Calculation of the photoelectric effect in silicon

J. Phys. Chem. Solids

Pergamon Press 1971. Vol. 32, pp. 1075-1086.

CALCULATION

OF

Printed in Great Britain.

THE PHOTOELECTRIC IN SILICON

EFFECT

L. R. SARAVIA and L. CASAMAYOU Instituto de Fisica, Facultad de Ingenierla y Agrimensura, Universidad de la Reptlblica, Montevideo, Uruguay

(Received 9 February 1970; in revised forrn 4 May 1970) A b s t r a c t - T h e contribution from direct interband transitions to the photoelectric effect in Silicon is calculated for photon energies up to 10 eV, Refined computational techniques for the calculation of the energy distribution curves allow a detailed comparison of the theoretical and experimental curves. The agreement for the higher electron energies is excellent and the structure is interpreted in terms of the details of the energy bands. 1. INTRODUCTION

THERE has been considerable interest on the photoelectric energy distributions in Si as a means of studying its energy band structure. There exist experimental results[l-4] for photon energies covering the range 3-10 eV and showing very interesting structure. Theoretical analysis has been performed for photon energies below 6 e V on the assumption that the bulk of the optical transitions are direct. Kane[5] has developed a theory for the critical point structure on the energy distributions. Brust[6] has performed a detailed calculation of the distributions using a model of the energy bands of Si based on a pseudopotential approach. Its resolution is not enough to allow a clear identification of the structure in terms of the energy bands. On the present work we improve the resolution of the photoelectric energy distribution calculations applying the numerical techniques introduced by Gilat and Raubenheimer [7] to study the critical point structure of phonon spectra and recently used in the study of the optical properties of Si in terms of its electron energy bands[8]. Also, the photon energy range for the calculation is extended to 10 eV, allowing the interpretation of structure of importance because of its relation with conduction bands not usually studied through optical experiments. It is expected that the

structure at higher energies will be easier to interpret "as several effects, like escape probability factors and transport effects, are not so important for those energies. The procedure to compute the photoemission energy distributions is described in Section 2. A direct transition model is adopted. The contribution from the different energy band transitions is calculated, but no detailed study of the scattering of electrons in the crystal is performed. The escape probability factor is calculated using the hypothesis adopted by Brust[6]. The model adopted for the energy bands of Si is explained in Section 3. They are calculated using a pseudopotential approach combined with a k.p extrapolation procedure. Spin-orbit effects are included. The theoretical results are given in Section 5. The agreement is shown to be excellent for the higher electron energies. The experimental structure shows considerable broadening, making difficult its interpretation in terms of individual critical points. All the same, the contribution from a whole interhand transition is in general quite localized, and it is possible to interpret the peaks in terms of them as well as indicate the region in the. Brillouin Zone responsible for the structure. For lower electron energies, especially for the higher photon energies, the discrepancies are large, showing that other effects should be

1075

1076

L.R.

S A R A V I A and L. C A S A M A Y O U

taken into account. Several types of transport processes [9-11 ] as well as contributions from non-direct transitions[12] have been considered in the past as possible sources for the difference. No attempt has been made to include them in the present calculation.

analytical calculation of integral[1J. If we consider oJ and E as independent variables, we can perform a change of variables in (1). Because of the delta functions, the integrations in E and oJ are immediate and we get

N(E, o2)=A ~ f ]P,,.~(k) ]='P(E.k) 2. CALCULATION OF THE ENERGY DISTRIBUTIONS

ll,.g C

We are dealing with photoelectrons produced in the volume of a semiconductor by direct optical excitation. If o2 is the frequency of the incident photon and E the final conduction band energy of the excited electrons, the energy distribution function N(E, co) is given by

N(E, o2) =A ~, f

]p,~(k)l-~P(E.k)8(w,s(k)

-- o2)~(E,, (k)

--E)d'Jk.

(1)

The sum is over all conduction bands n and valence bands s. The integral is performed over all the Brillouin Zone (BZ), p,,~(k) is the momentum matrix element between bands n and s, A is a normalization constant and hco,~(k) is equal to E,,(k) --E~(k). P ( E , k ) is the probability that the electron produced with energy E and wave vector k will escape. We are interested in the calculation of N(E, co). A method previously used[6] consists on the calculation of the energy bands E, (k) at a number of points over a cubic mesh in the BZ, and their classification according to energy and photon frequency. Because of the double classification, we need a large number of points in order to reduce the scattering on the histograms to a reasonable value. This was solved by Brust[6] increasing considerably the number of points initially obtained with the energy band calculation through a quadratic interpolation. All the same, the resolution obtained by this method is not enough to study in details the sharp structure produced by the two dimensional critical point structure related to the energy distributions. We develop here a method based on the

1 X ]VkE • V#co] dl.

(2)

The curve C in the BZ is defined by the surfaces En(k) ---- E, (O,,s(k) = ~

(3)

dl is an element of the curve C. In order to perform the integration in (2) we need an analytic expression for E,,(k) and co,,s(k). The energy band calculation gives numerical values for the energy bands over a cubic mesh on the BZ. We introduce a linear interpolation as explained elsewhere [8]. Briefly, each cube in the mesh is divided in six tetrahedrons. Inside a tetrahedron we adopt a linear expression for E, (k):

E,(k)

= a. k + b .

