < 2 k ~ .From Eq. (48.16) we see that this requires that r , << 4.5a*. Thus in the range of validity of Eq. (48.13) we see from (48.1Gb) that 9 << 0.25. This is the basis for the statement made earlier in this section that the difference between Eqs. (48.13) and (48.3) is not large. Since real metals have values of r8 of the order of 3a*, the validity of Eqs. (48.13) and (48.3) is a t best marginal for metals. On the other hand some semiconductors, because of their small effective masses aiid high static dielectric constants, have large values of a x , and thus may be in the high-density limit, with r, < a*, for quite moderate carrier densities. For example if m* = O.lm and K~ = 10, the carrier density for xhich r8 = a* is 1.6 X lo1* 1*4n
V. L. Bonch-Bruevich and Sh. M. Kogan, Fiz. Tverd. Tela 1, 1221 (1959); see Soviet Phys.-Solid State (English Transl.) 1, 1118 (1960).
ELEMENTARY OPTICAL PROPERTIES O F SOLIDS
403
~ m - ~which , is well within the range of common carrier densities in semiconductors. The condition for validity of Eq. (48.14) can be established in a similar way, and is S'-l >> 2kF, which implies rB>> 71a*. This condition is not met by any metal, and by few semiconductors. One might suspect that the small x and large z expansions of the static dielectric constant (48.9) exhaust the possibilities for obtaining simple results for the form of the potential around an impurity. We must, however, look carefully a t the region near z = l, where the derivative of Eq. (48.9) has a logarithmetic ~ingularity,'~~ although the dielectric constant itself is continuous and has the value 1 (2?raokF)-' there. (We return here to the free-electron gas. The expressions can be scaIed to apply to other solids by replacing a. by a*). The effect of the singularity in the derivative of the dielectric constant on the potential is easily brought out by integrating Eq. (48.8) by parts once. We find that the dominant term in the potential can be written:
+
(48.17) where x = 2kFr. For large values of x the greatest contribution to the integral comes from the immediate neighborhood of z = 1. If we introduce x = 1 y, we find that the leading term in the potential will be
+
#(r)
-
-[UokF2ze
COS
+ $)')>3
X/(?rroX2(nUok~
1
m
0
COS
x y In y dy. (48.18)
We have restricted the integration to positive values of y, and have added a factor 2, since cos x y is an even function. The term in sin x which arises x y ) vanishes in our approximation, since in the expansion of cos (x sin x y is odd. The integral in Eq. (48.18) has the value - ? ~ / ( Z X ) . ' ~ Thus ~ we find that the singularity in the derivative of the dielectric constant a t z = 1 gives rise to an oscillating potentia1126a
+
4(r)
-
+ 3)2~31,
aok2Ze cos X/[2€O(?raOkF
(48.19)
I am indebted to Dr. A. A. Maradudin for making available unpublished results (1959) by J. J. Quinn, R. A. Ferrell, and A. A. Maradudin on this subject. l Z 6 See Table 467 in Bierens de Haan.109 l Z B s Expressions for the oscillations in the potential far from an impurity are given, for example, by J. S. Langer and S. H. Vosko, Phys. Chem. Solids 12, 196 (1959), and by T. Murao, J . Phys. SOC.Japan 17, 341 (1962). For comparison with experimental results on the Knight shift in alloys, see A. Blandin, E. Daniel, and J. Friedel, Phil. Mug. [S] 4 , 180 (1959); and W. Kohn and S. H. Vosko, Phys. Rev. 110, 912 (1960). 'z6
404
FRANK STERN
with a corresponding oscillating charge distribution. Since Eq. (48.19) drops off more slowly than (48.13) with increasing T , (48.19) will be the dominant term in the potential far away from the impurity. At nonzero temperatures the oscillations will be damped, because the Fermi surface is no longer sharp.12Bb A treatment using a dielectric constant is essentially a linear treatment, valid for weak fields. Thus methods based on a wavelength-dependent dielectric constant cannot be expected to be valid in the immediate vicinity of the impurity, where the electrostatic potential and the electric field will be very strong. The wavelength-dependent dielectric constant does, however, give a more accurate description of the potential far from the impurity, particularly the oscillating term (48.19), than does the conventional result (48.3). 49. ENERGY Loss OF
A
MOVINGCHARGED PARTICLE
The wavelength-dependent dielectric constant provides a convenient formalism for describing the energy loss of a charged particle moving in a medium, We give here only the principal result of such a treatment, namely the dependence of the energy loss on the imaginary part of the reciprocal of the longitudinal dielectric c ~ n s t a n t . * * J ~ ~ J ~ ~ Consider a particle of charge Ze moving with velocity v in a medium. The fields produced by the particle lead to polarization of the medium, and this polarization in turn produces a field a t the position of the particle which will tend to oppose its motion. The power loss can be expressed as -d&/df = -ZeE.v, and the stopping power, or energy loss per unit length, is: (49.1) -v-1 d&/dt = -ZeE v/v, where E is the field a t the position of the particle. To find the fields associated with a moving point charge we need only find the appropriate expressions for and j e x t , and apply Eqs. (7.8b) and (7.10). The charge density for a particle of charge Ze which passes the origin with velocity v a t time t = 0 is p(r, t ) = Ze8(r - vt), and its Fourier components are :
p(k, u) = [2e/(8lra)]8(u Jext(k,u)
=
- k*v),
[Zev/(8na)]G(u - k - v ) .
(49.2a) (49.2b)
N. H. March and A. M. Murray, Proc. Phys. Soc. (London) 79, 1001 (1962); C.P. Flynn and R. L. Odle, ibid. 81, 412 (1963). 187 H. Frohlich and R. L. Plataman, Phys. Rev. 92, 1152 (1953). 1*6b
405
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
Note that these satisfy the conservation equation (7.9). If we choose the gauge 6 = 0, we find that:
iL(k,
w) =
&(k,
0)
=
- k.v)/[k2w'iik,
-[Ze/(8a3~o)]jk6(w [Zep~/(S+)][v
w)]],
- s(&.v)]b(u - k*v)/[k2 - (w2/c2)"Kr(k,a)]. (49.3b)
The electric field a t any point is then given by:
E(r, 1 )
where
EL(r,t )
=
(49.3a)
[-iZe/(Sa%o)]
=
EL
+ ET,
{ k exp [zk-( r - vf)]/k2K(k, kev) 1 d3k, (49.4)
with a corresponding expression for ET which is easily deduced from Eq. (49.3a). To find the energy loss of the particle in the medium we must use Eq. (49.1), i.e., we must find the field a t the point r = vf. The integral in Eq. (49.4) simplifies if we introduce new variables w' and t such that k - v = kvt = w', and d3k = 2ak2dkdt = 2ak dkdw'lv. The components of k which are perpendicular to v will cancel in the integration, so we can write k ---f v ( k . v ) / u 2 = w'v/u2. Thus we findsa:
E L (vt, t )
=
li-' dk [- iZev/ (4a2~Gv3) ]
[w'/K
0
( k , 0') ] dw'.
(49.5)
We noted in ( 7 . 4 ~ that ) the reality of E(r, t ) and D(r, f) implies:
'i(-k,
-0)
=
'i*(k, w ) .
(49.6)
For a medium with inversion symmetry, 'i(k, w ) = 'i(-k, w ) , so that for such media we have 'i(k, - w) = 'i*(k, w ) . Under these conditions it follows, as in Part 111, that the real part of K is an even function of w , while the imaginary part is an odd function of w . The same properties hold for the real and imaginary parts of l / K . Then we find that the stopping power (49.1) is: rm
- v-I d E/dt = [- Z2e2/ ( 2n2eov2)]
]0
rkv
k-' dk
w' 10
Im[ 1/K (k,w ' ) ] dw', (49.7)
plus another term derived in a similar way from (7.33b), which gives the energy loss to cerenkov r a d i a t i ~ n , ~and " J ~is~ usually much smaller. Equa-
406
FRANK STERN
tions (49.5) and (49.7) can be expressed in an alternative form if we integrate using cylindrical instead of spherical coordinate^."^ Equation (49.7) shows that the greatest part of the stopping power comes from regions of the spectrum for which the imaginary part of the reciprocal of the dielectric constant is large, i.e., from regions for which i ( k , w ) E 0. A similar result was found in Part IV, where we noted that a weak external disturbance of angular frequency wL could produce largeamplitude longitudinal oscillations of the lattice, and could result in substantial power dissipation in the medium. In an electron gas these longitudinal oscillations are the plasma oscillations, and occur at a frequency given by Eq. (47.10) for long wavelengths. The contribution of the plasma oscillations to the stopping power of an electron gas is sensitive to the form of L(k, w ) , and the theoretical results are modified when electron-electron interactions are considered.117 A very simple expression for the stopping power can be derived if we assume that all the absorption occurs a t the plasma angular frequency, which we take equal to up from Eq. (41.13), ignoring the dispersion given in (47.10). If we use the sum rule (23.4), we find that the w’ integral in Eq. (49.7) equals -7rNe2/(2meo), provided k > k m i n = wP/v. An upper limit to the integration over k is given by the maximum momentum transfer possible for an electron of velocity v, namely fik,,, = 2mv. With these drastic approximations the power loss of a charged particle moving through a free-electron gas becomes: -v-I
d&/dt .= [NZ2e4/ (47reo2mv2)]In (lcmaX/k,,,i,,) =
[NZ2e4/ (47reo2mv2) 3 In (2mv2/hw,) ,
(49.8)
a result given by Kramers.12*
50. FORCES BETWEEN CHARGED PARTICLES The electric field of a slowly moving charged particle is given by Eq. (49.4), plus a small term derived from Eq. (49.3), which can normally be neglected. If the particle of charge Ze is at rest at the origin, and a particle of charge Z‘e is at the point r at time t, the latter particle will experience a force Z‘eE(r, t )
=
[-iZZ’e2/(87r3eo)]
1{ k
exp (zk.r)/[k2L(k, O)]} d3k. (50.1)
If the wavelength dependence of the dielectric constant can be neglected, we can replace i(k, 0 ) by K ~ the , static dielectric constant, and can remove l*8
H. A. Kramers, Physica 13, 401 (1947).
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
407
it from the integral. The remaining factors are easily integrated if we put: k - r = lcrt, and exp (zk-r) = c i z ( 2 Z l ) j ~ ( k r ) P l ( t ) ,where l ~ ~ jz is a spherical Bessel function of order Z, and Pl is a Legendre polynomial. Only the component of k in the direction of r will survive the integration over the azimuthal angle, so we can write k + krE/r = lcrPl(E)/r. After the integration over angles, Eq. (-50.1) reduces, for a wavelength-independent dielectric constant, to:
+
Z’eE(r, t )
=
/om
[ZZ’e2r/(2a2eor3~s)] zjl(x) dx.
(50.2)
The value of the integral in Eq. (50.2) is +7r,l3O so the force on a particle of charge Z‘e a t the point r is:
Z’eE (r, t )
=
ZZ’e2r/(47reor3~,),
(50.3)
which is the rcsult we would have written down from elementary electrostatistics. We have used this lengthy procedure to find a-simple result in order to emphasize that it is the static dielectric constant which appears in Eq. (50.3) or in the more general wavelength-dependent expression (50.1), even though the particle experiencing the force may be moving.lal A simple model which leads to Eq. (50.3) is that the particle of charge Ze, which is static, polarizes the medium and attracts a compensating charge -Ze (I - K ~ - ~ ) somewhat , like the case of screening by free electrons which we discussed earlier in this chapter. In the more general case we are considering now, the screening may be the result of the displacement of free electrons, bound electrons, or ions, or a combination of all of them. A charged particle a t a distant point will see a force which is reduced by the effect of this screening, and which is given in the general case by Eq. (50.1). If both the charge Ze a t the origin and the charge Z‘e at the point r are stationary, one mi’ght ask why there are not two factors of K~ in the denominator of Eq. ( 5 0 . 3 ) , since both particles will have a screening 130
See p. 409 in Stratton.2 This result is a special case of the more general result
/o+m
z ~ , ( z ) da: = 2~--14r (3
+ +q + ;p)/r (1 + t q - t p )
valid for q p > -1, q - p > - 2 , which can be deduced from problem 32 a t the end of Chapter 17 in Whittaker and Watson.lZ4 131 On the other hand, the force exerted on the charge Ze by a moving charge will contain the frequency-dependent dielectric constant even though the charge Ze is stationary.
408
FRANK STERN
cloud around them. Only one factor appears because we are calculating only the force on the particle, and do not include the force on the screening cloud of electrons and ions. The screening ions can, however, modify the dynamics of the motion of the charge considerably.182 The forces between charged particles in a dielectric have been studied from a many-body point of view by Kohn,las who showed that results of the form given here will still be valid in the presence of electron-electron interactions. Note added in proof: Recently Berreman1a4has observed structure at the longitudinal optical mode angular frequency W L in the transmittance and reflectance of thin LiF films. This effect, like the reflectance peak for silver14JsJasmentioned on p. 326 and the photoemission near the plasma frequency of alkali metals found by Ives and Briggsts is observed only for obliquely incident light polarized in the plane of incidence. This light couples strongly to longitudinal modes of the system at frequencies for which both the real and imaginary parts of the dielectric constant are small. We have seen that the longitudinal modes may be electronic, ionic, or, as in Section 34,a combination of the two. Structure in the reflectance or transmittance at a frequency for which i? SJ 0 can be observed only in thin films because in thicker samples, as Ferrell and E. A. Stern14 have pointed out, such a frequency is an endpoint of a spectral region of nearly total reflection. Acknowledgments
I am indebted to many colleagues at the Naval Ordnance Laboratory, the University of Maryland, and the Thomas J. Watson Research Center for discussions of varidus aspects of this work, and to Dr. Jack R. Dixon, Dr. William Frank, Dr. Arnold J. Glick, Dr. Peter B. Miller, and Dr. Peter J. Price for comments on the manuscript. I particularly thank Dr. Richard A. Ferrell for many valuable suggestions, and Mr. Ralph F. Guertin for writing most of Part I1 and for reading and criticizing the entire text. The careful preparation of the manuscript by Mrs. Marcella Z. Anderson is gratefully acknowledged. Finally, I thank Dr. Louis R. Maxwell and Mr. Herbert W. McKee for unfailing support during the writing of the manuscript, and Dr. Robert W. Keyes and the Thomas J. Watson Research Center for facilitating the preparation of the final draft. See, for example, T. D. Schulta, Phys. Rev. 116, 526 (1959). W. Kohn, Phys. Rev. 110, 857 (1958). la4D.W.Berreman, Phys. Rev. 130,2193 (1963). 186 A dip in the transmittance of thin silver films at the plasma frequency has been observed by A. J. McAlister and E. A. Stern, Bull. Am. Phys. SOC.8, 392 (1963). la
1aa
...... . . 300 . . . . . . . . . . . . . . 300 1. Introduction. ......................... 2. Maxwell’s Equations. ....................................... 301 3. Conservation Equations. ...................... ..... . . . . . . . 304 305 4. Plane Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Inhomogeneous Case ........................ 307 6. Energy Density.. . . ........................ 308 7. Sources, Potentials, and the Wavelength-Dependeht Dielectric Constant. . 310 8. Alternative Treatments of the Magnetization. . . . . . . . . 11. Reflection and Transmission. . . . . . . . . 9. Boundary Conditions. . . . . . . . . . . . . . . . . . . . 10. Amplitude Reflection Coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 . . . . . . . . . . . . . . . . . 317 11. Snell’s Law and Generalizations. . . 12. Energy Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 13. Reflectivity. .............................. 14. Total Reflection.. ......................... 15. Transmission of Isotropic Radiation. . . . . . . . . . 16. Thin Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 111. Dispersion Relations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 328 17. Dispersion Relations for the Dielectric Constant.. . . . . . . . . . . . . . . . . . . . . . 18. Dispersion Relations for the Index of Refraction.. . . . . 19. Qualitative Consequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 20. Phase Shift Dispersion Relation. . . . . . . . . . 21. Numerical Procedures for Calculating the P 22. Extrapolation Procedures. . . . .................................... 338 23. Derivation of Sum Rules. . . . .................................... 340 IV. Optical Properties of Simple Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 24. The Effective Field. . . . . . ..... ... . 342 A. The Free-Electron Gas.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 25. General Considerations. . ................................ 344 26. Conductivity and Mobilit ................................ 346 27. Optical Properties in Various Frequency Ranges.. ..................... 347 28. Reflectivity Minimum. . . . . . . . . . . . . . . . ...... 350
I. Electromagnetic Preliminaries. . . . . . . . . . . . . .
.
* Present address: IBM Thomas J. Watson Research Center, Yorktown Heights, New York. 299
300 29. Energy Density.
FRANK STERN
...........
............................
B. Optical Modes in Ionic Crystals. ..........................
30. Phenomenological Discussion. . . . . . . . . . . . . . . . . . . . 31. Atomic Theory of Optical Modes. ...................... 32. Longitudinal and Transverse Modes; Effective Charge.. . . . . . . . . . . . . . . . 33. Reflectivity. .................................. .............. 34. Free-Carrier Effects.. .......................... .............. V. Quantum Theory of Absorption and Emission.. ..........................
350
356 359 361 364
.......... 37. Optical Absorption for a Simple Model. 38. Free-Carrier Absorption. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39. Oscillator Strengths .............................. VI. Wavelength-Dependent Dielectric Constant of a Free-Electron Gas 40. Introduction. ...... ........................ ................... 41. Calculation of the Co 42. Verification of the Su ........................................
370 374 375
45. Surface Impedance and Anomalous Skin Effect.. ...................... 46. Real Crystals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII. Interactions of Charged Particles.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47. Longitudinal Dielectric Constant of a Free-Electron Gas .......... 48. Screening of a Static Point Charge.. . . . . . . . . . . . . . . . . . 49. Energy Loss of a Moving Charged Particle. .......... 50. Forces between Charged Particles.. .................................. Acknowledgments............ .....................................
389
383
394 394 406 408
1. Electromagnetic Preliminaries
1. INTRODUCTION One of the most powerful tools for studying the properties of solids is the measurement and analysis of their optical properties. In this work some of the results required for such an analysis are presented, with emphasis on the detailed development of simple models. We shall express many of the results in numerical form, and this in part has dictated the use of mks units throughout. The treatment is elementary in the sense that no physics beyond Maxwell’s equations and simple quantum mechanics is used. Our detailed considerations are limited to materials for which the usual optical constants are scalars, i.e., to isotropic substances and to cubic crystals, and to the case of zero magnetic field other than that of the electromagnetic radiation itself. We neglect the distinction between
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
301
energy and free energy, a procedure which is strictly correct only a t absolute zero.’ Most of the material in the first two parts is a review of standard results of electromagnetic theory, and serves mainly to establish the notation used here, and to list results for later reference. Sections 7 and 8, which give a formal introduction to the wavelength-dependent dielectric constant, may be somewhat less familiar. Dispersion relations as applied to the analysis of optical properties are discussed in Part 111. We make frequent use of dispersion relations in later chapters. I n Part IV we summarize some of the classical results for two very important physical systems, the free-electron gas and the optical lattice vibrations in ionic crystals. These systems are sufficiently simple that detailed results can be obtained Yery easily, yet realistic enough that the results give quite a good representation of a t least some of the properties of real solids. Some of the more basic results of the quantum theory of absorption and emission of radiation are given in Part V. N o attempt is made to derive specific results for real solids, since this requires detailed calculations beyond the scope of this work. In the last two parts we consider the transverse and longitudinal wavelength-dependent dielectric constants for a free electron gas, for which detailed results can be obtained relatively easily. These results are applied in a number of examples, including the anomalous skin effect, the screening of a static point charge, and the energy loss of moving charged particles. 2. MAXWELL’S EQUATIONS
The theoretical framework for optical properties is given by Maxwell’s equations, which we give in mks units2:
V - B = 0,
(2.la)
V - D = p,
(2.lb)
V x E
=
-dB/dt,
(2.lc)
V x H
=
J
(2.ld)
+ aD/at,
where p is the externally introduced charge density and J is the current density including both the conduction current Jcond and the externally 1
*
H. Frohlich, “Theory of Dielectrics,” p. 9. Oxford Univ. Press, London and New York, 1949. J. A. Stratton, “Electromagnetic Theory.” McGraw-Hill, New York, 1941.
302
FRANK STERN
TABLE I. QUANTITIES ENTERINQ IN MAXWELL’S EQUATIONS Quantity
mks Units
E (electric field) D (electric displacement) p (charge density) J (current density) B (magnetic induction) H (magnetic field)
1 volt/m 1 coul/m* 1 coul/ms 1 amp/m2 1 weber/m* 1 amp-turn/m
F~
Practical units = 10-2 volt/cm = lo-‘ coul/cmz = 10-6 coul/cma = 10-4 amp/cma = 104gauss = 47 X 10-0 oersted
= 4u X 10-7 henry/m = 1.25664 X 10- henry/m
eo = ( M O C * ) - ~ = (go/eo)i
=
farad/m
8.8542 X
FOC =
( c o c ) - ~= 376.73 ohms
introduced current Jmt. The mks units for the quantities appearing above are given in Table I. In the presence of matter Maxwell’s equations can be written: (2.2a)
V*B = 0, e0V.E = p
v
x
E
+ aB/at
=
V x B - c-*dE/dt
We used the relations
- v.P,
0,
= po(Jex.t
+
Jcond
(2.2b)
+ aP/dt + V
(2.2c) x
M). (2.2d)
,
+ P,
H pO-lB - M, (2.3) where P is the polarization (the electric dipole moment per unit volume), and M is the magnetization (the magnetic moment per unit volume) of the matter present; po and e0 have the values given in Table I. The charge density can be thought of as the sum of an externally introduced charge density p and an effective charge density -7.P.Similarly, the current density contains the external current, the conduction current, and V x M. The absence of source terms in the other additional terms dP/dt two equations indicates that the electromagnetic field can be thought to originate entirely in changes and currents, since there are no free magnetic charges in nature.* When the fields vary with time there will generally be a phase shift
D
=
+
W. I(.H. Panofsky and hi. Phillips, “Classical Electricity and Magnetism,” p. 128. Addison-Wesley, Reading, hiassachusetts, 1955; see also P. -4.M. Dirac, Phys. Rev. 74, 817 (1948); H. A. Wilson, ibid. 76, 309 (1949); E. M. Purcell, G. B. Collins, T. Fujii, J. Hornbostel, and F. Turkot, ibid. 129, 2326 (1963).
ELEMENTARY OPTICAL PROPERTIES O F SOLIDS
303
between the polarization and the electric field. It is simplest in such cases to represent the fields as the real parts of complex quantities. For example, we represent a plane wave b ~ 3 ~ :
E
=
Re
e = Re [6:0exp (&or - i w t ) ] =
Re [go exp (zlrl.r
- id)]exp ( -kz*r),
(2.4)
where w = 21v is the angular frequency of the field and is the polarization vector. The symbol Re means that the real part is to be taken: Re A = +(A A*). I n general we will have a complex propagation vector f = kl zk~,where klis perpendicular to the planes of constant phase, while kg is perpendicular to the planes of constant amplitude. We can now easily include the possibility of a phase shift between D and E by defining
+ +
D
=
Re
m0exp (&-r - i w t ) ] = Re 6,
fi = &,
(2.5)
where fi, fi, and i may all be complex. Note that D and E, which are real quantities, are no longer proportional to each other a t arbitrary times. One often omits the symbol Re in writing the expressions for the fields. This does not lead to difficulties as long as relations linear in the fields are considered, since Maxwell’s equations are themselves linear in the fields. The current Joond which appears as a source term in Maxwell’s equation (2.2d) includes the current which flows in a conductor in the presence of an electric field, as given in the linear range by Ohm’s law Joond = uE. Furthermore, the displacement in phase with the field is given by D = EE, where both u and e are real. We can combine these terms if we note that for fields whose time variation is given by Eq. (2.4), Jcond
+ dD/dt = uE + =
e
dE/dt = Re [(u
Re { - i w [ c
- iwe)g]
+ i(u/w)]g].
(2.6)
In everything which follows we shall, unless otherwise indicated, combine the effects of the conduction and displacement currents in a complex dielectric constant X defined by X ( w ) = i(.)/€O, i(0) =
E(O)
+ i[u(w)/w]
=
€1
+ ies.
(2.7a) (2.7b)
Where it is desirable to distinguish complex quantities from real quantities, we indicate complex quantities with a tilde, and their real and imaginary parts by subscripts 1 and 2; for example, 6 = a1 ia~.
+
304
FRANK STERN
Alternatively, we can represent Eq. (2.6) as the real part of a complex current Z (a)8, where ".w)
- iwe(w).
= .(Lo)
(2.8)
Only at low frequencies will the conductivity equal the dc value uo. We uniformly use the time variation exp (-id). If we had used then i would be replaced everywhere by -i. In applying exexp (id), pressions taken from various sources, one should be careful to check which time variation was used. For the magnetic fields we can write by analogy:
H
= Re H =
Re [Hoexp
B
=
Reg,
P
=
fir,
p
=
- id)],
(&or
+ i/& =
(2.9)
where p is the permeability and Z,,, is the relative permeability of the medium. 3. CONSERVATION EQUATIONS
The fact that electric charge can be neither created nor destroyed is expressed by the conservation equation:
+ ap/at = 0.
V -J
(3.1) This equation is easily obtained from Maxwell's equations (2.lb) and (2.ld). A second conservation equation can be deduced from Maxwell's equations if we use the vector identity:
V*(E x H) = H.(V
x
E) - E*(V x H).
(3.2)
) (2.ld) we find: Substituting from Eqs. ( 2 . 1 ~ and
V. (E x H)
+ E-jext+ E-(jcond + aD/at) + H-(aB/at) = 0.
(3.3)
When we deal with fields varying sinusoidally with time we can make use of the easily proved relation#: Re A*Re B = Re A x Re B
=
4 R d - 8 * = 4 Re A**B
3 Re
(A x 8*) = + Re (A* x 8 )
(3.4a) (3.4b)
where the bar over the quantities on the left means that their time average 4
See p. 136 in Stratton?
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
305
over one period of the field is to be taken. Thus we find that for sinusoidally varying fields :
+
E*(Joond dD/at)
+ H-(dB/dt) = +(wezfi*g* + o p $ H * H * ) .
(3.5)
We have already identified WE^ with the electrical conductivity in Eq. (2.7). Thus $ ~ & - f i * gives the power dissipation due to Joule heating and, by analogy, 3wp2H*H* gives the magnetic power dissipation. I n both cases the power dissipation arises from the phase difference between the field and the polarization or magnetization, since it is the imaginary part of I or p which appears. This connection between phase shifts and power dissipation is quite general.6 The term E. Jext in Eq. (3.3) can be associated with the power dissipated by externally introduced currents. Note that since we are dealing here with sinusoidally varying fields, the time-averaged energy density in the field is constant. The power dissipated in the material is supplied by the external sources which maintain the fields. Conservation of energy requires that the first term in Eq. (3.3) give the net outflow of energy per unit volume. By analogy with Eq. (3.1), we see that S = E x H (3.6) is the power flow across a unit area normal to S. S is called the Poynting vector. 4. PLANE WAVES
For the plane waves defined in Eqs. (2.4) and (2.9), we find that in a homogeneous material containing no externally introduced charges or currents Maxwell’s equations can be written:
pf.H
=
0,
(4.la)
af.B
=
0,
(4.lb)
f xE
=
WpI3,
(4.1~)
ii x H
= -w&,
(4.ld)
If the real and imaginary parts of f are parallel to each other (a special case of this is a vanishing imaginary part) these equaiions show that for electromagnetic waves in isotropic media g, E, and H form a mutually orthogonal set of vectors. 5
C. Zener, “Elasticity and Anelaaticity of Metals.” Univ. of Chicago Press, Chicago, Illinois, 1948.
306
FRANK STERN
From Eq. (4.1) we find:
i;
x
(f
x
0) = w p f
x H = -dp&
=
(f.f)f- (f.f)B
=
-(f*f)B. (4.2)
Thus
f - f = pi02 =
[ kl
= "K"Kn'w2/cz = (n
l2 - I k2 l2
+ ik)W/C2
+ 22kak2.
(4.3)
We have introduced the index of refraction n, and the extinction coefficient k. The real and imaginary parts of Eq. (4.3) are sufficient to determine I kl I and I ka from n, k, and the angle between kl and k2.If that angle is zero, we have
I
1 kl I
If, in addition,
K~
= ~u/c;
1 k2 I
= ~u/c.
(4.4)
is real, we have: ?z
k
(1 2 I + Ki)]"', = [(~m/2)(1 2 I - K1)I1/'. = [(%/2)
(4.5)
The time-averaged energy flow is given by:
8
=
4 Re (g* x
H)
=
Re
[B*
x
(ff
x B)/2wp]
[(g**fi)ff- (g**ff)&] ( ~ C C ~ / ~ K(f*.E)k, ~ ) if COB (kl, kz) = 1.
= ( 2 ~ ~ w p o )Re -~
(4.6~~)
=
(4.6b)
We can express this nlimerically as:
9 (watts/cm2)
= 1.3272 X 1 0 - * ( n / ~ [I~ )B
I
( v o l t / ~ r n ) ] ~ ~( ~4 . 6 ~ )
Here 2, is a unit vector in the direction of kl_and kz, which are assumed to be parallel to each other. Substituting for E from Eq. (2.4) we find in this case :
8=
(nce0/2rc,)(80*-8~)k exp (-2k2-r).
(kl I / k2).
(4.7)
Thus the flux is attenuated by a factor exp ( - 2 J k2 I d ) on traveling a distance d through the material. The coefficient of -d in the exponent is called the absorption coefficient, and is given by .(w)
= 2
1 k2 I
= 2kw/c = 4nk/X,
(4.8)
where h is the wavelength in vacuum of electromagnetic radiation of angular frequency w, and k is the extinction coefficient for that frequency. When the real part of the dielectric constant is negative, k and a! can be quite large, corresponding to strong attenuation of the light beam, even
307
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
though K Z , which measures the power dissipation in the medium, is small. The attenuation in this case arises because the incident power is strongly reflected. I n metals the distanre k2 1-l = c/kw is called the skin depth 6. From Eq. (4.5) we see that when 1 c1 , << € 2 , i.e., wkl << U / E ” , we have: 6
(4.9 1
= [tL/(puw)]1’2.
