Theory of anisotropic photoelectromagnetic effects in germanium and silicon type semiconductors

J. Phys.

Chem. Solids

THEORY

Pergamon

Press 1966. Vol. 27, pp. 597-609.

OF ANISOTROPIC

Printed in Great Britain.

PHOTOELECTROMAGNETW

EFFECTS IN GERMANIUM

AND SILICON TYPE

SEMICONDUCTORS Yu. KAGAN I. V. Kurchatov’s

Institute of Atomic Energy,

Moscow, USSR

and V. SOBAKIN Moscow Engineering

Physical Institute,

(Received

27 February

Moscow,

USSR

196.5)

Abstract-The microscopic theory of anisotropic photoelectromagnetic effects in semiconductors is considered. General expressions are obtained which are valid in the case of an arbitrary value of the magnetic field and arbitrary crystal orientation. The case of germanium and silicon type semiconductors is analysed in detail. Results of numerical calculations are presented and a comparison of theory with experimental data is given.

1. INTRODUCTION

within the framework of a subsequent microscopic examination which took into consideration the actual form of the dispersion law for electrons and holes. A microscopic theory of anisotropic photoelectromagnetic effects that is valid in the case of arbitrary values of the magnetic field is developed in the present paper. Particular attention is devoted to semiconductors of the Ge and Si type, for which the carrier spectrum is well known. The results of the theory yielded practically the same pattern of anisotropy for n-Ge as that observed in.(s) On the other hand, it was found that under the same geometry of measurements the behaviour of the anisotropic even photoelectromagnetic effect in p-Ge must be quite different from that in n-Ge. This stimulated the carrying out of a new series of measurements for p-Ge, the results of which are presented in Ref. 7. Experimental and theoretical results were again found to be in excellent conformity. Qualitative conformity with the theory was also observed in paper(s) where ultra high magnetic fields were used. The theory of the anisotropic odd photoelectromagnetic effect is also examined in the present

A SHARPLY

pronounced pattern of anisotropy was detected by KIKOIN and BYKOVSKIY~) in 1957 during a study of the even photoelectromagnetic effect in cubic crystals of germanium. On the basis of the consideration of symmetry, one of the authors and SMORODINSKIY(~) subsequently provided a phenomenologic description of the phenomenon for a case when the magnetic field is small. A detailed study of the anisotropic events-5) and odd(s) photoelectromagnetic effects were subsequently undertaken by I. K. Kikoin and S. D. Laxarev and I. K. Kikoin and I. N. Nikolayev. It was found, that in the case of weak magnetic fields all the observed relationships are in excellent conformity with the results of the phenomenologic examination. However, a quite new and extremely non-trivial nature of the even photoelectromagnetic effect with respect to the magnitude of the magnetic field and of its direction with respect to the axes of the crystal was experimentally found in the case of a transition to large magnetic fields and low temperatures in n-Ge.(s) These results could only be explained 597

Yu.

598

KAGAN

and

paper. Unfortunately, there is little corresponding experimental data available at the present time. As far as the relationships observed in(“) are concerned, they are explained entirely within the framework of the theoretical results. I’roblems associated with the general isotropic photoclcctromagnetic ctl‘cct are not examined in the paper. ‘I’he corresponding theory was dcvcloped by van ROOSBROIXK.(~) 2. GENERAL EXPRESSION FOR THE ELECTRIC FIELD OF THE PHOTOELECTROMAGNETIC EFFECT

We consider a semiconductor plate in an arbitrary magnetic field with one of its sides illuminated continuously by light. The quasiclassical kinetic equation for the distribution function of particles of the p type under steadystate conditions has the following form:

In addition to the collision integral (8fa/Zt)c,,,, in the right part of (1) there are terms NB and MB describing respectively creation and recombination of particles. The creation of particles is associated with the absorption of light in the sample resulting in the excitation of electron-hole pairs. Thus, if the axis of a light fall direction is x and the thermalization path of excited carriers is small in comparison to the characteristic light absorption path A, it is possible to write N8

=

cge-@‘“)exp

is the correction to the Here jjl’ = f8-jj”’ equilibrium distribution function. We write the collision integral in a relaxation time approximation:

