Theory of fine structure on the absorption edge in semiconductors

.I. Phys. Chem. Solids

Pergamon Press 1959. Vol. 8. pp. 382-388.

SESSION EXCITONS

o.lTHEORY

Printed in Great Britain

0:

AND PHOTONS”

OF FINE STRUCTURE ON THE EDGE IN SEMICONDUCTORS R. J. ELLIOTT

ABSORPTION

and R. LOUDON

Clarendon Laboratory, Oxford, England Abstract-General equations are derived for the details of the direct absorption edge in a semiconductor when the effects of excitons and of an applied external magnetic field are included. The effects produced by magnetic fields and excitons separately are readily derived as special cases and these are briefly reviewed with reference to work already published. The problem including both simultaneously could not be solved in general but expressions for the continuous absorption are obtained in an approximation valid in large magnetic fields. Simple spherical bands are assumed throughout and spin kffects neglected. 1. IN’IRODUCTION

THE important properties of semiconductors

are largely determined by those one-electron energy states which lie immediately above and below the forbidden energy gap. A great deal of information about the nature of these states can be obtained from the study of the optical absorption in a crystal-in particular that associated with transitions in which an electron is transferred from the valence band to the conduction band, Ieaving behind a hole. This creation of a holeelectron pair requires a minimum energy roughly equal to the band gap and the absorption rises steeply as the photon energy is increased from this value. This absorption edge shows in high resolution a complicated shape which may form a series of lines. These are due to the formation of excitons, which are essentially bound holeelectron pairs, and they have been extensively studied in ionic semiconductors (notably CusO) by GROSS,(~) NIKITINE@) and others (the references give review papers). More recently lines of this kind have been observed in Ge by MACFARLANE et aZ.(s) and LAX et al.@) In the presence of a magnetic field the continuous absorption beyond the exciton lines changes shape and may form a series of peaks. This magneto-optic effect has been *Chairman: R. A. SMITH; Co-Chairman:

M. LAX. 382

observed by BURSTEIN,(5) Lox and co-workers and proves a powerful method of determining effective masses in the bands. The magnetic field also affects the excitons and when the magnetic energy is much less than the exciton binding energy, linear and quadratic Zeeman effects have been observed in the lines of CuzO by GROSS.(I) In larger fields more complex effects have been seen by LAX, ROTH and zWERDLING.(4) In this paper the theoretical form of this absorption will be discussed. An attempt is made to set up a general form of the theory within the framework of the effective mass approximation which includes exciton and magnetic field effects simultaneously. Simple spherical bands are assumed and spin effects neglected. This problem is only solved in an approximation valid for large magnetic fields to give the shape of the continuous absorption under various conditions of allowed and forbidden transitions between the band edges. The fine structure arising from excitons(7) alone and the theory of magnetooptic effect(*) without excitons have already been treated theoretically. These results are reviewed however as they are readily obtained as special cases of the general expression. This theory is confined to direct optical absorption and does not discuss the indirect transitions involving phonons investigated by MACFARLANE et al. (9) where exciton effects were also observed.@)

SESSION 2. WAVE

EXCITONS

FUNCTIONS

The single particle functions of form uk,l(r)

0:

wave functions exp@

are Bloch

-4

exp i(k,

’ re+kh

’ rh)

(2)

The effective mass approximation for one particle systems has been given its most complete exposition by KOHN and LWTINGER(~~) and applied to the two particle problem of excitons by DRESSELHAUS.(~~) The wave function of the two particle system including the Coulomb attraction of the particles and the magnetic field may be written as a linear combination.