Numerical expressions for a and 6 are given in Part IV. 5 . ISHOMOGESEOUS CASE Let us now consider the energy flow in the more general case in which
kland k2 are not parallel. Such waves are sometimes called inhomogeneous
waves. A thorough discussion of the inhomogeneous case is given in the article by Konip in the old Handbuch der Physilc. I n order to discuss Eq. (4.6a) in the inhomogeneous case, we must introduce a more explicit representation for the polarization. Let us define Such a choice can always be made for a field of the form (2.4). El and Ez are the directions of the major and minor axes of the ellipse traced by the end point of E during a complete period. For a linearly polarized wave, Ez will be zero. The phase factor exp (i&) will drop out in any expression in which E is multiplied by E*. We can cause it to vanish by a n appropriate choice of origin on our time scale, and will therefore ignore it henceforth. From the real and imaginary parts of Eq. (4.lb) we deduce that if # 0:
Elski - E2.kz
0;
=
E1*k2
+ Ez*k,
=
0.
(5.2)
Thus we can write: -
w
E*.k
= =
+ (E* - E)].k 2E2.kz + 2iE1*k2. [E
-
4
-
=
-2iE2.G
=
2Ez.kz - 2iEz.kl (5.3)
In the homogeneous case (kl , I kz) we have E * k * = 0, E*.k write : Re [- (E*.G)E]
5
=
0. We now
+ iEl.k2)(El + 2X2)]
=
- 2 Re [(E2*k2
=
-2[(Ez*h)Ei - (Ei.kz)Ez]
=
2k2 x (Ep x El).
(5.4)
W . Konig, in “Handbuch der Physik” (H. Geiger and K . Scheel, eds.), 5’01. 20, p . 194ff.Springer, Berlin, 1928.
308
FRANK STERN
Substituting in Eq. (4.6a) we find for the time-averaged energy flow in the inhomogeneous case :
S=
(21c,wpo)-’[(E*~E)k1
+ 2k2 x
(Ez x El)].
(5.5)
6. ENERGY DENSITY The conservation equation [Eq. (3.3)] for the energy in the electromagnetic field is the basis for the conventional electrostatic result that the energy density is +E.D 4H.B. This result cannot be carried over directly to the electromagnetic case if e or p are functions of frequency. The extension to the dispersive case has been given in a number of places708and is given below. At a fixed point in space, let the electric field vary with time as
+
E(t)
=
Re [EoG(t) exp (-iwot)]
=
Rea(t),
(6.1)
where the envelope function G ( t ) is a slowly varying function of time which we can write in the form ~ ( t= )
/
m
g”(w)
-m
g”(w) =
(2~)-’
m
J-m
exp (-iwt) dw,
(6.2a)
G(t) exp ( i d ) dt.
(6.2b)
The Fourier integral for E(t) is ~ ( t= )
Im ~ ( w exp )
-m
(6.3)
( - i w t > dw,
where B ( w ) = &(w - uo). The Fourier expansion for B ( t ) is similar to Eq. (6.3), with f i ( w ) = Z ( w ) B ( w ) . Thus, on taking the time derivative, we obtain
1
m
dD((t)/dt
=
-m
uZ(w)g”(w
If we introduce a new variable w’ c i i j ( t )/dt
=
-iEo
exp ( --coot)
= w m
J-m
(wo
-
-
UO) exp
wo,
we find
(-id)dw.
+ a’)~ ( w o+ w ’ ) ~ ( w ’ ) X exp (-iw’t)
7
(6.4)
dw’.
(6.5)
L. Brillouin, “Wave Propagation and Group Velocity.” Academic Press, New York, 1960.
* L. D.
Landau and E. M. Lifshita, “Electrodynamics of Continuous Media,” p. 253. Pergamon Press, New York, 1960.
309
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
Since G ( t ) is slowly varying, the range of values of o’ will be small compared with 00, and we can expand Eq. (6.5) in a power series, using (WO
+ o’)Z(WO + w’)
=
wOZ(WO)
+
u’[Z(UO)
+
+
UOZ’(WO)]
* * a ,
(6.6)
where Z’(w) = d a ( w ) / d w , and the series has been terminated with terms linear in w’. On substituting Eq. (6.6) into Eq. (6.5) we obtain: ~ ( t ) / d= t
goexp (-iwot) { -iooi(wo)G(t) + [Z(wo)
+ woZ’(wo)]G’(t))
(6.7) where we have omitted terms involving the second and higher derivatives of G ( t ) . We can now calculate the value of the term E ( t ) -D( t ) from Eq. (3.3). To suppress the rapid sinusoidal oscillations of angular frequency wo we use Eq. (3.4a), which is still valid when the sinusoidal functions are multiplied by a slowly varying function of time. We find:
+ $[rl(wo) + woei’(wo)](d)dt) [B*(t) * B ( t ) ] . (6.8) We have dropped terms which depend on higher derivatives of G ( t ) . The first term on the right in Eq. (6.8) has already been identified in Eq. (3.5) as the power dissipation. The second term represents the rate at which the energy density is being changed. We may therefore give the time-averaged energy density associated with an electromagnetic field in a dispersive medium, including magnetic field terms, in the form:
V(t>
= *[(q
+ wel’)B*(t)
*B(t) + (p1 + w p l ’ ) H * ( t ) - H ( t ) ] .
(6.0)
The relation between H and E is given by Eq. (4.lc), from which we can deduce that, for the case of homogeneous fields:
H*.H
=
(ii
=
[(Mi*) (E.E*) - (&.B*)(E*.E)]/(pp*w2)
=
I c I E*.B// p I.
x E).(ii* x E*)/(pp*w2)
(6.10)
We will consider the energy density for some simple examples in Part IV. If the material has a nondispersive real permeability and a real dielectric constant, then n2 = KK,,,, and we can write:
U
=
nro[n
+ w(dn/dw)]E*.E/(2~,)).
(6.11)
310
FRANK STERN
It follows from Eqs. (4.6b) and (4.4) that the velocity of energy transport in this case is 8 / 0 = c/[n 4-w(dn/dw)] = dw/dkl (6.12) i.e., the group velocity? 7. SOURCES, POTENTIALS, AND THE WAVELENGTH-DEPENDENT DIELECTRIC CONSTANT In this section we shall give formal expressions for the electromagnetic fields produced by given external sources p(r, t ) and Jext(r,t ) . We do this by Fourier-analyzing the sources, finding the response of the medium to the component waves, and integrating over these responses. For example we write:
Jext(r,t )
=
/
W
jext(k,w ) exp (zk-r - i w t ) dak dw,
(7.la)
J-W
for which the inverse Fourier transform is
jext(k, w )
=
J
m
( 2 ~ ) - ~ Jext(r,t ) exp (iwt - 2k.r) d3r dt. -m
(7.lb)
The externally introduced charge density p, and the fields E, B, D, and H can be similarly represented. In these expansions k and w are taken to be real quantities. Fourier components such as jext(k, w ) will generally be complex, but they must satisfy a condition of the form:
Bext(k, w ) ] *
=
j=t(-k,
(7.2)
-w)
to assure the reality of physical quantities such as Jext(r, t ) . We introduce the scalar potential C$ and the vector potential A by noting that Maxwell's equations (2.la) and (2.lc) are identically satisfied if we take:
B(r, t )
=
V x A(r, t ) ,
E(r, t )
=
-vC$(r, t ) - A(r, t ) ,
(7.3a)
which implies that the Fourier-transformed quantities satisfy :
g(k, w )
=
zk x A(k, w ) ,
B(k, w )
=
-zkd,(k, a)
+ iwA(k, w ) . (7.3b)
If we now generalize the complex dielectric constant and permeability of Eqs. (2.4) and (2.9) to permit them to depend on the wave vector k as well as on the angular frequency w, we can write: fi(k, w ) = Z(k, w)B(k, w ) ,
(7.4a)
Wk,
(7.4b)
w) =
w,w)/P(k,
w),
31 1
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
where the conductivity is included in Z(k, w ) , as in Eq. (2.7b). These definitions and the reality condition (7.2) imply:
[i(k, u)]* = i (-k,
(7.4c)
-u),
(7.4d) CP(k, .)I* = P(-k, - w ) . Maxwell's equations (2.lb) and (2.ld), which relate the fields to the external sources, can now be written:
k.k$ - wk-A
+ [wPZ$ - k*A]k
p * k - w2pi]A
= ?/Z,
(7.5a)
=
(7.5b)
PJextI
where each quantity with a tilde is a function of k and Lorentz condition
w.
If we use the (7.6a)
k.A(k, a) = wp(k, w)Z(k, w)$(k, a) to reduce the arbitrariness in the potentials, we find:
- w2p(k, w)Z(k, ~ ) ] $ ( k , a) [k*k - w2p(k, w)Z(k, w)]A(k, U ) [k*k
=
P(k, w)/i(k, o),
(7.6b)
=
p(k, W)Je.t(k,
(7.6~)
w).
Another way to simplify Eq. (7.5) is to separate the vector potential A and the current Jextinto longitudinal and transverse parts:
k
X
A(k, W )
=
L ( k , m ) 4-
Jext(k, w )
=
JL(k a)
+ JT(k,
w),
= 0,
k*&
k*JT = 0.
AL = k
X JL
=
w),
(7.7)
Then Eq. (7.5) reduces to:
k*k$ - wk*AL = z/Z,
[knk
(7.8a)
- o2pZ]A~ = PJT,
(7.8b)
provided the sources satisfy the conservation equation
k*J,,t(k, W ) = w?@, w ) , (7.9) as in Eq. (3.1). Thus the transverse part of the vector potential is determined by the transverse current sources. For the longitudinal part of the field we may choose either the gauge $ = 0,
or the gauge
AL = 0,
AL(k, U ) $(k,
=
W)
-kF(k, w)/[wk*ki(k, =
"pk, w)/[k*kZ(k,
w)],
w)].
(7.10) (7.11)
312
FRANK STERN
It is clear from Eq. (7.3) that Eqs. (7.10) and (7.11) lead to the same electric field, and that both of them lead to a vanishing magnetic field. The wavelength-dependent dielectric constant arises in quite a natural way in this formal treatment. The solution of Maxwell’s equations to find the fields produced by given sources in an homogeneous isotropic medium is reduced to the evaluation of Fourier integrals, provided i ( k , w ) and p(k, 0)are known. Of course the integrals will generally be quite formidable. For most ordinary optical problems one can neglect the dependence of the dielectric constant on k, but for some phenomena, such as the anomalous skin effect, the more complete expressions are required. In Parts III-V we shall assume that the dielectric constant is independent of k, but in the last two parts we return to the wavelength-dependent formalism for the particular case of a free-electron gas, with applications to the anomalous skin effect and to the screening of a static point charge. Some simple consequences of the relations given in this section can be deduced for waves propagating in the absence of externally introduced charges or currents. Then Eq. (7.8b) shows that a transverse wave of wave vector k and angular frequency w can exist only if: k-k
=
(7.12)
p(k, w ) i ( k , w ) w z .
This equation will not have a solution for real k and w if, as is generally the case, pZ is complex. But if we somewhat cavalierly disregard the restriction of k and w earlier in this section to real values, Eq. (7.12) is equivalent to the resul,t found for a single plane wave in Eq. (4.2). A longitudinal wave can exist in the absence of external sources only if
Z(k, w )
=
(7.13)
0
as is easily seen from Eqs. (7.10) and (7.11). Although Z(k, w ) cannot vanish in a real solid in equilibrium, since there will always be a t least a small positive imaginary part of Z, those values of k and w for which Z(k, w ) is close to zero characterize waves for which a fixed external source a(k, w ) will produce a large longitudinal response in the medium. Such waves are called plasma oscillations, and a frequency for which Eq. (7.13) is approximately satisfied is called a plasma frequency. We shall return to this subject later. 8. ALTERNATIVE TREATMENTS OF THE MAGNETIZATION
I n the expressions given so far we have described the induced magnetization through the permeability p&,, i.e. M(k, a)
=
[1
- Z(k, o)]B(k,
PO,
(8.h)
313
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
where we have introduced the reciprocal relative permeability (8.lb) because many equations take a somewhat simpler form in terms of (R than in terms of "K. Equation (&la) is not the only way in which magnetic effects can be described within the framework of Maxwell's equations. The reason for this is that the magnetization enters Maxwell's equations (2.2) only through the term V x M in (2.2d). We can consider this. term to be simply another current term, just as the conduction current and the displacement current are so considered. If we do this, we find by an argument similar to the one in Eq. (2.6) that the magnetic effects can be included in the complex conductivity or in the complex dielectric constant. Thus we can write: CCO-'~
+
+ CTE+ (aD/at) + V
M, (8.2s)
B
=
Jext
Jind
=
Re ( i w ( a -
=
Re [iwutr(k, w)i(k, w ) ] ,
(8.2~)
=
Re [wzeoP(k, w)i(k, w ) ] ,
(8.2d)
X
Jind
=
Jext
h e )
X
+ k * k ( l - %)po-l}L(k, w ) ,
(8.2b)
if no external charges are present and we choose the gauge t j = 0. In Eq. (8.2) the magnetic effects have been absorbed in Ztr or P, and we must then write : f8.2e) R(k, u) = &B(k, 0). The notation for the modified constants P and Fr follows that used by Lindhardsa in his paper on the properties of a gas of charged particles. We use Ztr or Ztr only for transverse fields, since B, H,and M vanish for longitudinal fields, It is possible to absorb only part of the magnetic effects into the conductivity. Thus we could absorb the imaginary part of % in the complex conductivity, and would then write = &p0. If we use Eq. (8.2), the condition for the propagation of an eleclromagnetic wave in the absence of sources is:
k*k = (wa/ca)Pr(k,w ) . But from Eq. (8.2) we see that P(k, w)
=
X(k, 0)
+ C % * k d ( l- %),
(8.3) (8.4)
for which Eq. (8.3) reduces to (7.12), as it should. A treatment based on a wavelength-dependent dielectric constant 88
J. Lindherd, Kgl. Dam& Videnskab. Selskab, Mat.=&s. Medd. 20, No. 8 (1954).
314
F K A S K STERK
must be used with some caution if a boundary between different media is presents*b,soOne example is given in the discussion of the anomaIous skin effect in Part VI. If one wanted to use the formalism of Eq. (8.2) for substances which are magnetic even in the absence of an external field, one would include the current associated with the spontaneous magnetization in Jext. The considerations of the present work, however, assume that there is no spontaneous magnetization or polarization in the media we deal with. II. Reflection and Transmission
9. BOUNDARY COSDITIOSS When electromagnetic waves pass from one medium into another, certain boundary conditions must be satisfied, giving rise to the phenomena of reflection and refraction. These boundary conditions, which may be obtained from Maxwell’s equations, are9:
n.(DA - DB)
= 7,
n x (EA - EB) = 0, n-(BA - BB)
(9.la) (9.lb)
0,
(9.lc)
n x (HA - HB)= j,
(9.ld)
=
where n is a unit vector, directed from medium B toward medium A at right angles to the boundary between them, and j and r represent the surface densities of current and of externally introduced charge, respectively. Equations (9.lb) and ( 9 . 1 ~ )express the continuity across the boundary of the tangential component of t,he electric field and the normal component of the magnetic indue tion. We shall be concerned only with radiation incident upon a boundary containing no currents or charges of external origin. Consequently, we may write : ,. n. (ZAEA - ZBEB) = 0, (9.2a)
-
nx(L -
EB)
=
0,
n.(,izAHA - E ~ H =~ 0,)
nx
-
HB)
=
0,
(9.2~) (9.2d)
I am indebted t o Dr. John C. Sloncxewski for an interesting discussion of this point. Boundary conditions for the case of excitons have been discussed by V. M. Agronovich and V. L. Ginzburg, Usp. Fiz. Nauk 77, 663 (1962) [Soviet Phys.-Usp. (English Transl.) 6, 675 (1963)], and by J. J. Hopfield (unpublished). See p. 185 in Panofsky and Phillips.3
8b 80
9
(HA
(9.2b)
315
ELEMENTARY OPTICAL PROPERTIES O F SOLIDS
These relations follow from the fact that we may set EA = F ~ E A and HA = Re HA as in Eqs. (2.4) and (2.9). Let radiation with the electric vector E = Re [Eoexp (2g.r - id)] be incident upon the plane boundary between two media. In order to satisfy the boundary conditions, we assume the presence of a reflected wave in the first medium and a transmitted wave in the second medium having, respectively, the electric vectors E” = Re [Eo” exp (z%”*r- id)] and E’ = Re exp (&.r - id)]. Similar equations may be written for the vectors representing the magnetic fields. We shall assume that the real and imaginary paits of k (k, and k2) and the normal n, which we shall take as directed into the first medium, are coplanar, referring to their common plane as the plane of incidence.
mot
Y
t
, .
I K2’ I
FIG.1. Wave vectors and angles in the plane of incidence.
The similar time dependence of all three waves follows from the boundary conditions. Furthermore, it is necessary that a t each point ro on the boundary, tern = E’era = E”-ro. (9.3) Therefore the components of k, k’,and k” in the boundary plane must be equal :
Cxn
=
ii”xn,
Cxn
=
ii’xn.
(9.4)
Equation (9.4) shows that fl and g’lie in the plane of incidence. If we take this plane to be the x-y plane, with the normal n in the y direction, as shown in Fig. 1, then E, = E,’ = k,”. (9.5)
-
From Eq. (4.3) we obtain:
k.E Pel;’
=
p.p’=
= (n’
(n
+ ik)2(02/c2)
= &,2
+ ik’)~(w~/cz)= &;z + fE,?
+ &,2 = &,”2 +
fl.,”Z,
(9.6)
316
FRANK STERN
+
Here n ilc is the complex index of refraction of the medium from which the wave is incident, and n‘ f ik’ is the complex index of refraction of the second medium. We find from Eqs. (9.5), (9.6), and Fig. 1 that:
li,“ 6,’
=
-kv,
= - [ ( w z / c 2 ) (n’
+ ik’)2 - Lz2]1’2.
(9.7)
10. AMPLITUDE REFLECTION COEFFICIENTS We are now in a position to determine the components of the reflected and transmitted waves in terms of those of the incident, wave. Equations (9.2) may be written n.[Eo E:’ - ( ~ ’ / ~ ) E P=, , I0, (1O.la)
+
nx
n.[Ho
(Eo +
- EP,,)= 0,
+ Hd’ - (Z/X,,JHP,,I = 0, n x (Ho+ Hi’ - Hot)= 0,
(1O.lb)
(1O.lc) (10.ld)
where we used i = “KO, p = Zmp0. To evaluate the amplitude of the reflected and transmitted waves, it is convenient to consider separately cases with the electric vector polarized perpendicular to and parallel to the plane of incidence. There is no loss of generality in doing so, since any electric or magnetic vector may be resolved into components normal to and in the plane. We shall employ the subscript s to refer to the normal component of a vector and p to refer to the parallel component. The s component will be considered positive in the +z direction. Equation (10.lb) then gives:
Eoa + Ell;‘
=
Eo;.
(10.2a)
Combining Eq. ( 4 . 1 ~ )with (1O.ld) we have
n x [(E xEo.)
+ (k” xEos”) - (XJk’)
(k xEo8’)] = 0.
(10.2h)
Expanding the triple vector product and noting that n9EoS = n.EO8’= n.Eol’ = 0, we find: &(Bob, -
EOa”)
=
(~~2n’)li:Eoa’.
(10.2c)
We define the amplitude reflection coefficient P, and the amplitude transmission coefficient t8by :
El8‘’= P*Eo,,
(10.3a)
Elh‘ = f # E 0 8 ,
(10.3b)
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
317
and find from Eq. (10.2) that: (10.4a)
la =
fEu
+
2fEU (zm/;rn’>G’
= 1
+ P,.
(10.4b)
We will usually assume that Zm = Zm’ = 1. That will be closely true for any material a t optical frequencies,’O and will hold quite well also for many materials a t lower frequencies. When the electric vector lies in the plane of incidence, it is convenient to carry out the calculations in terms of the magnetic vector, which is perpendicular to the plane. Proceeding as in Eq. (10.2) we find for this case : Roa” = P p R O a , (10.5a) R06’ =
Z S o I ,
where
P,
=
fE,
- (“K/“K’)JEy’
iu + (“.?)
JEu”
(10.5b) (10.6a) (10.6b)
11. SNELL’S LAWASD GESERALIZATIONS The various components of the propagation vectors can also be expressed in terms of the angles they make with the normal n. Let kl make an angle 0 with the normal, as in Fig. 1; we may call this the angle of incidence. Similarly, we may let kl”and kl’make angles of 0” and 0’ with the normal, referring to them as the angles of reflection and refraction respectively. The angles that kz, kz’, and k2” make with the normal will be referred to as $, $’, and $”, In terms of these angles we find, from Eq. (9.5), that:
I kl 1 sin 0 1 ke 1 sin $
= =
1 kl’ sin 0’, 1 4’1 sin $’,
which form a generalized form of Snell’s law of refraction. lo
See p. 252 in Landau and Lifshitr.8
(1l.la) (1l.lb)
318
FRANK STERN
Equation (4.3) allows us to solve for the magnitudes of kl and kz (or I 4’’I and 1 ki’ I). We find:
Here x = 0 - 6 (see Fig. l ) , and is assumed to be less than go”, and K,,, is assumed to be real in the second line of each equation. Equation (11.2) is the extension of Eqs. (4.4) and (4.5) to the inhomogeneous case. To find the wave vector in the second medium, we assume for simplicity that the first medium is a lossless dielectric with index of refraction n. Then, if K,,’ = 1, Eqs. (9.7) and (11.2) yield:
I 4I k,’
=
na/c,
= (nw/c) sin 0,
J kz I k’,
=
0,
= -(w/c)
[X’ - n2 sin20]1/2.
(11.3)
It follows that for the case we are considering, with a real wave vector in the first medium, ki is always normal to the boundary plane. This can also be seen from Eq. (11.1b) . The phase velocity V’ = w / /
k1’ 1
(11.4)
in the second medium is no longer a function only of the properties of the medium, but depends also on the angle of incidence. This is sometimes6 taken into account by defining an effective index of refraction d e f t = c/v’. The effective index equals the index defined in Eq. (4.3) for normal incidence, or for arbitrary incidence when the second medium is also lossless. 12. ENERGY FLOW
We now turn our attention to the energy flow in the two media, which is given by the Poynting vector [see Eq. (4.6a)l. We note that the field in the first medium is the sum of the contributions of the incident and reflected waves. Thus, when the first medium is a dielectric, and kz =
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
kit
=
319
0, the average energy flow is
8 (medium 1)
4 Re (8* + 8"*)x (H+ H = Re (fo*X H , + &'* xiid' =
*
)
- exp [-i(k1 - k:') ar] + go*xHd' + 80"*x% exp [i(k1 - kl") or]} (12.1) = s + s" + sin*. In evaluating the energy flow we take the general case of elliptic + f o p , and HO= Ho,+ Hop. We choose polarization, with 80= 208
and go, as the independent components, with the components in the plane of incidence determined from these by Eq. (4.1). For simplicity we assume that the first medium is a lossless dielectric, so that the incident and reflected beams are both homogeneous, and that Km = Km' = 1. &a and Ho.can both be chosen to be real (by an appropriate choice of zero on the time scale) if the incident wave is linearly polarized, but not otherwise. Straightforward substitution in Eq. (12.1) 'gives for the energy flow in the incident beam:
S
=
+ K-'[~Oa*~:,/(2€0w)]k1.
[Eoa*E0a/(2p0~)]kl
(12.2)
The energy flow in the reflected beam is
B"
=
~ a * ~ * [ ~ 0 , * ~ 0 a / ( 2 / 1 0 . ) (]?kp ~ * ?" p / ~ ) [g08*fl0a/(2WJ)]k1", (12.3)
where we used Eqs. (10.3a) and (10.5a). A somewhat lengthier calculation gives for the interference terms: sint
=
[&a*&/
+ 4 sin 28 b
( P O W ) lkiz
(Hoa*&[Pp*
Re
[Pa
exp ( -2iykiy) I
exp (2iyk1,)
-
?8
exp (-2dgk~]).
(12.4)
The interference terms give energy flow only parallel to the boundary plane. These terms arise from the overlap of the incident and reflected beams, and can be ignored in the typical reflectance experiment, in which collimated beams are used. The interference terms must, however, be considered near the boundary where the beams overlap. Since the incident beam normally contains a range of wave vectors, say between k and k Ak, phase cancellation will occur and we can expect Eq. (12.4) to go to zero at distances 2 (Ak)-' from the surface.
+
320
FRANK STERN
The energy flow in the second medium is
s’ = 3 Re ( E ’ * x H ) = i e x p (-2ki.r)
Re (E0’*xH0’),
(12.5)
which gives
s’ = { [ E o ~ * ~ o(2poW) : o . / ]f8*@’1
+
2’
[~oII*~o‘/(2EoW)](~p*tp/(
12)
(Kl’k,’
+ Kz’ki)
+ ( ~ ~ k z y ) ’ k l ~Re/ ~ (~-)i t , t , * E ~ , ~ ~ , * / i ?1* )exp ( -22ykzy).
(12.6)
Conservation of energy requires that the component of energy flow normal to the surface must be continuous across the boundary. This is seen to be the case if we compare Eq. (12.6) with (12.2) and (12.3), and use (10.4b) and (10.6b). The presence in Eq. (12.6) of a component of energy flow normal to the plane of incidence seems paradoxical until we consider that the overlap of the incident and reflected waves in the first medium also gives such a component of energy flow. 13. REFLECTIVITY The reflectivity of the material is defined for the general case by R = and can be calculated from Eqs. (12.2) and (12.3). When the electric field is polarized perpendicular to the plane of incidence, or in that plane, we have, respectively:
-S”.n/S.n,
R,
=
P,*P,,
R,
=
P,*Pp.
(13.1)
For unpolarized light, fbr light polarized at 45” to the plane of incidence, and for circularly polarized light, R = +(R, R p ) . Introducing 20’ = n’ ilc‘ = (2’)1/2, we can show from Eqs. (9.7), (10.4), and (10.6) that when the first medium is a lossless dielectric the equations for the reflection and transmission coefficients are :
+
+
(13.2a) (13.2b)
32 1
ELEMENTARY OPTICAL PROPERTIES O F SOLIDS
At normal incidence we find from Eqs. (13.1) and (13.2) that the reflectivity is: (13.2e)
If the first medium is also lossy, we must replace kI2 by (k - k ' ) 2 in the numerator and by ( k k') in the denominator. To understand better the phenomena which occur during reflection and refraction, it is helpful to discuss the simple case where the second, as well as the first medium, is a lossless dielectric. Then we can simply by n' in Eq. (13.2). If n' 1 72, we see that the square roots in replace Eqs. (13.2) and in (11.3) are real for all values of 8. Thus in this case Eq. (1l.la) can be written:
+
m'
n sin e
=
n' sin 8'
(13.3)
the familiar form of Snell's law of refraction. TABLE 11.AMPLITUDEREFLECTION COEFFICIEKTS Fa AN D P, FOR LIGHTWHOSEELECTRIC PERPENDICULAR TO AKD IN THE PLANE OF INCIDENCE, RESPECTIVELY FIELDIs POLARIZED
The coefficients are defined in Eqs. (10.3) and (10.5)) and explicit expressions are given in Eq. (13.2). The special values of the angle of incidence 0 are Brewster's angle Be = arctan (n'ln) and the angle for = arcsin (n'ln),where 72 and n' are the real onset of total reflection, indices of refraction of the medium from which the beam is incident, and of the second medium, respectively. The and - signs indicate real coefficients with magnitudes between 0 and 1, and correspond to phase shifts of Q" and 180" respectively in the reflected beam. The real phase n). shifts 6, and 6, are given in Eq. (14.1) ; we use ro = I n' - n I/(n'
+
+
n'
>n
n'
e
7,
i.,
0
- To -
+
o
+
-
-1
-1
eB
+
o
To
0
e
0
es
eT
eT
< e < 90" 90"
.Fa
ro
+
1 exp (is,) -1
FP
-ro
0
+1
exp (isp) -1
322
FRANK STERN
It is instructive to consider the phase changes which light undergoes on reflection. These are summarized in Table 11. It is easily verified from Eq. (13.2) that for a beam incident a t angle 8’ from the second medium the amplitude reflection coefficient is the negative of the coefficient for incidence a t angle 0 from the first medium, where 0 and 0’ are real angles related by Eq. (13.3). This opposition of signs is shown in the first five rows of Table 11. The table also shows that a t normal incidence the electric field has a 180” change of phase on reflection for waves incident from the medium of lower index, while it is the magnetic field which changes phase when the wave is incident from the medium of higher index. 14. TOTAL REFLECTION When n’ < n, the angle of refraction 0’ given in Eq. (13.3) becomes imaginary when the angle of incidence 0 exceeds OT = arcsin (n’/n). For greater angles of incidence the beam is totally reflected, but with a phase shift whose magnitude increases from 0, reaching 180’ a t glancing incidence. Thus a t glancing incidence the phase change on reflection is 180” in all cases; this is verified experimentally by the dark fringe observed a t the surface of Lloyd’s mirror.“ The phase shifts for total reflection, as determined from Eq. (13.2))are: f, =
exp (i6,),
=
exp (is,),
f p
6,
=
-2 arctan [(sin2 e - [n’/n12)1/2/cosel,
6, = - 2 arctan [(sin2e - [n’/n12) (n’/n)-2/cos el.
(14.1) I n the case of total reflection it is seen from Eqs. (9.5) and (9.7) that
k,’ lies in the boundary plane while kz’is normal to it. Thus the plane of
constant phase is perpendicular to the plane of constant amplitude, and the field in the second medium is an extreme example of an inhomogeneous field. There is no energy flow normal to the boundary plane, as is implied by the term total reflection. When the second medium becomes conducting, i.e., ~ 2 ’ # 0, it follows from Eqs. (11.3) and (12.6) that there mill be a component of energy flow normal to the boundary plane in the second medium, and total reflection is no longer possible. 1s. TRASSMISSION
O F ISOTROPIC
RADIATIOX
It is sometimes useful to know the value of the reflectivity averaged over all directions of incidence. To calculate the average, note that the 11
F. A. Jenkins and H. E. White, “Fundamentals of Physical Optics,’’ p. 66. McGrawHill, New York, 1937.
ELEMENTARY OPTICAL PHOPEltTIES OF SOLIDS
323
power striking an element of area d A a t an angle of incidence B and in a cone of directions covering solid angle ds2 is I cos 0 dAds2, where I measures the intensity of the radiation, as discussed for blackbodies in Part V. Thus the appropriate average for the reflectivity R is
R (0) cos B sin -9 &d+ R A= ~
1cos
(15.1) B sin 8 dBd+
where we introduced E = sin2&It is more convenient to consider the radiation transmitted through the surface, rather than that which is reflected, so we introduce T = l - R . (15.2) The average transmissivity will be TAv= J T ( 5 ) d& For this section we will restrict ourselves to materials with real indices of refraction. We will use n = n>/n<, (15.3) where n> and n< are, respectively, the larger and smaller of the indices of the media on opposite sides of the boundary plane. We will indicate as unprimed the reflectivity and transmission for beams incident from the medium with the lower index, and will label with a prime the corresponding quantities when the beam is incident from the material with a larger index. We will consider the radiation to be unpolarized. Thus T = $ ( T , T p ) .Using Eqs. ( E l ) , (13.2), and (15.2) we find:
+
(15.4a)
The primed quantities are obtained by replacing ‘n by n-1 in Ey. (15.4); By a straightforward transformation of both T,’ and Tpl vanish for > r2. integrals we can show that
T i , & = n-2T,,h,
=
n-2Tp,Av.