() at

coil =--*

SOBAKIN

We find the solution of equation

(1) in the form

f/g= f’A”+f ‘p”= fT’(l +x&,

(5)

where fk”’

=

exp[pP-k;(p) ]_

Here cp(P) represents the energy of a quasiparticle measured from the bottom of the conduction band for electrons or from the top of the valence band for holes. pti reprcscnts the chemical potential corresponding to the total number of particles of a given type per unit volume. In the solution of the problem in an arbitrary magnetic field we turn, as usual, from the variables pz, pY and pt to the variables PH, EB and #a, where pH represents the projection of the quasimomentum in the direction of the magnetic field, lfl represents the energy of a quasi-particle and tis = mat, where t represents the motion time along a closed trajectory in a magnetic field in momentum space, while wB = e/H/cm; represents the cyclotron frequency for a quasi-particle with an effective mass mf (see, for exampIe,( Then we obtain for x0 the equation

(2)

The recombination of particles in general case is due to the action of a series of capture mechanisms that are characteristic for the given model. Within the framework of the ordinary approximation we introduce the single lifetime l/X,. Then MS = A,&‘;‘. (3)

af5

V.

fs

(1)

78(P)

(4)

Under ordinary conditions $T@) < 1. Thercfore, the solution of equation (7), which is periodic with respect to +B, can be written in the form:

PHOTOELECTROMAGNETIC

EFFECTS

The electric current generated the B type, is determined by equation

.

Jp =

(2,nj3

IN GERMANIUM

by carriers of the following

s s

2eF d3Py8f/?@)

AND SILICON

SEMICONDUCTORS

599

model, the total current i = CjPS B

(13)

integrated over a cross section of the specimen, is equal to zero.(*) This leads to the following system of equations :

=- 2ea d3P‘V8fyxp (2743

d

(je)dz = 0,

s 0

In the expression (8) the second term obviously does not contribute to the electric current. In the first term we expand V, in a Fourier series in #B and we replace ~~(68, pH, tia) by the following value, which is averaged over the trajectory in momentum space.

d

s

(jm)dN- = 0,

(14)

0

Here Q represents the unit vector along the z axis (in the direction of the light fall) and e represents the unit vector along the axis of measurement of Then it is possible to perform the integrations over #” and #’ in an explicit form. Expanding V, in (9) also in Fourier series and performing the integration here over I/~, after some simple manipulations we finally find

Here

FIG. 1.

Geometry of the photoelectromagnetic experiment.

effect

the e.m.f. of the photoelectromagnetic effect, rn = [qx e]. (It was possible to remove the integration in the last relation in (14), inasmuch as jz is not a function of z, which follows directly from equation of continuity.) If the equation curl E = 0 is used, in the geometry under consideration Let us now turn to the determination of an electric field corresponding to the photoelectromagnetic effect. For this purpose we examine a monocrystal semiconductor plate, the width of which is small in the incident direction of light in comparison to the two other dimensions (see Fig. 1). Then all the physical magnitudes, especially &‘B and fp',will be a function of the coordinate z only. Under stationary conditions and in the case of disconnected ends of the

P

= const,

EY = const,

(15)

and Et, generally speaking, is a function of z. We solve the last of the equations of (14) with respect to Ez and insert the result in the first two equations. We limit ourselves to examination of the case of a weak signal (AC?,Ap < IZO,~0, where no and PO are the equilibrium values of the electron and hole concentrations, respectively, and An and Ap represent corrections to them due to the photoexcitation of the carriers). In addition, we assume,

600

Yu. KAGAN

and V. SOBAKIN

as usual, that the relative hole concentrations in those valence bands which participate in the conduction current are preserved and are equal to the equilibrium relation. Then, keeping in mind the equations of (15), we arrive at the following expression for the electric field of the photoelectromagnetic effect (a) for a rz type semiconductor (no < ~0):