(4)

The four constants in (4), a and b, are determined by matching the values of E,,(k) at the four corners of the tetrahedron, which are known from the numerical calculation. Inside each tetrahedron, the surfaces defined by (3) are planes and the curve C is a straight line. Its length and end points can be determined algebraically. If we also fit the other terms in the integrand of (2) by a linear form, the evaluation of the contribution from each tetrahedron is immediate. For the calculations to be performed in this work, the cubic mesh necessary to obtain a good resolution will have about 1600 points in the asymmetric part of the BZ. The energy distributions will be computed at intervals of energy equal to 0.01 eV. As an example, we show in Fig. 1 the result of a calculation for an

CALCULATION

OF THE PHOTOELECTRIC

E F F E C T IN S I L I C O N

1077

h2kT 2

Constant, ifE,,(k) > - 2m

(s) ,

hekr 2

i

(5)

0, i f E . ( k ) < 2---~-

it ii

I,

(s)

where kr is the component of k parallel to the surface of the crystal and Evac is the vacuum level. This factor is obtained on the supposition that the absolute value of the momentum k is conserved during the process and kr is continuous through the escape surface. Next, we average the escape probability over all the available states of energy E in the BZ, obtaining an escape probability factor P(E) independent of k.

tl H

'

I

9 '

:II i

,,,

(u)

"

iL

i

I

~,

5 5

6.0

7.0

E ,eV Fig. 1. Contribution of an interband transition in Si to the photoemission energy distribution curve for a value of photon energy equal to 5-39 eV. The calculated values are indicated as obtained from the computation (black dots). The scattering of the points is typical for this type of calculation. Structure produced by the three types of twodimensional critical points can be seen in the figure.

interband transition in Si where the three types of two dimensional critical points discussed by Kane[5], minimum (l), maximum (u), and saddle point (s), are present. We see that the scattering of the calculated points is small and the critical points are well defined. In (I) and (2) there appears the escape probability factor P ( E , k ) . To treat this function exactly would require having rather detailed information about the surface potential and scattering mechanisms. We will compute it adopting the hypothesis proposed by Brust[6]. We assume that an electron produced with an energy E, (k) has a momemturn k which is completely randomized by elastic scattering processes before the electron reaches the surface of the crystal. Electrons whose momentum is directed towards the surface are assumed to leave the crystal with an escape probability

3. CALCULATION OF THE ENERGY BANDS We need to compute the energy bands and the dipole matrix elements over a cubic mesh in the BZ containing a considerable number of points. We use a pseudopotential method to generate the bands and a k . p extrapolation procedure to increase the size of the cubic mesh as explained elsewhere[8]. Briefly, one computes eigenvalues and eigenvectors at a small number of points in a coarse cubic mesh (around 100 points in the asymmetric part of the BZ) by solving the secular equation generated by the semiempirical pseudopotential approach. Then, a refined mesh of points is generated from the previous one (around 1600 points) by a k . p extrapolation procedure. The combination has the advantage of greatly expediting the computational work. The convergence errors have been discussed elsewhere [6], and the same conclusions apply here. The spin-orbit terms have also been considered. We use the formalism developed for Ge [14], taking the orthogonalization coefficients proper to Si. The pseudopotential parameters used for Si are: V(l, l, I) = - 0 . 2 1 Ryd, V(2, 2,0) = 0-058 Ryd, V(3, l, I) = 0.078 Ryd. The spin parameter was selected to fit the experimental value of the spin-orbit splitting at the Fzs,' level [15]. The spin-orbit parameters in Si are rather small and have little importance from the point of view of the interpretation of the photoemission results. The splitting at some of the most important levels are given in Table I. The present pseudopotential model for the Si energy bands is very similar to one previously discussed by Saravia and Brust[8], and we refer to their work for a comparison with the energy bands calculated by other authors [ 16, 17]. The resulting energy bands along two principal directions in the BZ are shown in Fig. 2. The double group notation is not used. In the discussion that follows the energy value at a general point in the BZ is denoted by E~ ( i = 1,2 . . . . ), not taking into account Kramer's degeneracy, i.e., each value of i corresponds to two eigenvalues.

1078

L.R.