383

PHOTONS

Eat is the energy difference between the k = 0 states in bands a and c and pat is the momentum matrix element

(27d3 UO,*(-

(1)

where k specifies the wave vector and j the band. The ground state of the system is a determinantal function for N electrons which just fill the states available in the valence bands. The excited state consists of a similar function with one electron in a conduction band state ke, c and an electron missing in -kh, v. This latter may be regarded as a hole with kh, w so that the state may be regarded as two particles with a product function: ~~d,c(r&kb,v(~h)

AND

-

s

s2

ihv)uo&

and A is the vector potential, Q is the volume of a unit cell. The functions $ represent the modulation introduced by the atom cores on the total wave function. The overall motion of the two particles is given by the function @&K,(%

rh)

=

c

B, ,g(ke ,kh)ef(k~@k~rh)(6)

k&h In the effective mass approximation this function is found to be a solution of a Schrodinger-type equation with the effective masses m,, mh of the bands included and the perturbing potentials of the magnetic field and a Coulomb potential in a medium of dielectric constant E

[$(p,+~e)2++--(ph-~~)ac&-j @=E@ \yn,K (%

c

rh) =

(3)

B(k,, kh)e”(ke.=e+kh”h)~~,~(r~)~k~,~(ra>

bki,

where the functions $k are expanded as a linear combination of the complete set of functions formed by the Bloch functions of a specific k in all the bands. For convenience we assume that the conduction and valence band edges are at k = 0 and only spin degenerate. Then

Ck,e=

uop+z EC[fik,+$,] uoa

82

(7)

E gives the energy of the state relative to the band gap energy. The states which are of interest in optical transition are only those in which K = 0. This is because of a selection rule of conservation of momentum or wave vector and because photons in the optical range have essentially no momentum on the electronic scale. In this case equation (7) may be considerably simplified as shown by LAMBED)by changing the variables to the centre of mass co-ordinate p =

%!%-bhrh/%+mh

and relative position

@b!, where there is a summation directions u.

(5)

rh)

=

u(r)

f

1

(8)

r = re-rh

exp

-

2xp.r

over the three Cartesian Then ieh(m,-mh)

--GV~+~~ ~mh H.rxV+ L(Hxr)z8~~

U(r) = EU(r)

1 (9)

384

SESSION

0:

EXCITOKS

p is the reduced mass m,mh/m,+ma. Solutions this equation under various conditions will considered below. 3. ABSORPTION

AND

PHOTONS

of be

n = p+yA

COEFFICIENT

The transition probability that the crystal be excited from its ground state into an excited state with a wave function such as (3) is proportional to the square of the matrix element of o = e@r’.&’ where q, 6 are the photon wave and polarization vectors and j is the current operator to be taken between electron states. Since q is essentially zero, transitions are only possible to states with zero total momentum; i.e. K = ke+kh = 0. If the relative motion of the electron-hole pair is localized and U(r)-+0 as r-fco it is found that the absorption takes the form of discreet lines with f values

(11) where v is the photon frequency. If the relative motion is not localized there is continuous absorption with coefficient

an operator written as a function of Y. The direction z is that of H. There are clearly two quite distinct cases which arise according as pev#O or is 0. In the former case the first bracket of (13) gives the dominant effect and the second term of that bracket is negligible. This case is referred to as that of allowed transitions since transitions can take place between the states at the edges of the conduction and valence bands. If pcv = 0 these transitions are forbidden and in this case the results are given by the final term in (13) alone. Band to band transitions neglecting both coulomb and magnetic field effects are particularly simple. The overall wave functions are plane waves and the relative motion is simply ei”*r and the density of states is 2n(2p/A)*Ef. Here rU(O)=AK and ~U’(O)=(~~E/NQ)*. The expressiOnS for K are then

(12) where sn(E) is the density of excited states per unit volume per unit energy range. The matrix elements of j between the ground state and (3) are conveniently written in terms of the momentum matrix element (5). After some manipulation one finds <0/t * PI% 71) = iehH

5

-[4xpcvl, * pcv+ 2mcE cz,

&(O)+Mcv*nG43 (13)

where

(16)

C

Ipc#E*

(17)

(18) for the allowed

and forbidden

4. MAGNETO

cases

respectively.

