Transport and free carrier electromagnetic phenomena in semiconductor boundary layers

SURFACE

SCIENCE 24 (1971) 61-76 0 North-Holland

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Publishing Co.

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ELECTROMAGNETIC

IN SEMICONDUCTOR

BOUNDARY

LAYERS *

F. FLORES Gabinete

de Aplicaciones

Nucleares

de O.P., J.E.N.

Madrid,

Spain

and F. GARCIA-MOLINER Institute

’ ‘RocasoIano”,

and G. NAVASCUES Serrano

119, Madrid,

Spain

Received 22 June 1970 Standard semiconductor surface transport theory is rewritten in a classical Green function formulation physically equivalent to the path variable method. This makes it possible to account for surface boundary conditions, while improving upon the relaxation time solution of the Boltzmann equation itself. The extension to ac fields is trivial. The complex conductivity is written (in the relaxation time approximation) as a function of frequency, position in the inhomogeneous boundary layer, surface potential, polish parameter and external magnetic field. The way to study long wave electromagnetic properties of carriers in boundary layers is indicated and illustrated with applications to the theory of reflectivity as affected by boundary layers.

1. Introduction The behaviour

of carriers

in space charge

boundary

layers near surfaces

under external fields has been the subject of considerable attention for obvious reasons. Standard treatments have been long known l, 2, for dc fields, describing the bulk collision mechanism in the relaxation time approximation. The correlation between chemisorption studies and the ensuing changes in the electronic properties of the adsorbent can be a very useful research technique and it seems desirable to widen the scope as much as possible. It emerges from a review 3, of the current literature that the common practice in physicochemical research consists in resorting to the simplest dc transport measurements, namely, conductivity, Hall effect and sometimes thermopow-

* Work partially supported by the General Electric Company (through General Electrica Espafiola) and by the “Fundacion Pedro Barr% de la Maza, Conde de Fenosa”. The financial support of both these institutions is gratefully acknowledged. 61

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F. FLORES,

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AND G. NAVASCUES

er. While in standard semiconductor physics the study of magnetooptical and electromagnetic properties in general has been a most active and fruitful field, comparatively very little has been done for semiconductor surface systems. Metal surfaces4), on the contrary, and thin films as wells), have been studied much more extensively. The physical situation, of course, is very different. The problem of surface scattering is about the same, and remains almost equally obscure in both cases, but in semiconductors one can usually do with a local theory, except for rather extreme (although realizable) situations. On the other hand the semiconductors present a large scale inhomogeneity, namely the space charge boundary layer. These are the facts which make the semiconductor case different, and the purpose of this article is to indicate how to include them in an analysis of the standard kind extended to describe the behaviour of carriers in the boundary layer under the influence of ac fields. To this end the standard treatment is reformulated in a manner which lends itself to this, as well as to an extension beyond the relaxation time approximation. The treatment of the long wave electromagnetic properties is then outlined and applied to the study of reflectivity as a function of the potential barrier height.

2. Classical Green function formulation of standard dc surface transport We shall treat the surface scattering in the standard crude manner which consists in using a fudge factor p for the degree of polish of the surface. In spite of recent commendable attempts6) to improve upon this, it does not seem that real progress has been achieved, to the extent of being able to use a better method for actual explicit calculations. All counted, the old Fuchs method is still perhaps a practical empirical procedure when one considers the other aspects of the problem which remain to be studied. Even using a polish parameter, the boundary condition that this imposes on the distribution function at the surface complicates the solution of the Boltzmann equation rather considerably. The standard treatment is perfectly correct but has one feature which makes it unsuitable for easy extensions of the theory. This is a change of variable which is introduced in the differential equation, after which the intuitive picture is somewhat lost. The philosophy of the work reported here is that it is more convenient to rewrite the formalism in such a way that the boundary condition is automatically built in, while the treatment remains formally as simple as in the bulk problem. One can then see more clearly how to concentrate on further improvements (notably a more careful treatment of bulk scattering). To this end it is convenient to use a time-dependent formulation, even if one were interested only in dc fields. Thus, in the relaxa-