(15.5)
This old12 and important result is used in Part V in discussing thermal equilibrium in an isothermal enclosure containing two different media. 1*
P. Drude, “The Theory of Optics,” Part 111, Chapter 11, Article 6. Longmans, Green, New York, 1907;Dover Publications, New York, 1959.
324
FRANK STERN
Explicit evaluation of the integrals is somewhat more troublesome, but involves no unusual steps. We find128:
Ta.Av
=
Tp,Av=
+ 1) (n + I)-', 4n3(n2+ 2n - 1) (n' + 1)-2(n2 - l)-I
(15.6~3)
Q(2n
+ + 2n2(n2- 1)2(n2+ 1)-3In [n(n + 1) ( n - 2n2(n2 1) (ne- 1)-2In [n]
l)-I].
(l5.6b)
The transmissivity goes to 1 for all polarizations and angles of incidence when n = 1, since the two media are then identical and the boundary does not affect the propagation of radiation. This is easily verified for Eq. (15.6a), but cannot be checked directly for (15.6b), which is undefined for n = 1. If we put n = 1 u we find for u << 1 :
+
For all values of n the average transmission is higher for light polarized in the plane of incidence, as one expects from the behavior of the reflectivity. 16. THISFILMS An interesting application of the theory already discussed is to the phenomena of reflection from and transmission through thin solid films. In this case there are'multiple reflections a t both surfaces of the film, and interference effects are present in the reflected and transmitted light. In our discussion, we shall limit ourselves to light that is plane polarized in, or normal to, the plane of incidence. Assume that the film is bounded on one side by a dielectric, in which the incident beam is located, and o n the other side by any other medium. Let f l and tl be the reflection and transmission coefficients for light going from the first medium toward the second, 72 and tz the coefficients for light going from the second into the third, and fl' and 2,' the coefficients for light going from the second into the first. The light will undergo reflection and refraction a t the first surface, pass to the second where it will again be reflected and refracted, and so on. Thus there will be an infinite number of beams reflected back into the first medium by the film and an infinite number transmitted into the third. It is easy to see from Eqs. (9.5) and (9.7) that the propagation vectors are identical for each of the reflected beams in the first medium 12*
Some further details are given in F. Stern, A p p l . O pt ., t o be published.
325
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
and that they are also identical for each of the transmitted beams in the third. If there were no absorption in the film and no phase difference between reflected waves, the ratios of the amplitudes of the fields of the reflected waves would be given1* by the sequence: PI, Pz&B’, P1‘7~~&&’, 71’28~s&&’ * . However, upon passage one way through the film, the field of each beam is attenuated by a factor exp ( - I kz,’ 1 d ) where d is the thickness of the film. Furthermore a t a given value of z there is a phase difference of - 2kl,‘d between successive reflected beams, and between successive transmitted ones. We can write, using Eq. (11.3) :
6
= -&,’d
=
(2nd/X) (2’
-
sin2 0)1/2,
116.1)
where refers t o the first medium, i? to the second, and X is the wavelength in vacuum. The sum of the reflected fields will then give for the reflection coefficient F of the film:
P
=
fl
+ P2&f11 exp (22%) + P1’P~2flfl’ exp ( 4 4 + P1’2P~811&’exp (6i5) +
0
a .
(16.2) With the help of Eqs. (10.4b) and (10.6b) and of the fact that F1’ = we obtain tlB’ = 1 - TIz. Then Eq. (16.2) becomes
- pl,
(16.3)
Using the same approach, it is easily shown that the transmission coefficient t of the film is: (16.4) Let us first discuss the case in which there is air on both sides of the film. Then we may set 72 = -Pl in Eq. (16.3). We obtain for the reflection coefficient of a film when the electric vector is polarized normal to the plane of incidence :
P, =
la
{ [cos { [cos .e/(i?
e/(? - sin2 e)1/2] - [(Z’ - sin2 e)1/2/cos e l ) sin 5
- sin2 e p ]
+
[(;I
- sin2 e)1’2/cos e l ) sin 5
(1 - “K) sin5
(1
+ i’ - 2 sin2 e) sin 5 + 2i cos e cos
- sin2 e]l/z’
+ 2i COB 5 (1G.h)
0. S. Heavens, “Optical Properties of Thin Solid Films,” p. 56. Academic Press, New York, 1955.
326
FRANK STERN
and when the electric vector is polarized in the plane of incidence:
rp
=
-
-.
1 [itcos e/ (i’ - sin2 e) 1 / 2 1 - [(x‘- sin2 e) 1 / 2 / i ’ cos e l } sin i { [itcos e/ (x’ - sin2 e) 1/21 + [(2’ - sin2 e) 1 / 2 / z f cos e l 1 sin 5 + 2i cos 5 (P e - i’ + sin2 e) sin I + X’ - sin2 e) sin i + 2izt cos e cos i(xt- sin2 e>l/z’ COS~
(P cos2 e
(16.5b) The square of the magnitude of Eqs. (16.5a) and (16.5b) will give the expressions for the reflectivity of a thin film in air. Ferrell and Stern14note that there will be a sharp rise in the reflectivity of a thin film near the plasma frequency for light polarized in the plane of incidence when the imaginary part of the dielectric constant is small. Inspection of Eq. (16.5b) shows that FP + -1 when 2’ -+ 0, except for normal incidence. Numerical calculations by G ~ e r t i n ’show ~ that in the case of silver one may expect a small rise in the reflectivity for films of thickness 200 A or less; increasing the angle of incidence increases the effect. There is never any rise in the reflectivity of light polarized normal to the plane of incidence; therefore the effect disappears for normal incidence. Even when the theory of reflection at a single surface suggests that most of the incident energy.should be reflected at the first surface, there can be interference effects that considerably reduce the strength of the reflected beam when one has a thin film. This is due to the previous observation that, even at total reflection, the field does not vanish within the second medium, but decays exponentially with distance. Therefore, as the numerical results15 demonstrate, the thinner the film, the greater the interference effects one should expect. One could, however, experience difficulties because of the failure of the bulk optical constants to hold for thin films.16 Since it appears that for most metals films no more than a few hundred angstroms thick would be required in order to observe any rise in reflectivity at the plasma frequency, a backing will be required. When the incident beam is in air and the film has a backing, the expressions for the reflection coefficients of light polarized perpendicular to and in the plane of incidence, R. A. Ferrell and E. A. Stern, Am. J . Phys. 30, 810 (1962). See also the note added in proof on page 408. 16 R. F. Guertin, unpublished work, 1961. 16 H. Mayer, “Physik dunner Schichten,” Part I. Wiss. Verlagsges., Stuttgart, 1950. 14
327
ELEMESlARY OPTICAL PROPERTIES OF SOLIDS
P,
=
] cosb-i[ i’(iB-sin28)1/2 (;’Isin2,9)1/2
(XB - sin28)112
25 (Xl-sin28) 1 / 2 1
+
XI
case
sin8J
(16.6) where we have introduced XB for the dielectric constant of the backing. The expressions for the reflectivity obtained from the above two expressions are extremely complicated. Numerical calculations have been carried out for several metals with a fused quartz backing.ls For silver, which was the only metal investigated both with ,and without a backing, the calculations predict that for angles of incidence less than Brewster’s angle for the backing, the maxima in the reflectivity will be less pronounced than with no backing. At the Brewster’s angle the results are approximately the same, and for larger angles of incidence the maxima are more pronounced. 111. Dispersion Relations
In this chapter we discuss some properties of the optical constants of materials which can be derived without reference to specific models. We shall give a brief account of dispersion relations, often called KramersKronig relations, and shall show how they lead to several sum rules. The reader interested in further details should consult the books of Bode,” Frohlich,l* Landau and Lifshitz,lg MOSS,^^^ and Solodovnikov,lgb or the papers by TollJ20and Wu.)Oa H. W. Bode, “Network Analysis and Feedback Amplifier Design,” pp. 278, 303ff. Van Nostrand, Princeton, New Jersey, 1946. See p. 4 in Fr0hlich.l 10 See p. 256 in Landau and Lifshitz.8 lB. T. S. Moss, “Optical Properties of Semiconductors,” Chapter 2. Butterworth, London, 1959. I O b V. V. Solodovnikov, LLIntroduction to the Statistical Dynamics of Automatic Control Systems.” Dover, New York, 1960. 2O J. 9. Toll, Phys. Rev. 104, 1760 (1956). ma T. T. Wu, J. Math. Phys. 3, 262 (1962), 1’
328
FRANK STERN
17. DISPERSION RELATIONS FOR
THE
DIELECTRIC CONSTANT
We have already seen in Part I that the existence of a phase shift between the displacement D and the electric field E can be characterized by a complex dielectric constant 2 = Z/eO. The relation between the frequency components of fi and E is: fi(w)
=
Z(w)E(w).
(17.1)
For arbitrary time variations at a fixed point we takezob
J-lw
The physically required reality of D ( t ) and E ( t ) is achieved by requiring that f i ( - w ) = f i * ( u ) and E ( - w ) = E*(w). From Eq. (17.1) it therefore follows that i(-w) = i*(w). (17.3) These relations show that the real parts of fi (a), fi (a),and 0 (a)are even functions of w , and that the imaginary parts are odd functions of w. The response of the dielectric to an electric field is given by the polarisation P(t),where P(t) = D(t) - e O E ( t ) . Thus
J--0,
If E(t) is a square-integrable function, corresponding to a wave train of finite energy, we can invert Eq. (17.2b) and obtain ~ ( t= )
/
m
-m
J
-m
[~(w)
- EO]
m
dw
i,
~ ( t ’ exp )
[iw(t’
- t ) ] dt’/2n,
-m
(17.4b) Thus G ( t 2Ob
-
t‘) gives the polarization response a t time
t to a unit delta-
We use the same symbol both for D ( t ) and for its Fourier transform. The argument will usually be explicitly given.
ELEMENTARY OPTICAL PROPERTIES O F SOLIDS
329
function electric field pulse a t time t’, and is seen to be the Fourier transform of i ( w ) - e0. If the Fourier integral defining G can be inverted, then we can write : i(w)
-
m
e0 =
i,
G ( T ) exp
(iwT)
dT.
(17.5)
The causality requirement, that there can be no response before the field is on, means that G( T ) = 0 for T < 0. The integration in Eq. (17.;) can therefore be limited to positive values of T . Equation (17.5) then defines a function which is analytic for complex values of w having a positive imaginary part r. To show this we note that the derivative of Eq. (17.5) with respect to w will be independent of the path in the complex plane, and will converge in the upper half-plane because of the convergence factor exp (-I’T). Thus : ( w ) - e0, and therefore i ( w ) , itself, are analytic functions in the upper half of the complex w plane. It is well to point out
w
w’
FIG.2. Contour of integration in the complex a’plane.
that we consider functions whose time variation is given by exp ( - i d ) . Had we used exp ( i w t ) in Eq. (17.2), the roles of upper and lower halfplanes would be interchanged. To deduce the dispersion relations from the analyticity of ; ( w ) - €0, we make use of the result that the integral of an analytic function over a contour that encloses no singularities is zero. Thus we can write $[i(w’) - E O ] ( W ’ - w ) - I dw’ = 0 when the integral is evaluated along the closed contour shown in Fig. 2, where the pole a t w on the real w‘ axis is excluded by a small semicircle of radius 6. The integral along the real axis, from - ~0 to w - 6 and from w 6 to a, in the limit 6 + 0 is, by definition, the Cauchy principal value of the integral along the real axis. The large circle which closes the contour contributes nothing, since we assume that Z(w’) - eo goes to zero there. Finally, the small semicircle around the pole contributes - i ~ [ i ( w ) - eo]. Thus we find that
+
;(w)
-
e0 =
(in)-’P J
m
[;(w’) -m
- E ~ ] ( w ’- w ) - I dw’,
(17.6a)
330
FRANK STERN
where P stands for the Cauchy principal value. If we separate real and imaginary parts, we find that el(o)
- €0 ea(w)
=
(I/T)P
=
(l/r)P [m
Q(o’)[w’
J
[el(w’)
-m
- ~1-l dw’,
- e O ] ( w - w’)-l
(17.6b) dw’.
(17.6~) We can transform to integrals over positive frequencies only by noting that;
(17.7) Inserting this result in Eq. (17.6), and making use of (17.3), we find the following form for the dispersion relations between the real and imaginary parts of i : ~ ( u)
€0
=
l
(2/r)P
m
u’e2(w’)
(d2- w2)-l du’,
(17.811)
The constant co has been dropped in the integrand of Eq. (17.8b1, since it is easily verified that
P
lm -
( u ’ ~ wO2)-l dw’ = 0
(17.9)
for wo # 0. The conditions underlying the derivation of Eq. (17.8) have not been stated here with precision. In Toll’s paper,20 boundedness of i ( w ) was assumed. This led to the presence of the constant €0 in Eq. (17.8a), which we know to be the limiting value of i ( w ) for very high frequency. Toll points out that singularities at finite frequencies or divergences at infinite frequency can be treated in modified dispersion relations which contain additional arbitrary constants. One such case occurs for conductors, since, as we showed in Eq. (2.8), the imaginary part of i ( w ) is (u/w), and is not bounded as w + 0 if the material has a nonvanishing dc conductivity uo.
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
331
This is most easily dealt with by subtracting ao/o from ep(w) throughout the derivation of the dispersion relations. No change in Eq. (17.8a) is required, since this additional term contributes nothing to the integral. Thus the dispersion relations for the dielectric constant, in a form in which they are readily applied, are :
-1=
K~(w)
m
(2/n)P
(d2- 09)-~dw’,
O’K~(O’)
(17.10a)
where we have divided by eo to make the expressions dimensionless. The derivation of the dispersion relations (17.10) contains only three major assumptions : boundedness, linearity, and causality. Thus these dispersion relations will be valid for many other physical quantities which express a linear relation between an input and an output if the output cannot precede the input. Our interest here is in the optical properties of solids, but dispersion relations like those discussed here are of importance in many other parts of physics, as well as in electrical engineering.” 18. DISPERSION RELATIONS FOR
THE
INDEXOF REFRACTION
Tol120has defined B dispersion relation as “a simple integral formula relating a dispersive process to an absorption process,” and that is the sense in which we use the term here. Dispersion relations are often called Kramers-Kronig relations because of the work by H. A. KramersZ1and R. de L. Kronig,22*22a who gave dispersion relations for the dielectric constant and for the index of refraction, with application originally to the dispersion of X-rays. We have given the dispersion relations for the dielectric constant in Eq. (17.10). The analog to Eq. (17.10a) for the complex index of refraction n(w) a ( w ) is:
+
n(.)
-1=
1
0
(2/n)P
w’k(o’) (w’2
- w2)--1 dw’.
(18.1)
The derivation of this result proceeds along somewhat different lines from the derivation leading to Eq. (17.6), since the index of refraction H. A. Kramers, Atti del Congress0 Znternazionale dei Fisici, 11-30 Settembre 1967, Como-Pauia-Row (Published by Nicola Zanichelli, Bologna), 2, 545 (1928). R. de L. Kronig, J . Opt. SOC.Am. 12, 547 (1926). *I* R. de L. Kronig, Ned. Tijdschr. Natuurk. 9, 402 (1942). s1
332
FRANK STERN
does not express a linear relation between two physical amplitudes. A detailed derivation of Eq. (18.1) from the causality requirement has been given by Toll,” who also points out that the analog to Eq. (17.10b) can be used if u[n(w) - 11 is square-integrable. An alternative form for Eq. (18.1) is obtained if we substitute $ca(u’) for u’k(w’), where a is the absorption coefficient. 19. QUALITATIVE CONSEQUENCES Some interesting consequences of the dispersion relations (17.10) are best displayed by carrying out an integration by parts which puts them in the form: xI(u)
- 1 = (1/r) Im [dK4(w’)/dw’] 0
KZ(w)
=
In
- (1/r)/m[dK1(w’)/~w’~In 0
(1
u’2
([a’
- w2 1-1)
+
w’
(19.la)
dw’,
- w 1)
dw‘. ( 19.1b)
For simplicity we have assumed that we are dealing with a material whose dc conductivity vanishes. The logarithms in both Eqs. (19.la) and (19.lb) will have strong peaks in the neighborhood of w’ = w. Thus K ~ ( w )will tend ) large and positive, to peak at frequencies for which the derivative of K ~ ( w is and will have minima near frequencies for which the derivative of K Z ( W ) is large and negative. The behavior of K Z ( W ) depends in the opposite way on the derivatives of K ~ ( w ) ,maxima of K ~ ( w )occurring where the slope of K~ ( w ) is large and negative. An important application of Eq. (19.h) makes use of the result, derived in Part V, that K ~ ( w )is proportional to w - ~I m 12p(w), where p ( w ) is the density of final states for the transitions near energy hw, and 3n is the matrix element for the transition as given in (35.9). Our interest here centers on p (a), the density of final states. In a periodic crystal the energy difference Ruj, between initial and final states can be represented by a periodic function in the reciprocal lattice, and therefore must have certain critical points24 where VkWj, will vanish. At the critical points the density of states in three-dimensional crystals remains finite and continuous, but Thus maxima and minima its derivative has a square-root ~ingularity.~4 in K ~ ( w )may be associated with critical points in the energy difference between initial and final states. These critical points need not occur a t 88
J. S. Toll, Ph.D. Thesis, Princeton University, 1952 (unpublished), Chapter 1. I am indebted t o Professor Toll for several valuable discussions of this work. L. Van Hove, Phy.9. Rev. 89, 1189 (1953).
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
333
the same points in reciprocal space as the critical points for either the initial or the final bandlXabut they will often coincide with one or both. A particularly simple example in the dielectric constant of a semiconductor is the behavior near an allowed direct energy gap, where the density of states vanes as ( w - wu)1/2?4b If I 312 l2 is slowly varying, then drcz/dw varies as ( w - wu)-1’2 when w is greater than wg. Equation (19.la) shows that this singularity will result in a peak in the dielectric constant near wu. Furthermore, since the absorption is still weak near the edge, the reflectivity, (13.2e), is controlled mainly by the index of refraction, and will also peak near wg. In some materials, like germanium and the III-V semiconductors, the density of states near the lowest direct energy gap is very small, and the effect is not easily observed. The first pronounced reflectivity peak may occur near the energy gap for transitions at another point in the Brillouin zone. Beyond the first major energy gap the situation is less clear-cut, but it should still be possible, once a rough energy band structure is known, to correlate maxima and minima in K~ ( w ) with critical points in the energy difference between Our arguments to this point have been based’on a matrix element 312 which varies slowly near a critical point. For forbidden transitions, or for indirect transitions, for which the absorption coefficient increases as a higher power than ( w - wu)1/2, there is no singularity in dKa/dw, and no pronounced structure is expected near a band edge. On the other hand, if exciton2592Sa or magnetic field26 effects are important, the absorption coefficient near a band edge may rise more rapidly than for allowed direct transitions, and K ~ ( w )may tend toward a singularity near the band edge. Singular behavior may also occur if one or more of the second derivatives of the energy difference between bands goes to zero at a critical point.24
SHIFT DISPERSION RELATION 20. PHASE The dispersion relations between the real and imaginary parts of the dielectric constant are very powerful tools in the analysis of optical properties. One important use, the derivation of sum rules, is discussed later in D. Brust;,’J.C. Phillips, and F. Bassani, Phys. Rev. Letters 9, 94 (1962) for an example. *4b See Section 37 for a more detailed discussion. 240 B. Velick9, Czech. J. Phys. B11, 787 (1961). 25 R. J. Elliott, Phys. Rev. 108, 1384 (1957). 268 T. P. McLean, in “Progress in Semiconductors” (A. F. Gibson, ed.), Vol. 5, p. 53. Heywood, London, 1960. 26 See, for example, B.[Lax and S. Zwerdling, in “Progress in Semiconductors” (A. F. Gibson, ed.), Vol. 5, p. 221. Heywood, London, 1960. U* See
334
FRANK STERN
this Part. For direct application to experiments on optical properties, Equations (17.10a) and (17.10b) have only limited usefulness, since we usually know the real part or the imaginary part of the dielectric constant only in limited frequency ranges. Another useful relation is one involving the reflectivity a t normal incidence, since this is a quantity that can be directly measured. The reflectivity R is the ratio of reflected to incident intensities. It can be expressed in terms of 7, the complex ratio of outgoing to incoming electric or magnetic fields, by:
R(w)
= P(w)[P(w)]* = [ ~ ( w ) ] ~ .
(20.1)
For normal incidence the amplitude reflection coefficient P ( w ) is given, ilc,by in terms of the complex index of refraction n
+
~ ( w )= r ( w )
exp [ i e ( w ) ]
(n
=
+ ik - l ) / ( n + ilc + 1).
(20.2)
We have chosen the amplitude reflection coefficient for the magnetic field Fp from Eq. (13.2~).The amplitude reflection coefficient for the electric field at normal incidence P, differs from Eq. (20.2) only by a factor - 1. Since P ( w ) expresses a linear relation between two amplitudes, it will be analytic in the upper half of the complex w plane, and relations analogous to Eq. (17.8) exist between its real and imaginary parts. But what we really need in order to analyze reflectivity data is a relation between the phase e ( w ) and the magnitude r ( w ) of the amplitude reflection coefficient, since it is r ( w ) = R1I2.whichis given by the measurements. The optical constants are determined from e ( w ) and r ( w ) by the relations
+ r2 - 2r cos e), + r2 - 2r cos O ) ,
n
=
(1 - rZ)>(l
(20.3a)
k
=
2r sin e / ( l
(20.3b)
which are direct consequences of Eq. (20.2). Thus e ( w ) and r ( w ) specify the optical constants completely, and this shows the great value of a dispersion relation for e ( w ) . To show that the derivation of a phase shift dispersion relation cannot be as straightforward as the derivation leading to Eq. (17.8), we note that P(ij) could be multiplied by any function of ij which has absolute value 1 for real values of 6 without changing the reflectivity. Such a function, called the Blaschke product,20is B(S) =
IT [(S - Pn)/(Pn* n
- 41,
(20.4)
where the pn are complex constants. They must have non-negative imagi-
336
ELEMENTARY OPTICAL PHOPERTIES OF SOLIDS
nary parts, since, as we mentioned above, P ( i j ) is analytic in the upper half of the ij plane. Furthermore, any jZn whose real part is nonzero must be paired with another whose real part is of opposite sign, since f ( w ) , and therefore B ( w ) , must satisfy Eq. (17.3). TollZOhas shown that the dispersion relation for the phase shift e ( w ) is O(w) = ( 2 w / r ) P
Lrn
In
[r(w’)3(w2
- ~ ’ ~ ) - ~ d+ w ’[(a),
(20.5)
where two other terms given by Toll, which arise from pathological behavior at finite or infinite frequencies, have been dropped. The last term in Eq. (20.5) is the phase shift of the Blaschke product: [(w) =
-iln
[B(w)]
=2
+
C tan-l
[(u
n
- ~nr)//~ns],
(20.6)
where jZn = bnr ipni, and the tan-’ takes values between -3. and .3. We find that [ ( w ) is an increasing function of frequency, and that27
- €(O>
€(a>
= (2P
+ d.,
(20.7)
where p is the number of factors in the Blaschke ptoduct in which Pnr = 0, and p is the number of pairs in which llnr # 0. While the Blaschke product cannot be ruled out mathematically, it is possible to show on simple physical grounds that the Blaschke product cannot contribute to the phase of the amplitude reflection coefficient of a solid which is in thermal equilibrium or in its ground state. Under these conditions the extinction coefficient k is nonnegative, and one can then easily show that, both for conductors and for insulators, the phase of P(w) in Eq. (20.2) is zero when w = 0, and is restricted to the range 0 I 6 I ?r. When w passes through an absorption region, the phase shift increases, reaches a maximum less than or equal to ?r, and then decreases again. When the last absorption peak has been passed (this corresponds to excitation of the Is shell of the heaviest atom) we oan approximate the absorption by expressions for the photoelectric effect, for which the cross section (and therefore the absorption coefficient a) decreases approximately as w-a.28 Therefore the extinction coefficient k = c a / ( 2 w ) varies as w-4. On the other hand, at high frequencies K (1 - wp2w-2) , i.e., n 1+ ~ , ~ w - ~as , will be shown in Eq. (23.2). These asymptotic relations for n and k, together with Eq. (20.2), imply that O ( w ) -+ ?r as w -+ a. The
-
-
n N. G. Van Kempen, Phys. Rev. 89, 1072 (1953). 18 A. H. Compton and S.K. Allison, “X-rays in Theory and Experiment,” Chapter VII. Van Nostrand, Princeton, New Jersey, 1935.
336
FRANK STERN
amplitude reflection coefficient r ( w ) approaches awpaw-2 a t these high frequencies, and from this behavior it is not hard to show that the integral in Eq. (20.5) gives a canonical phase shift which approaches ir asymptotically. Thus the Blaschke product, which always adds to the phase shift, as we showed in Eq. (20.7) , cannot be present for the reflectivity of solids a t normal incidence. Our proof that the Blaschke product cannot be present must be modified in one respect. The amplitude reflection coefficient for the electric field a t normal incidence P g ( w ) is just the negative of Eq. (20.2)-This factor of - 1 is provided by a single factor in the Blaschke product (20.4), with p,, equal to zero. Except for the restriction just mentioned, the phase shift dispersion relation which applies to the reflectivity of solids a t normal incidence is 6 ( w ) = (w/ir)
la
[ln R(w’)
- In R ( w ) ] ( w * - d2)--1 dw’,
(20.8b)
where e ( w ) is called the canonical phase shift by Toll. We have subtracted In R(w) from In R(w’) in the integrand of Eq. (20.8b). Because of Eq. (17.9), the value of the integral is not affected, but the singularity for w’ = w is removed, and the principal value is no longer required. Equation (20.8b) is a dispersion relation giving the imaginary part of the function In [ P ( w ) ] as an integral over the real part, and is therefore the analog of (17.8b). Because of the different properties of In [ P ( w ) ] , the proof used to derive Eq. (17.8) will not be valid here, and, in particular, the analog of (17.8a)’ cannot be written down directly. Teutsch and Gottlieb%have given dispersion relations for the real and imaginary parts of In [ P ( w ) ] based on the vanishing of its derivative as w + a . They find an expression equivalent to (20.8b) , and give for the other dispersion relation a result which can be written: In [ r ( w ) / r ( v ) ] = (2/n)
/
m
w’e(W’)[(w’2
0
- U21-1 -
(U’2
- v2)-11
dw’.
(20.8a) The proof given by Teutsch and Gottlieb assumes that the derivative is analytic in the upper half-plane, and therefore ignores the possibility, represented by the Blaschke product, that P(&) may have zeros in the upper 2 plane. A proof of Eq. (20.8b) was first published by VelickQ,sO * 8 M. Gottlieb, Ph.D. Thesis, University of Pennsylvania, 1959 (unpublished) , Appendix 11. I am indebted to Dr. P. Miller and to Dr. W. Teutsch for information about this
so
work. B. VelickQ, Czech J. Phye. B11, 541 (1961),
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
337
who eliminates the Blaschke product in essentially the same way as above, by using Eq. (20.2) and the requirement that k > 0. 21. NUMERICAL PROCEDURES FOR CALCULATING THE PHASE Numerical evaluation of Eq. (20.8b) is quite easy if the reflectivity is known over a wide enough range of frequencies. The simplest case is one in which the reflectivity is known over a frequency range w1 5 w 5 WZ, and is constant outside this range. Examples of this are the reflectivity associated with lattice vibrations in LiF,29*81 and the reflectivity associated with free carriers in semiconductors, for example PbTe,82where one may generally take w1 = 0. In these simple cases the phase is easily found from Eq. (20.8b) to be (21.la) = ez(w) ~ 4 ,
e m+
e,(~)
=
+ ( 2 ~ ) - 'In [ R I / R ( u ) ]In [ ( w +
&(w) = (w/T) &(w)
=
C'
[ln R(o')
w>/l
~1
- 0 11,
- In R(w)](w2- d2)--1do',
(2?r)-'ln [Rs/R(w)] In
[I
w2
-
l/(wz
-I- w ) l ,
(21.1b) (21.1~) (21.Id)
where R1and R2are the constant values of the reflectivity for w 5 w1 and for o 2 02,respectively. If o becomes equal to w1, B l ( w ) is undefined. But we can expand Rl/R(w) about w = wl, and find that we must take el(w1) = 0. Similarly we find that 8 4 4 = 0. Note that the dimensions of w do not affect the values of the expressions in Eq. (21.1). Thus it is possibIe to express o in any convenient units, such as wave numbers or electron volts. The integration in Eq. (21.1~)is easily carried out numerically, particularly if the reflectivity is known for equally spaced values of d. The only special case arises for o' = w, in which case one takes for the integrand the average of the values at the grid point just preceding and just following. Thus provision must be made for evaluating the integrand one grid interval outside the range of integration in either direction. If w1 = 0, we put = 0, and the procedure is slightly simpler. One possible source of error in the phase & ( w ) arises if the minimum reflectivity is very small, since the reflectivity minimum will make a large contribution to the integral. Experimental errors can well increase near a reflectivity minimum, particularly if it occurs in a narrow frequency interval. A low reflectivity, however, implies that the extinction coefficient 81 8
M. Gottlieb, J . Opt. SOC.Am. SO, 343 (1960),[LiF]. J. R.Dixon, Bull. Am. Phys. Soc. 6,312 (1961); and private communications, 1961.