(b) for a p type semiconductor

(pa$,ne):

paper we are mainly interested in the anisotropic photoelectromagnetic effect and its dependence upon the magnitude and direction of H, and therefore, we omit the determination of the absolute values of the photoelectromagnetic effect (see, for example Ref. 10). We factor out the expressions for the e.m.f. of the even photoelectromagnetic effect (which does not change with the substitution of H by -H) and the odd photoelectromagnetic effect (which changes a sign for such a reversal) in an explicit form in (16) and (17). Whereupon for a n type semiconductor we have :

AT

(E+e) = - -.

ed

Here tensor

P~do)

- 4441 n0

#k = (&)-1 represents the resistivity for the corresponding component of the

current carriers, while or = $+ I&, where K = ps/pl = ApslApl. (Bearing in mind the application of the results to Ge and Si type semiconductors, we allow for the presence of two valence bands, whereupon the index “1” refers to the band of “heavy” holes and index “2” refers to the band of “light” holes.) It must be emphasized that it is possible to see from the expressions of (16) and (17) that the behavior of the electric field of the photoelectromagnetic effect as a function of the magnitude and direction of the magnetic field will be quite different in semiconductors of the n and p type, inasmuch as it is described each time by its combination of tensors; in the general case the dependence of these tensors upon H is different. The absolute determination of the magnitude of the photoelectromagnetic effect requires a knowledge of the difference [API(O) - API(~)] or [An(O)- An(d)]. These differences can be determined from the solution of the appropriate diffusion equation obtained from the equation of continuity for the individual components of the current carriers and the inclusion, in addition to the equations of (15), of the relation

dEz = 4rre(Apr+Ap,-An), dZ as well as the boundary for surface recombination.

conditions which allow However, in the present

(Ge)

=

_

~~[*~l(O)--*~Ml ed

nn

Here and below the indexes “ +” and “ - ” designate respectively the even and odd magnitudes with respect to the magnetic field (as consequence of the Onsager principle, the even tensors in this expression are symmetrical, while the odd tensors are antisymmetrical). Evidently, the same expressions will be obtained for a p type semiconductor by the substitution of the indexes p + n and -

An(O) -An(d)

PlO -

I*

Expressions (18) and (19) contain anisotropic, as well as isotropic parts of the photoelectromagnetic effect. It is of interest to factor out in an explicit form that part of the e.m.f. of the photoelectromagnetic effect which is due entirely to anisotropy of this effect in the crystal. In an isotropic semiconductor the general expressions for a second rank tensor in a magnetic field are of the following form

HiHk A”k = a(Hz)?Yk+ C(H?)T + (20) Aik = @$2)&$

11

PHOTOELECTROMAGNETIC

EFFECTS

IN GERMANIUM

It is easy to prove directly from (20) and (18-19) that if the magnetic field lies in a plane perpendicular to the vector e, the isotropic -part of an even photoelectromagnetic effect vanishes identically and nonzero value of the e.m.f. in that case is associated entirely with the anisotropic part of the effect. If the magnetic field lies in a plane formed by the vectors e and q, the isotropic part of the odd effect vanishes identically. Therefore, it is found possible to factor out purely anisotropic effects in the case of a completely arbitrary orientation of the crystal. In the subsequent analysis of the even and odd effects we will assume that the magnetic field lies precisely within the planes indicated above. Let us clarify the problem of the asymptotic behavior of the e.m.f. of the photoelectromagnetic effect in the case of large magnetic fields. As it appears, this problem can be solved in a general case for a crystal of arbitrary symmetry and for arbitrary orientation of the magnetic field. Indeed, for this purpose it is sufficient to use asymptotic values of the components of the conductivity tensor (and, consequently, the resistivity tensor) which can be obtained for closed particle trajectories in momentum space (as always occurs in semiconductors) in the most general case (see, for example Ref. 11). The appropriate analysis proves directly that, irrespective of the typeof crystal, (E+e)H,,

+ cdnst,

(E_e)H_+

Within the limit of small magnetic most general case &k +

- i.

(21)

fields in the

= Llk + LlkstHsHt,

On the basis of this, we immediately find from (18) and (19) that the expansion E+ begins with a term that is not dependent upon H and the expansion E_ begins from a term that is linear with respect to the field. However, in the case of a cubic crystal which is of primary interest for us, purely anisotropic effects (compare(s)) have the following dependence on the modulus of the magnetic field (E+e) - H2, We

(E-e) N H3.