ABSORPTION

The theory of the shape of the absorption edge in the presence of a magnetic field has been given in detail in reference 8. The results of that paper relevant to simple spherical bands can however be readily obtained by neglecting the Coulomb term in (10). The most useful form of these solutions is that given by DINGLE(~~) in cylindrical polar co-ordinates. The component of angular momentum I about the x axis is fixed and the motion along the z axis is free. The solutions are

__I

r

eHn! (19)

SESSION

h2kz2 ehH

E=.--+-

2P

Pn+Pl+l)+

0:

EXCITONS

AND

385

PHOTONS

eftH(%--mh)l (20)

2PC

2cm

mh e

Here 11 is eH(S+y2)/2hc, LI~ln+l~lan associated Laguerre polynomial and L, is the length of the specimen in the x direction. U(0) is only non zero for states with 1= 0 when LnO(0) = tz! The experimental observations@-@ appear so far to have been confined to the allowed kind of Thus allowed transitions take place only to I = 0 transition. states and the absorption consists of a series of edges beginning at energy Z&_,+ (ehH/2pc)(2n+ 1). The density of states associated with motion in one dimension is now 3LZ(2p/E)f/h so that the absorption in each sub-band has a peaked form.

For forbidden transitions the absorption coefficient is obtained from the second term in (13). (22)

l-I,U(O) = Ak,U(O)

and again only I = 0 states give an effect. Thus for radiation polarized along the direction of H the absorption has the form of a series of steps instead of peaks given by

5. EXCITONS

The

theory

of

the

absorption

edge

taking

account of the Coulomb attraction between holeelectron pairs is given in reference 7 in essentially the form-of this theory and will not be reproduced here. The solutions of (10) with H = 0 are hydrogen like states, bound when E < 0 and with unbound Coulomb wave functions if E > 0. The predicted hydrogenic like series runs together as

I (23)

For radiation polarized perpendicular to H the transitions take place to states of I = &l since Un,l # 0 only if I= 1 and

l/(n+ 1)Un+l,

the photon with the continuity. is given in

energy increases to Egap and merges continuous absorption without disThe form of this continuous absorption reference 7 as (a = (e2/2he)d(2p/E))

0

(24) d(n+l)G&,o

iraeTa

fC=lc*-

sinh Ta

na( 1+ a2)enc’

K = KF

where n* = ++?rl, This absorption is again peaked but at different energies to that of (21) because of the form of (20). In fact

sinh na

where ICAand Kg are the plane wave forms of K for allowed and forbidden transitions respectively given by (17) and (18) which neglected the effect of the Coulomb attraction. It therefore seems that a correct evaluation of the magneto-optic effects discussed in Section 4 should include the effect of the Coulomb attraction, which produces such a marked effect close to the band edge.

386

SESSION

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EXCITONS

6. EXCITONS IN A MAGNETIC FIELD The general problem therefore requires the solution of equation (lo), particularly for those states where the component of angular momentum I is 0 or f 1. For, the bound hydrogenic states in very small fields the second term in (10) is a dominant perturbation and gives a linear Zeeman splitting. The term in H2 gives a quadratic Zeeman splitting which becomes dominant as H increases. When the magnetic energy is of the order of the Coulomb energies such a procedure breaks down. The higher energy hydrogenic states are close together and are affected at smaller fields than the lower states. In atoms this has been investigated by SCHIFF and SNYDER.(13)The energy of the lowest bound state of an equation similar to (10) has also been considered by a variational technique by ADAMS, YAFET and KEYES. We shall be concerned here with an attempt to determine the wave functions of the unbound pairs which will still form a series of sub-bands beginning at energies given by (20) relative to the gap. Because of the localized nature of the Coulomb perturbation it does not alter the energy of unbound states as the magnetic field does. It is assumed along the lines of reference 14 that in large magnetic fields the wave functions in the direction perpendicular to z are determined entirely by the magnetic field and have the 17dependence of (19), f(~) say. The motion in the zdirection then takes place in an effective potential

AND

PHOTONS

where a = (+)- *. This has the same value at z = 0 and the same asymptotic form as (29) and has a discontinuity in slope at the origin. The solutions of the equation fia d%+b

can be written,

s For the form

lowest

band

(28)

&)l”M

n = 0, I=

0 this

has the

when

where

(29)

where /3 = eH/2Ac. Solutions of the x dependence of the wave function are of course impossible in closed form with (29). It is however possible to proceed by approximating (29) by -e2/~(~+1z1)

(30)

e_(k(a+z)k(a+z)

