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Transport and free carrier electromagnetic phenomena in semiconductor boundary layer
SURFACE SCIENCE 34 (1973) 773-790 © North-Holland Publishing Co.
TRANSPORT PHENOMENA
AND FREE CARRIER ELECTROMAGNETIC IN SEMICONDUCTOR
BOUNDARY LAYER
II. REFLECTIVITY WITH NON-NORMAL INCIDENCE AND SURFACE PLASMONS* F. FLORES and G. NAVASCUES Centro Coordinado de Fisica ( C.S.L C.-U.A.M.) , Departamento de Fisica, Facultad de Ciencias, Universidad Aut6noma, Canto Blanco, Madrid 34, Spain
Received 15 June 1972 A classical analysis is given of the reflectivity at oblique incidence on semiconductor surfaces. The theory includes the effect of a space charge boundary layer and yields formulae from which one can obtain the dispersion relation for surface plasmons, for which the effects of the boundary layer are particularly important. An application to experimental data on n-InSb gives satisfactory agreement while yielding a plausible estimate of the surface excess AN. The electroreflectance is also studied, including numerical evaluations. It is suggested that measurements at almost grazing incidence could yield valuable information on the surface scattering of carriers. 1. Introduction T h e study o f s e m i c o n d u c t o r b o u n d a r y layers has been the subject o f c o n s i d e r a b l e research. S t a n d a r d t r e a t m e n t s a p p l y especially to dc t r a n s p o r t properties1,2). I n a recent p a p e r 3) (hereafter called I), we also studied ac fields in o r d e r to investigate the m a g n e t o o p t i c a l properties o f the b o u n d a r y layer. In particular, we considered the effect o f this layer on the reflectivity at n o r m a l incidence. This analysis was later a p p l i e d to discuss the optical d e t e r m i n a t i o n o f the carrier r e l a x a t i o n time in semiconductors4). In this case, we c o n c l u d e d that the space charge layer can m o d i f y a p p r e c i a b l y the optical estimate o f the relaxation time. It is b e c o m i n g increasingly o b v i o u s 7) that, for a c o m p l e t e analysis o f electroreflectance d a t a in s e m i c o n d u c t o r s S, 6), one o u g h t to correlate changes in reflectivity with changes in the surface p o t e n t i a l o f the b o u n d a r y layer. The p u r p o s e o f this article is to e l a b o r a t e on this p o i n t a n d to c a r r y * Work partially supported by the General Electric Company (through General El6ctrica Espafiola) and by the "Fundaci6n Pedro Barri6 de la Maza, Conde de Fenosa". The financial support of both these institutions is gratefully acknowledged. 773
774
F. FLORES A N D G . N A V A S C U E S
on our previous research into the general problem of the effect of the space charge boundary layer on the free carrier electrodynamic properties of semiconductors such as, for instance, the important question of surface plasmons. Since this is an electronic excitation, which is basically confined to the neighbourhood of the space charge layer, it is natural to expect that its details may have a non-trivial effect on the dispersion relation of surface plasmons. This will be borne out in section 6, while reflectivity and electroreflectance are discussed in sections 3-5. In section 2, we study the response of the carriers in the boundary layer to an E.M. wave.
2. Study of the conductivity in the boundary layer In I we have calculated the conductivity within the boundary layer, with an ac field parallel to the surface. When this field varies in a distance much larger than the mean free path of the carriers, )~, a local conductivity can be assumed. This situation occurs when an electromagnetic wave arrives at the surface plane of the semiconductor (z = 0) with normal incidence or with non-normal incidence but s-polarization, because 2 ~ ~ (2 is the wavelength). With p-polarization, the electric field has two components, E x and Ez, parallel and normal to the surface. Assuming translational symmetry along the x axis, we can separate the effect of these two components, due to the fact that Ex/Ez induces only a current Jx/Jz. The j~ component is related locally to the field component Ex, as indicated previously, but the Jz component depends non-locally on Ez. In fact the current jz has to change from zero at the surface to some definite value in the bulk, inducing a charge near the surface which must be calculated self-consistently with the field Ez. The depth of this region is close to ~, which usually is larger than a characteristic length of the space charge, the Debye length L D [~/LD ~--~pZ, and usually COpZ> 18)]. We analyze the non-local relation between j~ and Ez, using eq. (7) of I, oo
f, (t) =
G(t, t'
- q - - V'E GI3
dt',
(2.1)
t'
where fl {t}, the deviation of the distribution function from equilibrium, is expressed by means of the local time, t, of the particle, using the path variable methodg). In our case, we have to use the following Green function [see eq. (5) of I] t
G(t,t')=O(t-t')exp{-f
dz~s)}. t'
(2.2)
TRANSPORT
AND FREE CARRIER
ELECTROMAGNETIC
PHENOMENA.
775
II
We choose p = 1 in eq. (5) of I for the following physical reason. In the calculation of Jx, P measures the effect of the surface on the carriers, as ( 1 - p ) gives the average momentum loss along the x axis due to surface scattering. Thus, in the diffuse case (p =0), the average momentum along x becomes zero after reflection. Note, however, that when the induced current j~ is considered, the overall average momentum of the carriers along z is specularly reflected from the surface. Hence, in order to make a reasonably simple assumption about the carrier surface scattering10), we take p = 1 in this paper. A more elaborated hypothesis can modify accordingly our final results, specially the ones related to the electroreflectance. If we write Ez (x, z, t) = Ez ( z) exp {i (tcx - cot)), in eqs. (2.1) and (2.2), assuming z to be a constant and x;[< 1, we get fl{t}=
Of°l e x p { i ( x x - cot)} x t3eAt
-q
×
f exp{-(t--t')(!-ico)}
[v,E,]t, at'.
-oo
Here [v~Ezlt, is a function of past local times, t', the inhomogeneous field, E~, and the initial coordinates (z, v~, t). The current, j~, is given by j= (x, z, t) = i
q f l {t} v, day = exp {i (xx - cot)} x
i/ -ct~ oo
t
63f°lF -oo
t')(1-ico)}{vzE.}t
exp{--
,
.
-oo
Replacing the variable of integration t' by z', we can write oo
L (x, z, t) = exp {i (xx - cot)} [
o~(z, z', co) Ez (z') dz'
(2.3a)
0
with
~x(z,z',co)=~fl-q2Vz~]texp{-(t-t')(~-ico)}d3v, s
(2.3b)
-oo
where ( t - t ' ) is a function of z, z' and v~. [ ( t - t ' ) as a function of z and z',
776
r . FLORES AND G. NAVASCUES
has at least two values, because the carriers can go from z to z' directly or reflecting on the surface. These posibilities are implied by ~s]. Eqs. (2.3) show the non-local relations between the current jz (x, z, t) and the field E~(z) exp {i(xx-cot)}. Observe that the effect on the conductivity of the variation of the field with x has been neglected. With this we can define a non-local dielectric function, e~z, by 4hi ez, (z, z', co) = eL 6 (Z -- Z') + - - e (Z, Z', CO), (2.4) CO where eL is the lattice dielectric function. This and the local dielectric function, ex,, relating Jx to E~, cf. I, allow us to calculate the effect of the boundary layer on the reflectivity.
3. Reflectivity analysis We start with Maxwell's equations 1 c~H VAE . . . . ,
c 8t
V ^ H-
1 8(rE)
c
8t
'
(3.1)
where 8.E= (e~xEx, e~zE~) represents symbolically
e'E=
f
e(x,t;x',t')E(x',t')dx'dt'.
z>0 --oo
Firstly let us study a p-polarized electromagnetic wave. We choose the geometry of fig. 1 :
E=(Ex(x,z,t),O, Ez(X,Z,t)),
H=(O, Hy(x,z,t),O).
With a dependence on x and t going like e x p { i ( x x - c o t ) } , we have explicitly, from eqs. (3.1) dE~ dz
ixE~
09 i-cHy,
dHy i co . . . . . exx (co, z) Ex, dz c co
ixHy = - 1"¢
[",J ezz (co, z, z')
(3.2)
dz'
For z>>~, we have e,~(co, Z)=eb=eL+4rdab/co (trb is the bulk conductivity) and Sezz(co, z, z') Ez(z') dz'=ebEz(z). Then we get from eqs. (3.2) dz 2 +
~e b-x 2 Hy=0.