(22)

note that in certain special cases, when the

AN’D SILICON

SEMICONDUCTORS

601

light falls along a high symmetry axis in a crystal, the quadratic term in the even effect vanishes and

(E+e) - Ha.

(224

A complete study of anisotropic photoelectromagnetic effects requires a knowledge of the tensor oik for arbitrary values of the magnetic field and arbitrary crystal orientation. For this, in turn, it is necessary to know the energy spectrum of the electrons and holes and a comprehensive solution of the problem of the motion of quasiparticles with such a spectrum in the magnetic field. We now turn to semiconductors of the germanium and silicon type. The structure of the energy bands of such semiconductors has been thoroughly studied in experiments on cyclotron resonance.(ls) The energy surfaces of the conduction band consist of a specific number (4 for Ge and 6 for Si) of isolated ellipsoids of rotation with a large ratio of the semiaxes and the necessary symmetry is provided by these ellipsoids only together, that is a reason of a very pronounced anisotropy of the conduction electron dispersion law. A maximum of the valence band is situated at p = 0. At the same time this point is the point of degeneracy in the hole spectrum which, together with anisotropy, leads to a nontrivial nonanalytic energy dependence of cr, on quasi-momentum. A strict examination of hole dynamics with such a dispersion low results in extremely complex calculations even in the case when the direction of the magnetic field coincides with a high order symmetry axis in the crystal (see, for example, Ref. 9, where the field was directed along the [O,O,lJ axis). It is clear that the angular relationships of the ordinary galvanomagnetic effects (such as, for example, the change of resistance in the magnetic field) in models containing only electrons or holes will be determined by the anisotropy of the dispersion law of the corresponding type of carriers. However, what is very essential is the fact that in any type of equilibrium conductivity, the photoelectromagnetic effect is determined by the diffusion of non-equilibrium carriers of both signs. And inasmuch as there occurs such an extremely pronounced anisotropy of the total energy surface for the electron band and a practically weak

602

Yu.

KAGAN

and

anisotropy of the valence bands that is associated only with “crimping” of the spherical energy surfaces, it is obvious that the effect of the anisotropy of the electron dispersion law upon the photoclcctromagnetic cffcct will be the predominant one. In view of this fact, it is found possible to USC the cxprcssion, avcragcd with rcspcct to angles, for the hole dispersion law of both types with isotropic ctfcctive masses. This approximation will be examined below. 3.

DETERMINATION

of:

OF THE AND uf

pH = const.

these

designations,

= 0.

(27)

Here (11 =

1-po(~,~)2,

a2 =

pO(naS)nf,

m>

a3 = l--&~z~)~-z~, where

?zt = (n,H)/H.

PJP,,

An analysis of equation (27) indicates that real positive solutions occur on the condition x, > .Z,O = (1 --~a)(1 -&&).2). We take the corresponding Z,O value yo = yon;t/ni as the origin and we introduce a polar system of coordinates with the variables r and $. We then have

(23)

CLo(P@l,

we find from (23):

yza~-2ya~+a~

5 =

The equations of (23) d escribe a curve in a plane perpendicular to the magnetic field vector along which the motion of the quasi-particle occurs in momentum space. First let us examine the motion of an electron belonging to one of the ellipsoids. We write the corresponding dispersion law in the form:

%dP> = 60+ &I+

Using

TENSORS

As is known, the energy magnitude and the quasi-momentum projection in the direction of the magnetic field are conserved in the case of the semiclassical motion of a quasi-particle with an arbitrary dispersion law in the magnetic field:

e(p) = const,

V. SOBAKIN

z, - x,0

r2 =

(29)

1 -cLo(%%)z’ where rzr = (cos +, sin (b) The carrier usual manner

effective

mass is determined

s

2n87-2

m

m* = CL

(24)

in the

2rr

-dcj. ax,

0

where EO represents the energy value of the point of minimum; n, represents the unit vector of the major axis of the ellipsoids; ~0 = (1 -(m&)), mt and ml represent respectively the transverse and longitudinal effective masses (p is measured from the minimum point). We introduce the dimensionless magnitudes

z, =

Qsa-EO ~

(25)