338
FHANK STEltN
is small. Therefore the absorption coefficient a = 41k/X may be small enough to be measured directly in transmission experiments. It may also be possible to determine the index of refraction in the neighborhood of the reflectivity minimum. Even if the index cannot be measured, it will generally be useful to know the extinction coefficient, since the minimum reflectivity is k2/4. This technique was used successfully in LiF,a8where the and would have been very minimum reflectivity was about 3 X difficult to measure directly. The phase e(w) given by Eq. (21.1) equals zero when w = 0, and approaches zero as w + 0 0 . It represents the phase associated with a single absorption band which is well separated in frequency from other absorption bands, since we have assumed that the reflectivity is constant both above w2 and below WI. Thus we completely neglect the contribution of other absorption bands to the phase e ( w ) in the region being studied. The validity of this procedure can be established in general, in much the same way that it was established for the free-carrier case.84 A numerical test of the phase shift dispersion relation has been made for the lattice vibrations of quartz by Spitzer and Kleinman,aSwho found that there were relatively large errors for small values of the phase or the extinction coefficient k. 22. EXTRAPOLATION PROCEDURES In many cases of interest the criteria for application of Eq. (21.1) are not met, since the reflectivity has not approached a constant limit at the highest frequencies for which experimental results are available. Reflectivity measurements become increasingly difficult with increasing photon energy. Above about 6 ev vacuum ultraviolet techniques must be used, and above about 18 ev the transmission of windows is sufficiently poor that the light source must be placed inside the vacuum system with the sample. No really satisfactory way to proceed is available when the reflectivity is still varying at the highest frequency attainable. However, several extrapolation methods have been used, often with very good results.a6a One common way86-asto extrapolate the reflectivity beyond the cutoff See Figures 4-7 in Gottlieb.20 F. Stern, J . A p p l . Phys. 32, 2166 (1961),Eq. (10). 86 W.G. Spitzer and D. A. Kleinman, Phys. Rev. 121, 1314 (1961). as. A power series expansion of O(w) has been used for extrapolation by Velickf.Plc 80 H. R. Philipp and E. A. Taft, Phys. Rev. 113, 1002 (1959),[Gel. 8' M. Aven, n. T. F. Marple, and B. Segall, J . A p p l . Phys. 32, 2261 (1961), [ZnSe]. 88 R.E. Morrison, Phys. Rev. 124, 1314 (1961),[ I n k , InSb, GaAs]. 88
M
339
ELEMENTARY OPTICAL PROPERTIES O F SOLIDS
of the experimental data is to assume that R
=
Rz(wz/w)P,
w
2
wz
(22.1)
where Rz is the measured reflectivity at w2, and p is an empirical parameter chosen to give best agreement with the experimental results for small W . A sensitive criterion is that the extinction coefficient, and therefore, because of Eq. (20,3b), the phase associated with interband transitions must vanish for energies less than the energy gap. For extrapolations based on Eq. (22.1) we must replace (21.ld) by:
(22.2)
Another extrapolation pro~edure3~~40 is to assume that R = Rzexp [B(wz - w ) ] ,
w
2
WZ.
(22.3)
The constant B is usually chosen to make the derivative of the extrapolated reflectivity equal the measured derivative a t w 2 , I n this case the integration for e3(u) must be truncated a t a n upper limit w3, or it will diverge. The cutoff frequency w 3 is determined by optimizing the agreement with experiment at low frequencies. In this case we find: &(w)
=
[In
~R2! w+) Bwz] In
+
( ~ 3
W)
1
~
-2 1
(w2+w);w3-w1
BW +-In 2n
1 ~ 3 '
- 0' 1
/uz'-wZ/'
(22.4) Neither of the two extrapolation procedures just described is fundamentally sound, although both give good results in many cases. An alternative approach would be to recognize that on the high-frequency side of an absorption band the extinction coefficient falls off approximately as w-4.z8 Thus, far above an absorption band the reflectivity is determined by n = K ~ ' ~where , K = K, - sW-2. (22.5a) Here 40
40.
K,
is the contribution to the dielectric constant of all absorption
M. P. Rimmer and D. L. Dexter, J . A p p l . Phys. 31, 775 (1960), [Gel. M. G. Doane, M.S. Thesis, University of Rochester, 1961, (unpublished), [KCl, KBr, NaCl]. I am indebted to Dr. K. Teegarden for making a copy of this thesis available. Perhaps the first application of the phase shift dispersion relation to optical properties of solids was made by T.s. Robinson, Proc. Phys. SOC.(London) B66,QlO (1952), [polythene].
340
FRASK STERS
processes a t energies above those of the band we are considering. From the dispersion relation (17.10a) we deduce that
8
= (2/T)
1
W’Kz(W’)
(22.5b)
dw’,
the integral being taken over the band being studied and all lower bands. Unfortunately, experimental measurements have not been made to high enough frequencies to make extrapolation procedures based on Eq. (22.5) applicable. In the event that data which included the highest absorption bands were available, we would have K , = 1, and the reflectivity would go to zero asymptotically as wW4. The procedures we have described are not hard to carry out. Equations (20.3) and (20.8b) have been used by many workers to determine the optical constants of ~ o l i d s . ~ ~ JThe ~ J ~advantage -~* of this method over other methods of determining optical constants is that measurements may be made near normal incidence, and do not require polarization of the light beam. There are disadvantages also. One is that measurements must be made over a very wide frequency range, and must often be extrapolated beyond the upper limit of experimental data, usually in the ultraviolet. Another disadvantage is that errors in reflectivity in a small wavelength range will affect the calculated optical constants over a rather wide range. For the experiments cited above, these disadvantages have not been so serious as to offset the advantages offered by use of the dispersion relations. 23. DERIVATION OF S U M RULES
At frequencies higher than those of the highest absorption band we can write, using Eq. (17.10a), Kl(W)
= 1
-
(2/T)U-’
[
aJ
W’Kz(W’)
dw’.
(23.1)
This expression becomes more accurate as w increases. (We restrict ourselves, however, to photon energies low enough that photoionization is the dominant process.) At very high frequencies we can neglect the periodic potential in the crystal, and even the binding of the inner atomic electrons, and can treat all the electrons as free. A very simple derivation, to be given in Part IV, shows that in this case the dielectric constant is K ~ ( w )= (1
1
- Ne2/(meow2)= 1 - wp2w-2,
(23.2)
E. A. Taft and H. R. Philipp, Phys. Rev. 121, 1100 (1961), [Ag]; H. Ehrenreich and H.R. Philipp, ibid. 128, 1622 (1962), [Ag, Cu].
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
341
where N is the total number of electrons per unit volume, and m is the free-electron mass. Identifying the coefficients of in the two equations above allows us to write the sum rule O ’ K (~w ’ )
dw’ = (7r/2)Ne2/ (meo).
(23.3)
Another sum rule which can be deduced in the same way applies to the imaginary part of [ ; ( w ) ] - ’ . The reciprocal of the dielectric constant obeys dispersion relations of the same form as does the dielectric constant. If we again apply Eq. ( I 7.1Oa), and take the reciprocal of (2:3.2),we find that
/om
w’
Im ([-Z((W)]-~I
dw’ =
(7r/2)Ne2/(meo).
(23.4)
+
The imaginary part of -[;(a)]-* equals K 2 / ( K 1 2 K?), and is of considerable importance in describing the interaction of charged particles with matter, to be discussed further in Part VII. Finally, we may write a sum rule, derived from the Kramers-Kronig relation (18.1), in the form:
/om
o ’ k ( w ’ ) dw’ = (7r/4)Ne2/(meo).
(23.5)
This result, in a slightly different form, was given by Kronig.22 The sum rules (23.3)-(23.5) are formally incomplete, in that we neglected the motion of the nuclei, which will also move as free particles at the high frequencies being considered. For an atom with atomic number Z and atomic weight A , the nuclear contribution to the dielectric constant is approximately Z/ ( 1 8 3 6 4 ) as big as the electronic contribution, and can be neglected. Less trivial is the use of the free-electron mass, instead of an effective mass, in Eq. (23.2). We justify this qualitatively by noting that for a very high-frequency disturbance the effect of the periodic potential is negligible compared with the kinetic energy of the electron. The binding of the inner electrons is ignored on the same basis. The sum rule (23.3) is formally equivalent to the Thomas-ReicheKuhn f-sum rule for atom^.^**^^ We defer a discussion of this equivalence until Part V, in which oscillator strengths will be defined. An interesting consequence of Eqs. (23.3) and (23.5) is that %Av = 1 if the average is computed with o k ( w ) as a weighting factor. 42
H. A. Bethe and E. E. Salpeter, in “Handbuch der Physik” (S.Fliigge, ed.), Vol. 35, Sect. 61. Springer, Berlin, 1957.
342
FRANK STERN
We note in conclusion that the derivations leading to the sum rules do not assume the system to be in its ground state, or in thermal equilibrium. If there is emission in certain frequency ranges, as in masers, it is associated with negative values of K ~ ( w or ) k ( w ) . The sum rule shows that this emission must be compensated by additional absorption in other parts of the spectrum. IV. Optical Properties of Simple Systems
I n this Part we shall give in some detail the optical properties for two simple physical systems, the free-electron gas and the lattice vibrations of ionic crystals. These models involve phenomenological constants whose exact significance requires a much more detailed analysis than that given here. The models do, however, describe the behavior of optical properties of real solids quite well in many cases, and provide a good basis for understanding these properties. 24. THEEFFECTIVE FIELD
One of the simplest systems whose optical properties can be calculated in detail consists of particles, each of charge q and mass m*, bound to a fixed center by a force -m*wo2x,where x is the displacement. If an effective field F = Re POexp (--id) is applied, and x = Re f, the equation of motion for g is: m*(d28/dP)
+ m*y
id%/dl)
+ m*oo28 = qpoexp (-id)= qP.
(24.1)
The middle term on the left in Eq. (24.1) is a damping term proportional to the velocity of the particle, and y is a phenomenological damping constant. Such a damping constant is generally introduced in simple models to represent the processes that dissipate energy or momentum in a system being driven by external forces, and return the system to thermal equilibrium when the external forces are removed. In the electronic case, y represents the combined effect of collisions with lattice vibrations, with impurities or lattice defects, with dislocations or boundaries, or with any other departure from the perfectly periodic potential of an infinite single crystal. I n very few cases can a single phenomenological damping constant adequately describe all these scattering processes for all the carriers. We use it nevertheless, because it gives simple results and because the results have a t least qualitative, and . sometimes quantitative, validity in describing the optical properties of free-carrier systems. We will refer in Part V to some quantum-mechanical treatments in which scattering processes are taken into account more accurately.
ELEMESTARY OPTICAL PROPERTIES O F SOLIDS
343
The solution of Ey. (24.1) is: f = (*P/rn*)/(wo2
-
- iyw).
w2
(24.2)
If there are N particles per unit volume, the polarization can be written:
P
=
Nqf
=
(Nq2P/m*)/ (uo2
- w*
- i y w )*
(24.3)
The effective field P is different for different types of carriers. If the carriers are free, as in metals or in the ionosphere, Darwin43found that the effective field is the electric field E. His criteria suggest that this will also be the effective field acting on conduction electrons and holes in a semiconductor. Thus we can write
E, P, = &,E,
(24.4a)
$, =
(24.4b)
where 2, is the electric susceptibility of the free charges. For bound charges, the effective field can be deduced by drawing an imaginary sphere around the point a t which the field is required, and summing the contributions to the field from the bound charges inside the outside the The material outside the sphere is treated as a continuum, and gives a field equal to P b / ( 3 ~ 0 ) inside. For crystals of cubic symmetry, or for amorphous materials, the field at the origin due to all the other particles in the sphere vanishes.44-d6 DarwinAghas shown that the free charges do not contribute to the field seen by the bound charges. Thus the effective field seen by bound charges is
8,
=
E + Pb/(363)
(24.5a)
If we write the relation between the polarization of bound charges and the effective field they see in the form
P,
(24.5b)
= eOxb8b,
then it follows from Eq. (24.5a) that: jib Ja
=
Pb/(&)
= zb/(l
- gxb),
(24.5~)
C. G. Darwin, Proc. Roy. SOC.A146, 17 (1934); A182, 152 (1943). Experimental
evidence concerning the effective field acting on electrons in the ionosphere is discussed by J. A. Ratcliffe, “The Magneto-Ionic Theory and its Applications to the Ionosphere,” p , 154. Cambridge Univ. Press, London and New York, 1959. 44 See p. 31ff in Panofsky and Phillips.8 46 M. Born and K. Huang, “Dynamical Theory of Crystal Lattices,” p. 103. Oxford Univ. Press, London and New York, 1956. 4 6 Other calculations on effective fields in dielectrics are described by W. F. Brown, Jr., in “Handbuch der Physik” (S. Fliigge, ed.), Vol. 17, p. 1. Springer, Berlin, 1956.
344
FRANK STERN
where Sib is the susceptibility of the bound charges. Equation (24.5~)gives the Clausius-Mossotti-L~rentz-Lorenz~~ expression for the dielectric constant of liquids and cubic crystals if we replace 2 b by Z - 1. When both free and bound charges are present, it follows from Eqs. (24.4) and (24.5) that their susceptibilities add. Thus the dielectric constant is : Z
=
1
+ %a + 2,.
(24.6)
A . The Free-Electron Gas 25. GENERAL CONSIDERATIONS When w is small compared with the frequencies wo that characterize the bound electrons, the contribution of the bound electrons to the susceptibility becomes independent of frequency. Thus for frequencies from the infrared down to dc, except for that part of the infrared where the contribution of the ions is important, we can restrict our attention to the free carriers, and replace the effect of the bound charges by a frequencyindependent dielectric constant which we shall call K,. For the general case in which there are several groups of free carriers, for which wo = 0, the dielectric constant deduced from Eqs. (24.3) and (24.6) can be written: i(w> =
K,{
1-
C [wii2/ j
.
(w2
+ yjz)I + ( i / w > C
Cwlj2Yj/ (wz
j
+ ~i")II , (25.1)
where y j is the damping constant for carriers of type j , and wlj =
[Nje2/(mjcoK,)]1/2.
(25.2)
Here Nj is the concentration of free carriers in the j t h group, and mj is their effective mass. We have replaced q2 by e2, since the carriers have charge f e . Some useful numerical expressions are: wlj
[sec-'1
=
5.641 X lo4 [Nj [~rn-~]rn/(rnj~,)]~/~,
vlj
[sec-'1
=
wlj/2~ = 8.978 X
Elj [ev]
=
fiwlj = 3.713 X lO-"[Nj
Xlj [microns]
=
c/vlj
=
lo3 [Nj [~ m -~ ] m/(mj~ ,)]~ /~ , [~m-~]m/(mj~,)]~'~,
3.339 X lOlo[Nj [~rn-~]m/(mj~,)]-'/~.
(25.3)
When the damping is sufficiently weak we can drop the y's in Eq.
345
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
(25.1), and obtain the simple result:
where
The plasma frequency w , / ~ x ,defined in general as a frequency for which the real part of the dielectric constant vanishes while the imaginary part is TABLE111. OPTICALPROPERTIES OF ALKALIMETALSNEARTHE PLASMA FREQUENCY
We give the approximate wavelength A, at which the real part of the dielectric constant equals zero, and the corresponding photon energy &., The damping is measured by K ~ ( W , ) , the imaginary part of the dielectric constant a t the plasma frequency. Also given are the lattice constant a, the carrier concentration N (assuming 1 electron per atom), and the value of m*K,/m deduced from Eq. (25.2) using the observed plasma frequency and carrier concentration. XP"
Metal [microns] Li Na K Rb CS
0.21 0.34 0.395 0.435
~ ~ [ e v ] KZ (w,) 5.9 3.65 3.15 2.85
0.012 0.1 0.23 0.35
a[A]* 3.509 4.291 5.32 5.70 6.16
N[cm-a] 4.63 2.53 1.33 1.08 8.56
X X X X X
m*k/ni
m*/mc
1.00 1.38 1.51 1.45
1.45 0.98 0.93 0.89 0.83
lozz 10"
loz2
1022 10"
Taken from Ives and Brigg~.~8
* The values are for 20"C,and are taken from Pear~on.~g The value for Cs is extrapolated. c
Taken from Pines.m
small, is one of the most striking features of the optical properties of a solid. Below the plasma frequency the dielectric constant is negative, and the imaginary part of the index of refraction will be greater than the real part. Above the plasma frequency the imaginary part of the index drops sharply, and the real part becomes dominant. This behavior is demonstrated by the experiments of Wood4' on transmission through thin films of alkali metals. He found a sharp onset of transmission as the wavelength decreased. More detailed studies of the optical constants of the alkali metals were made by 47
R. W. Wood, Phys. Rev. 44, 353 (1933); R. W. Wood and C. Lukens, ibid. 64, 332 (1938).
346
FRANK STEHN
Ives and B r i g g ~ and , ~ ~ qualitatively confirm the behavior predicted by Eq. (25.1), Some r e ~ u l t s ~on~ properties -~~ of alkali metals are summarized in Table 111. 26, COSDUCTIVITY ASD MOBILITY The conductivity of a free-carrier. system containing several groups of carriers is: U = QWK2 = Uoj/(l W2T?), (26.1)
+
j
where
~j
is the relaxation or scattering time: (26.2)
r j = yj-1.
The dc conductivity uoj is usually written in the form uoj = Njepj,
pj =
erj/mj,
(26.3)
where pj is the mobility of carriers in the j t h group, and is conventionally defined to be positive for both electrons and holes. (Kote that in our notation the charge of the electron is - e . ) Some examples of mobilities a t room temperature are : Cu, 40 cm2/volt-sec ; n-type InSb, 70,000 cm2/volt-sec; PbS, PbSe, PbTe, Ge, lo3to 4 X los cm2/volt-sec. Some useful numerical relations are : pj
[cm2/volt-sec]
=
1.7589 X 10'6(m/mj)7j [sec],
rJ [sec] = yJ-' = 5.085 X
(m,/m)pj [cm2/volt-sec],
u3 [ohm-' cm-l] = 1.6020 X 10-'9NJ [ ~ m - ~ ] p [cm2/volt-sec]. j
(26.4)
A few words are necessary to explain the significance of the effective mass m* and the scattering time r which we have used as phenomenological constants in our discussion of the optical properties. The effective mass has a unique meaning only for bands whose energy can be represented as a quadratic, isotropic function of the wave vector k. For cubic crystals whose energy bands are ellipsoids with principal effective masses ml, (8
40
60
H. E. Ives and H. B. Briggs, J. O p t . SOC.A m . 28, 238 (1936), [K]; 27, 181 (1937), [Na]; 27, 395 (1937), [Rb, Cs]. A relation between the plasma frequency and the photoelectric effect was found by H. E. Ives and H. B. Briggs, J. Opt. SOC.Am. 28, 330 (1938); see also the note added in proof on page 408. W. €3. Pearson, "A Handbook of Lattice Spacings and Structures of Metals and Alloys." Pergamon Press, New York, 1958. D. Pines, Solid State Phys. 1, 414 (1955).
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
347
m2, and ma, the effective mass to be used in describing the free-carrier optical effects is6‘ m* = 3(rnl-I mz-l msl)-l, (26.5)
+
+
while for a spherical but nonparabolic band in a degenerate sample with Fermi energy E p S 1 : m* = fi2k I d k / d E I E - E ~ . (26.6) The scattering time T which enters in the theory of transport properties is generally a function of carrier energy and in many cases cannot be defined at a11.62 Thus in characterizing a whole group of carriers by a single scattering time T we make a grossly oversimplified assumption. One can make a rough correction for this by multiplying expression (27.5) for the absorption constant by T A ~ T - ~ A ~ . ~ * However, for reliable results one must use a quantum-mechanical treatment. We will refer to such treatments in Part V, and have restricted our attention here to simple models which give the over-all features of the optical behavior. IN VARIOUS FREQ U E N C ~RANGES 27. OPTICALPROPERTIES
Simple expressions for the optical properties of a free-carrier system can be obtained from Eq. (25.1) in certain wavelength ranges. For this purpose we will suppose that there is only one type of carrier present, with effective mass m*.Then we can put A, = XI, using Eqs. (25.3) and (25.5). The results can be generalized to a many-carrier case without difficulty. The two characteristic wavelengths for the system are A,, already given by Eq. (25.3)) and A, = 2ac7, or A, [microns] = 1.0709 (.t*/m) p [cm2/volt-sec].
(27.1)
At room temperature A, M 40 microns for Cu, while A, M 1000 microns for pure InSb. Most semiconductor measurements are made at wavelengths short compared with both X, and A,. In that range we find 7i M
Km[l
-
-
(wp2/w2)
+
i(ywp2/w3)].
.(
>> UP, 7 )
(27.2)
The imaginary part of the dielectric constant is much smaller than the real (~~)1/2, and IC = ~ ~ / 2 nThe . optical properties for which part. Thus n W. G. Spitzer and H. Y. Fan, Phys. Rev. 106, 882 (1957). See, for example, C. Herring, Bell System Tech. J . 34, 237 (1955), Appendix A; C. Herring and E. Vogt, Phys. Rev. 101, 944 (1956). 6) R. A. Smith, “Semiconductors,” p. 219. Cambridge Univ. Press, London and New York, 1959. 61
61
348
FRANK STERN
numerical expressions are must useful are the absorption coefficient a, and the reflectivity at normal incidence R. From Parts I and I1 we find that: (27.3a) a = 4&/A = 2kw/c = O K ~ K , , , / ( ~ C ) = U K ~ / ( ~ C E ~ ) , a [cm-l]
=
1.014 X 106k&[ev],
(27.3b)
k
=
0.9866 X 10-sa [~m-~]/&[ev],
(27.3~)
R
=
[(n
- 1)' + k']/[(n
+ l ) z + k'].
(27.4)
In Eqs. (27.3b) and (27.3c), & is the photon energy, and K~ = 1. For short wavelengths we find, making use of the numerical expressions given earlier, that Eq. (26.2) leads to a [cm-I]
=
5.262 X 1~-17n-l[Nm2/(m*2~)](A[microns])2.
>> up,y)
(w
(27.5)
We will express N in cm-8 and p in cm2/volt-sec in all numerical expressions. The reflectivity is
R
- Ra{l-
1.794 X 10-21[Nm/(m*n(~, - l))](A micron^])^], (a
+
>> u p , 7)
(27.6)
where R, = (K,*/' - l)z(~,1/2 1)-' is the asymptotic value of the reflectivity at short wavelengths. Equations (27.2), (27.5), and (27.6) have ignored interband transitions, and thus are no longer valid if these transitions are present. For semiconductors, A, and A, are big enough that Eqs. (27.5) and (27.6) should hold for a fairly wide range of wavelengths, with the wavelength of the absorption edge as a lower limit and the upper limit determined by lattice vibration effects, which have also been ignored in (27.2). The region of validity of Eqs. (27.5) and (27.6) may be small or nonexistent for metals, in which interband transitions generally play a role at wavelengths comparable with A,. The second simple case, that of very long wavelengths, can be reached without difficulty for metals, and also for impure semiconductors with low mobility, in which A, and A, lie in the not-too-distant infrared. Thus when w << w,, y, Eq. (25.1) can be approximated by M M
In this case
K~
>> I ~1 1,
+ ~ , ( 1- w p 2 ~ . "+ ) iuo/(eow). K,[(1 - Wp2'?)
iOp'T/W]
(O
<< wp, y)
(27.7)
and we have
n M k M (K2/2)'/' M 5.4753 X 10-2(A [microns]
U O ) ~ / ~ .
(a <
ELEhfENTARY OPTICAL P R OP E R T I ES O F SOLIDS
849
We give the dc conductivity u0 in ohm-' cm-I in all numerical expressions. From Eq. (27.7) we find: a
<< u p , y) 1 - 36.327(A [ r n i c r o n ~ ] u ~ ) - ~( w~ < ~<. up,y )
[em-l] w 6.8803 X 103(u0/A [mi~rons])"~,
Rw 1
-
(2/n)
-
(w
(27.9) (27.10)
The second of these is the famous Hagen-Rubens formula,54which gives the deviation of the reflectivity from 1 with an error of 20: or less for a great variety of metals a t wavelengths beyond about 4 microns. An alternative to Eq. (27.9) is an expression for the skin depth, 6 = 2 / a , which equals : 6 [microns] = 2.9068 ( A [microns]/ao)
8[cm]
=
l/*,
5.0330( Y [bIc/sec]~~)-'/~.
(w
<< w p , r'r
(27.11)
For metals i t will generally be possible to find a region of wavelengths short in comparison with A,, but long in comparison with A,. That will be true because WpT
=
3.207
Up/?
x
lo-11[NWZ*/(WZK,)]1/2/L
(27.12)
is generally very large for metals. When y << w << w, the complex dielectric constant is given by Eq. (%7.2),with ~2 << ' K~ ,. Now K~ is negative, and therefore li >> n. \Te find that both the absorption coefficient and the reflectivity are approximately independent of wavelength in this case, with the values [Cm-l]
R
=
2WpKm1"/C
=
3.763
x
10-'(Nm/?7Z*)'/*, (y << 0 << U p ) (27.1:3)
W 1
- ( 4 n / k 2 )X 1 -
X 1
- 6.236 x 1010(m*N/m)-"*p-'.
2/(OprK,112)
(y
<< w << w,)
(27.14)
The only remaining case, which applies if wPr << 1, can be realized for very pure germanium, and perhaps in a few other semiconductors. Then for w p << w << y the complex dielectric constant will be given by Eq. (27.7), but with ~2 << K', and k << 1. We find: a
R
FZ
R,[l -
[em-'] w :376.75n-1~o, ~K,W~*T'/(K,
<< w << y)
(up
(27.13)
- I)]
M Rcl{l- 2.037 X l o - * ' [ ( ~ ~ * / m ) . V ~ ' ( ~ , l)-']], (w,
where R, 54
= (K~'/*-
1)"~ m
<< w << y)
+ l)-', as for Eq. (27.G).
E. Hagen and H. Rubens, Ann. Physzk [4] 11, 873 (1903)
(27.16)
350
FRANK STERE
28. REFLECTIVITY MINIMUM The reflectivity of a free-carrier system a t normal incidence has been shown in Eq. (27.6) to decrease with increasing wavelength a t short wavelengths and, in (27.10), to approach 1 asymptotically a t long wavelengths. Thus there will be a minimum in the reflectivity, whose spectral position gives a good measure of (N/m*). I n the simplest case of negligible damping the dielectric constant is given by Eq. (25.4) and it is clear from (27.4) that the reflectivity a t normal incidence will have a minimum value of 0 when n = 1, which occurs at the angular frequency w = w~[K,/(K, - 1)] l i z . From Eq. (27.4) it can be shown without difficulty that the condition for a reflectivity extremum is (K1
- 1) (dn/dW)
+ Ki(dk/dW)
=
0,
(28.1)
where we have considered the material to be nonmagnetic. From Eq. (28.1) it is possible to obtain an implicit equation for the frequency a t which the reflectivity is a minimum. The equation has a simple solution if we assume that only one type of carrier is present and if wPr >> K,. I n that case we find ;
where w p r is given by Eq. (27.12), and X, by (25.3). The position of the minimum is given fairly accurately by Eq. (28.2a) if wpr >> K,, but the magnitude of the minimum reflectivity given by (28.2b) should be considered only a crude estimate. When w P r >> K,, the predicted value of Rminis of the order of 15; or less, and experimental difficulties may prevent an accurate measurement from being made. I n the case of metals the plasma frequency is in the ultraviolet or the visible, where interband transitions can occur, so the damping constant y = 7-l deduced from the dc conductivity no longer characterizes the actual losses. 29. EKERGY DENSITY The energy density in a material medium can be considered as the sum of the energy in the field and the energy in the particles of the medium. We illustrate this for a free-carrier system, ignoring damping for simplicity. Then the dielectric constant is given by Eqs. (24.10) and (24.11),
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
35 1
and the time-averaged energy density given by (6.9) is:
+
T J = r~ & * * E [ K I w ( d K l / d u ) which gives:
O 0
+1
11,
K
. . I
(29.1)
=
$c~E**EK,,
w
=
iE$*'EK,(Wp2/W2),
w
1 up; 5 up.
(29.2a) (29.2b)
The kinetic energy of the free-carrier motion is found from Eqs. (24.2) and (24.11) to be:
0 (carriers)
=
$
c Nlml(dxj/dt)2 4 c NjmjW29,*'fj =
j
I
= $ & * - E K , ( w ~ ~ / w ~(29.3) ).
The remaining energy density can be written in the form
0 (field)
= =
+ tp$t**H t€&*'E(K, + I 1). +&**EK,
(29.4)
K
The sum of Eqs. (29.3) and (29.4) is equal to 'the total energy density (29.2). We find that for frequencies below the plasma frequency the kinetic energy of the carriers accounts for half of the total energy. In this frequency range & is imaginary, and the fields decay exponentially. At higher frequencies the kinetic energy of the carriers is a fraction uP2/(2u2)of the total energy, dropping off as their contribution to the dielectric constant becomes less important. When dissipation is introduced, the separation of the energy density into two parts like Eqs. (29.3) and (29.4) can no longer be made. In the extreme case for which w << y, the kinetic energy of the carriers becomes unimportant, and the energy density is mainly that of the magnetic field. The time-averaged energy density in this limiting case is 0 = ( u / u ) B * * R . The power dissipation is &uB*.E,and if we define the Q of the system to be w 0 divided by the power dissipation, we find that Q 3 in the limiting case of high conductivity.
-
B. Optical Modes in Ionic Crystals 30. PHENOMESOLOGICAL DISCUSSION
The lattice vibration spectrum of a crystal with 2 atoms per primitive cell contains 6 branches, 3 of which correspond a t long wavelengths to sound waves in the crystal, and are therefore called the acoustical branches. At long wavelengths the remaining 3 branches, with frequencies of the
352
FRAPiK S T E R S
order of 1013 sec-l in most crystals, correspond to motion of the 2 atoms in opposite directions. I n ionic crystals these modes produce a strong polarization, and interact strongly with electromagnetic radiation (transverse modes) and with charged particles (longitudinal modes). They are called the optical modes. Interaction between photons and phonons requires that both energy and momentum be conserved. This condition can be met for single-phonon processes only for the long-wavelength optical modes in a crystal, since the wave vector of a photon with frequency sec-l is 2 X lo3 cm-l, and is very small compared with the range of wave vectors (=los em-l) allowed for lattice vibrations. There are also weaker interactions involving 2 or more phonons. These have been extensively studied recently in highresolution absorption measurements, and can give detailed information about key points in the lattice vibration spectrum.55We will not consider the multiphonon processes here. In this section we shall make use of the result, to be established in a later section, that the absorption of infrared light by an ionic crystal is strongly peaked at the angular frequency W T of the long-wavelength transverse optical modes. From this fact alone we can predict the frequency dependence of the dielectric constant, making use of the dispersion relation (17.lOa), which we rewrite here in the form: K ~ ( w )=
K,
1
$. ( ~ / R ) P o ’ K ~ ( u ’ )
-
w2)-l
dw’,
(30.1)
where the integral is extended only over the absorption associated with the lattice vibrations. The effect of all absorption mechanisms a t higher frequencies is included in K,, which is assumed to be real and frequencyindependent in the frequency range being considered. If we approximate the absorption peak a t wT by a delta function, W’KZ(W’) = ( n C / 2 ) 6 ( w ’ - C O T ) , then Eq. (30.1) leads to: Ki(U)
-
K,
=
~ ( W T ’
-
d-’.