(3.3)
TRANSPORT AND FREE CARRIER ELECTROMAGNETIC PHENOMENA. II
777
This equation describes the propagation o f a wave along the positive z-direction. Therefore, for z > (~, LD) , we have !
[(°:
Hy = eb~ E 3 exp iz c 2 eb
):1
-/~2
(3.4)
.
To get the reflectivity in a first approximation, we look for an approximate solution to eqs. (3.2). Let us follow the same m e t h o d used in I, assuming 2 >>7,. Consider the incident El, reflected E 2 and transmitted E 3 waves. In the region z < 0 , we have E = E1 exp {i(qz + ~x - o90} + E 2 exp { i ( Hy = E , exp {i ( q z
+ xx
o90} - E2 exp {i (--
-
qz + tcx -
cot)}
qz + xx
cot)},
--
(3.5)
and for z > (~, LD) E = E 3 exp {i ( q ' z
cot)},
+ tcx -
Hy = eb~ E a exp {i ( q ' z
+ xx
-
cot)},
(3.6)
where q,2 + t¢2 = (092/c2) %.
B'
/ / /
,/
//
t
,/Hy (z)
IEx (z)
i
Ez(Z )
tlI
/
/
/
/
/
/
/
z=0 Fig. 1. Geometry for the application of Stokes theorem to the reflectivity problem. AA' and CC' are on the outer face of the surface; BB' and CC' are essentially in the bulk [L > (Ln, A)I.
778
F. F L O R E S A N D G . N A V A S C U E S
If 0 is the angle between the incident wave and the surface normal, we can write the vector E 3 a s follows E3 = E3 (cos O'xo + sin O'zo), where sin0=e~ sin0'. Let us consider now the contour CCD'C'C of fig. 1, and evaluate
fH'dl=~f(V^H)'nds=-i~f; For H r we have eq. (3.5) on CC' and eq. (3.6) on DD'. For E x we have eq. (3.5), neglecting its variation along the boundary layer. Thus we get E1-E2-ebE3=-i-E
co ¢
sii0)2f,xdz 1 L
/
~- t
t (
a 1---
\
(3.7)
0
where El = E3 exp {iq'L} and (X, LD) < L ~ 2. Next we choose ABB'A'A and evaluate
~ E.di= f f (V ^ E)'nds + ico f f This yields L
sin 2 ½ inOf Ea, ( 1----O)eb --(E'+E2) c°sO+i s e~ Ez(z)dz= 0
co
½ ,
= i -¢ LebE3,
(3.8)
with Ez(x,z, t)=E~(z)exp{i(xx-~ot)}. We can evaluate IoLE z(z) solving the equation div (8"E) = 0, which can be deduced from (3.1). Notice that x,~ (1/LD, 1/)I); thus we approximate this equation by
dz
by
OlOx~OlOz because
oo
0 f
(z, z', co) E z (z') dz'
0.
(3.9)
0
Eq. (3.9) determines Ez(z) across the boundary layer as a function of its value at z= L, Ez ( ~ ) . From the eqs. (3.7) and (3.8) we obtain the reflectivity E2 2
R =
IAI2 = IBI
.
TRANSPORT AND FREE CARRIER ELECTROMAGNETIC PHENOMENA. II
where
779
oo
A=(eb--sin20)½--ebc°sO--ico--sin2VJc
o ~Ez-Ez(~)
Ez-(~
dz+
0 oO
°
+ i -- (eb -- sine 0) ~cos0
;
(e~x--eb) dz
C
o
B=(eb--sin20)½
+ebc°s0--i~sin20c
i~-ez(~) ~-(-~
(3.10)
dz-
0
- i -
(~b - s i n e 0 ) ~ c o s 0
(~x -
8b) O z .
C 0
The upper limit of the integrals is written oo because the integrands are negligible for z>L. Notice that, in order to apply eqs. (3.10), we have to solve (3.9) previously. This problem will be considered in the next paragraphs. Here appears the physical effect described in section 2, namely the charge accumulation near the surface due to the field component, E z. For an s-polarization wave, we can follow the same method. However, we need know only the local dielectric function ex~(z), because the electric field is parallel to the surface. For completeness, we give here the result
R = IAI2/IBI2, A = cos 0 - (eb - sin 2 O)~ +
i CO_ci (exx - eb) dz, 0
(3.11)
oo
B = cos 0 +
(eb -- sin 20) ~ -- i CO-fc
(exx --
eb) dz.
0
4. Reflectivity for flat bands With flat bands and a specular surface, the term ~o ( 8x ~ - 8 b ) dz of eqs. (3.10) or (3.11) is zero. However, that is not the case for the third term in A or B, eq. (3.10). In fact, E z varies near the surface, as given by eq. (3.9). To solve now this equation, consider the electric field E z for z > 0 , and its mirror plane image for z<011). In these conditions we can write
~z
f --o0
811
(z -- z'" co) Ez (z') dz' = 4rcQo6 (z), '
(4.1)
780
F. F L O R E S A N D G . N A V A S C U E S
where we have extended eq. (3.9) to the whole space, and changed e~z(z, z'; co) by ell (z-z'; co), which is the longitudinal dielectric function of the infinite crystal. Here, Qo is related to the electric field at z = 0
[Ez( +O)=Z~Qo]. Fourier-transforming eq. (4.1) gives:
Ez (q) -
2~Qo { 1 1 } iell (q, co) q + it/+ q - it/
(4.2)
(t/is a positive infinitesimal quantity). We evaluate the third term of A or B [eq. (3.10)] using eq. (4.2) and the identity
if(z)dz=
1 i qf(q) +it/
0
(4.3)
--oo
[ f ( q ) is the Fourier-transform off(z)]. Thus o0
oo
~ Ez(z)- Ez(oo)
J
~;(~))
dz = - 2ieb
0
f
dq (q + it~)z ell (q, CO)"
(4.4)
--oo
The contribution to this integral of the zeros of elf represents the excitations of volume plasmons by the incident wave lZ). Eq. (4.4) is to be used in the reflectivity formula [eq. (3.10)]. In order to use eq. (4.4), the dielectric function ell (q, co) is needed. This function has been calculated following the formulation of section 2. From eqs. (2.1) and (2.2) f l {t} =
- q Oe .It exp ( - icot) exp (ik. r) x t
X fexp{ik'(r'-r)}exp{(t'-t)(!-ico)}(vr'Et)dt, -o0
assuming an electric field going as exp (it:-r')exp (-icot'). With flat bands, vt, = vt, and r' = r + vt(t'-t). Then, the longitudinal conductivity, all (k, co), is given by (v'k) 2 ~
air (k, co) =
_ qa ~fo) ~
exp[ik.vt'] x
t?e J (
-oo
-oo
Oe ] k z 1/z + i(k'v-co) day" -oo
TRANSPORT AND FREE CARRIER ELECTROMAGNETIC PHENOMENA. II
781
From this and the definition ell = eL + 4hi trll/O9, we get, for effective mass m*
4rcnqZ(m*)ico-1/z ell (q' og) = eL -- i - - - ~ - k r " q2 2
1
,3_
.4rtnq 2 (2) 7 (m*~ 2 (ico-- l/z) - ' rn~*-
\~]
gq i
J r n* (--1/'c + ico) a] exp 2k-T q2 x
1
X {I - err [~ (i~ -- ico) (2~T)2]}.