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we start with the differential

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equation

(1) which so far is exact. Here dldt means the total hydrodynamic derivative. As in the path variable method49 ‘) we use the symbols r(t’) and k(t’) which describe the collision free trajectories and, at the local (present) time t, take on the values r and k of the independent variables in the time-independent formulation. The equilibrium distribution f. depends on z (the coordinate perpendicular to the surface) because of the inhomogeneous boundary layer, and so doesf. We assume in (1) an arbitrary magnetic field (although we do not consider quantization effects in this paper) and an arbitrary electric field parallel to the surface. Now, the differential operator on the left-hand side of (1) describes the time evolution of the distribution function f (r, k, t), which in the same spirit we shall write as f {t}. It is also the differential operator of the linearized eauation fi = - q ‘;;

WE,

(2)

where fi is f-f0 and d is the energy of the carrier. Both, eqs. (1) and (2) are completely defined after the boundary conditions have been specified. We define the corresponding Green function by G(t, t’) = d(t - t’) plus the given boundary

conditions.

The form of G is immediately

(3) found to be:

G(t, t’) = A(3(t - t’) exp .f’ where the step function expresses the principle of causality, and A is a factor which only takes the values zero or one according to the boundary conditions on the differential equations. Consider three cases of interest: a) A bulk system. Then A= 1 and the Green function solution of either eq. (1) or eq. (2) reproduces the well known Chambers formula. b) A specular surface. Then A = 1. c) A diffuse surface. In this case A=O(t’-tB), where t, is the time in the past at which, going backwards along its collision free trajectory, the particle was at the surface. We define the specular and diffuse Green functions, G, and G,, with A= 1 and A= f?(t’ - tB) respectively, and then an effective

64

Green

F. FLORES,

F. GARCIA-MOLINER

AND

G. NAVASCUES

function

G,=pGs+(l

-P)%

f

= O(t - t’) [p + (1 - p) o(t’ - te)] exp

is

f’

25 r 6) I

(5)

and the formalism is all set to study surface effects with one adjustable empirical parameter. Part of the hard work is thrown into the calculation of t, = t- tn. This is the length of time taken up by the carrier under the sole influence of the fields, having started from the surface at t’ = t,, to reach the local time 1 with position r = r(t) and k = k(t). This includes the effect of the inhomogeneous internal field, so that t, is a functional of the electrostatic profile of the boundary layer. But the form of the Green function solution is the same as if there were no surface. The boundary condition is built into G,. At this stage one may note the following points: (i) The Green function of eq. (5) formally inverts the time evolution operator. Suppose one is interested in studying surface transport beyond the relaxation time approximation. Then D, is the sum of two linear operators, the second one being given by the collision integral. The point is that such a sum can be formally inverteds) treating one of them as a constant parameter. That is to say, we can take 7-r to stand for the linear integral collision operator and the formal inversion of D, is exactly the same. Then one has to work out in detail what 7-r really is. This is the standard problem of inverting the collision operator of the Boltzmann equation, which can be treated as a separate problem, for which there exist well known procedures. But the surface boundary condition is already included in the effective Green function which inverts D,. Herein lies the hope of solving the surface transport problem while taking advantage of better treatments of the bulk scattering. This point is under current investigation and will not be further mentioned in this paper. ii) The same Green function solves the linear and the non-linear cases. One can thus study non-ohmic effects in the boundary layer. This is also under current investigation and will not be treated in this paper, which will be concerned with the linear case. Since we are also using a relaxation time and in fact the most plausible models of collisions tractable in this way yield a r which depends only on energy but not on the vector k, we shall also simplify the treatment to this case. Then the exponential in the Green funcf tion is simplified to exp{-Jr:J

=exp{-c
which will be used everywhere henceforth. [Notice plied by E in eq. (7) to be written down presently.]

that this is to be multi-

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iii) What physical

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we have written

solution

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in eq. (5) is a causal

of eq. (2) is written m

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response

in the standard

function.