(30.2)
The constant of proportionality C is easily found in terms of K ~ ,the lowfrequency limit for the dielectric constant in the lattice vibration range. Then we find: K ~ ( C O ) = K,
+
-
( K ~
K,)uT~(wT~
-
w2)-l.
(30.3)
This result will be derived again in the following from a microscopic treatO5
See, for example, the paper of W. G. Spitzer, J . A p p l . Phys. 34, 792 (1963), which discusses phonon assignments for GaAs and CdS and refers to experimental results for a number of other semiconductors.
ELEMENTARY OPTICAL PROPERTIES O F SOLIDS
353
ment of lattice vibrations. Comparison of Eq. (30.3) with (30.1) shows that
1 w'K~(u')
dw' = ( n / 2 )( K ~
K,)wT',
(30.4)
where the integration is limited to the absorption peak near COT. Since ~2 is nonnegative for a system in its ground state or in thermal equilibrium, it follows that K~ 2. K., We have already seen that the frequency for which the dielectric constant vanishes has a special significance. I n particular, a longitudinal electric field can exist at that frequency even in the absence of external sources. This was shown in the discussion leading to Eq. (7.13), and also follows from Eq. (4.1), which shows that H = 0 if the dielectric constant vanishes, and that E is parallel to the wave vector k if H = 0. It is therefore natural to suppose that the long-wavelength longitudinal optical mode will occur a t the frequency for which K = 0. This result is confirmed by the detailed theory to be given below. We then find from Eq. (30.3) that (30.5) where w l ; / 2 a is the longitudinal optical frequency. Equation (30.5) is often called the Lyddane-Sachs-Teller relation.66 In crystals having two identical particles in the primitive cell, such as crystals with the diamond structure, the long-wavelength optical modes produce no polarization, and cannot give the one-phonon absorption at COT which ordinarily dominates the optical absorption. Other absorption mechanisms, such as multiphonon processes, will be present, but they will be relatively weak. Thus Eq. (30.4) shows that K~ - K, will be small in crystals with the diamond structure. It follows from Eq. (30.5) that the longitudinal and transverse optical modes of long wavelength are degenerate in these crystals. The experimental value for the microwave dielectric constant of germanium is 16.0 f 0.3 at 4.2°K,5' while the optical value at 16 microns is 16.01 at room temperatureu and 3%, to 4% smaller at very low temperature^.^^ I n gallium arsenide the static and optical dielectric constants differ by a considerably larger amount, the room temperature values being K~ = 12.5, and K, = 10.9.60 R. H. Lyddane, R. G. Sachs, and E. Teller, Phys. Rev. 69, 673 (1941); see also p. 155 in Frohlich,' and p. 86 in Born and Huang.46 KT F. A. D'Altroy and H. Y. Fan, Phys. Rev. 103, 1671 (1956); see also W. C. Dunlap, Jr. and R. L. Watters, ibid. 92, 1396 (1953). 68 C. D. Salzberg and J. J. Villa, J . O p t . Soc. Am. 47, 244 (1957); 48, 579 (1958). M. Cardona, W. Paul, and H. Brooks, Phys. Chem. Solids 8, 204 (1959). 6o K. G. Hambleton, C. Hilsum, and B. R. Holeman, Proc. Phys. Soc. (London) 77, 1147 (1961).
354
FRANK STERN
For angular frequencies between W T and W L the dielectric constant (30.3) is negative. If damping is confined to a very narrow frequency range near wT1 we can ignore the imaginary part of the dielectric constant, and find that between W T and W L the real part of the index of refraction vanishes. Thus the crystal will be totally reflecting throughout this range. In real crystals there will be some damping, and the reflectivity will be less than 100%:. There will be a peak in the reflectivity between W T and W L ; several reflections from the surface of an ionic crystal will give a fairly narrow range of infrared frequencies. The reflection peak is at the Reststrahl frequency, important both because of its origin in the vibration spectrum of the crystal and as a practical source of relatively monochromatic infrared radiation. We have seen that the main features of the optical properties of ionic crystals in the lattice vibration frequency range can be found very simply if we assume that absorption is strongly peak near LOT. I n the more detailed considerations below we show how the parameters in Eq. (30.3) are related to the atomic properties of the medium. 31. ATOMICTHEORY OF OPTICALMODES The polarization in an ionic crystal with one ion pair per primitive cell is given by
P
=
NQI
+ be$,
(31.1)
where Q is an effective charge, s' = (5, - ii-) is the relative displacement of the positive ions with respect to the negative ions, N is the number of ion pairs per unit volume, p is the effective field, and b is a constant describing the combined polarizability of the two ions. If we substitute p = E P'/(3to) from Eq. (24.5a), as appropriate for the motion of bound charges, we find:
+
P
=
(NQI
+ b&)/(l
- Qb).
(31.2)
For high-frequency oscillations the displacement of the ions is small, and the polarization is given in terms of the high-frequency dielectric constant K, by P = ( K , - l)&. Thus b / ( l - +b) = K , - 1, and Eq. (31.2) can be rewritten :
-
P
=
NQ's'
+
where
Q'
=
(K,
(K,
- 1)~&
+ 2)Q/3.
(31.3) (31.4)
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
355
The equations of motion of the ions can be written in the form
+ QP,
M+(d%+/dt2)
=
-K(fi+ - ii-)
M-(d2ii-/dt2)
=
K(ii+ - ii-) -
QP,
(31.5a) (31.5b)
where M+ and M- are the masses of the two ions, and K+ and Ei_ are their displacements. Born and Huang61 show that for crystals with tetrahedral or higher cubic symmetry the force constant K is a scalar. If we multiply Eq. (31.5a) by M - and (31.5b) by M+, subtract, and divide by (M+ M - ) , we obtain:
+
d4,(d2P/dt2)
+
=
+ QP,
-KI
(31.6)
M-) . Replacing P by E where M , = M+M-/(M+ and using Eq. (31.3) to eliminate p, we find:
+ p/(3e0) as before,
M,(d21/dt2) = - ( K - [NQQ'/(3eo)])I
(31.7a)
+ Q'E.
If we define M*T2 to be the quantity in braces in Eq. (31.7a), and introduce a phenomenological damping constant y, we can rewrite the equation of motion in the form :
+ y(ds"/dt) +
(d2S/dt2)
WT%
=
(Q'/Mr)E.
(31.7b)
Equations (31.3) and (31.7), together with Maxwell's equations, are the fundamental relations describing the long-wavelength optical modes. They apply to waves whose wavelength is large compared with the lattice constant (so that the force constant K in Ey. (31.5) is independent of wavelength) but small compared with sample dimensions. To proceed further, let us consider a plane wave I
(31.8)
I. exp (iq-r - i d ) ,
=
with similar expressions for p and k. Equations (31.3) and (31.7) show t,hat s', E, and are parallel. Substituting (31.8) into (31.7b) leads to: =
(&'/M,)
(UT2
-
fJJ2
- i'yu)-'E.
(31.9)
If Eq. (31.9) is substituted into (31.3) we find:
-
P/Q =
[K,
-1
+ (NQ"/(M,eo))
(COT'
-
u2 - iyu)-']E. (31.10)
K , ) w T ~ ( w T ~-
w2
- iyw)-l,
The dielectric constant is: Z((w) =
where
K,
+
K8 61
-
( K ~
-
K,
See p. 105 in Born and Huang.46
=
NQ'2/(MreoW~2).
(31.11) (31.12)
356
FRANK STERN
We have thus verified the dielectric constant found in the previous section, and can rewrite Eq. (30.4) in the form:
/
W ’ K ~( w ’ )
dw’ = (7r/2) NQ’2/ ( M,Q) .
(31.13)
A numerical expression for evaluating Eq. (31.12) for crystals with rocksalt or zincblende structure is: K~
-
K,
=
1.968(&’/e)2(X~ [rnicroi~~])~A,-~(a[A])-3, (31.14)
where e is the magnitude of the charge of the electron, a is the lattice constant, A , is the reduced atomic weight of the ions on the C’* scale, and A T = 27rc/wT is the wavelength in vacuum corresponding to angular frequency W T . For a CsCl structure the numerical coefficient in Eq. (31.14) is 0.984. The force constant K in Eq. (31.5) is related to the elastic properties of the crystal, as well as to the optical mode frequencies. This leads to the relation6*Ia B = ~ N L ~ IK,, T ~2)~ ((K , ~)-‘wT*, (31.15a)
+
+
where B is the bulk modulus (the reciprocal of the compressibility), and T O is the nearest neighbor distance. I n numerical form this gives:
B [dyne/cm2] = 5.891
x
1 0 1 4 C A r (f ~ 82) ( K ,
+ 2)-’(a[Al)-’(X~
[microns])-*
a
(3I. l5b)
where C is 4 for the rocksalt structure and for the zincblende and CsCl structures, and the other quantities have the same significance as in Eq. (31.14). For many alkali halides the calculated and observed bulk moduli agree within about lot;, 32. LOSGITUDISAL .4ND TRASSVERSE J I O D E S ; EFFECTIVE CHaRGE
A simple physical picture of the significance of the transverse optical mode angular frequency W T is best obtained from Eq. (31.9), which gives the relative displacement s produced by an electric field of angular frequency w. We see that in the absence of damping a displacement can be present without a driving field only a t the angular frequency w T . Furthermore, an external field with w near w T will produce large displacements 62 63
B. Szigeti, Proc. Roy. Soc. A204, 51 (1960). Equations (9.29) and (9.30) on p . 111 in Born and Huang46 are incorrect; the factor d l should be deleted.
ELEMESTARY OPTICAL PROPERTIES O F SOLIDS
357
of the ions, limited by the damping constant y. Thus a transverse field with w near W T will excite large-amplitude oscillations. The energy transfer from these modes to other modes, caused by anharmonicity of the effective potential, by other phoiion scattering mechanisms, and by the nonlinear dependence of the electric moment of the crystal on the lattice displacements, is responsible for the energy absorption peak near W T . Calculations for several damping mechanisms have been made for the combination bands of GaP by Kleinman and Spitzer,6*who conclude that the anharmonic potential is the major source of damping in that case. The physical discussion for longitudinal modes is similar to the one we have just given for transverse modes, except that the driving force arises from a lionstatic charge distribution. If we assume the presence of a source p =
Re
[po
exp ( i q - r - i w t ) ]
=
Re F,
(32.la)
we have a n electric displacement D given by
D = Re [Do exp (iq-r - id)]= Re D, D
qgq2,
=
(32.lb) (32.lc)
where Eq. ( 3 2 . 1 ~ follows ) from blaxwell’s Eq. (2 .lb ). If we now take the divergence of Eq. (31.7b), and use the auxiliary results
E WL*
=
(D - NQ’S)/(QK,),
-
WT*
=
NQ’*/(i1fr~o~,),
(32.2a) (32.2b)
which follow from (31.3) and (31.12), we find: v . [ ( d 2 6 / d t 2 ) f y(ds’/dt)
+
WL2S]
=
[&’/K,]V’D,/(hl,Eo)
For the transverse part of E, V ~ S ‘ T= 0. Thus Eq. (32.3) shows that the Fourier components of an external charge are sources for longitudinal lattice vibrations. This is consistent with Eq. (7.10), which shows that an external charge component F(q, w ) is the source for the corresponding component of the longitudinal vector potential. Equation (32.3) also shows that in the absence of a source F, we can have a longitudinal lattice vibration only at the angular frequency wL, and only if there is no damping. In a real crystal, for which y is not zero, a fixed source j(q,W) with angular frequency near W Lwill give rise to large longitudinal lattice displacements, and these will lead to large pou-er dissipation in the medium. The discussion in the preceding paragraphs gives the physical basis for 64
1). A. Meinman and W. G. Spitzer, Phys. Rev. 118, 118 (1960).
358
FRAEK STERN
the identification of W T and W L as transverse and longitudinal optical angular frequencies, and also for the assumption that the absorption of electromagnetic radiation is strongly peaked near w T , One interesting feature must be added to our discussion of transverse waves. It follows from Eq. (4.3) that the relation between the wave vector and the frequency of a plane wave is
where we have assumed the material to be nonmagnetic. For a given w this equation uniquely determines the real and imaginary parts of the propagation vector q (but not its direction) if we assume that the wave is homogeneous. On the other hand if we fix q, there will be two solutions for w . When 1 q I >> W T K , ~ / ~ / C ,the solutions are w N W T and w N c 1 q 1 corresponding respectively to a pure lattice vibration of angular frequency W T and to a pure electromagnetic wave traveling a t the velocity K,-% On the other hand, for longer wavelengths there is substantial coupling of electromagnetic and mechanical energy in both frequency branches. This is discussed in some detail by Huang.E6 When the wave vector becomes much smaller than WTK,'/'/C the two solutions of Eq. ( 3 2 . 4 ) approach 0 and W L . Thus a t these long wavelengths the 3 lattice vibration modes approach the same frequency. The longwavelength optical lattice vibrations have been discussed recently by several authors.66nfJ a . Efective C h a r g e
The effective charge Q which was introduced in Eq. (31.1) is called the SzigetP charge; it is a measure of the polarization associated with the relative displacement of the positive and negative ions. I n ionic crystals it is generally smaller than the static charge per ion that one would use in a calculation of the cohesive energy of the crystal. Only if the ions are small and rigid will the two effective charges be the same. A material in which the Szigeti charge is surprisingly large is Sic, for which Q = 0 , 9 4 e . @ One might characterize S i c as a covalent crystal, and expect it to have zero static charge. The large value of Q in S i c and its deviation from the nominal value in alkali halides arise from the distortion a K. Huang, Proc. Roy. Soc. A208, 352 (1951); see also pp. 90-98 in Born and Huang.dE (6 H. B. Rosenstock, Phys. Rev. 121, 416 (1961). 87 A. A. Maradudin and G. H. Weiss, Phys. Rev. 123, 1968 (1961); T. H. K. Barron, ibid. 125, 1995 (1961). (8 W. G.Spitzer, D. Kleinman, and D. Walsh, Phys. Reo. 113, 127 (1959) ; W. G. Spitzer, D. A. Kleinman, and C. J. Frosch, ibid. 113, 133 (1959).
359
ELEMENTARY OPTICAL PHOPEHTIES OF SOLIDS
of the atoms when they are displaced. Szigeti,B2Matossi,6Band Cochran’o have discussed the meaning of effective charge in some detail. The electric field accompanying longitudinal lattice vibrations can also be described by an effective charge. To show this we note that if y = 0, the electric displacement D vanishes at the longitudinal mode angular frequency W L , and P = -eOE. If we substitute this relation in (31.3), we find EL
= -N[&’/K,]SL/EO
= -N[(K,
+ 2)&/(3K,)]s~/Eo
(32.5)
to be the electric field associated with a longitudinal wave in which the relative displacement of the positive and negative ions from their equilibrium positions is SL. The expression in brackets in Eq. (32.5), which also appears in (32.3), is called the Callen7’charge. It is the effective charge used in transport calculation^,^^ since the electric field of a longitudinal optical phonon, as given in Eq. (32.5), will scatter charge carriers. A numerical expression for the Callen charge, &’/K,, in units of the electronic charge is: &’/ex,
=
0.7128{A,(a[A])8(~,-*-
) 1 / 2 ( X ~C.mi~rons])-~,
(32.6)
K~-I)
where X L = 2rc/uL, and the other quantities are the same as in Eq. (31.14). For some materials XL?* and K, may be known even though AT and K* are not known. I n such cases Eq. (32.6) gives an upper limit for the Callen charge if we let K#-’ + 0. The Szigeti charge Q is easily obtained from Eq. (32.6) if we remember that Q’ was defined in (31.4) to be ( K , 2)&/3. Thus in materials with a high optical dielectric constant the Callen charge is about one-third of the Szigeti charge.
+
33. REFLECTIVITY
The most commonly used experimental method for studying the optical properties of ionic crystals in the lattice-vibration region of the spectrum is through measurement of reflectivity. Absorption measurements can also be used if thin samples are available. The reflectivity maximum is of considerable interest, since this gives the position of the Reststrahlen, the band of wavelengths obtained after several reflections F. Matossi, 2.Nuturforsch. 14a, 791 (1959). W. Cochran, Nature 191, 60 (1961). H. B. Callen, Phys. Rev. 76, 1394 (1949). T S See, for example, H. Ehrenreich, Phys. Chem. Solids 2, 131 (1957). 78 The energy of longitudinal optical phonons in several semiconductors has been deduced from tunnel diode characteristics by R. N. Hall and J. H. Racette, J . Appl. Phys. 32, 2078 (1961). (0
70
360
FRANK STERN
from an ionic crystal. The maximum is not very sharp, and thus cannot be used for reliable determination of optical parameters of the crystal. A maximum in the reflectivity R, given in Eq. (27.4), occurs where there is a maximum in (1 - R)-I, or in (n2 IC’ l ) / n , which, for K:, << I ~1 1, implies a maximum in ( 1 ~1 ,312 I ~1 i1”)/K2. If we define
+ +
+
-
2 = (W2
and assume that y
WT2)/(WL2
-
<< w T , we find from the dielectric K1
K2
(1
+
(33.2a) (33.2b)
UZ) ‘ I 2 / (UWTZ‘),
where a =
constant (31.11) :
- 1)/2,
= K,(Z
= YK,
(33.1)
W?),
b
( K ~- K , ) / K , ,
=
(
K
-~ l ) / ~ , .
(33.3)
The position of the Reststrahl peak, urnax,is given by ~ m a x= W T (
where zm is a root of 3abz3
1
+ azm)
(33.4)
l”,
+ (46 - u - 2ab)z2 - (3b f 2 ) +~ 1 = O
(33.5)
lying between 0 and 1, since this is the range of 100% reflectivity when y = 0. The solutions of Eq. (33.5) can be given in simple form around the edges of the range of values taken by a and b : a > 0, 0 < b < 1. These solutions are :
+ 2) - ((3b + 2 ) ’ - 16b)”2]/(8b), = [a(2b + 1)]--1’*, = [(l + - l]/u, zm = [(4 + 3a)”2 - 2]/(3a), Z,
=
[(3b
2,
2 ,
a)1’2
0;
(33.6a)
m;
(33.6b)
a
=
a
-+
b
=
0;
(33.6~)
b
=
1.
(33.6d)
To facilitate the determination of z,, the solution of Eq. (33.5) for a rather coarse grid of values of a and b is given in Table IV. Rough interpolation in the table can be used to give a good trial root to substitute in Eq. (33.5) for determination of an accurate value. It is somewhat disconcerting that this procedure, when applied to a number of alkali halides, gives slightly poorer agreement with experiment than the approximation zm = (4b 2)-l given by Havelock.74 These considerations concerning the reflectivity maximum are of some formal interest, but do not always describe the observed reflectivity
+
74
T. H. Havelock, Proc. Roy. Soc. A106, 488 (1924) ; see also p . 123 in Born and H ~ a n g . ~ 6
361
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
TABLEIV. REFLECTIVITY MAXIMUM IN IONIC CRYSTALS
We tabulate the solution of Eq. (33.5) lying between 0 and 1 for a range of values of a = ( K , - K , ) / K , and b = ( K , - 1)/~,. The angular frequency of the reflectivity maximum is given in terms of the tabulated value z, by wmsx = o T ( 1 az,)”2, where W T is the transverse optical mode angular frequency.
+
a
b: 0 . 0
0.25
0.5
0.75
1 .o
0.0 0.5
0.500 0.449 0.414 0.366 0.232
0.431 0.385 0.353 0.310 0.193
0.360 0.323 0.297 0.262 0.165
0.298 0.269 0.251 0.224 0.144
0.250 0.230 0.215 0.194 0.128
.o
2.0 10.0
accurately. This is because the reflectivity near the peak is rather slowly varying, and can easily be changed by fairly small deviations from the simple model we have used, based on the dielectric constant (31.11) with negligible damping. Reflectivity curves showing’ one or more subsidiary maxima between W T and W L are not uncommon. The determination of the reflectivity minimum is more straightforward. When the imaginary part of the dielectric constant is small, the minimum is close to the point K~ = 1. If we define w0 by K ~ ( w O ) = 1, then the reflectivity minimum occurs for Wmin
M
WO
+ [(
KZ/K~’)
(4.1’
- (K ~ ’ / K ’ )
(33.7a)
}]w=wg,
where the prime indicates the derivative with respect to the reflectivity at the minimum is
w.
The value of (33.7b)
Rmin N [ K Z ( W O ) ] ’ / ~ ~ .
If we use Eq. (31.11) for the dielectric constant, we find:
Rmin
(Ks
- 1) ( K , - l)ay2/[16(K~-
K,)’WT2].
(33.813)
EFFECTS 34. FREE-CARRIER In most ionic crystals the concentration of free electrons and holes is negligibly small. But in semiconductors the free carrier concentration can be appreciable. We shall see that this modifies our conclusions about longitudinal lattice modes, but not about the transverse modes.
362
FRASK S T E H S
In a system that does not contain externally introduced charges, Maxwell’s equation [Eq. ( 4 . l b ) l can be written K(w)k.E = 0, for which the solutions are E = 0, K ( W ) = 0, or k - E = 0. The first of these is uninteresting, since it follows from Eq. (4.1) that all the fields will be zero. In the second case, Eq. (4.1) shows that H = 0, and k /I El which is a longitudinal mode. The third case is a transverse mode. If we neglect damping, the equation of motion for the lattice vibrations, (31.7b), shows that the modes at the angular frequency W T will not produce an electric field. Thus the transverse optical modes do not excite the free carriers, and are unaffected by their presence. The longitudinal modes can exist without a driving force at frequencies for which K ( O ) = 0, a condition which can be exactly satisfied only if there is no damping. Making use of the discussion of effective fields a t the beginning of this chapter, in particular of Eq. (24.6), we find that the polarization can be expressed as the sum of the contributions of the free carriers] the ions, and the bound electrons:
P
=
NeqX
+ N&’s +
(K,
-
l)~cE,
(34.1)
where N , and N I are the concentrations of free carriers and of ions, respectively, and the other symbols are defined in Eqs. (24.3) and (31.3). If we assume that there is only one group of free carriers] characterized by effective mass m* and damping constant ye, we find from Eqs. (24.6), (25.1), and (31.11) that ;(a)
- w * ~ ( w ?+ iysw)-’
= K,[1
+ ( w 2 - w$) (w$
- w2 - i7rw)4], (34.2)
where 71 is the damping constant for the lattice motion. If both damping constants vanish we find that the dielectric constant will have 2 zeros, given by : w2
= +(w,~
+
WL.”)
* [t(wp2 +
WL*)2
- WP*WT2-J1’2.
(34.3)
These two longitudinal modes of the system will in general consist both of ionic and of free-carrier motion. Equation (34.3) was derived by Yokota’j directly from the equations of motion. The longitudinal mode frequencies are easily understood in the limiting cases when up << w L or w,, >> w L , If the free-carrier effects are minor, then the high-fiequency mode will be close to W L , while the low-frequency mode will be a t w p ( w T / w L ) = w ~ ( K , / K ~ ) ~ / ~This , reduction in the free-carrier plasma frequency arises because the low-frequency dielectric constant has , of the value K, which it would have in the absence of the value K ~ instead 75
I. Yokota, J. Phys. SOC.Japan 16, 2075
(1961).
ELEMENTARY
owrcfi
363
PROPERTIES OF SOLIDS
the ions. If wP >> WL, the free-carrier plasma frequency is affected very little by the lattice, as we would expect. But the lattice mode now occurs at WT, instead of at WL. This happens because the free-carrier dielectric constant is large and negative at low frequencies. Thus the dielectric constant can vanish only if the ionic contribution is large and positive, which occurs only in the region of maximum dispersion near up. This zero in the dielectric constant will probably not occur when damping effects are considered. The effect of the lattice vibrations on the dielectric constant must be taken into account if optical properties are used to determine the freecarrier effective mass. The necessary corrections for a general case are best obtained from Eq. (34.2), but a simpler expression is possible at short wavelengths. We find :
- cA2, R ( w ) = R,[1 - 2Km-'12(K, - 1)-'CA2], K(W)
=
>> up,W L ) (W > > wP, W L )
K,
C [P-~] = 8.97 X 10-22N [~rn-~](rn/rn*)
(34.4a)
(W
+ (K#
K,)
(34.4b)
(AT [P])-~,
(34.4c) where R, is the short wavelength reflectivity as in Eq. (26.6), and wavelengths are expressed in microns. Thus the effective mass calculated without considering the lattice vibrations will be smaller than the actual effective mass. The effect of the lattice vibrations on the position of the reflectivity minimum is easily calculated if up>> W L and damping is neglected. We find: Wmin
=
WpK,I/'(
K,
- 1)-'"[1
+$
(Ka
-
K,)
Km-1W~2Wp-2]l
(Up
>> W L ) (34.5)
which is to be compared with Eq. (28.2a). When the carriers cause only a small perturbation of the longitudinal lattice mode, we find that in the absence of damping
which is to be compared with Eq. (33.8a). The effect of screening on the interaction of free carriers and lattice vibrations has been discussed by Ehrenrei~h.7~ 70
H. Ehrenreich, Phys. Chem. Solids 8, 130 (1959).
364
FRANK STERN
V. Quantum Theory of Absorption and Emission
35. ABSORPTION The interaction of a system of charged particles with electromagnetic radiation is described in nonrelativistic quantum mechanics by the Hamiltonian”: i-1
a-1
where 4 and A are the scalar and vector potentials of the applied field, p i and mi are the charge and mass of the ith particle and the ( p , . - q , A , ) /mi are its velocity components in rectangular coordinates. ‘Uois the potential energy of the system when the external fields are removed. The relation between E and B and the potentials is given in Eq. (7.3a). We shall consider here a single plane wave with 4 = 0 and with vector potential
A = Re
[A, exp (z1.r - i d ) ] = Re A,
(35.2)
and shall not restrict ourselves to the usual case of transverse fields, because we want to apply the results to the longitudinal case in Part VII. We saw in Eqs. (7.8b) and (7.10) that we can always choose the gauge 4 = 0 even if the field is not purely transverse. From Eq. ( 3 5 . 2 ) and the relation p = -ifiV, we find: , Asp
+ p * A = 2A. (p + ihk) = 2A.p+.
(35.3)
The additional term 6Ak in p+ will have no effect for transverse fields, for which k e i = 0, but must be included when we discuss longitudinal fields. The interaction between the field (35.2) and the charges is thus XI=
0=
0 exp
- iwt)
+ Q* exp ( i w t ),
- C (q;/2mt)A0 exp (z1.r) -p,+.
(35.4a) (35.4b)
t=I
To obtain &* from &, replace io by i o * , k by -k, and p,+ by pi - ink. One of these terms will induce upward transitions, and the other will induce downward transitions at the same rate?* Considering only one of these terms, we find that if the system was initially in the state Qjl the 77
78
J. H. Van Vleck, “The Theory of Electric and Magnetic Susceptibilities,” p. 20. Ox-
ford Univ. Press, London and Sew York, 1932. L. I. Schiff, “Quantum Mechanics,” 2nd ed., p, 252. McGraw-Hill, New York, 1955.
365
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
probability of finding it in state 9, a time t later is given by78a
1 Cj 1 qj) i2 sin2 [ + ( u m j - ~ ) t ] / [ f i ( ~ m j - ~ ) ] 2 , (35.5) where fiu,j = Em - G j is the energy difference between the upper and lower states. If the number of states for which Em - G j lies between liw 1
(qm
+
and fiu dG is transition rate
p(&)
d&, an integration of (35.5) over these states gives a = (21/fi)
1
(qm
i 0 i qj) 1 2 p ( & ) -
(35.6)
If the initial and final states are discrete, then a constant transition rate will be obtained only if the perturbation X I contains a band of frequencies which includes u m j , 7 8 The rate at which the interaction Hamiltonian (35.4)induces transitions is thus given by TO
= [ire2
1 Lo j2/(2m2h)]1 ( q m I iio exp
N
(zk-r)
C pi+ I \kj)
(35.7)
12p(E),
i-1
where we have taken the electronic case (qi/mi) = -e/m, and have introduced a unit polarization vector &, Induced transitions will take place both from j to m and from m to j . The net rate of transitions from state m to state j is given by rind
=
To{ f ( E m )
- f ( E j ) 3 - f ( E j ) C1 - f ( E m ) 11 - f(M1,
[1
(35.8) roc f ( E m ) where f(&) is the probability that the state of energy & is occupied. If the lower state, with energy E j is more likely to be occupied, as in thermal equilibrium, then our sign convention for r i n d gives a negative net rate of induced transitions, and corresponds to absorption. If the upper state can be populated more than the lower state, induced emission will result, as in masers, and r i n d will be positive. To this point we have talked about transitions of the entire system. In most cases, however, a one-electron approximation is adequate, and we can describe the transition as a change in the state of a single electron. If the total wave function \k can be taken to be a determinant of oneelectron wave functions $ i , then it is easily shown7gthat the induced (or stimulated) transition rate is given by =
rind
m ‘*a 79
= Lire2
I
=
I Po exp
($m
1 2 / / ( 2 m 2 f1 im )~( ~ P ( E ) C J F ( E~ )f
(zknr) ‘p+ I
$j)j
(35.9a)
( ~ j ) ~ ,
(35,gb)
See Eq. 29.17 in Schiff.78 E. U. Condon and G. H. Shortley, “The Theory of Atomic Spectra,” p , 171. Cambridge Univ. Press, London and New York, 1953.