(4.6,
If q < I(1/z-ico) (m*/2kT)½l, the erf function in eq. (4.6) can be approximated by an asymptotic expansion, yielding
2
e,(q, co)=eL
1
cop
3 (kT/m*) co,q2
co(co+i/z)
co(co+i/'r) 3
} +""
'
where cop2= 4xnq2/eLm.. If q>> ](1/z-ico) (m*/2kZ)~l, eq. (4.6) gives directly
ett (~, ,o) = eL {1 + co~ (o~ + i/r)/m*'x
coq2
~kT) +'"} •
These two limits suggest using as an approximate dielectric function,
eln(q, co) = eL
1
co(co + i/z) 2
_
coqZvZJ,
(4.7)
with v2=kT/m *. Using eqs. (3.10), (4.4) and (4.7), the reflectivity for flat bands becomes
R = IAIZ/IBI 2 with 1
A = (eb -- sin 2 0) ~ -- eb cos 0 +
veosin20(
c eL
co
)7
~
CO/COp
[co(co+ ilz)lco~
-
1] ~'
(4.8) 1 co/coP B = (eb - sin 2 O)~ + eb cos 0 + v eb sin20 (og___~f~) g c eL [CO(co + i/z)/cop2
-
1] ~"
In these expressions, the third term in A or B gives the effect of using a non-local dielectric function. Its relative importance is, however, small. Its highest contribution appears at co = cop, the semiconductor plasma frequency.
782
F. F L O R E S A N D G . N A V A S C U E S
At this frequency, the relation between the third and second terms is close to sin 2 0 v (~OpZ)~ cos
0c
10- 3 sin2 0 (COpZ)~
eL
cos 0
eL
which is negligible for normal values of ~OpZand eL, except for 0 very close to 90 ° .
5. Non-fiat bands Consider now a small bending of the bands produced by a surface potential us. Firstly, we have to solve eq. (3.9) to get E~ across the boundary layer. Similar to the previous paragraph, we consider the electric field E= for z > 0 and its mirror-plane image for z < 0 . Then, we can write, instead of eq. (3.9), its equivalent oo
of
O~z
e~ (z, z') E= (z') dz' = 4~Qo6 (z),
(5.1a)
-oo
where e~l(z, z'), the longitudinal dielectric function, can be written e~l (z, z') = ell (z -- z') + e, (z, z').
(5.1b)
Here el (z, z') is a small perturbation of the longitudinal dielectric function ell (z--z'), which can be obtained following section 2. Eqs. (5.1) can be solved by a perturbation method. Write E~(z')= =Eo(z)+El(z), where Eo(z) is the solution of eq. (4.1) for flat bands. Using eqs. (4.1) and (5.1) and neglecting terms of second order in eI (z, z'), we get
O~z ~ i eo(Z-z')Ea(z')dz'-
t3z ~3 i ex (z, z') E o (z') dz'.
oo
-
oo
This equation gives E 1 (k), the Fourier-transform of E 1 (z),
E, (k) = - F (k)/eo (k),
(5.2)
where F ( k ) = ~ exp (ikz) e, (z, z') Eo(z' ) dz' dz. The variation of the third term of A and B [eq. (3.10)] as a result of the bending of the bands, can be obtained using this value. In fact, from identity (4.3)
~ Ez(z)-E,(OO)dz= IE,(Z) dz= E~(oo i 0
J E~(~) 0
[E~(k)/QOldk"
ieu &
k+it/ -~
(5.3)
783
TRANSPORT AND FREE CARRIER ELECTROMAGNETIC PHENOMENA. II
We can get a more explicit result assuming the surface potential us to be so small that the potential at z is given 2) by u (z) = u s exp ( - z/Lo),
z > 0.
(5.4a)
Then, we approximate e~l(z, z') by the dielectric function ell ( z - z ' ) electronic density at point ½ (z + z'). Then ~, (z, z') =
An [(z'+ z)/2] no
{~o (z - z') - ~L~ (z -- z')} =
z,q
2DL j {~0 (Z -- Z') -- eL6 (Z -- Z')}.
= u s exp
for the
(5.4b)
With this value, eq. (5.2) and finally eq. (5.3) can be evaluated. The result can be obtained in a straightforward way, but will not be given here because it is rather involved. In order to get the reflectivity, we also need to calculate the term S~ {exx (z) -- eb}dZ. Within the approximation given by eq. (5.4a), and putting the Fuch's parameter p = 1, we have oo
f
COP {exx (z) -- eb} dz = - UseLLD (CO + i/z)"
(5.5)
0
With these results, the reflectivity for non-flat bands is given by R'=
1AI21IBI '-, 1
A = (eb -- sin2 0) ½ -- eb COS 0 + -
X
¢
~L o9
CO/COp
x [CO(CO + il~)lCO~
-
-
COieb sin 2 0 1" [El (k)/Qo] k + iq 1] ~ + C 4/t 2 J x -oo
CO
xdk+i
2
(eb--sin 20) ~cos0useLLoCO
c
v,sin20( )2
COP (CO + i/~)'
1
B = (eb-- sin 20) ~ + eb COS0 +
C
m/COp
x [CO(CO + ilr)lCO~ (.0
x dk + i
X
~L
-
-
COie b sin 2 0 i 4n 2
1] ~ + c
[El (k)/Qo] x k + iq 2
(eb -- sin 20) ~ cos OUseLL D CO(CO~P C + i/z)"
(5.6)
784
F. FLORES A N D G . N A V A S C U E S
The new terms appearing in these equations are negligible when compared with the two first ones, because us has been assumed small. However, in an electroreflectance measurement these terms, which represent the effect of the bending of the bands, are essential. When the bending of the bands is not small, our approximation fails. However, in these conditions, the third term in A or B [eq. (3.10)] is small compared with the fourth. Then, the reflectivity can be obtained neglecting the effect of a non-local dielectric function.
6. Surface plasmons Let us now apply the previous results to obtain the effect of the boundary layer on the dispersion relation for surface plasmons. In the reflectivity analysis (section 3), we have considered the incident, El, reflected, E2, and transmitted, E3, waves. From Maxwell's equations, the reflected and transmitted amplitudes have been obtained as a function of the incident one [eqs. (3.7) and (3.8)]. Suppose there is no external stimulus (E 1 =0). Then, E 2 and E 3 are zero unless eqs. (3.7) and 3.8) are compatible. This is the physical situation corresponding to a surface plasmon. Thus, the compatibility of eqs. (3.7) and (3.8) with E l = 0 , gives the dispersion relation for surface plasmons. Now the same result follows from the reflectivity, noticing that R
=
IE'~Ie/IEll ~.
For E 1 = 0 (no incident field), the reflectivity has a pole. We can say that the poles of the reflectivity give the dispersion relation for surface plasmons. In order to apply this result, we must interpret carefully the meaning of the variables appearing in the formula. The reflectivity depends on the frequency co and the incident angle 0. The dispersion relation follows writing this angle as a function of the frequency, ~o, and the wave-length parallel to the surface, to, sin 0 = xc/co. This and eq. (3.10) yield the dispersion relation for surface plasmons, where boundary layer effects are included
(
,,c,,, ,
1
i
c
~(~5
dz-i
o
c x
0
2c2) (11
l
m
(6.1a)
TRANSPORT AND FREE CARRIER ELECTROMAGNETIC PHENOMENA. II
785
When us is small, the two last terms can be obtained from section 5 [see eqs. (5.6)]. When the bending of the bands is important, the third term can be neglected compared with the fourth one. In this case, the main effect of the boundary layer on the surface plasmons is due to the excess conductivity parallel to the surface. If the last two terms in eq. (6.1) are neglected, we get
c°2 :
2
(6.1b) a well-known dispersion relation 14) obtained using a local dielectric function in the whole crystal.
7. Applications, results and conclusions Firstly, we have applied eq. (3.10) to the same model studied by Berz15), using an algorithm which, as such, constitutes a very accurate computational procedure for the situation to which it is applicable. This provides a reliability test for formulae here derived, after approximations from physical theory. As in Berz' work, we shall study a model of accumulation layer on n-InSb assuming exx and to be local and equal. In these conditions it is easy to show by means of eq. (3.9), that
ezz
f Ez-E~(°°) dZ=eb (e~zl(Z)_e~'}dz. 0
(7.1)
0
With this relation, using the same input data as Berz, we evaluate eq. (3.10). The results are shown in fig. 2, and compared with those of ref. 15. The two methods give very close results, which can be taken as a positive test. We now apply our method to the study of some measurements which yield the dispersion relation for surface plasmons in InSb 16). On comparing the experimental data with a theoretical calculation essentially based on eq. (6.1b), where all boundary layer effect are ignored, one finds 16) good agreement at low k; but with increasing k, the experimental frequencies lie below the theoretical values. The authors suggest that this small deviation is due to the effect of the boundary layer. With our eq. (6.1a), we can substantiate this suggestion. Fig. 3 shows the experimental points, the theoretical curve given by eq. (6.1 b), and some theoretical curves deduced from eq. (6.1 a) with different space surface charge. We have assumed a strong inversion layer in the semiconductor surface. Thus, the third term of eq. (6.1a) happens to be negligible as compared to the fourth term. Furthermore,
786
F. FLORES AND G. NAVASCUES
6R!en% 0.3.