The

form

-CO

which is of course the Chambers formula itself with surface-boundary conditions incorporated. Also, for semiconductors it is easily seen that this formula, when used to calculate transport coefficients, reproduces all the well-known formulae192) for surface mobility, etc., from standard dc surface transport theory. If numerical considerations advised it so, there would be no essential difficulty in extracting from eq. (7) non-local formulae for semiconductors, as is usually done in metals. In this paper we shall be concerned only with the local approximation for semiconductors, putting the emphasis instead on the large scale inhomogeneity. The program to be presently developed has the following steps: (a) Calculation of ac conductivity and (b) insertion into Maxwell equations (section 3); (c) derivation of a formula for the reflectivity (section 4), and application to some examples (section 5). 3. AC fields and long wave electromagnetic

analysis

We have remarked that the Green function of eq. (4) is a causal response function. Thus its Fourier transform has all the analytical properties granted by standard dispersion theory, and this is an incidental advantage of the reformulation expounded in section 2. We shall be concerned with wavelengths much larger than the thickness of the boundary layer, a situation often met with in practice. For a semiconductor, disregarding non-local effects, we have to evaluate the integral of eq. (7) with an electric field of the form E, exp( -iwt). Also, being in the long wave limit, we shall not consider differences between longitudinal and transverse fields. Including a static magnetic field perpendicular to the surface, the result of evaluating eq. (7) is fi = g1 (0, z> J% exp (-

iwt) ,

(8)

with ”

= -

afO (7-l - iw) V, - wcVY 88 ’ 0,2 + (7-l _ iw)’ 1 -(l

x

1

-P)exp{-&(z-‘-iiw)} wcVx +

(7-l - iw) V

cos WJ, + ___ wcVy - (7-l

’ sin@, - io) V,

II

.

(9)

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The dependence

F. GARCIA-MOLINER

AND

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in f0 and in t,, which is a functional

on z is contained

of the

electrostatic profile q(z) in the boundary layer. From this one evaluates o(H,, w z) in the standard manner, H, being the external magnetic field which enters eq. (9) through 0,. For H, = 0, p = 0 and flat band conditions, the result is equivalent to a formula used by Yi-Han Kaos) to study optical size effects in metals. With the full formula for a(H,, w, z) one can proceed to study magnetooptical phenomena in the boundary layer. Clearly some average of this function over z will appear in the formulae for the macroscopic phenomenological coefficients, but each case has to be studied separately. We shall consider explicitly the case H, =O. We start from Maxwell’s equations 1 aH V/YE=--

m,

c at

where sL is the lattice dielectric constant, and we write explicitly the current as the conductivity times the field. It is here that the physics which went into the calculation of d enters the analysis, i.e., we describe (though in a simple minded way) the linear response of carriers in an inhomogeneous layer and under the possible effect of surface scattering. Setting E equal to the electric field associated with fluctuations in the carrier system we can obtain from here dispersion relations for long wave surface plasmons (propagating parallel to the surface). Work on this problem is in progress and will not be further discussed here. We shall discuss instead the response to an external electromagnetic field. We choose the geometry as shown in fig. 1, i.e. we take

E=

(E,(z, t), CO> and

With time dependence

H = (0, H, (z, t), 0).

going like exp( -iot),

we have to deal explicitly

with

=x =iyHy, C

aZ

aH, aZ

47l

0

c

C

=-o(o,z)E,-i

Hence the propagation of an electromagnetic given by the usual equation a2E,

co2

dz2

--ll c2

sL+iF--

i.e. we have an inhomogeneous

47Lo(w, z) Co

dielectric

.zLEx.

wave through

1

E,--%(w,z)E,,

function.