366
FRASK STERN
where p ( & ) now refers to the density of one-electron states per unit energy, and &j and &, are now one-electron energies. Our considerations here assume that k is real, although we shall later apply the results occasionally to cases in which k is complex. When the imaginary part of f is small we can factor out exp ( -2kz.r) in Eq. (35.9a), which shows that the transition rate is proportional to the local field intensity. I n such cases the matrix element (35.9b) should involve only k,, the real part of G. The absorption coefficient CY is the power absorbed per unit volume divided by the incident flux. The power absorption per unit volume is given by fiurind provided p ( & ) in Eq. (33.9) is referred to unit volume. The incident flux is
s=
3 Re [(-id*) x (& x i ) / p ] = ( w / 2 p ) Re [ ( i * . i ) f- (6 * .G)i] = [nwz/(2cp)] I I2k1, (35.10) where n is the index of refraction, f - i *is assumed to be zero (i.e., we now Re (E* x H )
=
restrict our attention to homogeneous transverse waves), and Eq. (4.4) was used. fl is a unit vector in the direction of propagation. The absorption coefficient for light polarized along Po is therefore given by a(w) =
[ncpe2/(nwm2)] 1311 j2p(&)[f(&j)
- f(Em)].
(35.11a)
We can also express the power dissipation per unit volume, AwT,,~, as
$wezE.fi*= 1* w3 E ~ K z ~ - ~using * , Eqs. (3.5) and (35.2), and noting that 9 = 0. Then, from Eq, (35.9), we find: KZ
=
[ne2/(m2eow2)1I3n I2p(E)[f(&,) - f(Em)].
(35.llb)
One-electron wave functions in a periodic lattice have the Bloch form u,(r) exp (zk,-r) where k, is the wave vector of the state, and u,(r) has the periodicity of the lattice. The matrix element 311, defined in Eq. (35.9b), will therefore vanish unless k, = k, k. For visible or infrared radiation the wave vector k is negligibly small compared to the size of the Brillouin zone (the ratio is of the order of the lattice constant divided by the wavelength), and can be ignored. Thus direct radiative transitions in a solid are described by vertical lines on an & versus k diagram. For cubic crystals the absorption coefficient will not depend on the direction of the polarization vector lo, and can be written J/)
=
+
a(u) =
Cnwez//3nwmZ>11 (hI P
I
+j)
I2p(&)Cf(E3)
- f(Em)l.
(35.12)
This equation can be transformed from mks to Gaussian units by replacing p by ~TK,/C*; a t optical frequencies K, = 1. If several processes contribute to the absorption, we must sum Eq. (35.12) over them.
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
3 87
We have assumed throughout this section that the field acting at a point is just the macroscopic field at that point. We saw in Section 24 that this is valid for free or nearly free electrons, but not for localized charges. For the localixed case we must include corrections to Eq. (35.1), the socalled local field corrections.80We shall not attempt to incorporate these in the results given here, but shall discuss them in Section 39 in connection with the oscillator strengths of F centers in ionic crystals. 36. BLACKBODY RADIATION
AND SPONTANEOUS EMISSION
A medium in thermal equilibrium at temperature T contains blackbody radiation which will induce transitions between states of the system, Thermal equilibrium is maintained by spontaneous emission from the excited states of the system into lower states. We can use this fact to calculate the rate of spontaneous emission.80*z80b In a sample of volume V the number of plane waves modes having wave vectors in an element dk,dk,dk, of k space is (2?r)-*Vdk&k,dk.,81 and does not depend on the shape of the sample or on the boundary conditions82 provided the dimensions are large compared to the wavelength. Thus for electromagnetic radiation, if we take k = 1 k I = nw/c, from Eq. (4.4), and note that two independent directions of polarization are possible for each wave vector, we find the density of modes per unit volume between w and w dw to be G ( w ) dw, where
+
G(w)
=
(27r)-*(47rk2)(dlc/dw) (2)
=
n2w2/(?r2c2~,).
(36.1)
We have introduced the group velocity v, = dw/dk, and consider only frequency regions where there are no strong absorption peaks with their resulting anomalous dispersion. In a system in thermal equilibrium at temperature T, the average See, for example, R. L. Adler, Phys. Rev. 126, 413 (1962); N. Wiser, ibid. 129, 62 (1963). 80s A. Einstein, Physik. 2.18, 121 (1917). gob Our treatment assumes that the electronic states involved in the optical transitions interact only with the radiation field. In real solids there will be interactions with lattice vibrations which can change the state of the system between absorption and emission, and which can substantially change the relation of emission and absorption rates from the one described here. This is discussed by W. B. Fowler and D. L. Dexter, Phys. Rev. 128, 2154 (1962). I am indebted to Dr. Dexter for sending me a preprint of this pa.per. 8 1 N. F. Mott and H. Jones, “The Theory of the Properties of Metals and Alloys,” p. 56. Oxford Univ. Press, London and New York, 1936. 82 See p. 45 and Appendix IV in Born and Huang.46 80
368
FRANK STERN
energy of a mode with angular frequency w ism
-
E(w) = fiw [exp (fiw/KT)
- l]-I,
(36.2)
where K is Boltzmann’s constant. Thus the energy density of blackbody dw is u(w)dw, with radiation in a dielectric in the range w to w
+
u (w ) = [n%w8/ (T ~ C ~ U ,][exp )
(fiw/K T )
- 134.
(36.3)
The velocity with which energy flows in a dielectric is the group velocity, as was shown in Eq. (6.12). We assume that the radiation is isotropic. Then the time-averaged flux of radiation with wave vector k lying in an element of solid angle dn, with polarization vector do lying in an angular interval dc9 in a plane perpendicular to k, and with angular frequency in a range dw, is
ISI
= u(w)v,(dn/4r) (dO/2r) dw = [n2fiws/(&2)][exp (fiw/KT)
- 1]-1(dn/4r)
(d0/2?r) dw.
(36.4)
We can use Eqs. (36.4) and (35.10) to eliminate the vector potential from (35.9). We find that the transition rate induced by blackbody radiation is: rM
=
ne2pw
-1 m 1 2 p ( ~ ) [ f ( ~ m ) Tm2c
- f(~j)][exp (fiw/KT) - I]-’ X (dn/4r) (dt9/27r) dw,
(36.5)
where 3n is the matrix element which appears in Eq. (35.9). For a system in thermal equilibrium a t temperature T the probability that a one-electron state of energy E i is occupied is the familiar expression for Fermi-Dirac statisticsaan: fo(Ei) = {exp [(Ei
where Ej
&F
- &F)/KT] + l}-l,
(36.6a)
is the Fermi energy. If we consider two states such that Em =
+ fiw, we find fo(Ej)
- fo(&m)
=
fo(L>[1
- fo(Ej)][exp
(fiw/KT)
- 11.
(36.613)
In thermal equilibrium the induced rate rbb is negative, corresponding to absorption, and the induced transitions from E j to Em must be balanced by spontaneous transitions from E m to Ej. From Eqs. (36.5) and (36.6b) we See, for example, C. Kittel, “Introduction to Solid State Physics,” 2nd ed., p. 56. Wiley, New York, 1956. 888 T his expression must be modified for impurity levels in semiconductors. See p. 87 in Smith.“ 88
369
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
find that the rate of spontaneous emission from states is rspont= [ne2w/(.rrm2c)] I 3n2 I p ( G j ) f ( G m ) [ l
$m
to
and adjacent
- f(E,)](dfl/4~)(dO/2~)dw. (36.7)
This result was derived for a system in thermal equilibrium, for which we should put fofor f. But the result (36.7) is valid even when the system is not in thermal equilibrium, since f(&,) [l - f(&,)]is the probability that the initial state is occupied and the final state empty, and the remainder of the expression depends only on the properties of the states themselves. I n a system not in thermal equilibrium, the radiation density may exceed the blackbody value, and the rate of stimulated emission will then exceed the rate given by Eq. (36..5), since we know from Eq. (35.7) that this rate is proportional to the energy density. If we let X stand for the number of photons in a radiation mode, we can combine the induced and spontaneous emission rates, and find:
(36.8) where f m stands for f(Gm).The spontaneous part of Eq. (36.8) is equal to Eq. (36.7)) and the stimulated part reduces to Eq. (36.5) if X has its thermal equilibrium value, [exp (fiw/KT) - 13-'. The spontaneous emission rate can be expressed in terms of the absorption coefficient. For a cubic crystal we can average Eq. (36.5) over all directions, and can eliminate the matrix element by using (35.12). We find that in thermal equilibrium: TBpont =
[nzwz~(w)/(n2c2)][exp (RwlKT) - 11-l dw.
(36.9)
If we integrate over all values of w we find that the total rate of spontaneous emission per unit volume is given by
where we introduced x = Rw/K T . The coefficient in brackets in Eq. (36.10) has the numerical value 2.53 X 1011T 3~ m sec-I - ~ = 6.83 X lo1* ( T/300)8 cm-3 sec-l if LY is given in cm-I. Equation (36.10) is equivalent to the expression given by Van Roosbroeck and Shockley in their calculation of the radiative recombination of electron-hole pairs in semiconductors.& 84
W. van Roosbroeck and W. Shockley, Phys. Rev. 94, 1558 (1954).
370
FRANK S T E R N
If the absorption coefficient near the absorption edge is given by a(.) = A ( A w - & G ) P , and the energy gap &G is >>K T , then the total emission rate is given approximately by (Ro=
[ K aT3/(7r2Aac2)]n2A( KT )p [ (&G/KT )
+ p I 2 exp ( -&a/KT )I? ( p + l), (36.11)
and equals the rate of radiative recombination of electron-hole pairs in thermal equilibrium. Our results for spontaneous emission can easily be extended to the more familiar atomic case. The density of states can be written p ( & ) = k l p ( w ) = NA-'G(w - w m j ) , where N is the number of atoms per unit volume and 6 is the Dirac delta function. The matrix element can be written in more familiar form if we use the dipole approximation, exp (zkar) = 1, and the resultss
( h n I P I $i)
=
imUmj(#m
I r 1 rClj)
1
(36.12)
where #j and $m are the one-electron wave functions of the initial and final states. Then the spontaneous emission rate per atom between states m and j is given by substituting these expressions for the density of states and for the matrix element in Eq. (36.7) and integrating over solid angle d0, over polarization directions do, and over a range of frequencies near w j . We find TRpont
1 (h1 r
= CneZwa/(3nch)]
$j)
;2.ff(Sm)[1- f(&j)].
(36.13)
This agrees with the usual result86 if we put n = 1 and replace 1 by 47r/c2. An interesting incidental point related to blackbody radiation inside a dielectric is that the energy flux given in Eq. (36.4) is greater by a factor n2 than the corresponding flux in a vacuum. One might a t first suspect that thermal equilibrium could not be maintained near an interface between a dielectric and a vacuum in an isothermal enclosure. But the necessary balance in the energy flow across the interface is provided by the ordinary laws of reflection,12which lead to Eq. ( 1 5 . 5 ) . 37. OPTICALABSORPTION FOR
A
SIMPLEMODEL
The optical absorption is easily found for direct transitions from Eq.
(36.11), We consider transitions between conduction and valence bands
whose band edges occur a t k = 0, snd assume that &, and & j are functions only of k = I k 1. Then the density of states per unit volume and per ed 88
See the equation preceding (35.20) in Schiff .7* See Eq. (36.22) in Sohiff
ELEMESTARY OPTICAL PROPERTIES OF SOLIDS
371
unit energy whose wave vectors have magnitudes near k is given by p(&) =
(k?/T*) 1 (d€,/dF;) - (d&,/dk)
1-1.
(37.1)
The derivation of Eq. (37.1) is closely analogous to that of (36.1), except that the factor 2 here refers to spin, and the wave vector k in (37.1) is determined by the relation &,(k) = &,(k) Aw. If we introduce conduction band and valence band effective masses mc and m,,and write :
+
&,(k) = &,a
+ (fiW/2mc),
(37.2)
we find that the density of states is:
(37.3b) Iiote that if m, >> m,,m, FZ m,. The model Tve shall use to calculate the band structure and matrix elements needed to calculate the optical absorption is based on the assumption that the conduction band is s-like, and the valence band is p-like. ITe shall use the k - p perturbation methods7to find the energy dependence of these bands near k = 0, and shall neglect interactions with all other bands. The Schrodinger equation for a Bloch wave function #k(r) = uk(r) exp (zk-r) can be written:
+
[- ( f i 2 / 2 m ) v 2 ih/m)k-p
+ (fi2k2/2m)+ V - Ek]uk(r) = 0,
(37.h)
n-here p is the momentum operator - i f i V ) and Ti is the periodic potential energy in the crystal. When k = 0 we can take the solutions to be
( H o - &)u8(r) = 0,
(Ho- &)up(r) = 0,
(37.4b)
where u p(r) is one of the three functions uz(r), u,(r), u.(r), whose symmetry properties are much like those of atomic p functions with angular dependence ( z / T ), (y/r) and ( z / T ) , respectively. If we use these functions 8’
E. 0 . Kane, Phys. C h e m Solids 1, 249 (1957).
372
FRANK STERN
as a basis and neglect the free electron energy h2k2/2m,the matrix elements of the bracketed expression in (37.4a) are:
-
E,
k,P
E
k,P
-
&,
&
kuP
k,P
0
0 0
p
0
&,-E
kJ’
0
0
I
0 = (% I Pu
kU
(37.5a)
E, -
where
P
=
(fi/m) (7-b I P z
I Uz)
I Ps I
(37.5b)
=
0)
(37.6a)
+
k2P2]1’2,
= (Us
The secular equation for the eigenvalues is:
-
(G,
- C)(EP -
&)2[(&,
whose solutions are: & = E,,
E,,
$(E8
W &,,
&,,
&,
-
+ Ep)
E)
- k2Pq
f {[&(C8 -
k2P2/&G,
&,
&,)I2
fk2P2/&~.
(37.6b)
The four bands in (37.613) are the two “heavy hole” bands, the “light hole” band, and the conduction band. If we compare these with (37.2), we find that the effective masses for the last two are I
m,
=
m,
=
h2&G/2P2.
(37.6~)
The results are quite simple, because we have neglected the free electron contribution to the energy, and the interactions with other bands which lead to the warped energy surfaces found in real and which determine the effective mass of the heavy hole band. Within the accuracy of our approximation, ( 3 7 . 6 ~ is ) consistent with the general relation=: (m/mj)zz =
+ (2/m) C
1
z
(&i
-
&*I-’ 1
1
(+i ~z
/+j)
1.’
(37.6d)
We now have all the information necessary to calculate the absorption using Eq. (35.11). Near k = 0 the matrix element between the conduction band and each of the three valence bands, when averaged over all polarizations of the incident light, is given by
j 3n
la
=
m2P2/3fi2= m2&~/6m,.
(37.7)
To take spin-orbit coupling into account very crudely, we assume that 88
A. H. Wilson, “The Theory of Metals,” 2nd ed., p. 47. Cambridge Univ. Press, London and New York, 1953.
373
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
one of the valence bands is well below the other two, and that near the absorption edge only transitions between the conduction band and the light hole band and one heavy hole band enter. We then find:
+ 2a12)~~~2(12rrnli2)-1m,112(liw -
(37.8a)
3.38 X lo5 n-1(m,/m)1~2(&~/~w) (Aw -
(37.8b)
a(w) = (1
a[crn-l]
=
&G)~/~(&G/~~u),
where n is the index of refraction. The energy in Eq. (37.8b) is expressed in electron volts. We have assumed that f(&i)- f(Ern) = 1, i.e., that the valence band states are occupied and the conduction band states are empty, which applies to pure materials a t low temperature. We have taken the material to be non-magnetic; to write Eq. (37.8a) in Gaussian units, replace P Oby 47r/c2. Equation (37.8) gives a fairly good fit to the optical absorption of the 111-V intermetallic semiconductors InAslSa InSb18' and room-temperature GaAsssbfor photon energies near their absorption edges, even though a very crude model was used to derive it. A detailed theory of the band structure and optical absorption of the 111-V semiconductors has been given by Kane.87 I n many semiconductors the bottom of the conduction band and the top of the valence band are not a t the same point in the Brillouin zone, and the treatment given here, based on Eq. (35.11), is not applicable. Another interaction mechanism must be present to provide the crystal momentum required for a transition from one of these states to the other. Lattice vibrations can provide this momentum, but other scattering mechanisms may sometimes be involved. The theory of this type of transition has been reviewed by M ~ L e a n For . ~ ~a~participating phonon group with energy near KO the absorption coefficient contains terms proportional to (Aw - &a f KO)2,where &G is the indirect energy gap. The positive sign indicates phonon absorption, which has a temperature dependence given by A( T) = [exp (el?") - l]-l, while the negative sign indicates phonon 1 = emission, which has a temperature dependence given by A( T) [l - exp (-O/T)]-l. An important effect, neglected so far, is the interaction between the electron and the hole which are produced when optical absorption causes a transition from a valence band to a conduction band. This interaction can lead to bound states of the electron-hole pair, called excitons, which give characteristic absorption spectra at low temperature^.^^ They also
+
asa 88b
J. R. Dixon, Proc. Intern. Conf. Semicond. Phys., Prague, 1960, p. 366 (1961).
I. Kudman and T. Seidel, J . A p p l . Phys. 33, 771 (1962).
See, for example, E. F. Gross, Usp. Fiz. Nuuk 78, 433 (1962);see Soviet Phys.-Usp. (English Trunsl.) 6 , 195 (1962).
374
F R A S K STERN
affect the absorption coefficient near the absorption edge, causing an abrupt increase a t the edge, followed by a gradual rise which merges a t energies far from the edge into the absorption coefficient that would be calculated without the interaction, using Eq. (35.11). The theory of this effect was given by Elliottz5and has been reviewed by & I ~ L e a n . ~ ~ ~
38. FREE-CARRIER ABSORPTION The classical theory of free-carrier absorption, for frequencies such that (27.5) and shows that the absorption is proportional to the square of the wavelength and to the reciprocal of the mobility. Physically one can think of the scattering processes as removing energy from the electron motion and transferring it to other modes of the system. This energy must be supplied by the electromagnetic wave that is maintaining the electronic motion, and the wave is attenuated. A quantum-mechanical description of free-carrier absorption requires that we consider not only the interaction (35.4) between the radiation and the carriers, but also the interaction between the charge carriers and the lattice vibrations or scattering centers. The transition from the initial state to the final state proceeds by a two-step process through a virtual intermediate state. A good description of the quantum theory of freecarrier absorption has recently been given by Dumke.go He finds that for photon energies less than K T the quantum-mechanical result reduces to the classical value, and that the leading correction term is of second order in (&w/KT).For degenerate systems K T is replaced by the Fermi energy E F . Dumke shows that intermediate states in other bands can be neglected provided the photon energy is small compared with the energy separation of the interacting states. The most detailed quantum-mechanical free-carrier absorption calculations have been made for n-type g e r m a n i ~ m , ~ and ~ J ' ~have taken into account the band structure and deformation potentials of this material. The results are in good agreement with experiment, though not for all wavelength and temperature ranges. Expressions for free-carrier absorption when the principal scattering mechanisms are optical mode scattering or impurity scattering have been given,93 and predict a wavelength dependence stronger than X2 a t short wavelengths. Free-carrier processes need not always involve scattering within a WT
>> 1, is given by Eq.
W. P. Dumke, Phys. Rev. 124, 1813 (1961). H. J. G. Meyer, Phys. Rev. 112, 298 (1958). 92 R. Rosenberg and M. Lax, Phys. Rev. 112, 843 (1958). 93 S. Visvanathan, Phys. Rev. 120, 376, 379 (1960). 90
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
375
single band. A calculation of the absorption that arises because of scattering between nonequivalent valleys in a conduction or valence band has been made by Risken and Meyerg4for the case of nondegenerate statistics. One of the early problems of free-carrier absorption was the unusual absorption spectrum of p-type g e r m a n i ~ m The . ~ ~ absorption was found to be proportional to the carrier concentration, but had considerable structure in no way resembling a X2 behavior. The explanation of the effect was given by Kahnlg6and in more detail by Kane,g7in terms of direct interband transitions between different valence bands, and gives good agreement with the observed absorption. 39. OSCILLATOR STREXGTHS
It is often useful to discuss the strength of the transition between two atomic levels in terms of the oscillator strength, defined as fji
=
2
1
I P,
(*j
~
qi)
12/(mfiuji),
(39.1)
where p , is the 5 component of the total momentum of the electrons, and qi are the wave functions of two atomic states, and fiw,i is the difference of their energies. It can be shown on quite general that C fji = 2, (39.2)
\kj
j
where 2 is the total number of electrons on the atom (or ion). The sum is over all possible final states j , and is independent of the initial state i. Because of the presence of coji in the denominator of Eq. (39.1), the oscillator strength is positive for upward transitions, in which the final state j has higher energy, and is negative for downward transitions. The rigorous sum rule (39.2) , often called the Thomas-Reiche-Kuhn sum rule, applies only to transitions of the atom as a whole. It applies exactly to one-electron transitions only for hydrogen or for ions with only one electron. I n solids, however, it is often a good approximation to think of the transitions between states of the whole system as involving only a change in a single electronic stat,e. The oscillator strength can then be approximated by fji
where $i and O4 O5 O6 O7
#j
=
2
I
(#j
I pz I
#i)
12/(mfiuji),
(39.3)
are one-electron wave functions, and p, is the x component
H. Risken and H. J. G. Meyer, Phys. Rev. 123, 416 (1961). W. Kaiser, R. J. Collins, and H. Y. Fan. Phys. Rev. 91, 1380 (1953). A. H. Kahn, Phys. Rev. 97, 1647 (1955). E. 0. Kane, Phys. Chem. Solids 1, 83 (1956).
376
FRANK STERN
of the momentum of a single electron, Thus Eq. (39.3) bears the same relation to (39.1) that (35.9) does to (35.7). The simplified expressions are somewhat easier to deal with, but one must bear in mind that they are approximations to the more exact relations (39.1) and (35.7). One important difference between the matrix element in Eq. (39.3) and the matrix element in (35.9) is the absence in (39.3) of the factor exp (zk-r). Neglect of this factor is called the dipole approximation in atomic transitions, since the neglected terms in the expansion of exp (zlr-r) lead to quadrupole and higher multipole transitions. For solids, taking exp (zk-r) m 1 corresponds to the long wavelength limit. When the states of the system lie in a continuum, we define fji(w) dw to be the sum of all the oscillator strengths (per unit volume of the crystal) dw. The initial condition of the for which wj; lies in the range w to w system is specified by the function f(&), which gives the probability that a one-electron state of energy & is occupied. The sum of all the oscillator strengths in a given angular frequency range is then found by multiplying the oscillator strength for a single transition by p ( w ) , the density of final states per unit angular frequency interval, and by f(&J - f ( & j ) , which is the probability that the initial state is occupied and the final state is empty, as in Eq. (35.8). If we note that p(w) = A p ( & ) , we find that
+
fii(w)
=
[2/(mwji)l
I (+i I P. I +J 12~(&)Cf(&i) - f(&i)I-
(39.4)
The oscillator strength is positive, corresponding to absorption, if the state j has higher energy than the state i and is less populated. If we specify that the state j in Eq: (39.4) shall always be the state of higher energy, we can remove the subscripts on wji, and can deal only with positive angular frequencies. For solids the subscripts on fji(w) can be thought to label the various bands that contribute to transitions of angular frequency w. Where more than one pair of bands contributes, the total oscillator strength, ftot ( w ) , is the sum over all contributing f j i ( w ) . Thus the imaginary part of the dielectric constant is given in terms of ftot by: KZ ( w )
= [?re2/
( 2 m w d I ftot (a) ,
(39.5)
which is obtained from Eqs. (35.11b) and (39.4) if we make the approximationexp (zk-r) m 1. The sum rule (23.3) for the imaginary part of the dielectric constant takes a very simple form if we use Eq. (39.5) to express tcZ(w) in terms of the oscillator strength. We find: (39.6)
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
377
where N is the total number of electrons per unit volume. This is the natural analog of the sum rule (39.2), since our oscillator strength was defined for a unit volume. The integrated oscillator strength is the total number of electrons, as in the atomic case. Thus we have established that the sum rule (23.2) and the oscillator strength sum rule (39.2) are equivalent for a simple case, using the long-wavelength approximation. The long-wavelength restriction is imposed so that the amplitude of the field should be roughly constant over the volume which contributes to the matrix element for the transition. A strict interpretation would therefore require that the wavelength be comparable to atomic dimensions, or larger, which limits the validity of the dipole approximation to photon energies of the order of lo3-lo4ev or less. An example will show that this is in an unnecessarily severe restriction. Consider a photon whose energy is just below 1.2 X lo6ev, the threshold energy for removing an electron from the 1s shell of lead.98 The corresponding wavelength is 0.11 A, which is much smaller than the size of a lead atom or ion. But such a high-energy photon will not interact strongly with the outer electrons, since the photoelectric absorption cross section Thus the main interfa.llsoff approximately as u3beyond the action of the photon in our example will be with the L shell, the shell with principal quantum number n = 2. As a rough approximation for the radius of this shell we take 2ao/(Z - s), where a. is the Bohr radius and s is a screening parameter. For the L shell of lead we have 2 = 92 and s M 20g9 and the radius of the L shell is m0.015 A, which is much smaller than the wavelength of the photon in our example. Thus the long-wavelength approximation holds for the dominant interaction with the atom. The example we have chosen is actually a conservative one, since a photon of slightly higher energy would interact mainly with the 1s shell, whose radius is smaller, while a photon of lower energy would have a longer wavelength. The argument we have given, which has also been given by Jarnesl1Oo is qualitative, but suggests that the dipole approximation is a good approximation up to quite high photon energies, at least insofar as it concerns the wavelength dependence of the dominant matrix elements for the interaction of photons with atoms and crystals. Before concluding this section we will consider the absorption associated with F centers in alkali halides, since this is conveniently discussed in terms of oscillator strengths. This absorption is associated with excitation of an electron bound to a negative ion vacancy, and leads to a well-defined See Appendix VI in Compton and Allison.% See p. 626 in Compton and Allison.*s 100 R. W. James, “The Optical Principles of the Diffraction of X-Rays,” p. 141. Bell, London, 1948.
9s
99
378
FRANK STERN
absorption band in the visible portion of the spectrum. We find, on using Eqs. (35.1la) and (39.4), that the integrated absorption can be represented by :
J a (w)
l~fii,
dw = ~re2//2nmeoc)
(39.7)
where n is the index of refraction of the host crystal at the frequency of the absorption peak, N is the number of contributing defects per unit volume, and f j i is the integrated oscillator strength per defect for the transition that leads to the absorption peak. We have assumed that K,,, = 1. In deriving Eq. (39.7) we assumed that the electric field acting at the defect site is the average field in the medium, since this is essentially what was done in deriving (35.11). But the discussion at the beginning of Part IV showed that for the case of bound charges the effective field at a lattice site in a cubic material is given by
F
=
E
+ P/(3eo) = i(n2 + 2 ) E ,
(39.8)
where we have assumed that K M na. Thus, since the field enters quadratically into the absorption, the integrated absorption of the F-center line can be expressed, on considering the local field correction, in the form:
which, if CY is expressed in em-', N in cm-', and fiw in ev, gives:
/
a(w)
+ 2 ) 2 N [cm-*]f&~
d(fio)[cm-lev] = 1.22 X 10-17(n2
(39.913)
Equation (39.9) can be used to determine the concentration of F centers if the oscillator strength is known, or to find the oscillator strength if the defect concentration is known. The oscillator strength is found to be of the order of unity for F centers in the alkali halides that have been studied. The paper by Doyle1'' should be consulted for experimental details, and for reference to earlier experimental work. A number of authors have studied the validity of Eq. (39.9) as applied to defects in ionic crystals and have given corrections to this equation. References to several theoretical treatments can be found in Doyle's paper.'O1 101
W. T.Doyle, Phys. Rev. 111, 1072 (1958).
379
ELEMESTARY OPTICAL PROPERTIES OF SOLIDS
VI. Wavelength-Dependent Dielectric Constant of a Free-Electron Gas
40. INTRODUCTIOX The wavelength-dependent dielectric constant introduced in Section 7 provides a very powerful tool for discussing nonlocal effects in solids. An example of such an effect is the response of carriers to an external field which varies rapidly in a distance comparable to the mean free path I of the carriers. This mean free path is related to the electronic damping constant y of Part IV by: 1 = B/y, where 0 is a mean velocity whose exact value depends on the degree of degeneracy of the carriers and on the scattering mechanism. A carrier a t a given point in the crystal will have a velocity determined by the electric fields a t points it has passed on its trajectory since the time of its last randomizing collision. If the field is varying rapidly in a mean free path, the effective field along the trajectory may not correspond a t all closely to the local field. It can be shown that undei these circumstances the current flowing is related to the electric field byIo2:
J(r, 1)
=
1
( 3 N e 2 / [ 1 6 n 2 m e ~ v ~ ] )t[t.E(r’, 1’)]t-4exp
(-t/O
d 3 t , (40.1)
for degenerate statistics, where V F is the velocity of carriers a t the Fermi surface, t = r - r’, 4 = F 1, and t’ = f - ( E / v F ) is a retarded time. We see that Eq. (40.1) describes a nonlocal responsf: to the applied field. We shall derive below and in Part VII the wavelength-dependent dielectric constant of a really free-electron gas, by which we mean a system in which the electronic damping constant y, associated with collisions with lattice defects or phonons, goes to zero, and the mean free path for such collisions becomes very long. Such a system is sometimes called a n ideal conductor. It provides the extreme case for nonlocal effects, since there is no scattering to help confine the “memory” of the electron to a small region in space. In this treatment we shall neglect the spin magnetic moment of the electron. I n this Part we restrict ourselves to transverse fields, with a vector potential ~
=
i ( k , u ) exp (zk-r - id),
k-i
=
0.
(40.2)
We defer discussion of the longitudinal case until Part VII. lo2
See, for example, D. C. Mattis and J. Bardeen, Phys. Rev. 111,412 (1958) ; J. L. Warren and R. A. Ferrell, ibid. 117, 1252 (1960).