In Sb
p-polarization t~'r=2.3 I=5.10" seg.
/ /
O/
//~'"
.../ _t"
/ ~ " ~'"fO;~ f /
0.2
0.1
0.0
-0.1
0'5
1'.0
.1'.5
u s (potential barrier
of surface boundary layer)
Fig. 2. Variation o f the reflectivity as a function of the surface potential, Us, for an accumulation layer on InSb at different angles o f incidence. Contribution o f Reststrahlen is taken into account as in Berz' paper15). Full lines have been calculated from eqs. (3.10) and (7.1). Dotted lines are taken o f ref. 15.
1.(
t
,*t~o.=u.b O =
0.9.
0,8,
0.7.
o
~
~
klkp
Fig. 3. Dispersion o f surface plasmons in InSb. Experimental points ( O ) are taken o f ref. 16. Theoretical curves have been calculated from eq. (6.1a) for different surface excess charge AN. Here c~= e L(AN/n) (ogp/c). Case ct = 0 represents no boundary layer effect.
787
T R A N S P O R T A N D FREE CARRIER E L E C T R O M A G N E T I C P H E N O M E N A . II
we assume a specular surface. In those conditions, the only relevant value of the boundary layer is its surface excess charge, AN, appearing through the magnitude AN COp c
We have drawn three curves for three values of a: 0.5, 1 and 2 (for n = 7 x 1018 electrons/cm 3, AN is 0.5 1 or 2 x 1014 electrons/cm2). Experimental points fall between the curves for a = 0 . 5 and 1, but cannot be fitted by a single curve. This seems reasonable because different doping are used, and in each case a different layer can be expected. The surface excess charge, AN, of the two limiting curves (ct = 0.5, I) is close to 1014 electrons/cm 2. We have shown, in I, that with this order of magnitude it is reasonable to expect an appreciable effect of the boundary layer in the electromagnetic properties of semiconductors surfaces. Furthermore, in analyzing different experimental data a, 4), we have found important effects of the
I
60* ~,r =3 eL=16
I
60 °
=)p~=10 SL=16
l
I I !
I
I I I
J
¢ .. .~._.j
0.316
.
!
i
\\\
"...._.__'~-~.
x\._/I
1.0
0 il
1.0 Wp
Fig. 4. Electroreflectance as a function of frequency for 0 = 60 °. The dashed line gives the electroreflectance curve obtained if one includes only the us dependence of e~x, while assuming e~z unaffected by us. The dash-dotted line gives the curve for the opposite assumption, i.e., only the us dependence of ezz taken into account.
r. FLORES AND G. NAVASCUES
788
boundary layer for a similar surface excess charge. All this substantiates our point of view, namely the role of a space charge boundary layer in analyzing electromagnetic properties of semiconductor surfaces. Another interesting physical feature of the present analysis is the different effect of the surface on the two dielectric functions, exx and ezz, as stressed in section 2. Within the frame of the Fuchs' model for surface scattering, we argue that p may range from 0 to 1 for the calculation of exx, but may be taken as 1 for ez~. The dependence of this parameter on the incident angle of carriers on the surface has been discussed in the literaturea7). It can be expected from our analysis that this angular dependence of the Fuchs parameter affects ex~ and ezz differently. However, until now, the angular dependence of p has been only investigated through the values of e~, frequently surmised from some galvanomagnetic measurements. In this situation it seems advisable to investigate the possibility of obtaining some data about e~z through appropriate reflectivity experiments, in order to increase our information about the angular dependence of the Fuchs' parameter. We have accordingly studied the electroreflectance by way of illustration in a material where eL= 16 and O)pZ ranges from 3 to 10, with two incident angles, 0 = 60 ° and 86 °, and changing the surface potential from zero to a value u~ less than 0.5 kT. Applying the results of sections 4 and 5, we have obtained the curves shown in figs. 4 and 5. These figures demonstrate how
86*
•i
/
/ ', \..\
',\ \A~ \\
,.,,
/
"-..
/ ,~ ,
i/
/
I
\
"-
"<,,~ \
-%
!
,
\ \ \X
\\~, ,
/i
-:\
/
/
'.2
'.3
'.~
~~.......~ t / '.s
~2
~3
'.4
Fig. 5. Same as fig. 4, for grazing incidence, 0 = 86°
~p
T R A N S P O R T A N D FREE C A R R I E R E L E C T R O M A G N E T I C P H E N O M E N A . II
789
the behaviour of exx and ezz affects the electroreflectance (see caption to figs. 4 and 5). With appropriate scaling in the intensity of electroreflectance, these curves are independent of v/c and us. For 0 = 6 0 ° the electroreflectance should be essentially determined by the behaviour of exx. Only for O)pz = 10 does the role of e= become appreciable. Furthermore, the influence of exx is quite similar to the one deduced by Dmitruk and Tyagai 7) for 0 = 0 °. Therefore, there is small variation in the electroreflectance values for a normal angle of incidence or 0 = 60 °. However, for 0 = 8 6 ° we find rather different results. For cop'C=3, the influence of ezz and exx are quite similar. Upon increasing COp'C, the influence of e~ on the electroreflectance is greater, predominating for COpZ=10. Moreover, we can observe in our results a signal around CO/COp= 0.25, which is narrowing for COpZgreater. These results suggest how experimental information about e~ can be obtained by measuring the electroreflectance at grazing incidence. Our theoretical values have been calculated by assuming specular surface scattering of the carriers. A more elaborated treatment of the surface scattering must modify the electroreflectance calculation. We believe that the comparison between these more elaborate calculations and suitable electroreflectance measurements can give valuable information on the surface scattering of carriers, which could complement the information obtainable from galvanomagnetic experiments.
Acklowledgments It is a pleasure to thank Professor F. Garcia-Moliner for his stimulating help and many fruitful discussions.
References 1) D. R. Frankl, Electrical Properties of Semiconductor Surfaces (Pergamon Press, Oxford, 1967). 2) A. Many, Y. Goldstein and N. B. Grover, Semiconductor Surfaces (North-Holland, Amsterdam, 1965). 3) F. Flores, F. Garcia-Moliner and G. Navascues, Surface Sci. 24 (1971) 61. 4) G. Navascues and F. Flores, Solid State Commun. 9 (1971) 1267. 5) J. D. Axe and R. Hammer, Phys. Rev. 162 (1967) 700. 6) J. Zook, Phys. Rev. Letters 20 (1968) 848. 7) N. L. Dmitruk and V. A. Tyagai, Phys. Status Solidi 43 (1971) 557. 8) J. F. Black, E. Lanning and S. Perkowitz, Infrared Phys. 10 (1970) 125. 9) R. G. Chambers, in: The Physics of Metals, Ed. J. M. Ziman (Cambridge University Press, 1969). 10) R. W. O'Donnell and R. F. Green, Phys. Rev. 147 (1966) 599. 11) F. Flores, to be published in Nuovo Cimento. 12) F. Forstman, Z. Physik 203 (1967) 495.
790
F. FLORES AND G.NAVASCUES
13) M. Abramowitz and I. A. Stegun, Handbook o f Mathematical Function (U.S. Department of Commerce, 1966). 14) R. H. Ritchie and H. B. Eldridge, Phys. Rev. 126 (1962) 1935. 15) F. Berz, Surface Sci. 2 (1964) 75. 16) N. Marshall, B. Fischer and H. J. Queisser, Phys. Rev. Letters 27 (1971) 95. 17) R. F. Green, in: Proc. Batelle Colloquium, Molecular Processes on Solid Surfaces (McGraw-Hill, New York, 1969).