CP;/24 and y=P,,

(26) PH

where p, represents the quasi-momentum ponent that is normal to the direction magnetic field.

of

comthe

After

the

calculation

we

obtain,

as should

be

expected: 7n* @-

mt X0(1

-Po)+Po(fN*

(30)

In order to find the Fourier components of velocity we need the explicit form of the relation between the magnitude # introduced in the previous section and the polar angle 4. Using the known relation for motion time along a trajectory in momentum space (see, for example Ref. 13), we have

PHOTOELECTROMAGNETIC

Keeping in mind relation-directly

(29),

EFFECTS

we find

the

IN GERMANIUM

following

(31)

t.& = cos f+&,+ sin #nz*(~on~n$J

where nz’ a’ ny’ d represent the Cartesian components of the vector ni. Then for the quasi-particle velocity components V,” = &,,/8pa we have

SEMICONDUCTORS

603

in an explicit form, we substitute the expressions of (33) and (30) into (11). It is necessary to know the explicit form of ena and pa-dependence of Q,, in order to calculate the integral in (11). However, even in the case of an extremely simple form of this dependence integration can only be carried out each time numerically for a particular direction of the magnetic field. In order to analyse the problem in a general case, we replace +:na(~nor,pa) by (7%) , which will be considered a function of the temperature and orientation of the magnetic field with respect to the axes of the crystal. (In an anisotropic case the differential character of the averaging of T over the whole phase volume will correspond to the various orientations of the magnetic field.) Introducing the dimensionless magnitude

and performing the integration in (11) in an explicit form, after some manipulations, we find in this approximation that for an arbitrary system of coordinates

X ~~cosp~~~+.:.(~)sin~~~~). [ Applying to (32) the Fourier the variable $J~~, we find

AND SILICON

transformation

in

e2(98>

&k n

= _.A.[(sik-v”k_tElkZ~Z~Xs)

mt (34)

+y(sik+Erkz,z+XIXk)]. v:’

= $J(

1 :~;(:;)a)

xW

The following designations

-r.co(n$)2l(s,,l+s,,-l):,

1---

A+,=~

(s )

I

a

v:=g2%o%yo-J( 1 -+)boC) [

np,J+s,,_+in~

x (~~,l-S~,-l) In order

o1 l.+y(l+s”)+(xn,)”

u.

mt

II

to calculate

(1 -L

.

the tensor

otk = c 2 R a

’

I

(35)

(6&1-s5,_1) )

x(%J+%,-l)-i

x

\

are used here:

&k

=

-

1

c

A r+

n a nki+

1 +y(l +X2)+(X&

Proceeding quite analogously, .it is easy to find the expression for the tensor G; as a function of the dimensionless magnitude x in an isotropic approximation :

Yu. KAGAN

604

where forj hP

we find:

= 0, 1, 2

@P,Y +gp @PJ

=

I

and V. SOBAKIN

1-I- x2(ApJ2



mt APr = -.-, %I

1+x2(ApJ2’

(37)

]-?J = [fy ‘[16yc

i = 1,2;

*,+I:

c A,AB @#B

a



-&(1+X2)

2 2 f/‘,j

A,A,A,],

Ge;

X

4. ANISOTROPIC PHOTOELECTROMAGNETIC EFFECT. DISCUSSION OF RESULTS. COMPARISON WITH EXPERIMENTAL RESULTS

The results obtained in the previous section enable to find an explicit expression for the electric field of an anisotropic photoelectromagnetic effect. Actually, we compute the inverse tensors in (34) and (36) and we factor out the symmetrical and antisymmetrical sections of the direct as well as the inverse tensors in an explicit form. Substituting the results in (18) and (19) we find after some manipulations: (a) for the n type semiconductor (E+e)

= ---kT

[*PI(O)-4

d

[9r 2 A,+ ; 2 &A, Y. s+B --+(1+x2)