S A R A V I A and L. C A S A M A Y O U

(L)

a

'

>~ I

tO

E2 E3,E4

EsiE6

I

E2,E~,E 0

-4

-4

-6

-6

K

W

L -'---K

1~

K--'-

X

U,K

d

I~

Fig. 2. Energy bands of Si along some principal symmetry lines. They were computed using the pseudopQtential method together with a k.p extrapolation procedure. The energy bands are labeled by E~ (i = 1,2 . . . . ), with i increasing with energy. The single group notation is given for some of the points of the BZ.

Table 1. Si: theoretical spin-orbit splittings in eV Levels

Splittings

F25, Fx~,

0.044 0"031

~,

0.030

/_,3

0"012 4. RESULTS

(a) Escape probability Calcott [2] has published energy distribution curves measured on surfaces covered with Cesium films of different thicknesses: one monolayer, 0.3 of a monolayer, and no monolayer. The film thickness affects the escape probability function, since the position of vacuum level is changed. Also, the dynamic processes related to the emitted electron, especially the randomization of the crystal momentum accepted as one of our hypothesis, are. altered. We are not going to attempt a study of the last point and the same hypothesis are used throughout all the calculation. We adopt different vacuum levels for the three

cases: 5.15, 3.0 and 2.2 eV over the top of the valence band respectively. The resulting escape probability curves are shown in Fig. 3. In the discussion that follows all the electron energies are referred to the top of the valence bands. (b) The photoelectric energy distribution curves One of the calculated curves, not taking into account the escape probability factor P (E), is shown in Fig. 4. We include interband transitions involving valence bands 2 - 4 and conduction bands 5-12. The results are extremely rich in details and the structure is in general quite sharp. This is produced by the fact that the structure due to saddle points is logarithmic, and the discontinuities related to maxima and minima points can take large values, even approaching infinite. A direct interpretation of the experimental peaks is not possible because they are much broader than the calculated ones, as can be seen from Figs. 5-7. This is expected because effects such as electron-

CALCULATION

OF THE PHOTOELECTRIC

0-5 Escape probobility Vocuum l e v e l J,,,/ 0.4 2.2 eV jr .-',. . . . 3.0 eV ~ ,-'" ...... 5"15eV / 0.3 ~ ~J ~.

/"

0'2

~

/' t'

/

,..i

3

,5

9

~f

4

5

6

7

6

r

7

8

E,eV Fig. 3. Shows the three different escape probability functions used in the present calculation. The differences are produced by the position assumed for the vacuum level in each case.

~

9

~'---"

/st

, I

8

E,eV

ht~=8-16 e7,~tV8 ~

3

.I"

I

2

- - Theory 1

LO0 MonoloyerCs I0

l .I

,."

1079

.""

i,//

II

/'

,,--'--.~'-'//

/" tl i /~""J l /t/./.i,, i t

0"1

E F F E C T IN S I L I C O N

9

5

6

7

E, eV

qO

8

Lhte=8.92ev / ' ~ " / J l ' l ~ '

hr =7.56 eV 5

6

/ ==963eV

a I-

3

4

5

6

7

E ,eV Fig. 4. Photoemission energy distribution curves of Si for a photon energy of 7.56eV. Escape probability and broadening factors have not been included.

phonon and electron-electron scattering, interaction of the electrons with the Cesium film and band bend!ng have not been considered. In order to perform a comparison, we introduce a phenomenological broadening of the form

N'(E',to) = f : N(E',to)L(E,E') dE'

(6)

where I

r

L(E,E') = 7r F e + ( E _ E , ) 2

(7)

F is the broadening constant. We adopt different values of F according to the thickness of the Cesium film: 0-15 eV, for the case with no Cesium, 0.20 eV for 30 per cent of a monolayer, and 0.25 eV for a full monolayer. The energy distribution curves for different

5

6

E,eV

7

8

8

7 .~-~,

7

9

I I

lo

B

(3

E ,eV Fig. 5. Shows the calculated (solid lines) and the experimental (broken lines) energy distribution curves in Si for several values of photon energy corresponding to a sample with no Cs on its surface. The experimental curves are those given by Callcott (Ref.[2]). Their peaks are identified by numbers, from 1 to 11, as used by Callcott (Ref.[2]). The calculated curves include an escape probability factor with the vacuum level at 5.15 eV and a broadening factor equal to 0.15 eV. Partial results indicating the contribution from some of the interband transitions are also shown. They are identified by the numbers labeling the bands involved in the transition (e.g. label 4 - 7 indicates the contribution from transitions between valence band E4 and conduction band E 0 . F o r the sake of clarity only partial contributions important from the point of view of the interpretation of the peaks were included.

photon energies, including the effects of the escape probability factor and broadening, are given in Figs. 5-7. The experimental curves (2) are included. Relative scales have been selected since the absolute one has not been measured. Partial results from the most important interband transitions are also indicated.

1080

L.R.