TRANSPORT PHENOMENA
AND FREE CARRIER ELECTROMAGNETIC IN SEMICONDUCTOR
BOUNDARY LAYER
II. REFLECTIVITY WITH NON-NORMAL INCIDENCE AND SURFACE PLASMONS* F. FLORES and G. NAVASCUES Centro Coordinado de Fisica ( C.S.L C.-U.A.M.) , Departamento de Fisica, Facultad de Ciencias, Universidad Aut6noma, Canto Blanco, Madrid 34, Spain
Received 15 June 1972 A classical analysis is given of the reflectivity at oblique incidence on semiconductor surfaces. The theory includes the effect of a space charge boundary layer and yields formulae from which one can obtain the dispersion relation for surface plasmons, for which the effects of the boundary layer are particularly important. An application to experimental data on n-InSb gives satisfactory agreement while yielding a plausible estimate of the surface excess AN. The electroreflectance is also studied, including numerical evaluations. It is suggested that measurements at almost grazing incidence could yield valuable information on the surface scattering of carriers. 1. Introduction T h e study o f s e m i c o n d u c t o r b o u n d a r y layers has been the subject o f c o n s i d e r a b l e research. S t a n d a r d t r e a t m e n t s a p p l y especially to dc t r a n s p o r t properties1,2). I n a recent p a p e r 3) (hereafter called I), we also studied ac fields in o r d e r to investigate the m a g n e t o o p t i c a l properties o f the b o u n d a r y layer. In particular, we considered the effect o f this layer on the reflectivity at n o r m a l incidence. This analysis was later a p p l i e d to discuss the optical d e t e r m i n a t i o n o f the carrier r e l a x a t i o n time in semiconductors4). In this case, we c o n c l u d e d that the space charge layer can m o d i f y a p p r e c i a b l y the optical estimate o f the relaxation time. It is b e c o m i n g increasingly o b v i o u s 7) that, for a c o m p l e t e analysis o f electroreflectance d a t a in s e m i c o n d u c t o r s S, 6), one o u g h t to correlate changes in reflectivity with changes in the surface p o t e n t i a l o f the b o u n d a r y layer. The p u r p o s e o f this article is to e l a b o r a t e on this p o i n t a n d to c a r r y * Work partially supported by the General Electric Company (through General El6ctrica Espafiola) and by the "Fundaci6n Pedro Barri6 de la Maza, Conde de Fenosa". The financial support of both these institutions is gratefully acknowledged. 773
774
F. FLORES A N D G . N A V A S C U E S
on our previous research into the general problem of the effect of the space charge boundary layer on the free carrier electrodynamic properties of semiconductors such as, for instance, the important question of surface plasmons. Since this is an electronic excitation, which is basically confined to the neighbourhood of the space charge layer, it is natural to expect that its details may have a non-trivial effect on the dispersion relation of surface plasmons. This will be borne out in section 6, while reflectivity and electroreflectance are discussed in sections 3-5. In section 2, we study the response of the carriers in the boundary layer to an E.M. wave.
2. Study of the conductivity in the boundary layer In I we have calculated the conductivity within the boundary layer, with an ac field parallel to the surface. When this field varies in a distance much larger than the mean free path of the carriers, )~, a local conductivity can be assumed. This situation occurs when an electromagnetic wave arrives at the surface plane of the semiconductor (z = 0) with normal incidence or with non-normal incidence but s-polarization, because 2 ~ ~ (2 is the wavelength). With p-polarization, the electric field has two components, E x and Ez, parallel and normal to the surface. Assuming translational symmetry along the x axis, we can separate the effect of these two components, due to the fact that Ex/Ez induces only a current Jx/Jz. The j~ component is related locally to the field component Ex, as indicated previously, but the Jz component depends non-locally on Ez. In fact the current jz has to change from zero at the surface to some definite value in the bulk, inducing a charge near the surface which must be calculated self-consistently with the field Ez. The depth of this region is close to ~, which usually is larger than a characteristic length of the space charge, the Debye length L D [~/LD ~--~pZ, and usually COpZ> 18)]. We analyze the non-local relation between j~ and Ez, using eq. (7) of I, oo
f, (t) =
G(t, t'
- q - - V'E GI3
dt',
(2.1)
t'
where fl {t}, the deviation of the distribution function from equilibrium, is expressed by means of the local time, t, of the particle, using the path variable methodg). In our case, we have to use the following Green function [see eq. (5) of I] t
G(t,t')=O(t-t')exp{-f
dz~s)}. t'
(2.2)
TRANSPORT
AND FREE CARRIER
ELECTROMAGNETIC
PHENOMENA.
775
II
We choose p = 1 in eq. (5) of I for the following physical reason. In the calculation of Jx, P measures the effect of the surface on the carriers, as ( 1 - p ) gives the average momentum loss along the x axis due to surface scattering. Thus, in the diffuse case (p =0), the average momentum along x becomes zero after reflection. Note, however, that when the induced current j~ is considered, the overall average momentum of the carriers along z is specularly reflected from the surface. Hence, in order to make a reasonably simple assumption about the carrier surface scattering10), we take p = 1 in this paper. A more elaborated hypothesis can modify accordingly our final results, specially the ones related to the electroreflectance. If we write Ez (x, z, t) = Ez ( z) exp {i (tcx - cot)), in eqs. (2.1) and (2.2), assuming z to be a constant and x;[< 1, we get fl{t}=
Of°l e x p { i ( x x - cot)} x t3eAt
-q
×
f exp{-(t--t')(!-ico)}
[v,E,]t, at'.
-oo
Here [v~Ezlt, is a function of past local times, t', the inhomogeneous field, E~, and the initial coordinates (z, v~, t). The current, j~, is given by j= (x, z, t) = i
q f l {t} v, day = exp {i (xx - cot)} x
i/ -ct~ oo
t
63f°lF -oo
t')(1-ico)}{vzE.}t
exp{--
,
.
-oo
Replacing the variable of integration t' by z', we can write oo
L (x, z, t) = exp {i (xx - cot)} [
o~(z, z', co) Ez (z') dz'
(2.3a)
0
with
~x(z,z',co)=~fl-q2Vz~]texp{-(t-t')(~-ico)}d3v, s
(2.3b)
-oo
where ( t - t ' ) is a function of z, z' and v~. [ ( t - t ' ) as a function of z and z',
776
r . FLORES AND G. NAVASCUES
has at least two values, because the carriers can go from z to z' directly or reflecting on the surface. These posibilities are implied by ~s]. Eqs. (2.3) show the non-local relations between the current jz (x, z, t) and the field E~(z) exp {i(xx-cot)}. Observe that the effect on the conductivity of the variation of the field with x has been neglected. With this we can define a non-local dielectric function, e~z, by 4hi ez, (z, z', co) = eL 6 (Z -- Z') + - - e (Z, Z', CO), (2.4) CO where eL is the lattice dielectric function. This and the local dielectric function, ex,, relating Jx to E~, cf. I, allow us to calculate the effect of the boundary layer on the reflectivity.
3. Reflectivity analysis We start with Maxwell's equations 1 c~H VAE . . . . ,
c 8t
V ^ H-
1 8(rE)
c
8t
'
(3.1)
where 8.E= (e~xEx, e~zE~) represents symbolically
e'E=
f
e(x,t;x',t')E(x',t')dx'dt'.
z>0 --oo
Firstly let us study a p-polarized electromagnetic wave. We choose the geometry of fig. 1 :
E=(Ex(x,z,t),O, Ez(X,Z,t)),
H=(O, Hy(x,z,t),O).
With a dependence on x and t going like e x p { i ( x x - c o t ) } , we have explicitly, from eqs. (3.1) dE~ dz
ixE~
09 i-cHy,
dHy i co . . . . . exx (co, z) Ex, dz c co
ixHy = - 1"¢
[",J ezz (co, z, z')
(3.2)
dz'
For z>>~, we have e,~(co, Z)=eb=eL+4rdab/co (trb is the bulk conductivity) and Sezz(co, z, z') Ez(z') dz'=ebEz(z). Then we get from eqs. (3.2) dz 2 +
~e b-x 2 Hy=0.