2 A,A@A,], n+B+l

1

(b) for a p type semiconductor: kT [An(O)-An(d)] (E+e) = -. ed PlO x (ey -hf+P

Si. (40)

A ‘h,,

+A

%$“)Q”)

(Xfy+X2(hy

’

(41)

:41 e52 -. 2 mtm%

‘20

(42) The expressions obtained are too cumbersome to be easily amenable to analysis in a general case. It seems more practical to us to begin an examination of the results for limit cases corresponding to weak and strong magnetic fields. In the case of a small magnetic field, when x4 1, expansion of (38) and (39) yields for n type semiconductors : kT

where +tj = v~l[l+y(l+xs)]+(v~r)-lj~I+V~SV~kXdXk Here ]a] is a determinant of the tensor At**, in order to find this determinant we use some general formula which is easily verified in a direct fashion:

(E+e) = --. ed

P%(O)

x x2{eQ

-*PMI

9*,1(1 +A

* N”(2 + 3y)2

no

rthfz: cos2B,)qk),

(38a)

OT (E-e)

= !$.[Apl(o)~Apl(d)l.

NZrJjy)”

]A] = 9[(T~~)a+2(T~~~~)--(T~~)(Traa)]. Taking

into consideration

that

&[46,,,- (- l)a+fi], (%zns) =

6 afl

,

Ge; Si;

a (39a)

I’NOTOELECTROMAGNETIC

and for a

(E+e)

EFFECTS

IN GERMANIUM

AND SILICOri

a p type semiconductor:

type kT

[An(O)-An(d)]

ed

PI0

= -x

605

SE%lICONDUCTORS

Ai1

‘(1-M2v +gP)

xycyc?+l; COS”e&Jf,

(E,.e) -+ z.

[An(o~~oAn(d’l .(a_t:: Pa

A_‘) P

Yl

(414

a

(E-e)

=

2. ‘An(o)pTbe(d)l +y~~~gp) at1

x x3(ef [ -

cfkJ(2

nic0s38,)

a

(424 Here h = H/H, cos 8, = (hn,) and N represents the number of the constant energy surface ellipsoids. The magnetic field dependence in the above relation conforms to that predicted in the general case of (22) (see also(s)). It is interesting to note that the even photoelectromagnetic effect in conformity to (38a) and (4laj, as well as the odd photoelectromagnetic effect, according to (39a) and (42a) possess within this limit a similar angular dependence for n and p type semiconductors. The fact that the anisotropic odd effect begins with the term ,. Ha in the case of small magnetic ticlds was experimentally ascertained in paper(s) on p-Ge models. Let us examine a case of a large magnetic field, when x & 1. From the general expressions of (38-39) and (4142) ne find: (a) for a II type semiconductor:

z& =

n”d a a z: y + cosv; CL

u=-2 J

1 y + cos2ea’

Let us first examine the even effects. As is seen from (38b) and (41b), the e.m.f. of the even photoelectromagnetic effect approaches a constant value in n, as well as in p semiconductors when x 9 1, which, of course, is found to be in conformity with (21). What is most noteworthy is the fact that in this limit &+ and En+ always possess an opposite sign, since when x < 1, both magnitudes possess the same sign. Therefore, it is possible to affirm that in the case of similar geometry (crystal orientation, direction of the magnetic field) the effect must change sign in one of the types of semiconductors as a function of H. In depending upon concrete conditions, principle, such a situation can be realized in n, as well as in p semiconductors. It must be noted that in the case of transition to an asymptotic limit, the e.m.f. of the even photoelectromagnetic effect passes through a maximum (minimum) not only in those cases when a change of sign occurs, but also, as a rule, when the effect retains its sign. The last fact appears most graphically in the case when the vector of the magnetic field is perpendicular to the direction along which light falls. It follows from (3Sb) and (41b) that the e.m.f. vanishes for that direction of-the magnetic field in the case of any crystal orientation.