380
FRkUK STERN
The results of this Part are most easily obtained if we work with the transverse conductivity, defined in Eq. (8.2) :
-
qtr
= Z
+ ik2(%
- l)/(pow),
(40.3)
where 8l = [%,(k, w ) ] - I is the reciprocal relative permeability, and Z = u - iwe is the ordinary electrical contribution to the conductivity. The additional term (40.4) AZ = i k 2 ( ( R - l ) / ( p o w ) represents the effect of the current V xM associated with the magnetization. We shall assume here that (R is real. Thus, for real k and w this term ir, i.e., we consider a wave is purely imaginary. If we replace w by w growing slowly with time with amplitude proportional to exp (-id) exp pi), and then let r + 0, there will also be a contribution to the real part of A;, This follows from the result that in an integration
+
+
ir)-1 = PW-1- i r S ( w ) , (40.5) lim ( W r+o where 6 ( w ) is the Dirac delta function, and P denotes the principal value. This result is easily proved by noting that the imaginary part of Eq. (40.5) is - r / ( w 2 r2),which vanishes for all w # 0 as F + 0, and whose is - ir. Thus we find that the integral over all values of w from - co to magnetic current V x M contributes
+
+
AUI =
[irk2((R
-
(40.6)
l)/p0]8(0)
to the real part of the cbnductivity. Note that Eq. (40.6) is an even function of w . We shall later use the integral of Ztr over positive values of w , and must therefore take only one-half of the integral of Eq. (40.6) over all w . This delta function contribution to the conductivity will be important when we check the sum rule. 41. CALCULATION OF
THE
CONDUCTIVITY
It is a straightforward matter to calculate the conductivity for a freeelectron gas in a transverse electromagnetic field if we use the relation gi(k, W )
= C T e 2 / / m 2 w )1
I (hI go ~
X P(5k-r)'P
I
+j)
- f ( L )1,
I2p(G) Cf(&j)
(41.1) which is based on Eq. (35.11). Note that for the transverse vector potential (40.2) used here, Zo*k= 0, so that & * p + = &-p. We consider a light wave of angular frequency w and wave vector k, propagating in the z direction and polarized in the z direction. The wave function of the free-electron state of wave vector kj, normalized in volume V , is V-1/2 exp (ikj.r), Thus
381
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
the matrix element in Eq. (41.1) is h2kjxz,We must conserve momentum and energy, which, for a transition from kj to k, means:
+ k, = k j 2 + 2mwh-‘.
k, k,?
=
(41.2)
kj
(41.3)
Squaring Eq. (41.2) and subtracting (41.3) gives (41.4)
where we have used the fact that k is in the z direction. The density of states per unit energy is easily found from Eq. (41.4) by noting that changing w by A-’ dE changes k,, by (m/fi*k)d&. Thus the density of states per unit volume and unit energy for which the final states are in a range dk,’dk,’ is found, by proceeding as in Eq. (36.1), to be: p (G) =
[m/ ( 4a3A2k)] dk,’dk,’.
(41.3)
The conductivity is then found from Eq. (41.1) to be:
al(k,w)
=
[e2/(4a2mwk)]
where E,
=
G, =
[ v z w ~ / ( ~~ ~QAw )] Gj
+ Aw,
/Im -m
(41.6)
k , ’ 2 [ f ~ ( G l) f o ( E m ) ] dk,’dk,’,
+ [Azk2/(8m)] + [(fiz/(2m)3(kz’* + ky’2)1 (41.7)
and fo is the occupation probability for thermal equilibrium, as given in Eq. (36.6). The integral in Eq. (41.6) can be simplified if we integrate over angles in the k,‘k,‘ plane, which allows us to replace the factor k,‘2dk,’dk,‘ by ahJ3dk’. Kext, following LindhardlEame introduce the dimensionless parameters 2 = k/(2ko), u = w/(kz!o), (41.8) where we define : vo = fiko/m, Go = fi2k02//(2m). (41.9)
For the time being we will consider ko and the associated energy arbitrary. Then we find that the conductivity becomes:
where Gj
=
+
&[(u - z ) ~ 4.P],
&,
=
+ z ) +~ 4d2].
&o[(u
&o
to be
(41.11)
382
FRANK STERN
To proceed further we must make simplifying assumptions about the statistical distribution lo, since the integration in Eq. (41.10) cannot be carried out simply for the case of general statistics. b7e first consider degenerate statistics, for which Go in Eq. (6.15) becomes the Fermi energy &F, and ko is taken to be k ~ the , wave vector a t the Fermi surface. For degenerate statistics we can use the relation kpS =
3a2N,
(41.12)
where N is the number of electrons per unit volume. The coefficient of the integral in Eq. (41.10) then becomes 6m0wp2/(w~), where up2=
Ne2/(meo).
(41.13)
For a nonzero contribution to Eq. (41.10) we must have the initial state inside the Fermi sphere, and the final state outside, i.e., (U
- 2)’
+4
~ <’ 1, ~
(U
+
2)’
+ 42” > 1.
(41.14)
From the integration limits which follow from Eq. (41.14) we find: a,(k, u) = [Ne2/(mu)](3au/4)(1 =
-
u2 - z 2 ) ,
[Ne2/(mw)][3n/(32~)][1 -
(U
-
u
2)*12,
+z
< 1,
I U - 2 1
=o
1u
- z
< I
I>
1.
< U + Z ,
(41.15)
To this we must add the delta-function contribution in Eq. (40.6). We have found the conductivity for the degenerate case by straightforward application of Eq. (35.11) to the free-electron gas. The nondegenerate case is even easier. In that case we have f t
where
kT
(&) = ~ T ~ ‘ ~ exp N ~( T &/K - ~ T ),
(41.16)
is defined by fi2k~’/(2m) = K T .
(41.17)
For the nondegenerate case we shall let the parameter ko in Eq. (41.8) equal k r , which means that so = K T . Thus, using the relationfo(&j) fo(&,) = f o ( E j ) [l - exp (-fiw/KT)], we find that Eq. (41.10) becomes:
~,(k, W)
= [Ne2/(mw)][x1’2/(4z>] exp
[- (u- ~ ) ~ ] [-1exp (-fiw/KT)], (41.IS)
to which we must again add Eq. (40.6).
383
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
42. VERIFICATION OF THE SUMRULE
We are now in a position to check the sum rule for the conductivity, which is easily deduced from Eq. (23.3):
lm
(42.1)
al(k, u) du = (n/2)Ne2/m.
Here the wave vector k is considered to be a fixed parameter. If we treat the degenerate case first we find that for both z z > 1 integration of Eq. (41.15) gives:
<
1 and
(42.2)
to which we must add the contribution from Eq. (40.6), which can be written
Lm
Acr(k, u) dw
= nk2(6i
-
Let us first consider the case of long wavelengths, since in the long wavelength limit the contribution (42.3) to the sum rule will vanish. We find, on expanding t,he logarithm in Eq. (42.2) in powers of z, that
I, q ( k ,
u) dw = (n/2) (Ne2/m)[1
- z2 + (z4/5)],
z
<< 1.
(42.4)
If we assume lhat Eq. (42.3) exactly makes up the difference between (42.1) and (42.4), we can deduce, using (41,12), that for long wavelengths 1
-
R(k, 0) E - (rokp/37r) [I TO
-
(k2/20kp2)],
k
<< kF,
= poez/(4nm) = e2/(4aeomc2) = 2.818 X 10-13 cm.
(42.5) (42.6)
We can compare Eq. (42.5) with the conventional result for the diamagnetism of a free-electron gas’m: Km
where P 10)
=
-1
=
- (NP2r~/3KT)[F1/2(&E’/KT)/F1/2’(&P/KT)],(42.7)
efi/2m is the Bohr magneton, F1n(x) is one of the standard
See p. 167 in Wilson.**
384
FRANK STERN
Fermi-Dirac integrals defined by104 (42.8a) and the prime denotes differentiation with respect to the argument of F . We note for later reference the asymptotic results:
= 1,
F l ~ ( z M) +d/2e2,
F1,2’(x)/F1,2(2)
FI,z(z) M (2/3)9/’,
F l p ’ ( x ) / F l , z ( z ) NN
<< -1; 2 >> 1.
2
3/(2~),
(42.8b) (42.8~)
Equation (42.8b) applies to a nondegenerate carrier distribution, and Eq. (42.8~)to a degenerate distribution. In the latter case the bracketed factor in Eq. (42.6) equals 3KT/(2€p), Thus the Landau diamagnetism of a degenerate free-electron gas is: K~
- 1 = - NP2po/(2&~)=
-poe2A2N/(8m2&p)= -poe2kp/(12a2m) =
-?‘okp/(3a),
(42.9)
which agrees with the long-wavelength limit of Eq. (42.5) if we neglect the difference between K,,, - 1 and 1 - 6%. That difference is small if 1 - 6% itself is small, as will be the case for the systems considered here. This gives us some confidence in the assertion that Eq. (42.3) represents the deficiency in the sum rule for all values of z. With this assumption we find, from Eqs. (42.2) and (42.3)) that: 1 - R(k, 0) = [?‘ok~/(16rZ~)][ (1 - z*)21n[(l +%)/I
1-z
I] - 22 - 2z8}. (42.10)
This agrees with the corresponding expression obtained by Bardeenlo5if his result is evaluated for the case of an ideal conductor (Lee,one with no superconducting energy gap), For large z, Eq. (42.10) leads to the following expression for the diamagnetism of a free-electron gas: 1 - R(k, 0) S
-[?‘o~F/(~HZ~)][~
-
(k2)-’], z
>> 1.
(42.11)
When z = 1, Lea,k = 2kp, the approximate expressions (42.5) and (42.11) both give a diamagnetic susceptibility which is 80% of the long wavelength limit, while (42.10) gives 7’5% of the limit. J. MoDougall and E. C. Stoner, Phil. Trans. Roy. SOC.London A237, 67 (1938); A. C. Beer, M. N. Chase, and P. F. Choquard, Helv. Phus. Acta 28, 529 (1955). 106 J. Bardeen, in I‘Handbuch der Physik” (S. Fliigge, ed.), Vol. 15, p, 303. Springer, Berlin, 1956; see also 0. Klein, Arkiv Mat., Astron. Fysik 31A, No.12 (1945) I
385
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
We can carry out the same analysis for the case of the nondegenerate free-electron gas, using Eq. (41.18). We find in that case also that Eq. (40.6) correctly supplies the deficiency in the sum rule for long wavelengths. The general expression for the diamagnetism in the nondegenerate case is: 1 - R(k, 0)
=
- exp ( - z 2 )
-[p&e2/(mk2)][1
m
+
( ~ ~ ~ " / ( 2 nl)n!]]] (42.12)
where z2 = A21C2/(8mKT).For small z this becomes 1
- &(k, 0) NN
-[N@2po/(3KT)][1
- (2z2/5)],
z
<< 1,
(42.13)
which agrees with Eq. (42.7) in the long-wavelength limit, since, in the nondegenerate case, FlI2(z)= Fl,z'(x) = *d/zez. For large values of z we 2) in the expression in braces in Eq. can replace (2% 1) by (2n (42.12), and find very roughly that
+
1
- R(k, 0) M
+
-[Nb2po/(2KT~2)](1 - (22')-'),
z
>> 1.
(42.14)
For z = 1 the complete expression (42.12) gives 0.693 of the long-wavelength limit, while the approximate results (42.13) and (42.14) give 0.6 and 0.75 of the limiting value, respectively. The qualitative behavior of the diamagnetic susceptibility deduced from the foregoing results is similar for both the degenerate and nondegenerate cases: a deviation proportional to z2 from the classical result, dependence. followed at large values of z by a r2 43. CALCULATION OF THE DIELECTRIC CONSTANT
We can now use the dispersion relation to calculate the real part of the dielectric constant from the conductivity. The appropriate dispersion relation, corresponding to Eq. (17.lOa), is: Klt'(k, a) - 1
=
[2/(~eo)]P
q(k, w')[o'~ -
w2]-'
dw'.
(43.1)
For the case of degenerate statistics we take al(k, w) from Eq. (41.15) for nonzero frequencies, and add the delta-function contribution given by (40.6) and (42.10). The integration of Eq. (43.1) is tedious but straightforward. It must be carried out separately for z < 1 and for z 2 1, but
386
FRANK STERN
leads in both cases to the result:
Kltr(k,W )
=
+ [l
1
+ [ 3 / ( 3 2 ~ ) ] ( ~ , ~ / d )[1 -
(U
-
Z)2]21n
u-2-1 u-z+l
+ z ) ~ In] ~
- (u
which agrees with the value obtained by Lindhard,sa who calculated the dielectric constant by finding the total current induced by a vector potential like that in Eq. (40.2). For nondegenerate statistics we find:
Kltr(k,W )
=
1-
+
( ~ p 2 / ~ 2 )
up2exp
(-9)
/om
O?ZG
u' sinh (2u'z) exp ( - d 2 ) du', u'2 - 7.42
(13.3)
where u and z have the values given in Eqs. (41.8) and (41.17). While Eqs. (43.2) and (43.3) are rather formidable, they can be simplified for several cases of physical interest. One of these is the case u z >> 1, for which we obtain, for degenerate statistics:
+
Kltr(k,W )
M 1
+
- ( w ~ ~ / w ~ ) [ (1/5)(U2 ~ -
Z2)-1].
(43.4%)
This differs slightly from the corresponding result given in Lindhard'ssa Eq. (3.19). For nondegenerate statistics we find
Kltr(k,W )
W
+
1 - ( W ~ ~ / / . ~ ) [ (2u2)-' C~
Another case of interest is the case u,x degenerate statistics, the result
P(k, W) M 1 -
+ (3
2z2)/(4u4)]. (43.4b)
<< 1. In that
case we find, for
+ 3u2 - (3niu/4) (1 - u2 - z 2 ) ] ,
(W,~//W~)[Z~
(43.5a)
which agrees with Lindhard's Eq. (3.18). We have included the imaginary part of the dielectric constant in Eq. (43.5a), using (41.15). I n Eq. (43.4) there is no corresponding contribution to the imaginary part. For nondegenerate statistics we find from Eqs. (41.18) and (43.3) that for small u and z
"Kr(k,W )
FZ
+
1 - ( w , ~ / w ~ ) [ ( ~ z ~2u2 / ~ )- i t h ( 1 - u2 -
44. DIELECTRIC CONSTANT WITH
NO
2')).
(43.5b)
EXTERNAL CURRENTS
The expressions given in the previous section give the response of a free-electron gas to an electromagnetic wave whose wavelength dependence
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
387
and frequency dependence are both specified. But in many cases we are interested primarily in the simpler case of a freely propagating wave of fixed frequency. Then we know from Eqs. (4.3) and (7.12) that when no external currents are present, hIaxwell’s equations lead, for transverse waves, to the result:
&*f=
(44.1)
(W2/C2)KXrn.
As we have shown above, it is possible to include the magnetic effects in the dielectric constant “Kr(&, w ) . Then we find, from Eq. (8.3):
f .f
P(&, w)*
= (w’/c’)
(44.2)
This is a n implicit equation which must be solved for f . f . When the resulting value for &.& is substituted in P ( f , w ) , we obtain the wavelengthindependent dielectric constant that best describes the behavior of freely propagating waves. Note that this will lead to complex wave vectors in most cases. Thus the term “propagating” as used here implies only that these are solutions of Maxwell’s equations in the absence of externally introduced currents, and does not imply absence of damping. From Eq. (43.4) we see that as long as 1 u2 1 >> I, the dielectric constant is accurately given by the classical value Ktr
= 1-
(
(44.3)
J
as in Eq. (25.4). But if we make use of both Eqs. (44.2) and (44.3), we find that
u2
= w’/(uo’k2)
x (c’/uo2) Thus the condition 1 u21
=
C2/(2)02Kt’)
[w’/(w2
>> 1 is satisfied
- w,’)].
provided
(44.4) w
>> uowp/c.
I n metals,
vo is the Fermi velocity, and vo/c is less than 0.01 ; in nondegenerate systems
a t ordinary temperatures vo/c will be even smaller. There is therefore a considerable frequency range in which vowp/c << w << wpJ for which the dielectric constant is given by Eq. (44.3), and is negative. I n that range, Eq. (44.2) leads to the relation
L x i ( w p 2 - w2)1’2/c x iw,/c,
vowp/c
<< w << u p .
(44.5)
In this case the field decays by a factor e in a distance k2-1
= c/wp,
(44.6)
called the London penetration depth. We now consider the limiting case of very low frequency, and will assume that both u and x are much less than 1, so that Eq. (43.5) can be
388
FRANK STERN
used. For degenerate statistics Eq. (44.2) then takes the form: c2IE2
=
4C2kpV m w,2[(37riu/4)
- 221,
(44.7)
where only the lowest order terms in u and 2 have been retained. This equation implies that IE, and therefore u and %, are complex. For real metals ckp >> wp, so that 9 on the right in Eq. (44.7) can be neglected. We then solve for 2, and find that a t low frequency
k M [37riWp2W/ (4c2vF)]'/'.
(44.8a)
For nondegenerate statistics we use Eq. (43.5b) and obtain:
IE M
[T'~?iWp2W/
(c2vo)1118,
(44.8b)
where vo is given by Eqs. (41.9) and (41,17), and equals (2KT/m)1'2. The results can be put into another, more commonly used form if we note that for degenerate statistics the mean free path 1 is given by UFT, where r is the relaxation time. Then Eq. (44.8a) can be written
IE M
[3TiuOo/ (4eoc21)]'/8,
(44.9)
where (TO is the dc conductivity, Ne2r/m, We can put Eq. (44.813) into a similar form, although for nondegenerate statistics we must consider the energy dependence of the relaxation time. For the particular case of an energy-independent mean free path, which applies to acoustic mode lattice scattering in semiconductors, we findl06 that 1 = ( 9 ~ K T / 8 r n ) ' / ~=r (37r1/2/4)v0r,so that Eq. (44.813) can also be written in the form (44.9). For other energy dependences a factor of the order of unity is required in Eq. (44.9) for the nondegenerate case. Equation (44.9) is misleading, however, since the procedure we have used, based on (44.2), is valid only in an infinite homogeneous medium without external sources. When P ( k , 0) depends on wave vector very slightly or not at all, the effect of a boundary will not affect the validity of the results. But in the low-frequency region the dominant term in the dielectric constant Eq. (43.5) varies as it-', and a procedure based on (44.2) will not give correct results. A more exact treatment for the particular case of specular reflection is given in the next section, and shows that Eqs. (44.8) and (44.9), while qualitatively correct in their dependence on carrier concentration, mean carrier velocity, and frequency, are in error by a numerical factor of order unity, and by a phase factor. 108
W. Shockley, "Electrons and Holes in Semiconductors," p. 277. Van 'Nostrand, Princeton, New Jersey, 1950.
ELEMENTARY OPTICAL PROPERTIES O F SOLIDS
389
IMPEDASCE AND AXOMALOUS SKINEFFECT 45. SURFACE The misleading results obtained by the method of the preceding section arise when the dielectric constant is a strong function of wave vector. In such cases the current a t a point will depend on the field at other points, and the effects of a boundary can be felt at substantial depths. This leads to a nonexponential decay of the field as it penetrates into the medium, and to other deviations from simple classical behavior. The phenomenon is called the anomalous skin effect, and has led to important information about the properties of metals.lo7 An experimental quantity which characterizes the properties of a medium is the surface impedance for normal incidence, 2, given bylo8 E=
Z(H xn),
(45.1)
where E and H are the values of the electric and magnetic fields a t the surface of the medium, and n is a unit vector along the direction of propagation of the wave. For simplicity we will consider a wave propagating in the positive z direction, in vacuum in the lower half-space z < 0, and normally incident on the medium in the upper half-space z 1 0. We assume that the electric field is in the x direction. With these conventions we find:
z=
k ( 2 = O)/Hu(2
= 0).
(45.2)
The reflectivity is easily derived from the surface impedance. If we assume that the wave is incident from vacuum, and assume a common time dependence given by exp (-id)throughout, we can write the magnetic field for z < 0 as the sum of an incident wave and a reflected wave, or R,(z) = Ri exp ( i w z l c ) R, exp (-iwz/c). (453a)
+
Then, using Eq. (4.lc), we find that &(z)
=
Zo[Ri exp ( i w z l c ) - 8, exp (-iuz/c)],
(45.3b)
where Zo =
(w)-l
= poc =
376.73 ohm
(45.4)
is a constant, sometimes called the impedance of the vacuum or the impedance of free space. If we define the amplitude reflection coefficient i: by i: = 101
108
a,/s,,
(45.5)
A , B. Pippard, Advan. Electron. Electron Phys. 6 , 1 (1964);Rept. Progr. Phys. 23, 176 (1960). See p. 284 in StrattonQ2
390
FRANK STERN
and solve Eqs. (45.2) and (45.3) for P in terms of i: =
(20- Z)/(ZO
z",we find that
+ 2).
(45.6)
If the medium from which the wave is incident is not a vacuum, we need only replace Zo by the corresponding value for that medium. If the electric and magnetic fields of the transmitted wave are exponentially damped with wave vector &, then Maxwell's equation ( 4 . 1 ~ ) ~ H = f xE/(pow), combined with (45.2) leads to the result
z = pow/fE, I
(45.7a)
which, if we use Eq. (44.2), is equivalent to =
Zo(p-1/2.
(45.7b)
We see on comparing Eqs. (45.6) and (45.7) with (10.6a), that the present results are consistent with those of the conventional treatment of reflectivity given in Part 11. The reflectivity for a wave normally incident from a vacuum is given by
R
=
I f 1 2
=
1 zo - 2 \'/I zo + z" 12.
(45.8)
We now proceed to calculate the surface impedance for a solid with a wavelength-dependent dielectric constant, for which Eq. (45.7) does not apply. To calculate the surface impedance for this case we first find the spatial Fourier components of the vector potential for z 1 0, which we write &(z)
=
exp ( - i w t )
/
m
A,(k) exp (ikz) dk.
(45.9)
J-ca
We have assumed that the amplitude is independent of x and y, and therefore integrate only over the z component of the wave vector, dropping the subscript z. The relation between the vector potential and an externally introduced transverse current Jext is [k2 -
(W2/C2)"Kr(k,
w)]i(k,
w) = poJext(k, w),
(45.10)
which is analogous to Eq. (7.8b). I n the problem considered here, a wave normally incident on a plane surface, we would not need any external current sources if we used the conventionaI boundary condition treatment of Part 11. With the present formalism, however, we must introduce an effective surface current at the boundary to simulate the proper conditions there. We shall treat the simplest case, that in which the carriers of the medium are assumed to be scattered specularly when they strike the boundary.
391
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
For specular reflection we imagine that any particle incident on the plane z = 0 from above will have its z component of velocity reversed, while its component of velocity in the x - y plane remains unchanged. We can replace this model by another, equivalent to it, by assuming that particles incident from above pass through the plane z = 0, and that the reflected particles are represented by particles which cross the plane from below. I n order that this alternative model should be equivalent to the actual specular reflection, we must assume that the medium in the lower half-space has the same properties as the other medium, and that the fields in the lower half-space are the mirror images of those above. This means that & ( - z ) = & ( z ) . For the magnetic field the relation is flu(-2) = - f l u ( z ) , since the magnetic field is a pseudovector which changes sign on reflection through a plane containing H.Thus we have a discontinuity in the y component of the magnetic field, which implies a surface current density, according to Eq. (9.ld)) which we can write in the form J,(z) = -2flu(0)6(z), (45.11) where Thus
R,(O)is the
J,(k)
= (27r)-1
limiting value of
Jm
-m
flu as z
s , ( z ) exp ( - i k z ) dz
approaches 0 from above,
=
-T-~R~(O). (45.12)
If we introduce this current as an external current source into Eq. (45.10), and use the relation B(k, u) = iwi(k, o),we find that &(z)
Jm
= [~oo/(i?r>]~~(O [k2) -W
- (o2/cZ)>tr(k,
u)]-1
exp (ikz) dk. (45.13)
Note that in our example k is in the z direction, and k = k,. To make our model self-consistent we must assume that P (-k, o) = P ( k , u), since this will assure that &( -2) = BZ(2). To find the surface impedance we evaluate Eq. (45.13) at z = 0, and use (45.2), with the result:
Z
= [2poo/(&)]
/= 0
[ka - ( w z / c z ) P ( k ,o ) ] - ’ d k .
(45.14)
When i t r does not depend on k, this integral can be evaluated very easily, and gives just the result (45.7b) which we deduced from more elementary considerations, and which includes the normal skin effect. The nontrivial application of Eq. (45.14) is to the anomalous skin
392
FRANK STERN
effect. We shall apply it here only to the case of an ideal conductor, for which we find, from Eqs. (43.5) and (44.9), that ( ~ ~ / c ~ ) > ~u) ~ (= ( k[37ricr~/(4~&7k)] , =
i@/k,
(45.15)
where p has been introduced as a temporary abbreviation for the parameters in Eq. (45.15). We find, then, that
= [2pOw/(~i)]
lrn 0
k d k / ( k 3 - ip).
(45.16)
We can simplify this expression by substituting for k the value Ic = ( - i p ) 1 / 3 s , and find that the surface impedance is
2
= (2pow/~i)( -ip)-’l3
(45.17)
The value of the integral in Eq. (45.17) is 27r/3a‘2.109Thus, substituting for p from Eq. (45.15) we find that,
2 = 4 ~ 0 3 - 3 1 2 [ ~ o ~ u 2 / ( 6 n ~(1~ )-] ’ i ~ 8G).
(45.18)
This is the surface impedance for the extreme anomalous skin effect for specular reflection. TO obtain the usually quoted value in Gaussian 1 - iG), simply replace namely (8/9) 31~6~*~3[ZwZ/(c4cr~)]1~3( po by 4?r/c2, and €0 by (47r)-’. Note that we use a time dependence exp ( -id))while both Jones1l0and Pippard’O’ use exp (iut). There is an ambiguity in taking the cube root in Eq. (45.17), but this is easily resolved if we note that the wave must be propagating in the +x direction and damped, which shows, by analogy with (45.7a), that the physically acceptable solution for 2 in our case must have a positive real part and a negative imaginary part. For diffuse scattering at the boundary, which better describes real solids, there appears to be no simple device like the one that led to Eq. (45.12). The calculations”‘ are rather formidable, and lead to a surface impedance bigger than Eq. (45.18) by a factor 9/8. We note in passing that our simple D. Bierens de Haan, “Nouvelles Tables d’Int6grales D6finies,” Table 17. Stechert, New York, 1939. 110 H. Jones, in “Handbuch der Physik” (S. Flugge, ed.), Vol. 19, p. 308. Springer, Berlin, 1056. 111 G. E. H. Reuter and E. H. Sondheimer, Proc. Roy. SOC.A196, 336 (1948); R. B. Dingle, Physica 19, 311 (1953). 109
ELEMESTARY OPTICAL PROPERTIES O F SOLIDS
393
arguments which led to Eq. (44.9) would have given, via (45.7), a surface impedance equal to (45.18) times a factor (3v3/4)i1'*, i.e., a result too big in magnitude by 30%, and in phase by 30'. Let us summarize our discussion of the surface impedance of a freeelectron gas by giving the values for the simpler limiting cases. At low frequencies we must take the conductivity into account. If we use Eqs. (45.7) and (27.7), and note that in the free electron case K, = 1 and up2= Ne2/(meo),we obtain the classical skin effect result: (45.19a) (2 U P T ) -'I2(1 - i) . 2/20 = ( . / U p ) As the frequency increases the classical skin depth (4.9) decreases. When it becomes comparable to the mean free path of the carriers we reach the nonlocal regime described a t the beginning of this chapter. When the classical skin depth is much smaller than the carrier mean free path we have the extreme anomalous skin effect, given in Eq. (45.18), which we rewrite here for degenerate statistics in the form:
Z/2o
=
2'3"''
(V F / ~ T C I/* ) ( 1 - ifl)
(u/w,) '1'
I
At still higher frequencies we reach the range we can use Eqs. (44.5) and (45.7) to find:
qzo
= -iw/w,.
VOWJC
(45.1913)
<< w << wp, for which (45.19~)
Finally, when w >> w p , the surface impedance of a free-electron gas approaches Zo. At sufficiently low values of w , we can take the bound electrons into account and generalize these results somewhat if we replace ZOby ZOK,-~, and let up2 = Ne2/(m*K,eo) as in Section 25. More detailed consideration of the anomalous skin effect leads to information about the transition between the classical skin effect and the extreme anomalous region,lll and to information about details of the shape of the Fermi surface.lW 46. REALCRYSTALS The results obtained in this part apply only to a freeelectron gas, and neglect both the short-range interactions between the electrons, and the periodic potential which is present in a real crystal. Few detailed calculations of the wavelength-dependent dielectric constant have been carried out for specific solids,112but there are many papers which take electronelectron interaction into account, and relate the dielectric constant to the matrix elements or oscillator strengths that would describe any particular crystaLsO 112
D.R. Penn, Phys. Rev. 128, 2093 (1962)
I
394
FRANK STERN
If the wavelength dependence of the dielectric constant is weak, then one can expand the dielectric constant in a power series in the rectangular coordinates of the wave vector k. To second order we would have: where repeated indices are to be summed. For crystals with tetrahedral or higher cubic symmetry the fourth-rank tensor in Eq. (46.1) has only 3 independent components. Furthermore, if the crystal has inversion symmetry all coefficients of odd powers of k will vanish in the expansion,'l8 and the crystal will be optically inactive.114 Thus for cubic crystals with inversion symmetry Eq. (46.1) takes the form115:
+
+
+
D(k, U ) = ~ o ( X ( 0u)B , 6(k*k)B E(k*E)k dT*EI1 (46.2) where T is a second-rank tensor whose diagonal components, measured with respect to the cubic axes of the crystal, are ka2, kU2,and kS2,and whose off-diagonal components vanish. The constant d describes the anisotropy of the dielectric constant. If d vanishes, then 6 gives the dependence of the E dielectric constant on k to second order for transverse fields, while 6 gives this dependence for longitudinal fields. A review of the electrodynamics of media with spatial dispersion which describes the consequences of the wavelength dependence of the dielectric constant in some detail has been given by Rukhadze and Silir~.~l'
+
VII. Interactions of Charged Particles
When charged particles are present as sources, the electromagnetic field is largely or entirely longitudinal, We shall first derive the dielectric constant of a free-electron gas for the longitudinal case, and shall then give applications of the longitudinal dielectric constant to screening, to energy loss, and to the forces acting on charged particles.
47. LOSGITUDINAL DIELECTRIC CONSTANT OF
A
FREE-ELECTROS GAS
To calculate the longitudinal dielectric constant for an ideal free-electron gas we proceed as in Part VI. We first find the conductivity, cr1 (k,u) , and This differs from the formally similar expansion of the conductivity or resistivity in powers of the components of the magnetic field where the term linear in H, which gives the Hall effect, does not vanish because H is a pseudovector. 114 A. A. Rukhadze and V. P. Silin, Usp. Piz. Nauk 74, 223 (1961); see Soviet Phys.Usp. (English Transl.) 4, 459 (1961). 111 The linear term also vanishes for the cubic point group T d (83m) [W. A. Wooster, "A Text-Book on Crystal Physics," p. 160. Cambridge Univ. Press, London and New York, 19381 which has six reflection planes along the face diagonals of the cube, but no inversion center. Thus the linear term vanishes for the sphalerite (zincblende) structure, which has Td symmetry. 113
395
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
then use the dispersion relation (43.1) to calculate the real part of the dielectric constant. If we retain the gauge $C = 0, the relation between the Fourier components of the longitudinal vector potential and the external charge density is given by Eq. (7.10) :
A(k, w ) where k
=
=
-kp(k, w)/[eok2wK(k, w)],
k
=
0,
(47.1)
I k 1.