(3.3)
TRANSPORT AND FREE CARRIER ELECTROMAGNETIC PHENOMENA. II
777
This equation describes the propagation o f a wave along the positive z-direction. Therefore, for z > (~, LD) , we have !
[(°:
Hy = eb~ E 3 exp iz c 2 eb
):1
-/~2
(3.4)
.
To get the reflectivity in a first approximation, we look for an approximate solution to eqs. (3.2). Let us follow the same m e t h o d used in I, assuming 2 >>7,. Consider the incident El, reflected E 2 and transmitted E 3 waves. In the region z < 0 , we have E = E1 exp {i(qz + ~x - o90} + E 2 exp { i ( Hy = E , exp {i ( q z
+ xx
o90} - E2 exp {i (--
-
qz + tcx -
cot)}
qz + xx
cot)},
--
(3.5)
and for z > (~, LD) E = E 3 exp {i ( q ' z
cot)},
+ tcx -
Hy = eb~ E a exp {i ( q ' z
+ xx
-
cot)},
(3.6)
where q,2 + t¢2 = (092/c2) %.
B'
/ / /
,/
//
t
,/Hy (z)
IEx (z)
i
Ez(Z )
tlI
/
/
/
/
/
/
/
z=0 Fig. 1. Geometry for the application of Stokes theorem to the reflectivity problem. AA' and CC' are on the outer face of the surface; BB' and CC' are essentially in the bulk [L > (Ln, A)I.
778
F. F L O R E S A N D G . N A V A S C U E S
If 0 is the angle between the incident wave and the surface normal, we can write the vector E 3 a s follows E3 = E3 (cos O'xo + sin O'zo), where sin0=e~ sin0'. Let us consider now the contour CCD'C'C of fig. 1, and evaluate
fH'dl=~f(V^H)'nds=-i~f; For H r we have eq. (3.5) on CC' and eq. (3.6) on DD'. For E x we have eq. (3.5), neglecting its variation along the boundary layer. Thus we get E1-E2-ebE3=-i-E
co ¢
sii0)2f,xdz 1 L
/
~- t
t (
a 1---
\
(3.7)
0
where El = E3 exp {iq'L} and (X, LD) < L ~ 2. Next we choose ABB'A'A and evaluate
~ E.di= f f (V ^ E)'nds + ico f f This yields L
sin 2 ½ inOf Ea, ( 1----O)eb --(E'+E2) c°sO+i s e~ Ez(z)dz= 0
co
½ ,
= i -¢ LebE3,
(3.8)
with Ez(x,z, t)=E~(z)exp{i(xx-~ot)}. We can evaluate IoLE z(z) solving the equation div (8"E) = 0, which can be deduced from (3.1). Notice that x,~ (1/LD, 1/)I); thus we approximate this equation by
dz
by
OlOx~OlOz because
oo
0 f
(z, z', co) E z (z') dz'
0.
(3.9)
0
Eq. (3.9) determines Ez(z) across the boundary layer as a function of its value at z= L, Ez ( ~ ) . From the eqs. (3.7) and (3.8) we obtain the reflectivity E2 2
R =
IAI2 = IBI
.
TRANSPORT AND FREE CARRIER ELECTROMAGNETIC PHENOMENA. II
where
779
oo
A=(eb--sin20)½--ebc°sO--ico--sin2VJc
o ~Ez-Ez(~)
Ez-(~
dz+
0 oO
°
+ i -- (eb -- sine 0) ~cos0
;
(e~x--eb) dz
C
o
B=(eb--sin20)½
+ebc°s0--i~sin20c
i~-ez(~) ~-(-~
(3.10)
dz-
0
- i -
(~b - s i n e 0 ) ~ c o s 0
(~x -
8b) O z .
C 0
The upper limit of the integrals is written oo because the integrands are negligible for z>L. Notice that, in order to apply eqs. (3.10), we have to solve (3.9) previously. This problem will be considered in the next paragraphs. Here appears the physical effect described in section 2, namely the charge accumulation near the surface due to the field component, E z. For an s-polarization wave, we can follow the same method. However, we need know only the local dielectric function ex~(z), because the electric field is parallel to the surface. For completeness, we give here the result
R = IAI2/IBI2, A = cos 0 - (eb - sin 2 O)~ +
i CO_ci (exx - eb) dz, 0
(3.11)
oo
B = cos 0 +
(eb -- sin 20) ~ -- i CO-fc
(exx --
eb) dz.
0
4. Reflectivity for flat bands With flat bands and a specular surface, the term ~o ( 8x ~ - 8 b ) dz of eqs. (3.10) or (3.11) is zero. However, that is not the case for the third term in A or B, eq. (3.10). In fact, E z varies near the surface, as given by eq. (3.9). To solve now this equation, consider the electric field E z for z > 0 , and its mirror plane image for z<011). In these conditions we can write
~z
f --o0
811
(z -- z'" co) Ez (z') dz' = 4rcQo6 (z), '
(4.1)
780
F. F L O R E S A N D G . N A V A S C U E S
where we have extended eq. (3.9) to the whole space, and changed e~z(z, z'; co) by ell (z-z'; co), which is the longitudinal dielectric function of the infinite crystal. Here, Qo is related to the electric field at z = 0
[Ez( +O)=Z~Qo]. Fourier-transforming eq. (4.1) gives:
Ez (q) -
2~Qo { 1 1 } iell (q, co) q + it/+ q - it/
(4.2)
(t/is a positive infinitesimal quantity). We evaluate the third term of A or B [eq. (3.10)] using eq. (4.2) and the identity
if(z)dz=
1 i qf(q) +it/
0
(4.3)
--oo
[ f ( q ) is the Fourier-transform off(z)]. Thus o0
oo
~ Ez(z)- Ez(oo)
J
~;(~))
dz = - 2ieb
0
f
dq (q + it~)z ell (q, CO)"
(4.4)
--oo
The contribution to this integral of the zeros of elf represents the excitations of volume plasmons by the incident wave lZ). Eq. (4.4) is to be used in the reflectivity formula [eq. (3.10)]. In order to use eq. (4.4), the dielectric function ell (q, co) is needed. This function has been calculated following the formulation of section 2. From eqs. (2.1) and (2.2) f l {t} =
- q Oe .It exp ( - icot) exp (ik. r) x t
X fexp{ik'(r'-r)}exp{(t'-t)(!-ico)}(vr'Et)dt, -o0
assuming an electric field going as exp (it:-r')exp (-icot'). With flat bands, vt, = vt, and r' = r + vt(t'-t). Then, the longitudinal conductivity, all (k, co), is given by (v'k) 2 ~
air (k, co) =
_ qa ~fo) ~
exp[ik.vt'] x
t?e J (
-oo
-oo
Oe ] k z 1/z + i(k'v-co) day" -oo
TRANSPORT AND FREE CARRIER ELECTROMAGNETIC PHENOMENA. II
781
From this and the definition ell = eL + 4hi trll/O9, we get, for effective mass m*
4rcnqZ(m*)ico-1/z ell (q' og) = eL -- i - - - ~ - k r " q2 2
1
,3_
.4rtnq 2 (2) 7 (m*~ 2 (ico-- l/z) - ' rn~*-
\~]
gq i
J r n* (--1/'c + ico) a] exp 2k-T q2 x
1
X {I - err [~ (i~ -- ico) (2~T)2]}.