606

Yu. KAGAN

and V. SOBAKIN

Insofar as odd effects are concerned, it follows from (39b) and (42b) that they decrease as l/H with increase of H. At the same time, depending upon the orientation of the crystal, the e.m.f. of the odd photoelectromagnetic effect can, as a function of H, either change sign or retain it throughout the entire change interval of the field. This again refers to tl, as well as to p semiconductors. Keeping in mind (39a), (42a), (39b) and (42b), it is easy to conclude that the e.m.f. of the odd photoelectromagnetic effect always passes through a maximum (minimum). In order to demonstrate the nature of the results obtained within a wide interval of change of the magnitudes of the magnetic field and its orientation, let us examine a particular case when the light falls along the [l, 1, 11 axis. This case offers special interest inasmuch as the predominant number of experiments(s>sps) were performed for this orientation of the crystals. Figure 2 presents the results of calculations of the anisotropic even photoelectromagnetic effect for n-Ge (formula (38)). At this point the angle between the direction of the magnetic field and the [l, 1, l] axis is designated by 0 (as we noted from the beginning, the magnetic field lies in a plane perpendicular to the direction of measurement), while the angle between the direction of measurement and the projection of one of the crystal axes on the plane of the model is designated by 4. The following values for the effective masses of the carriers were used in the calculations: mt = 0.082~ mp, = O-344m,, ml = 164m,,

The comparison of theoretical and experimental results for large values of the magnitude x is of special interest. For this purpose KIKOIN and NIKOLAvEV(5) undertook a study of E+ in n-Ge in the case of superhigh fields (2’ = 77°K). In this paper it was clearly ascertained that when 0 = 90”, the asymptotic value of E+ actually approaches zero, as is graphically seen from the corresponding curve of Fig. 2. The results of analogous calculations for the case of p-Ge are presented in Fig. 3. The sharp difference of the nature of the H and &dependences of the effect for n-Ge.and p-Ge which graphically demonstrates the general relationships described above attracts attention. I. K. KIKOIN and S. D. LAZARJXV undertook a special verification of these results. A comparison of the data obtained by them (see Figs. 20 and 21 in Ref. 7) with the curves of Fig. 3 indicates that there again occurs an excellent qualitative conformity of the measured and computed results. Analogous calculations were also performed for the case of odd effects. We will not present them here. We note, however, that in the case of p type semiconductors relation (42) is transformed into a comparatively simple form for the selected direction along which light falls. Thus, for p-Ge we have

m p, = O-043me.

The concentration of “light” holes K was assumed to be equal to O-02. In addition, considering the conditions corresponding to liquid nitrogen temperature and sufficiently pure models, we use the results obtained in paper,(l4) 2.0 and 1.88 respectively, for the relations (T~~)/ (TV) and I , while th e value ( 711> corresponds to the H orientation along the [l, 1, l] axis. If these curves are compared with the data obtained by I. K. KIKOIN and S. D. LAZAREV (see Figs. 16 and 17 in Ref. 7), it is immediately seen that a very good qualitative conformity occurs between the experimental and calculated results within a comparatively wide interval of parameter changes.

(43)

where $1 =

+2 = case

1 +y(l +Xs)+Xs(9-coss8),

I+Y(I+~~)+xL

6 - 7 cos2e 27

-x27

442

I

sins8 cos 3+.

It can be immediately ascertained from (43) that there is a two-fold relationship to the angle +, appearing for not very large fields as a-sin 3++/39 sin 64, which is found to be in confosmity with the nature of the relationship observed in paper.@)

PHOTOELECTROMAGNETIC EFFECTS IN GERMANIUM AND SILICON SEMICONDUCTORS

r

R

0

YI

0

607

608

Yu. KAGAN

and V. SOBAKIN

-.

P

+

-w

P

PHOTOELECTROMAGNETIC

EFFECTS

IN GERMANIUM

The correspondence between the theoretical and experimental results within a sufficiently wide interval of parameter changes compels one to believe that the approximations assumed in the present paper are quite satisfactory. This, in turn, permits the utilization of a direct quantitative comparison of theoretical and experimental results as a source for obtaining information about the kinetic parameters of a semiconductor. In particular, it is possible to determine
Acknowledgemenis-The authors are grateful to I. K. KIKOIN and S. D. LAZAFUZV for numerous discussions of the experimental aspects of the study.

AND SILICON

SEMICONDUCTORS

609

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