C2

the medium

is

(12)

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67

Now, with eqs. (12), or (11) and (9) we can study the electromagnetic properties of carriers in boundary layers. We shall assume the long wave limit in the sense A%&,

(13)

where L, is some effective measure of the thickness of the boundary layer. This condition is usually met with in practice in a typical semiconductor. In fact L, is something like a Debye-Hiickel length. If we define k, from the plasma frequency wP by oP=c k,, we have

k,L,

N v G 1 c

(14)

(here P is an average thermal velocity). This condition means essentially that the absorption of the electromagnetic wave on crossing the distance LD is negligible. However, if there is a strong surface current, then this results in a large change in the tangential component of the magnetic field. We shall therefore assume that E, varies negligibly but HY may vary appreciably. This can be illustrated by studying the reflectivity under the stated conditions.

Fig. 1. Geometry for the application of the 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.

F. FLORES,

F. GARCIA-MOLINER

AND

0. NAVASCUES

4. Study of the reflectivity

We consider the incident (E,), reflected (EJ and transmitted tudes. On the outer face of the surface we have

(E3) ampli-

E, = E, exp - (iwt) + E, exp( - iwt), H,,=E,exp(-iot)-E,exp(-iot),

(1%

and inside, for z 2 L,, E, = E, exp {i(kz - wt)>, Hy = &Es exp {i (kz - ot)j .

(16)

Here and henceforth E,, means the bulk dielectric function sL +

i47ccrb(0)/o.

We choose the contour CDD’C’C of fig. 1 and evaluate ~~-d~~jj

(V A H)*rr ds = ;jj

5(w,

z)

EL a

E*n ds + - c at jj

E-n ds.

(17)

For H,, we have eqs. (15) on CC’ and eqs. (16) with ZN L, on DD’. Using eq. (16) for E, we can write E, - E, - EWE; N 4” L,o;E; c

- i 0 EILEEN, c

fW

where IS, is defined by LO Lg,

=

s

o(o, z) dz,

(19)

0

and E; = E3 exp(ikL,,), i.e., practically E,. Next we choose ABB’A’A and evaluate (20)

This yields exp (- iot) [Ej -(E,

+E,)]=-

:;jjE&nds.

(21)

Assuming also negligible variation of HYfrom AA’ to BB’ we have Ej - (E, + E,) = ikL,E; _

GYJ

A word of caution is in order here. The damping of the electromagnetic wave due to absorption on crossing the length L, inside the material may be negligible, but if we contemplate appreciable surface effects with a large

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then this results in a large change

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in the tangential

compo-

nent of the magnetic field. On going from eq. (21) to eq. (22) we have replaced H,,(z) by H,(L,). Under conditions in which H,(-

0) g

H,(L,j ,

(23)

this procedure would seem to be in error. It is easily seen that this is not likely to change the results very much. At most on the right-hand side of eq. (22) we should have E; - (El + I?,) = i F L&,(O). Now, if the inequality tely

(24)

eq. (23) holds, then from eq. (18) we have approxima-

H,,(O) = E, - E, = f? L,a,Ej, c i.e., substituting

(25)

in eq. (24) Ej - (E, + E,) = ikLD

E;

.

In actual practice (as can be verified in the numerical example to be considered presently) a strong surface current, capable of causing an appreciable decrease in H,,, means that the magnitude of (47~ LD o,/c $) is at most of order unity and, because of the inequality eq. (13), both eqs. (22) and (26) mean that Ej-(E,+E,)
(27)

so in both extreme cases either equation expresses the same physical fact, namely, that E, remains practically unchanged. We shall therefore use the simpler eq. (22). With this and eq. (18) we evaluate the reflectivity

(28)

This can be written

in a more transparent

form as

[l - {&&oj>‘] [I - ikL, {E&B)}‘] R(w, cp,) = ~____

_-

I[1 + {E&J))+] where

- 2

~~-___ [I - ikL, {s&$)-l]

47r Aa(o,

4%) ’ ,

-~~ + ?