(4.6,
If q < I(1/z-ico) (m*/2kT)½l, the erf function in eq. (4.6) can be approximated by an asymptotic expansion, yielding
2
e,(q, co)=eL
1
cop
3 (kT/m*) co,q2
co(co+i/z)
co(co+i/'r) 3
} +""
'
where cop2= 4xnq2/eLm.. If q>> ](1/z-ico) (m*/2kZ)~l, eq. (4.6) gives directly
ett (~, ,o) = eL {1 + co~ (o~ + i/r)/m*'x
coq2
~kT) +'"} •
These two limits suggest using as an approximate dielectric function,
eln(q, co) = eL
1
co(co + i/z) 2
_
coqZvZJ,
(4.7)
with v2=kT/m *. Using eqs. (3.10), (4.4) and (4.7), the reflectivity for flat bands becomes
R = IAIZ/IBI 2 with 1
A = (eb -- sin 2 0) ~ -- eb cos 0 +
veosin20(
c eL
co
)7
~
CO/COp
[co(co+ ilz)lco~
-
1] ~'
(4.8) 1 co/coP B = (eb - sin 2 O)~ + eb cos 0 + v eb sin20 (og___~f~) g c eL [CO(co + i/z)/cop2
-
1] ~"
In these expressions, the third term in A or B gives the effect of using a non-local dielectric function. Its relative importance is, however, small. Its highest contribution appears at co = cop, the semiconductor plasma frequency.
782
F. F L O R E S A N D G . N A V A S C U E S
At this frequency, the relation between the third and second terms is close to sin 2 0 v (~OpZ)~ cos
0c
10- 3 sin2 0 (COpZ)~
eL
cos 0
eL
which is negligible for normal values of ~OpZand eL, except for 0 very close to 90 ° .
5. Non-fiat bands Consider now a small bending of the bands produced by a surface potential us. Firstly, we have to solve eq. (3.9) to get E~ across the boundary layer. Similar to the previous paragraph, we consider the electric field E= for z > 0 and its mirror-plane image for z < 0 . Then, we can write, instead of eq. (3.9), its equivalent oo
of
O~z
e~ (z, z') E= (z') dz' = 4~Qo6 (z),
(5.1a)
-oo
where e~l(z, z'), the longitudinal dielectric function, can be written e~l (z, z') = ell (z -- z') + e, (z, z').
(5.1b)
Here el (z, z') is a small perturbation of the longitudinal dielectric function ell (z--z'), which can be obtained following section 2. Eqs. (5.1) can be solved by a perturbation method. Write E~(z')= =Eo(z)+El(z), where Eo(z) is the solution of eq. (4.1) for flat bands. Using eqs. (4.1) and (5.1) and neglecting terms of second order in eI (z, z'), we get
O~z ~ i eo(Z-z')Ea(z')dz'-
t3z ~3 i ex (z, z') E o (z') dz'.
oo
-
oo
This equation gives E 1 (k), the Fourier-transform of E 1 (z),
E, (k) = - F (k)/eo (k),
(5.2)
where F ( k ) = ~ exp (ikz) e, (z, z') Eo(z' ) dz' dz. The variation of the third term of A and B [eq. (3.10)] as a result of the bending of the bands, can be obtained using this value. In fact, from identity (4.3)
~ Ez(z)-E,(OO)dz= IE,(Z) dz= E~(oo i 0
J E~(~) 0
[E~(k)/QOldk"
ieu &
k+it/ -~
(5.3)
783
TRANSPORT AND FREE CARRIER ELECTROMAGNETIC PHENOMENA. II
We can get a more explicit result assuming the surface potential us to be so small that the potential at z is given 2) by u (z) = u s exp ( - z/Lo),
z > 0.
(5.4a)
Then, we approximate e~l(z, z') by the dielectric function ell ( z - z ' ) electronic density at point ½ (z + z'). Then ~, (z, z') =
An [(z'+ z)/2] no
{~o (z - z') - ~L~ (z -- z')} =
z,q
2DL j {~0 (Z -- Z') -- eL6 (Z -- Z')}.
= u s exp
for the
(5.4b)
With this value, eq. (5.2) and finally eq. (5.3) can be evaluated. The result can be obtained in a straightforward way, but will not be given here because it is rather involved. In order to get the reflectivity, we also need to calculate the term S~ {exx (z) -- eb}dZ. Within the approximation given by eq. (5.4a), and putting the Fuch's parameter p = 1, we have oo
f
COP {exx (z) -- eb} dz = - UseLLD (CO + i/z)"
(5.5)
0
With these results, the reflectivity for non-flat bands is given by R'=
1AI21IBI '-, 1
A = (eb -- sin2 0) ½ -- eb COS 0 + -
X
¢
~L o9
CO/COp
x [CO(CO + il~)lCO~
-
-
COieb sin 2 0 1" [El (k)/Qo] k + iq 1] ~ + C 4/t 2 J x -oo
CO
xdk+i
2
(eb--sin 20) ~cos0useLLoCO
c
v,sin20( )2
COP (CO + i/~)'
1
B = (eb-- sin 20) ~ + eb COS0 +
C
m/COp
x [CO(CO + ilr)lCO~ (.0
x dk + i
X
~L
-
-
COie b sin 2 0 i 4n 2
1] ~ + c
[El (k)/Qo] x k + iq 2
(eb -- sin 20) ~ cos OUseLL D CO(CO~P C + i/z)"
(5.6)
784
F. FLORES A N D G . N A V A S C U E S
The new terms appearing in these equations are negligible when compared with the two first ones, because us has been assumed small. However, in an electroreflectance measurement these terms, which represent the effect of the bending of the bands, are essential. When the bending of the bands is not small, our approximation fails. However, in these conditions, the third term in A or B [eq. (3.10)] is small compared with the fourth. Then, the reflectivity can be obtained neglecting the effect of a non-local dielectric function.
6. Surface plasmons Let us now apply the previous results to obtain the effect of the boundary layer on the dispersion relation for surface plasmons. In the reflectivity analysis (section 3), we have considered the incident, El, reflected, E2, and transmitted, E3, waves. From Maxwell's equations, the reflected and transmitted amplitudes have been obtained as a function of the incident one [eqs. (3.7) and (3.8)]. Suppose there is no external stimulus (E 1 =0). Then, E 2 and E 3 are zero unless eqs. (3.7) and 3.8) are compatible. This is the physical situation corresponding to a surface plasmon. Thus, the compatibility of eqs. (3.7) and (3.8) with E l = 0 , gives the dispersion relation for surface plasmons. Now the same result follows from the reflectivity, noticing that R
=
IE'~Ie/IEll ~.
For E 1 = 0 (no incident field), the reflectivity has a pole. We can say that the poles of the reflectivity give the dispersion relation for surface plasmons. In order to apply this result, we must interpret carefully the meaning of the variables appearing in the formula. The reflectivity depends on the frequency co and the incident angle 0. The dispersion relation follows writing this angle as a function of the frequency, ~o, and the wave-length parallel to the surface, to, sin 0 = xc/co. This and eq. (3.10) yield the dispersion relation for surface plasmons, where boundary layer effects are included
(
,,c,,, ,
1
i
c
~(~5
dz-i
o
c x
0
2c2) (11
l
m
(6.1a)
TRANSPORT AND FREE CARRIER ELECTROMAGNETIC PHENOMENA. II
785
When us is small, the two last terms can be obtained from section 5 [see eqs. (5.6)]. When the bending of the bands is important, the third term can be neglected compared with the fourth one. In this case, the main effect of the boundary layer on the surface plasmons is due to the excess conductivity parallel to the surface. If the last two terms in eq. (6.1) are neglected, we get
c°2 :
2
(6.1b) a well-known dispersion relation 14) obtained using a local dielectric function in the whole crystal.
7. Applications, results and conclusions Firstly, we have applied eq. (3.10) to the same model studied by Berz15), using an algorithm which, as such, constitutes a very accurate computational procedure for the situation to which it is applicable. This provides a reliability test for formulae here derived, after approximations from physical theory. As in Berz' work, we shall study a model of accumulation layer on n-InSb assuming exx and to be local and equal. In these conditions it is easy to show by means of eq. (3.9), that
ezz
f Ez-E~(°°) dZ=eb (e~zl(Z)_e~'}dz. 0
(7.1)
0
With this relation, using the same input data as Berz, we evaluate eq. (3.10). The results are shown in fig. 2, and compared with those of ref. 15. The two methods give very close results, which can be taken as a positive test. We now apply our method to the study of some measurements which yield the dispersion relation for surface plasmons in InSb 16). On comparing the experimental data with a theoretical calculation essentially based on eq. (6.1b), where all boundary layer effect are ignored, one finds 16) good agreement at low k; but with increasing k, the experimental frequencies lie below the theoretical values. The authors suggest that this small deviation is due to the effect of the boundary layer. With our eq. (6.1a), we can substantiate this suggestion. Fig. 3 shows the experimental points, the theoretical curve given by eq. (6.1 b), and some theoretical curves deduced from eq. (6.1 a) with different space surface charge. We have assumed a strong inversion layer in the semiconductor surface. Thus, the third term of eq. (6.1a) happens to be negligible as compared to the fourth term. Furthermore,
786
F. FLORES AND G. NAVASCUES
6R!en% 0.3.