Ez = Ek2/2p,

g(x) is a linear

(31)

cc = pe2/h%k (32)

combination

of solutions

of

(33) One of function

these

is

a

confluent

and the other

‘I” is irregular

hypergeometric

1, z, x)

xrF$x+ is defined

(34)

by ERDBLYI(~~)as

xY(ior+l,

2, X)

at the origin

where

(35) it has the form

1 ___ rFr(io(+ T’(ia)

1, 2, X) log “r+ (36)

+

A+(1 -&)+0(X) Lx

.l I(za+l) alternative

rGl(l+ia,

function

2, X) = ilFl(l+ia,

1

is defined

as

2, x)+

2iecrla + Il?(l

-2e2B?e”“zErfc(B:lzl)E~1

= -&#

++Ixl)

eik”g(2ik[u+jxl])

A convenient V&z) = -

e2*

2/~ dz2

-ia)I

Y( 1+ ia, 2, X)

(37)

when the solutions e-“kzkx$r(ia+l, 2, 2&z) and e-t”ZkxrGr(ia+ 1, 2, 2&z) have asymptotic forms like the imaginary and real parts respectively of exp i(kx+ a log 2kz -

arg I( 1+ ia))

(38)

In these terms there are two orthogonal solutions of (31) which are respectively even and odd for reflection in the origin.

[l~l(a+~)lG~‘(a)--lGl(a+~)l~lr(a)l (rG~‘(~)2+rE’(~)2)’

(39)

SESSION

0:

EXCITONS

e_“=~~+z)k(a+~)

yu = &nor

AND

PHOTONS

[lFl(a+z)lGl(a)--Gl(a+z)lFl(a)l (1Gr(~)2+#i(a)2)’

where lFl’(a)

= (l-~ka)lFr(a)+arF1’(a)

and a similar definition holds for 1Glr. For allowed transitions we need the value of the function at the origin substituted in (13) and (12). Only the even function (39) gives an effect which, very close to the edge when LX> 0 (i.e. at energies, where Ez is less than the Coulomb energy) is /(==

t&H aWk% ~ ( mc 1 7rumch2

~FI(u)~ ___ &(a)2 (41)

lPc”la ’ (log2ku)s(l-e-s”a) For forbidden transitions with light polarized along H the second term in (13) is non zero for a#/& of the odd function (40). Here a4k%IMC,&F~(a)2 numc(log 2ku)2( 1 -e-29

(42)

The results for other sub-bands with n#O beginning at E = ehH(2n+ 1)/2pc may be obtained in a similar way. The radius of the orbit around the magnetic field is different and hence so is the effective potential (28) and (29). They may however be approximated by (30) with larger values of a. As a result the absorption in these bands will, under the above conditions, have forms (41) and (42) with different values of a. The approximate expression (41) and (42) will hold over a smaller .range of k. For florbidden transitions with light polarized perpendicular to H one again needs Y(O) for even functions like (39) together with the properties of f(y) given by (24). However the effective potential for I = fl states is different from (28) and can reasonably be approximated by replacing a by 2~. The shape of each absorption contribution is then of the same form as (41) and there are two series of peaks as in the magneto-absorption case (25).

387

(40)

Thus the Coulomb attraction is seen to sharpen considerably the peaks of the magneto-absorption from a form E-t to E-l(log E)+ in the allowed and perpendicular forbidden cases. For the parallel polarization in the forbidden case it induces in this approximation peaks of this same form, whereas in the absence of Coulomb terms there was a series of E* steps. This result is probably less reliable than the others since it depends on the slope of Y at the origin obtained from an approximate potential. The general formulae of this section tend to the results of the magneto-absorption calculation@) at large values of k. Unfortunately the experimental data at present available is not sufficiently accurate to allow a check on the detailed form of the absorption, but confirms the existence of sharp peaks in the allowed case. It is clear from this work that the effect of the coulomb attraction is of prime importance in considering the shape of the continuous absorption. It is planned to extend this work; in particular towards a more detailed study of the bound exciton states and the absorption from impurities in a magnetic field which has so far been treated only with the Coulomb effects neglected. REFERENCES *1. GROSSE. F., Nuovo Cim. Suppl. 3, 672 (1956). *2. NIKITINES., Phil. Mug. in press (19.58). G., MCLEAN T. P., QUARRINGTON J. $3. MACFARLANE and ROBERTS V., Proc. Phys. Sot. 71, 863 (1958). *4. LAX B., ROTH L. and ZWERDLINGS., Phys. Rev. 109, 2207 (1958). 5. BUR~TEINE. and PICUS G. S., Phys. Rev. 105, 1123 (1957). 6. LAX B., ROTH L. and ZWERDLINGS., Phys. Rev. 106, 51 (1957); 108, 1402 (1957). 7. ELLIOTTR. J., Phys. Rev. 108, 1383 (1957). 8. ELLIOTTR. J., MCLEAN T. P. and MACFARLANE G. G., Proc. Phys. Sot. 72, 553 (1958). 9. MACFARLANE G., MCLEAN T. P., QUARRINGTONJ. and ROBERTS V., Phys. Rm. 108, 1377 (1957). Footnote Correction: All formulae for absorption coefficients K should contain a factor n’/e where n’ is the refractive index.