47~Aa(o,

VP,)

(29)

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F. FLORES,

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is the excess surface conductivity. We write in cpSto recall that this incorporates the effects of a boundary layer of height qp,. This reduces to the standard formula in the limit L,-+O (no boundary layer) or do-+0 (no surface effects) as it should be. The term in k&,&t is in fact negligible under typical conditions in the infrared. Aspnes and FrovarO) (henceforth AF) have studied essentially the same problem, though in a slightly different context, and derived a formula similar to our eq. (29). It is worth discussing in some detail the relationship between the two formulae, because it turns out that in a way they complement each other. Let us formulate the problem in the spirit of AF, but solving it under the assumptions of the present work. We have to solve eq. (11). Therefore we Put

&(,,Z)~E1(0)+&Z(W,Z),

(31)

where si

is the (homogeneous) bulk dielectric function. Next E, = F1 + Fz, where F1 satisfies eq. (11) with si instead of E, i.e.

we put

(32) The boundary condition is that F2 vanishes as z+co. Instead of using a W.K.B. method we seek F, as the function which obeys the residual equation d2F2

o ’

dz2 --+(

c1 (

Ed + .s2) F2 = -

The term in F, is negligible (under the approximations of the present work) bacause F2 N 0 beyond z z L,, and its increment on going from z=L, to z=O is of the order of F, (0) = Li (d2F/dz2). This makes the second term on the left-hand side of eq. (33) everywhere smaller than the first one. Introducing this simplification we have, after solving eq. (33), m c)‘! Furthermore,

putting

s2 (z”) dz” .

dz’

F,(O) = C, 0

(34)

s

z’

Hy = B, + B,, and using eqs. (34) and (10): a, s2 (z) dz .

(35)

Now, for z>O we have E, (z, t) = (F, + F,) exp (-

ior),

Hy(z, t) = (B, + B,) exp (-

iot) .

(36)

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Setting

z=O,

matching

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to eqs. (15) and using eqs. (35) and (34) we obtain m (37)

and

The second term inside the last bracket is in fact of order (k2 L’, C2/q) and therefore negligible compared with unity. This finally yields

‘1

-&+iz

s2(z)dz I

s O

I-

R =

2

m

I

’.

cc

(1+ E?- i 0

s2 (z) dz~

s co

(

(39)

I

This is, in fact, the same as eq. (29) if the term in kL, is neglected, and the same as eq. (5) of Aspnes and Frovals) if the said approximation is used there. In short, the restriction of our work which was not present in AF’s work is the assumption of negligible absorption of the electric field on crossing the boundary layer. On the other hand we derive the same formula even allowing for fast variations in e(z) [ or g(z)] which, in the free carrier case, means essentially a large surface potential barrier. The significant feature of this formula, as pointed out by AF, is that the roles of the real and imaginary parts of the dielectric function are interchanged in the extra term which represents the effect of the boundary layer. An incidental comment on the thin film problem is also in order, since this is a situation in which surface effects can be of paramount importance. To use a differential equation with a z-dependent E would then be rather clumsy, due to extra boundary conditions, but the Stokes theorem method is readily adapted. It suffices to define the third region as the vacuum, i.e., putting Eb = 1 in eq. (28) we can write directly, for a film of thickness D,

’ [ 0

D

&C-_ 2-00 c The definition

_

4w (4

[ -4&+i(t,+ 0

2 +i(eL,--

1) (40) 1l)]i

.

of o’s is still the same as before, with L,n replaced

by D.

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5. Numerical examples We shall consider two cases. The first one is a calculation performed by Berzlr) for a model of accumulation layers on n-InSb. The same problem was faced, namely, to study the effect of a boundary layer on the reflectivity. The method used was in fact an algorithm, rather than an attempt at a physical theory. It is therefore interesting to compare it with the results of a calculation performed for the same situation with eq. (29). We refer specifically to fig. 6 of Berz, and to the curve for normal incidence. We have performed a calculation ab initio, using the same input data as Berz, and evaluating a(o, z) from our eq. (8). The reflectivity was then calculated at a fixed frequency (as indicated in the legend to our fig. 2) as a function of cp,. SubSR

0.006

I

1

[

;-_

0005.

/

LLr

0.5

1.0

1.5

2.0

.~~

2.5

J

Lp5, “RT

~.