With a longitudinal vector potential we must use: p+
=
P
+ 4%
(47.2)
as in Eq. (35.3). Thus the matrix element (35.9b) is: 312
= (#j
I exp (2k.r) [go* (p + %k)l 1 h).
(47.3)
Except for this change, we can follow the procedures and notation of Part VI, and find without difficulty that the conductivity in the longitudinal case is:
For degenerate statistics we find, using Eqs. (41.11)-(41.14), that u1
(k, w ) = [3Ne2w/ (mk2v02 j ] (7ru/2) , =
u+z
[ 3 N e 2 w / ( m k 2 u ~ 2 ) ] [ ~ / ( 8 z) ] ([u 1 - z]~),
ju-z~
1 u - z I > 1.
0,
(47.5)
In the transverse case we included a delta-function term in the conductivity to account for the current V xM,and chose the coefficient of that term to be of such magnitude that the sum rule (42.1) was satisfied. No such term is needed in the longitudinal case, since a longitudinal vector potential does not give rise to a magnetic induction B, as is easily seen from Eq. (7.3b) , and therefore does not induce any magnetization. Furthermore, the conductivity given in Eq. (47.5) satisfies the sum rule (42.1) exactly. For nondegenerate statistics we find that the expression corresponding to Eq. (41.18) is ul(k, w )
=
[ ~ " ~ N e ~ 1 i ~ w / ( 3 2 m z ~ exp [ K T[) ~ )(u ] -z)~] X [l - exp (-fiw/KT)],
which also satisfies the sum rule (42.1).
(47.6)
396
FRANK STERN
To find the real part of the dielectric constant from the conductivity, we need only integrate Eq. (43.1).For degenerate statistics the integration leads to the result :
This agrees with the result obtained by Lindhard.118He showed that one
can take the scattering of the electrons into account phenomenologically by replacing u by (u i r ) / ( k v F ) in Eq. (47.7)and eliminating the absolute value signs. The imaginary part of the resulting expression reduces to Eq. (47.5)in the limit r + 0. The phenomenological introduction of a relaxation time into quantum-mechanical expressions for the dielectric constant is discussed by Ehrenreich and Philli~p.4~ For nondegenerate statistics we find :
+
(47.8) The integral in Eq. (47.8)is exactly the same as the one in (43.3).This allows us to use our eaqlier results for K~~ (k, u) and to express the dielectric constant for longitudinal polarization and nondegenerate statistics in the form: Kl(k, W ) = 1 2U2[~lt'(k,W) - 1 (W~~/W')]. (47.9)
+
+
We can expand the expressions for the real part of the dielectric constant for various ranges of interest, and have summarized in Table V the results of such expansions for both degenerate and nondegenerate statistics for 1, z << u,u,z << 1, and u, 1 << z. For the first 2 of these ranges we have also listed the corresponding values for the transverse dielectric constant, given in Eqs. (43.4)and (43.5). The frequencies for which 1 ?(k, u) 1 % 0 have particular significance, since for these a longitudinal mode can most easily be excited in the medium by a source p(k, u). The damping of the mode will be small, and we refer to it as a plasma oscillation. From the first line of Table V we can show that for both degenerate and nondegenerate statistics the angular frequency of 116
Equation (3.7)in Lindhard?'
397
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
TABLE V. APPROXIMATE EXPRESSIONS FOR THE LONGITUDINAL AND TRANSVERSE DIELECTRIC COXSTANTS
We tabdate in each case the value of xl(krw ) , using the parameters k = 1 k 1, vo = liko/m, and vo is given by ( 2 & ~ / m ) for degenerate statistics and by (2KTlm) for nondegenerate statistics. EF is the Fermi energy, and up2= Ne2/meo. z = k/2k0, and u = w/kv0, where
Approximation
Nondegenerate
Degenerate
Longitudinal dielectric constant [see Eqs. (47.7)-(47.9)] 1U)z<
1
+ 3wv2 [l - (u2 + k2v02
I +
+22)]
2 4
- [l -
k2v02
(2242
+ Q.81
Transverse dielectric constant [see Eqs. (43.4) and (43.511
u, z<
0
UP2
1 - - [3u2 0 2
+
221
This entry is based on an expansion containing one more term than in Eq. (43.4a).
plasma oscillations in a free-electron gas is given by:
[w(k)l2 =
up2
+ k2[v2]~v)
(47.10)
where wv2, as before, is Ne2/(meo). We have used the fact that for degenerate statistics (47.1l a ) [v2]Ay = 6Ep/5m = #vo2, while for nondegenerate atatistics
[vZlAy = 3KT/m
=
tv2.
(47.11b)
398
FRANK STERN
Interactions between electrons will result in a longitudinal dielectric constant rather different from the one we have found for a free-electron gas in Eqs. (47.7) and (47.8). The reader is referred to the work of Glick1’7 for further details. 48. SCREESIKG OF
A
STATICPOISTCHARGE
The longitudinal dielectric constant for static fields (a = 0, u = 0) describes the effect of the electrons in screening the Coulomb field of an external charge introduced into the medium. Before discussing this effect in terms of the wavelength-dependent longitudinal dielectric constant, let us give the conventional theory of the screening.ll* Consider a free-electron gas with N electrons per unit volume, to which we add a point charge Ze a t the origin and, to keep the system neutral, Z free electrons. I n the neighborhood of the charge there will be an electrostatic potential 4 ( r ) , which will add -e$(r) to the energy of each electron. Since in thermal equilibrium the Fermi level must be constant throughout the sample, the electronic charge density near the impurity will be given by
+ ed (r)] / K T ),
-en (r) = [- e/ ( 2a2)](2mKT/fi2) a ’ z F( [GF ~~
(48.1)
where F,(z) is the Fermi-Dirac function defined in Eq. (42.8), and GF is the Fermi energy measured with respect to the bottom of the electron energy band far from the impurity. The total charge density also includes a term Ne for the uniform background of positive charge and a term Ze6(r) for the added charge at the origin, where 6(r) = 6(z) 6(y) 6 ( z ) is a threedimensional delta function. Thus Poisson’s equation takes the form:
+
V24(r) = [ - e / ( ~ ~ ~ ) ] [ Z 6 ( r ) N
-
(27~~)-~(2mkT/fi*)a’2
+
X F I I ~ ( C E F e9(r)l/KT)l, (48.2)
where K~ is the static dielectric constant of the medium in which the electrons move. We must solve Eq. (48.2) subject to the boundary condition r$ (r) + 0 as r CQ , since the system is neutral. At large distances from the impurity n(r) + N , and the right-hand side of Eq. (48.2) goes to zero. Equation (48.2) is easily solved if we expand the right-hand side in powers of +(r), and drop second and higher powers. This approximation will be valid if we are not too close to the impurity. Under these conditions it is easy to show, using V2(l/r) = -4d(r),1l9 that --f
9 (r)
3 exp ( -r / L ),
= [Ze/ (47rK&oT)
(48.3)
A. J. Glick, Phys. Rev. 120, 1399 (1963). G.5008, “Theoretical Physics.” Hafner, New York, 1934; see also p. 87 in Mott and Jones.8l 118 See p. 3 in Panofsky and Phillips.? 117 118
ELEMENTARY OPTICAL P R OP E R T I E S OF SOLIDS
399
where the screening length L, sometimes called the Debye length or the Debye-Huckel120length, is given by:
L-2
=
CNe2/ WOK T )] [ F I I z ’ ( x lFllz ) ( x )] z = E p l K T +
(48.4)
For a strong electrolyte*1st120 in which there are several types of ions, with charges Z,e and concentrations N , , contributing to the screening, the factor Ne2 in Eq. (48.4) must be replaced by x , N , Z , 2 e 2 . We can use nondegenerate statistics, for which FlI2’/F1I2 = 1, in this case. A simple extension of the theory to the case of semiconductors121with many conduction and valence bands leads again to the potential given by Eq. (48.3), with L given by
where n; and p j are the concentrations of electrons and holes in various bands, &,; and &,,j are the corresponding band edges, and x i = ( I F - &,;)/KT,z j = ( & , j - & F ) / K T .Our use of the function F I p ( x ) is based on the assumption that the density of states in each band varies as the square root of the energy from the band edge. It is easy to find the spatial dependence of the charge density around the added point charge directly from Eq. (48.3), using
p(r)
=
-eEoKaV2qb(r) = Zes(r) - [ Z e / ( 4 x r L 2 ) ]exp ( - r / L ) .
(48.6)
The first term represents the added charge at the origin, and the second term represents the screening electrons. The electrons are attracted toward an added positive charge, and are repelled from a negative charge. The integral of Eq. (48.6) over all space gives zero, which confirms that the screening charge just cancels the added charge. We have so far considered only the terms in Eq. (48.2) which are linear in +(r). For small enough values of T , however, the potential will become large in comparison with K T/e, or with &/e, and can no longer be treated as a small quantity. Solutions of Eq. (48.2) which do not make the linear approximation have been carried out numerically for a number of cases.122 The solutions for positive Z and those for negative Z differ markedly, while our linearized solution (48.3) only changes sign. The screening of a static charge distributed on a plane is an important aspect of the description of surface properties of semiconductors and metals. A solution of the equation corresponding to (48.2) for planar P. Debye and E. Huckel, Physik. 2.24, 305 (1923). R. B. Dingle, Phil. Mag. [i46, ] 831 (1955). 112 L. C. R. Alfred and N.H. March, Phil. Mag. [8] 2, 985 (1957). 120
IZ1
400
FRANK STERN
geometry has been given for nondegenerate statistics by Garrett and Brattain.12a Let us now consider the screening of an external charge using the wavelength-dependent dielectric constant of a free-electron gas. If we choose a gauge in which the vector potential vanishes, the relation between the Fourier components of the scalar potential and the charge density is given by Eq. (7.11) : $(k, w) = F(k, w)/[eok2z(k, .)lo For the particular case of a static point charge, p(r, 1 ) = zes(r),
we have
1
- z1.r)
p(k, 0) = ( 2 ~ ) - 4 p(r, t ) exp (iwt =
(48.7a)
dar dt
[Ze/(8r*)]6(u).
(48.7b)
Thus, if we use the spherical symmetry of this case, we find that the scalar potential is given formally by:
$(r, t )
=
1$(k,
w)
exp (2ls.r
- iwt)
d*kdw
Only for degenerate statistics is a simple closed form available for K(k, 0). In that case we have from Eq. (47.7) : Kl(k,
0) = 1
+ [3wp2/(4k5p2~)][ (1 -
2')
In
z + l
It-11
+
2zl,
(48.9)
but integration of Eq. (48.8) is still formidable. The situation is much simpler if we ask only about the behavior of the potential where it is slowly varying, that is, if we restrict ourselves to small values of k. Then we find from the second line of Table V that
K(k, 0)
= 1
- 7 + L-2k-2,
~<
(48.10)
where, for degenerate statistics,
L-2 12s
=
3ive2/(2EOEF),
C. G. B. Garrett and W.
9 =
W,2/(4kFSF2),
H.Brattain, Phys. Rev. 99, 376 (1955)
I
(48.1la)
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
401
and for nondegenerate statistics
L-2
= Ne2/(e&T),
q = wp2/(3ko2v02).
(48.11b)
If we neglect the term q which appears in Eq. (48.10), we find, using methods to be described below, that substitution of Eq. (48.10) into Eq. (48.8) leads to exactly the result (48.3) given by the conventional treatment. Furthermore, the screening length L given in Eq. (48.11) agrees with the value given in (48.4) if we take the appropriate limits for degenerate and nondegenerate statistics, and take x8 = 1. Thus in first approximation the wavelength-dependent dielectric constant of a freeelectron gas leads to the same description of the screening as that given by the conventional treatment. Let us now look more closely a t the quantities in Eq. (48.10). This is most easily done for the degenerate case. We find, using Eqs. (41.12), (41.13))and (48.11a),that L-’
=
4 k ~ (/~ a o ) ,
7 = 1/ ( ~ T u o ~ F ) ,
(48.12)
0.2, where a. is the Bohr radius, 4neoh2/(mez). For a typical metal q and should not be neglected outright in the integration for the potential using Eq. (49.10). If we insert Eq. (49.10) into (48.8) and rearrange terms we find:
4(r)
=
r
[ Z e L z / ( 2 ~ z e o r ) ] k sin kr(Czkz
+ 1)-* dk,
(48.13a)
where 2 2 = (1 - q ) Lz.This integral can be evaluated using standard methods of contour integrati~n,’~~ and gives:
4(r) = [Ze/(4se0[1
- q l r ) ] exp
(-r/d:).
(48.13b)
This differs from the conventional result (48.3) through the appearance of the factor ( 1 - q ) in the denominator, and through the presence of d: = (1 - q)l/zL,instead of L, in the exponent. We shall establish below that Eq. (48.13b) is valid only for high carrier density, where q is small, so that the difference between (48.1313) and (48.3) is not great. Furthermore, Eq. (48.13b) cannot be correct near r = 0, where r4(r) must equal Ze/(4aeo) for all electron densities. If we use the formal expression for the static dielectric constant in the (4w,2m2)/(fi2k4),we find that limit of large x, given in Table V to be 1 Eq. (48.8) yields a potential:
+
4 (r)
=
[Ze/ (4reor)] cos (rid:') exp ( -r/d:’),
(48.14)
See, for example, E. T. Whittaker and G . N. Watson, “A Course of Modern Analysis,” 4th ed., Sect. 6.222. Cambridge Univ. Press, London and New York, 1950.
402
FRAKK STERN
where 2’ = [ f i / (mwp)]1‘2for both degenerate and nondegenerate statistics. This agrees with the result given by Bonch-Bruevich and Koga111?4~for this case. To estimate the validity of the potentials (48.14) and (48.13) or (48.3), it is convenient to introduce the Kigner-Seitz radius rs, the radius of a sphere containing 1 electron in an electron gas of density N :
rs
=
[3/(4aN)]lj3
=
(9a/4)l13kp-l
=
1.9192kp-l.
(48.15a)
We take the unit of distance for the free-electron gas to be the Bohr radius ao. We can give the results somewhat wider validity by using an effective Bohr radius a* = ( 4 7 T € 0 K , 6 ~ )/ ( m*ez) = (Ksm/m *) Uo, 148.15b) where m* is the effective mass and K~ is the static dielectric constant (apart from the contribution of free carriers) of the medium. The theory given here will apply to such a medium if we replace m by m* and e2 by e 2 / K , everywhere, so that a* represents a scaling factor. For some purposes, therefore, we can characterize a free or nearly free-electron gas by the dimensionless parameters ?-,/a* or a * k ~ . We can now write the following expressions for the quantities which enter in Eqs. (48.3), (48.13), and (48.14) :
L/a*
=
7 =
(7r/12)1~2~r,/a*)1~2 = 0 . 6 4 0 ( r , / ~ * ) ~=’ ~ 0 . 8 8 6 ( a * k ~ ) - ” ~(48.16~~) 2L’33-513a-4’3(rs/a*) = 0.0553(r8/a*)= O.lOG(a*kF)-l, (48.16b)
C’/a* = 3 - 1 / 4 ( ~ s / a * ) = 3/4 0 . 7 G O ( r , / ~ * ) ~=’ ~1 . 2 3 9 ( a * k ~ ) - ~ / ~ ,
(48.16~)
where degenerate statistics have been assumed. Equation (48.13) was derived using the expansion of the static dielectric constant valid for x << 1 or X: << 2X.p. Thus a condition for the validity of Eq. (48.13) is that a typical wave vector in its Fourier expansion, say 2-l, is < < 2 k ~ .From Eq. (48.16) we see that this requires that r , << 4.5a*. Thus in the range of validity of Eq. (48.13) we see from (48.1Gb) that 9 << 0.25. This is the basis for the statement made earlier in this section that the difference between Eqs. (48.13) and (48.3) is not large. Since real metals have values of r8 of the order of 3a*, the validity of Eqs. (48.13) and (48.3) is a t best marginal for metals. On the other hand some semiconductors, because of their small effective masses aiid high static dielectric constants, have large values of a x , and thus may be in the high-density limit, with r, < a*, for quite moderate carrier densities. For example if m* = O.lm and K~ = 10, the carrier density for xhich r8 = a* is 1.6 X lo1* 1*4n
V. L. Bonch-Bruevich and Sh. M. Kogan, Fiz. Tverd. Tela 1, 1221 (1959); see Soviet Phys.-Solid State (English Transl.) 1, 1118 (1960).
ELEMENTARY OPTICAL PROPERTIES O F SOLIDS
403
~ m - ~which , is well within the range of common carrier densities in semiconductors. The condition for validity of Eq. (48.14) can be established in a similar way, and is S'-l >> 2kF, which implies rB>> 71a*. This condition is not met by any metal, and by few semiconductors. One might suspect that the small x and large z expansions of the static dielectric constant (48.9) exhaust the possibilities for obtaining simple results for the form of the potential around an impurity. We must, however, look carefully a t the region near z = l, where the derivative of Eq. (48.9) has a logarithmetic ~ingularity,'~~ although the dielectric constant itself is continuous and has the value 1 (2?raokF)-' there. (We return here to the free-electron gas. The expressions can be scaIed to apply to other solids by replacing a. by a*). The effect of the singularity in the derivative of the dielectric constant on the potential is easily brought out by integrating Eq. (48.8) by parts once. We find that the dominant term in the potential can be written:
+
(48.17) where x = 2kFr. For large values of x the greatest contribution to the integral comes from the immediate neighborhood of z = 1. If we introduce x = 1 y, we find that the leading term in the potential will be
+
#(r)
-
-[UokF2ze
COS
+ $)')>3
X/(?rroX2(nUok~
1
m
0
COS
x y In y dy. (48.18)
We have restricted the integration to positive values of y, and have added a factor 2, since cos x y is an even function. The term in sin x which arises x y ) vanishes in our approximation, since in the expansion of cos (x sin x y is odd. The integral in Eq. (48.18) has the value - ? ~ / ( Z X ) . ' ~ Thus ~ we find that the singularity in the derivative of the dielectric constant a t z = 1 gives rise to an oscillating potentia1126a
+
4(r)
-
+ 3)2~31,
aok2Ze cos X/[2€O(?raOkF
(48.19)
I am indebted to Dr. A. A. Maradudin for making available unpublished results (1959) by J. J. Quinn, R. A. Ferrell, and A. A. Maradudin on this subject. l Z 6 See Table 467 in Bierens de Haan.109 l Z B s Expressions for the oscillations in the potential far from an impurity are given, for example, by J. S. Langer and S. H. Vosko, Phys. Chem. Solids 12, 196 (1959), and by T. Murao, J . Phys. SOC.Japan 17, 341 (1962). For comparison with experimental results on the Knight shift in alloys, see A. Blandin, E. Daniel, and J. Friedel, Phil. Mug. [S] 4 , 180 (1959); and W. Kohn and S. H. Vosko, Phys. Rev. 110, 912 (1960). 'z6
404
FRANK STERN
with a corresponding oscillating charge distribution. Since Eq. (48.19) drops off more slowly than (48.13) with increasing T , (48.19) will be the dominant term in the potential far away from the impurity. At nonzero temperatures the oscillations will be damped, because the Fermi surface is no longer sharp.12Bb A treatment using a dielectric constant is essentially a linear treatment, valid for weak fields. Thus methods based on a wavelength-dependent dielectric constant cannot be expected to be valid in the immediate vicinity of the impurity, where the electrostatic potential and the electric field will be very strong. The wavelength-dependent dielectric constant does, however, give a more accurate description of the potential far from the impurity, particularly the oscillating term (48.19), than does the conventional result (48.3). 49. ENERGY Loss OF
A
MOVINGCHARGED PARTICLE
The wavelength-dependent dielectric constant provides a convenient formalism for describing the energy loss of a charged particle moving in a medium, We give here only the principal result of such a treatment, namely the dependence of the energy loss on the imaginary part of the reciprocal of the longitudinal dielectric c ~ n s t a n t . * * J ~ ~ J ~ ~ Consider a particle of charge Ze moving with velocity v in a medium. The fields produced by the particle lead to polarization of the medium, and this polarization in turn produces a field a t the position of the particle which will tend to oppose its motion. The power loss can be expressed as -d&/df = -ZeE.v, and the stopping power, or energy loss per unit length, is: (49.1) -v-1 d&/dt = -ZeE v/v, where E is the field a t the position of the particle. To find the fields associated with a moving point charge we need only find the appropriate expressions for and j e x t , and apply Eqs. (7.8b) and (7.10). The charge density for a particle of charge Ze which passes the origin with velocity v a t time t = 0 is p(r, t ) = Ze8(r - vt), and its Fourier components are :
p(k, u) = [2e/(8lra)]8(u Jext(k,u)
=
- k*v),
[Zev/(8na)]G(u - k - v ) .
(49.2a) (49.2b)
N. H. March and A. M. Murray, Proc. Phys. Soc. (London) 79, 1001 (1962); C.P. Flynn and R. L. Odle, ibid. 81, 412 (1963). 187 H. Frohlich and R. L. Plataman, Phys. Rev. 92, 1152 (1953). 1*6b
405
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
Note that these satisfy the conservation equation (7.9). If we choose the gauge 6 = 0, we find that:
iL(k,
w) =
&(k,
0)
=
- k.v)/[k2w'iik,
-[Ze/(8a3~o)]jk6(w [Zep~/(S+)][v
w)]],
- s(&.v)]b(u - k*v)/[k2 - (w2/c2)"Kr(k,a)]. (49.3b)
The electric field a t any point is then given by:
E(r, 1 )
where
EL(r,t )
=
(49.3a)
[-iZe/(Sa%o)]
=
EL
+ ET,
{ k exp [zk-( r - vf)]/k2K(k, kev) 1 d3k, (49.4)
with a corresponding expression for ET which is easily deduced from Eq. (49.3a). To find the energy loss of the particle in the medium we must use Eq. (49.1), i.e., we must find the field a t the point r = vf. The integral in Eq. (49.4) simplifies if we introduce new variables w' and t such that k - v = kvt = w', and d3k = 2ak2dkdt = 2ak dkdw'lv. The components of k which are perpendicular to v will cancel in the integration, so we can write k ---f v ( k . v ) / u 2 = w'v/u2. Thus we findsa:
E L (vt, t )
=
li-' dk [- iZev/ (4a2~Gv3) ]
[w'/K
0
( k , 0') ] dw'.
(49.5)
We noted in ( 7 . 4 ~ that ) the reality of E(r, t ) and D(r, f) implies:
'i(-k,
-0)
=
'i*(k, w ) .
(49.6)
For a medium with inversion symmetry, 'i(k, w ) = 'i(-k, w ) , so that for such media we have 'i(k, - w) = 'i*(k, w ) . Under these conditions it follows, as in Part 111, that the real part of K is an even function of w , while the imaginary part is an odd function of w . The same properties hold for the real and imaginary parts of l / K . Then we find that the stopping power (49.1) is: rm
- v-I d E/dt = [- Z2e2/ ( 2n2eov2)]
]0
rkv
k-' dk
w' 10
Im[ 1/K (k,w ' ) ] dw', (49.7)
plus another term derived in a similar way from (7.33b), which gives the energy loss to cerenkov r a d i a t i ~ n , ~and " J ~is~ usually much smaller. Equa-
406
FRANK STERN
tions (49.5) and (49.7) can be expressed in an alternative form if we integrate using cylindrical instead of spherical coordinate^."^ Equation (49.7) shows that the greatest part of the stopping power comes from regions of the spectrum for which the imaginary part of the reciprocal of the dielectric constant is large, i.e., from regions for which i ( k , w ) E 0. A similar result was found in Part IV, where we noted that a weak external disturbance of angular frequency wL could produce largeamplitude longitudinal oscillations of the lattice, and could result in substantial power dissipation in the medium. In an electron gas these longitudinal oscillations are the plasma oscillations, and occur at a frequency given by Eq. (47.10) for long wavelengths. The contribution of the plasma oscillations to the stopping power of an electron gas is sensitive to the form of L(k, w ) , and the theoretical results are modified when electron-electron interactions are considered.117 A very simple expression for the stopping power can be derived if we assume that all the absorption occurs a t the plasma angular frequency, which we take equal to up from Eq. (41.13), ignoring the dispersion given in (47.10). If we use the sum rule (23.4), we find that the w’ integral in Eq. (49.7) equals -7rNe2/(2meo), provided k > k m i n = wP/v. An upper limit to the integration over k is given by the maximum momentum transfer possible for an electron of velocity v, namely fik,,, = 2mv. With these drastic approximations the power loss of a charged particle moving through a free-electron gas becomes: -v-I
d&/dt .= [NZ2e4/ (47reo2mv2)]In (lcmaX/k,,,i,,) =
[NZ2e4/ (47reo2mv2) 3 In (2mv2/hw,) ,
(49.8)
a result given by Kramers.12*
50. FORCES BETWEEN CHARGED PARTICLES The electric field of a slowly moving charged particle is given by Eq. (49.4), plus a small term derived from Eq. (49.3), which can normally be neglected. If the particle of charge Ze is at rest at the origin, and a particle of charge Z‘e is at the point r at time t, the latter particle will experience a force Z‘eE(r, t )
=
[-iZZ’e2/(87r3eo)]
1{ k
exp (zk.r)/[k2L(k, O)]} d3k. (50.1)
If the wavelength dependence of the dielectric constant can be neglected, we can replace i(k, 0 ) by K ~ the , static dielectric constant, and can remove l*8
H. A. Kramers, Physica 13, 401 (1947).
ELEMENTARY OPTICAL PROPERTIES OF SOLIDS
407
it from the integral. The remaining factors are easily integrated if we put: k - r = lcrt, and exp (zk-r) = c i z ( 2 Z l ) j ~ ( k r ) P l ( t ) ,where l ~ ~ jz is a spherical Bessel function of order Z, and Pl is a Legendre polynomial. Only the component of k in the direction of r will survive the integration over the azimuthal angle, so we can write k + krE/r = lcrPl(E)/r. After the integration over angles, Eq. (-50.1) reduces, for a wavelength-independent dielectric constant, to:
+
Z’eE(r, t )
=
/om
[ZZ’e2r/(2a2eor3~s)] zjl(x) dx.
(50.2)
The value of the integral in Eq. (50.2) is +7r,l3O so the force on a particle of charge Z‘e a t the point r is:
Z’eE (r, t )
=
ZZ’e2r/(47reor3~,),
(50.3)
which is the rcsult we would have written down from elementary electrostatistics. We have used this lengthy procedure to find a-simple result in order to emphasize that it is the static dielectric constant which appears in Eq. (50.3) or in the more general wavelength-dependent expression (50.1), even though the particle experiencing the force may be moving.lal A simple model which leads to Eq. (50.3) is that the particle of charge Ze, which is static, polarizes the medium and attracts a compensating charge -Ze (I - K ~ - ~ ) somewhat , like the case of screening by free electrons which we discussed earlier in this chapter. In the more general case we are considering now, the screening may be the result of the displacement of free electrons, bound electrons, or ions, or a combination of all of them. A charged particle a t a distant point will see a force which is reduced by the effect of this screening, and which is given in the general case by Eq. (50.1). If both the charge Ze a t the origin and the charge Z‘e at the point r are stationary, one mi’ght ask why there are not two factors of K~ in the denominator of Eq. ( 5 0 . 3 ) , since both particles will have a screening 130
See p. 409 in Stratton.2 This result is a special case of the more general result
/o+m
z ~ , ( z ) da: = 2~--14r (3
+ +q + ;p)/r (1 + t q - t p )
valid for q p > -1, q - p > - 2 , which can be deduced from problem 32 a t the end of Chapter 17 in Whittaker and Watson.lZ4 131 On the other hand, the force exerted on the charge Ze by a moving charge will contain the frequency-dependent dielectric constant even though the charge Ze is stationary.
408
FRANK STERN
cloud around them. Only one factor appears because we are calculating only the force on the particle, and do not include the force on the screening cloud of electrons and ions. The screening ions can, however, modify the dynamics of the motion of the charge considerably.182 The forces between charged particles in a dielectric have been studied from a many-body point of view by Kohn,las who showed that results of the form given here will still be valid in the presence of electron-electron interactions. Note added in proof: Recently Berreman1a4has observed structure at the longitudinal optical mode angular frequency W L in the transmittance and reflectance of thin LiF films. This effect, like the reflectance peak for silver14JsJasmentioned on p. 326 and the photoemission near the plasma frequency of alkali metals found by Ives and Briggsts is observed only for obliquely incident light polarized in the plane of incidence. This light couples strongly to longitudinal modes of the system at frequencies for which both the real and imaginary parts of the dielectric constant are small. We have seen that the longitudinal modes may be electronic, ionic, or, as in Section 34,a combination of the two. Structure in the reflectance or transmittance at a frequency for which i? SJ 0 can be observed only in thin films because in thicker samples, as Ferrell and E. A. Stern14 have pointed out, such a frequency is an endpoint of a spectral region of nearly total reflection. Acknowledgments
I am indebted to many colleagues at the Naval Ordnance Laboratory, the University of Maryland, and the Thomas J. Watson Research Center for discussions of varidus aspects of this work, and to Dr. Jack R. Dixon, Dr. William Frank, Dr. Arnold J. Glick, Dr. Peter B. Miller, and Dr. Peter J. Price for comments on the manuscript. I particularly thank Dr. Richard A. Ferrell for many valuable suggestions, and Mr. Ralph F. Guertin for writing most of Part I1 and for reading and criticizing the entire text. The careful preparation of the manuscript by Mrs. Marcella Z. Anderson is gratefully acknowledged. Finally, I thank Dr. Louis R. Maxwell and Mr. Herbert W. McKee for unfailing support during the writing of the manuscript, and Dr. Robert W. Keyes and the Thomas J. Watson Research Center for facilitating the preparation of the final draft. See, for example, T. D. Schulta, Phys. Rev. 116, 526 (1959). W. Kohn, Phys. Rev. 110, 857 (1958). la4D.W.Berreman, Phys. Rev. 130,2193 (1963). 186 A dip in the transmittance of thin silver films at the plasma frequency has been observed by A. J. McAlister and E. A. Stern, Bull. Am. Phys. SOC.8, 392 (1963). la
1aa