388

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AND

IO. KOHN W. and LUTTINGER J., Phys. Rev. 97, 869 (1955). 11. DINCLE R., Proc. Roy. Sot. Mll, 500(1952). 12. LAMB W., Phys. Rev. 85, 259 (1952). 13. SCHIFF L. and SNYDER H., Phys. Rev. 55, 59 (1939). 14. ADAMS E., YAFET Y. and KEYES W., J. Chem. Phys. Solids 1, 137 (1956). 15. ERDBLYI A., MACNUS W., OBERHETTINCER F. and

J. Phys. Chem. Solids

0.2

Pergamon

*16.

PHOTONS

TRICOMI F. G., Higher Transcendental Functions (1953). BOWDEN H. and WALLIS R. J, Chem. Phys. Solids 7, 78 (1958).

*See also papers by these authors this Conference.

Press 1959. Vol. 8. pp. 388-392.

T. P. MCLEAN,

IN THE ABSORPTION AND SILICON

J. E. QUARRINGTON

Royal Radar Establishment,

of

Printed in Great Britain

EXCITON AND PHONON EFFECTS SPECTRA OF GERMANIUM G. G. MACFARLANE,

in Proceedings

Malvern,

and V. ROBERTS

England

Abstract-Evidence for exciton creation in Ge and Si by indirect and direct transitions has been obtained from high resolution absorption spectra. At low levels of absorption fine structure is observed, which is characteristic of indirect transitions involving the emission and absorption of phonons. For Ge four different phonons with energies of 90”K, 320”K, 350°K and 420°K and for Si four phonons with energies 212”K, 670”K, 1050°K and 1420°K contribute to the absorption. They correspond with the TA, LA, LO, and TO vibrational modes respectively. Each component is interpreted as being due to the formation of excitons and free electron-hole pairs. The exciton binding energy is found to be about 0.0027 eV for Ge and about 0.010 eV for Si. Absorption to the first excited state of the exciton is also observed. This state is estimated to be about 0.0010 eV above the ground state for Ge and 0.0055 eV for Si. The structure of the exciton absorption is smoothed out as the temperature is raised. The effect is explained as due to a temperature dependent relaxation time for the excitons. Direct exciton transitions have been observed as a line structure at high levels of absorption in Ge. By taking into account the coulomb interaction of the electron and hole in the theory of the direct band-to-band absorption, the energy dependence of the absorption above the exciton line can be explained and a value of about 0.0012 eV obtained for the exciton binding energy.

1. INTRODUCTION

In recent publication&2) we have shown that the low-level absorption edge spectra of Ge and Si can be interpreted in terms of indirect tran-

found corresponding to wave vectors in the (100) and (111) directions respectively. At temperature T and frequency v the absorption constant can be described by the formula

1 [ag(/tv--Ed,,

sitions involving phonons. In general each branch of the vibrational spectrum makes two contributions, one in absorption, the other in emission. In Si and Ge four phonon energies have been

T)+pc exp(W’)aa(hv---Ete, T)L

(1)

where EQ--E~~ = 2kt$. The factor p depends on the difference in energy of the intermediate and final states.(s) Its theoretical maximum value is about 2.2 for Ge and little