3.0

Fig. 2. Absolute change in reflectivity, 6R = R(& - R(O), in fraction of incident energy. Accumulation layers on InSb at T = 300 K. Curves calculated from eq. (29) including Reststrahlen effects in EL. w = 4.6 x lOl3 secl. Bulk relaxation time: rr, = 5 x lo-l3 sec. Relaxation time used for each curve: t = F x r&O, for F = l(1) 10. The figure shown for each curve gives the value of Fused in its calculation. Berz’ calculation corresponds to F= 1.

tracting the reflectivity at zero surface potential one obtains 6R, which is plotted here in fig. 2 as a function of cp,lkT. A comment is in order here. In an attempt to take some account of possible surface scattering, Berz used a relaxation time equal to one tenth of the bulk relaxation time, and then considered specular surfaces. We have therefore performed the calculation for a specular surface. To use throughout the bulk relaxation time divided by ten seems in fact arbitrary and rather too drastic, so the calculation was

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performed for ten different input values of r, up to the full bulk value. It is seen that this does change the results quite considerably. The curve with F= 1 corresponds to the same input data as used by Bet-z. The results with both methods are rather similar in absolute magnitude, although the derivative with respect to cp, is quite different. Perhaps such differences could be measured with current electroreflectance techniques, which in any case would be necessitated by the very small magnitude of 6R under these conditions. The second example concerns some measurements12) carried out with the purpose of studying the effect of carriers in the boundary layer on the reflectivity. There are several reasons why a naive use of eq. (29) would be unwarranted. The samples were sufficiently doped that the boundary layer is in fact rather degenerate, a proper account of electric quantization might be somewhat necessary, and the data show a temperature dependence in the high frequency region whose origin is not entire clear. There is also, for the actual samples used, too much uncertainty in the input data which would be necessary for a calculation entirely ab initio. Nevertheless it is interesting to see what comes out of an empirical fitting. An attempt was first made to fit the experimental data without invoking any surface effects at all, using the standard text-book formula for the reflectivity. There was no possible way of reproducing the experimental curves. Even the nearest possible fit (which was still considerably poor) was at the expense of fudging in impossible values for parameters like bulk relaxation time and average effective mass, which were off by at least two orders of magnitude with respect to the standard sort of values to be expected. Then eq. (29) was used, relying on a purely empirical best fit, and exploring whether it was capable of reproducing the shape of the experimental curves and the significant way in which they change with temperature. (We shall presently comment on the shape, i.e., the frequency dependence of the curves.) The results are shown in fig. 3. The dots are experimental and the curves are theoretical. Here are some of the input parameters which turn out to give the best fit: cL = 12.7, z is of the order of lo-l4 set, and varies but little in this temperature range, n/m* is of the order of 104’ cmd3 g -’ and again varies little. If a standard value of order 1O-2g g is used form”, then this yields a bulk carrier concentration of order IO” cm-3. This roughly agrees with the indication about impurity contents of the samples and a statistical estimate of the carrier concentration to be expected. All these values seem quite plausible. From the empirical best fit we also surmise a surface excess of about 5 x 1Or4 cme2 for curve (6) going down to about a third of this value for curve (2) and then decreasing abruptly to a comparatively negligible amount for curve (1). Now, with such carrier concentrations, the thickness of the boundary layer is of the order of a few tens of A, whereas the light penetrates into a

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much larger distance, and it might seem at first unreasonable to expect such large surface effects from such a very narrow layer. The point, however, is that the concentration of carriers in this layer is so high that in a depth of up to 1 urn, with a bulk concentration of order 10” cm-3, the light wave interacts with about lOi electrons/cm’, and this is of the same order as the excess number of electrons/cm’ in the boundary layer, so that it is not really unreasonable to expect a large surface effect. A final comment is in order as regards frequency dependence. The term in Q,(O) is quite clear, the question R (%)

l/k(cm-I)