In Sb
p-polarization t~'r=2.3 I=5.10" seg.
/ /
O/
//~'"
.../ _t"
/ ~ " ~'"fO;~ f /
0.2
0.1
0.0
-0.1
0'5
1'.0
.1'.5
u s (potential barrier
of surface boundary layer)
Fig. 2. Variation o f the reflectivity as a function of the surface potential, Us, for an accumulation layer on InSb at different angles o f incidence. Contribution o f Reststrahlen is taken into account as in Berz' paper15). Full lines have been calculated from eqs. (3.10) and (7.1). Dotted lines are taken o f ref. 15.
1.(
t
,*t~o.=u.b O =
0.9.
0,8,
0.7.
o
~
~
klkp
Fig. 3. Dispersion o f surface plasmons in InSb. Experimental points ( O ) are taken o f ref. 16. Theoretical curves have been calculated from eq. (6.1a) for different surface excess charge AN. Here c~= e L(AN/n) (ogp/c). Case ct = 0 represents no boundary layer effect.
787
T R A N S P O R T A N D FREE CARRIER E L E C T R O M A G N E T I C P H E N O M E N A . II
we assume a specular surface. In those conditions, the only relevant value of the boundary layer is its surface excess charge, AN, appearing through the magnitude AN COp c
We have drawn three curves for three values of a: 0.5, 1 and 2 (for n = 7 x 1018 electrons/cm 3, AN is 0.5 1 or 2 x 1014 electrons/cm2). Experimental points fall between the curves for a = 0 . 5 and 1, but cannot be fitted by a single curve. This seems reasonable because different doping are used, and in each case a different layer can be expected. The surface excess charge, AN, of the two limiting curves (ct = 0.5, I) is close to 1014 electrons/cm 2. We have shown, in I, that with this order of magnitude it is reasonable to expect an appreciable effect of the boundary layer in the electromagnetic properties of semiconductors surfaces. Furthermore, in analyzing different experimental data a, 4), we have found important effects of the
I
60* ~,r =3 eL=16
I
60 °
=)p~=10 SL=16
l
I I !
I
I I I
J
¢ .. .~._.j
0.316
.
!
i
\\\
"...._.__'~-~.
x\._/I
1.0
0 il
1.0 Wp
Fig. 4. Electroreflectance as a function of frequency for 0 = 60 °. The dashed line gives the electroreflectance curve obtained if one includes only the us dependence of e~x, while assuming e~z unaffected by us. The dash-dotted line gives the curve for the opposite assumption, i.e., only the us dependence of ezz taken into account.
r. FLORES AND G. NAVASCUES
788
boundary layer for a similar surface excess charge. All this substantiates our point of view, namely the role of a space charge boundary layer in analyzing electromagnetic properties of semiconductor surfaces. Another interesting physical feature of the present analysis is the different effect of the surface on the two dielectric functions, exx and ezz, as stressed in section 2. Within the frame of the Fuchs' model for surface scattering, we argue that p may range from 0 to 1 for the calculation of exx, but may be taken as 1 for ez~. The dependence of this parameter on the incident angle of carriers on the surface has been discussed in the literaturea7). It can be expected from our analysis that this angular dependence of the Fuchs parameter affects ex~ and ezz differently. However, until now, the angular dependence of p has been only investigated through the values of e~, frequently surmised from some galvanomagnetic measurements. In this situation it seems advisable to investigate the possibility of obtaining some data about e~z through appropriate reflectivity experiments, in order to increase our information about the angular dependence of the Fuchs' parameter. We have accordingly studied the electroreflectance by way of illustration in a material where eL= 16 and O)pZ ranges from 3 to 10, with two incident angles, 0 = 60 ° and 86 °, and changing the surface potential from zero to a value u~ less than 0.5 kT. Applying the results of sections 4 and 5, we have obtained the curves shown in figs. 4 and 5. These figures demonstrate how
86*
•i
/
/ ', \..\
',\ \A~ \\
,.,,
/
"-..
/ ,~ ,
i/
/
I
\
"-
"<,,~ \
-%
!
,
\ \ \X
\\~, ,
/i
-:\
/
/
'.2
'.3
'.~
~~.......~ t / '.s
~2
~3
'.4
Fig. 5. Same as fig. 4, for grazing incidence, 0 = 86°
~p
T R A N S P O R T A N D FREE C A R R I E R E L E C T R O M A G N E T I C P H E N O M E N A . II
789
the behaviour of exx and ezz affects the electroreflectance (see caption to figs. 4 and 5). With appropriate scaling in the intensity of electroreflectance, these curves are independent of v/c and us. For 0 = 6 0 ° the electroreflectance should be essentially determined by the behaviour of exx. Only for O)pz = 10 does the role of e= become appreciable. Furthermore, the influence of exx is quite similar to the one deduced by Dmitruk and Tyagai 7) for 0 = 0 °. Therefore, there is small variation in the electroreflectance values for a normal angle of incidence or 0 = 60 °. However, for 0 = 8 6 ° we find rather different results. For cop'C=3, the influence of ezz and exx are quite similar. Upon increasing COp'C, the influence of e~ on the electroreflectance is greater, predominating for COpZ=10. Moreover, we can observe in our results a signal around CO/COp= 0.25, which is narrowing for COpZgreater. These results suggest how experimental information about e~ can be obtained by measuring the electroreflectance at grazing incidence. Our theoretical values have been calculated by assuming specular surface scattering of the carriers. A more elaborated treatment of the surface scattering must modify the electroreflectance calculation. We believe that the comparison between these more elaborate calculations and suitable electroreflectance measurements can give valuable information on the surface scattering of carriers, which could complement the information obtainable from galvanomagnetic experiments.
Acklowledgments It is a pleasure to thank Professor F. Garcia-Moliner for his stimulating help and many fruitful discussions.
References 1) D. R. Frankl, Electrical Properties of Semiconductor Surfaces (Pergamon Press, Oxford, 1967). 2) A. Many, Y. Goldstein and N. B. Grover, Semiconductor Surfaces (North-Holland, Amsterdam, 1965). 3) F. Flores, F. Garcia-Moliner and G. Navascues, Surface Sci. 24 (1971) 61. 4) G. Navascues and F. Flores, Solid State Commun. 9 (1971) 1267. 5) J. D. Axe and R. Hammer, Phys. Rev. 162 (1967) 700. 6) J. Zook, Phys. Rev. Letters 20 (1968) 848. 7) N. L. Dmitruk and V. A. Tyagai, Phys. Status Solidi 43 (1971) 557. 8) J. F. Black, E. Lanning and S. Perkowitz, Infrared Phys. 10 (1970) 125. 9) R. G. Chambers, in: The Physics of Metals, Ed. J. M. Ziman (Cambridge University Press, 1969). 10) R. W. O'Donnell and R. F. Green, Phys. Rev. 147 (1966) 599. 11) F. Flores, to be published in Nuovo Cimento. 12) F. Forstman, Z. Physik 203 (1967) 495.
790
F. FLORES AND G.NAVASCUES
13) M. Abramowitz and I. A. Stegun, Handbook o f Mathematical Function (U.S. Department of Commerce, 1966). 14) R. H. Ritchie and H. B. Eldridge, Phys. Rev. 126 (1962) 1935. 15) F. Berz, Surface Sci. 2 (1964) 75. 16) N. Marshall, B. Fischer and H. J. Queisser, Phys. Rev. Letters 27 (1971) 95. 17) R. F. Green, in: Proc. Batelle Colloquium, Molecular Processes on Solid Surfaces (McGraw-Hill, New York, 1969).