Fig. 3. Reflectivity versus 1-l in n-Si at different temperatures. Theoretical curves calculated by fitting eq. (29) to Sato’s experimental data shown in dots. Temperatures: (1, n ) T = 295 K; (2, l ) T = 373 K; (3, A) T = 423 K; (4, 0) T = 473 K; (5, 0) T = 523 K; (6, A) T = 573 K.

concerns Ao, as a function of w. In a true aprioristic calculation this would have to come from using eqs. (8) and (9), in the first place, in evaluating ~(w, .a). For the reasons stated above this was not warranted in this case, and the attitude taken was totally empirical. The effects of surface scattering would undoubtely change somewhat the form of the frequency dependence, but such effects were neglected in fitting the data. We simply assumed Aa, to go like some empirical value of AoS(0), at zero frequency, times a factor (1 - io 7) -I. This is evidently making virtue of necessity, the hope being that at this stage of crudeness one may expect minor details to be fudged into the empirical estimate of AaS( The important point is that we are faced with a very large surface conductance : For sample No. 6 we estimate

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LDAos(0)-6

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x 103umho.

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It is from hAas

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that we estimate

75

the values of

AN quoted above. With such high accumulation layers, it would not be unreasonable to suspect the onset of electric quantization. If this were the case the tendency then would be to decrease the probability of surface scattering for the conduction electrons, which would tend to make more plausible the assumption about the frequency dependence of da,. On the face of it the agreement between theoretical curves and experimental data is, in fact, surprisingly good, actually better than anything one might expect from overstretching a really crude picture, and it might be to some extent fortuitous. Nevertheless the point is made; the overall picture one ends up with, and the numerical values of the parameters which fit it, are everywhere plausible, although it would be very wrong to take this as an accurate estimate or proof of anything.

6. Conclusion As stated in the introduction, there were several motivations for undertaking this work. One was to reformulate surface transport theory in a way in which one can see how to go beyond the relaxation time approximation. Another one was to study the electrodynamics of carriers in the boundary layer. This might help to broaden the scope for physicochemical studies of chemisorption, by estimating the effects to be expected from different measurements aimed at probing into the linear response of surface carriers, under various experimental conditions, as chemisorption proceeds. But there is also a purely physical question. All measured electromagnetic properties can in principle be affected by the state of affairs at and near the surface. In metal physics one is forced to pay attention to this because of the skin effect, but in semiconductor physics the possible role of the boundary layer is seldom mentioned. It is not impossible that a fair number of experimental data could admit a fruitful reinterpretation keeping such effects in mind. An example might be the reflectivity data just discussed.

Acknowledgments It is a pleasure to thank A. Frova and several colleagues at the 1970 Winter College on the Theory of Imperfect Crystalline Solids (ICTP, Trieste) for stimulating discussions. References 1) A. Many, Y. Goldstein and N. B. Grover, Semiconductor land, Amsterdam, 1965). 2) D. R. Frank], Electrical Properties of SemiconductorSurfaces

Surface Physics (North-Hol-

(Pergamon, Oxford, 1967).

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AND

G. NAVASCUES

3) F. Garcia-Moliner, Catalysis Rev. 2 (1968) 1. 4) R. G. Chambers, in: The Physics ofMet&, Ed. J. M. Ziman (Cambridge Univ. Press, 1969). 5) P. R. Baraff, Phys. Rev. 178 (1969) 1155. 6) R. W. O’Donnell and R. F. Greene, Phys. Rev. 147 (1966) 599; R. F. Greene, in: Molecular Processes on Solid Surfaces, Eds. E. Drauglis, R. D. Gretz and R. I. Jaffee (McGraw-Hill, New York, 1969). 7) H. Budd, Phys. Rev. 158 (1967) 798. 8) B. Friedman, Principles and Techniques of Applied Mathemathics (Wiley, New York, 1956). 9) Yi-Han Kao, Phys. Rev. 144 (1966) 405. 10) D. E. Aspnes and A. Frova, Solid State Commun. 7 (1969) 155. 11) F. Berz, Surface Sci. 2 (1964) 75. 12) K. Sato, J. Phys. Sot. Japan 27 (1969) 89.