3

3.3 Monoloyer Cs u..~3 ~ ' ~

S A R A V I A and L. C A S A M A Y O U

3

--The~

. I.tJ I

1.0 Monolo~erCs

z"

J

t/i ~ --Theory (~=3.65eV) ~--Exp, (~to=3-65eV)

-'J~'4i 2

~

~d"

,,

.- 3

"'l-. . . . "

",6X~\ ,

~

,

4

5

6

it~'~3

;

/

~--Theory(h~

//_\\---Exp

=4"41ev)

(TI~=4-14eV)

. 4 . ~ 7 ~ T h e o r y (ht~ ~ . - . . . . Exp. (f~=B.16eV)

//

3

4

-

5

6

7

I

8

2

_ 4 R --Theory(t~=8"92eV) .. ,~/(. -F. . . . . Exp ~co=886eV) ,.'i- " - - - - " -.. 8 "

3

,/ ~

3

[~. " ' ~ /

3

17

//'~ I \

I

f

5

6

Theory

]

("~:~3eev~l ---

Exp

7 I

3

4_

~

6

7

2 ,,,h---'" 2 ~

8

~ ""4 " . . . . f, --Theory~tu=9.31eV) / 9" " L-,

3

5

61

6 "~--Exp. (h~=9-19eV)

J

LtJ

~ 7

-,

Theory

I0 I

2

"

4"

3

-'p

4

~'l"

I

5

I|

6

E ,eV

E,eV Fig. 6. Shows the calculated (solid lines) and the experimental (broken lines) energy distribution curves in Si for several values of photon energy corresponding to a sample with 0.3 of a monolayer of Cs on its surface. The experimental curves are those given by Callcott (Ref.[2]). Their peaks are identified by numbers, from 1 to 11, as used by Callcott (Ref.[2]). The calculated curves include an escape probability factor with the vacuum level at 2"8 eV and a broadening factor equal to 0.20 eV. Partial results indicating the contribution from some of the interband transitions are also shown. They are identified by the numbers labeling the bands involved in the transition (e.g., label 4 - 7 indicates the contribution from transitions between valence band E~ and conduction band E7). For the sake of clarity only partial contributions important from the point of view of interpretation of the peaks were included. Particularly, partial contributions producing peak No. 10 are not included since they were discussed in connection with the clean surface experiments.

S. DISCUSSION OF THE STRUCTURE TWO general remarks can be made about the results shown in Figs. 5-7. First, the structure at higher electron energies is well reproduced both in position and relative intensity. We will be able to interpret it in terms of the details of the energy bands. Second, the energy distribu-

Fig. 7. Shows the calculated (solid lines) and the experimental (broken lines) energy distribution curves in Si for several values of photon energy corresponding to a sample with 1-0 of a monolayer of Cs on its surface. The experimental curves are those given by Callcott (Ref.[2]). Their peaks are identified by numbers, from I to 11, as used by Callcott (Ref.[2]). The calculated curves include an escape probability factor with the vacuum level at 2.2 eV and a broadening factor equal to 0.25 eV. Partial results indicating the contribution from some of the interband transitions are also shown. They are indicated by the numbers labeling the bands involved in the transition (e.g., label 4 - 7 indicates the contribution from transitions between valence band E4 and conduction band Er). For the sake of clarity only partial contributions important from the point of view of the interpretation of the peaks were included.

tions at lower energies show a large contribution, increasing as the electron energy decreases, which is not reproduced at all by our calculation. Part of the direct interband structure is superposed on this contribution. The pair production scattering mechanism [9, 13] or contributions from indirect transitions[12] could be possible sources for a secondary distribution of this type, although this is not definitive and the possibility is open

CALCULATION

OF THE

PHOTOELECTRIC

for errors in the experiment, the band calculation or the escape probability factor. On the following we study the interpretation of the different peaks in terms of the direct interband transition contributions. We use the notation adopted by Callcott [2] for the experimental peaks. We will not perform a detailed study of the critical point structure appearing in the theoretical calculation because it would be too lengthy. We find that each peak is produced by contributions from a few interband transitions at most. They are indicated in Table 2. In general, the energy distributions for each

--

"h~=8"16eV

.j~ ~,

\ii

....

4

-....

W

z

5

7

6

8

EFFECT

IN SILICON

1081

transition shows a dominant peak. This is shown for some photon energies in Figs. 8-10. A plot E-hoJ for these peaks is shown in Fig. II together with the experimental ones as given by Callcott[2]. The regions of the BZ producing most of the contribution to these peaks are shown in Figs. 12-19, where the electron energy and optical energy contours in the principal symmetry planes of the BZ are given. Regions were both types of contours are parallel give the large contributions to N(E, hoJ). This data establishes the connection between the experimental structure and energy bands and it has been summarized in Table 2. Peak No. 10 is well reproduced by the present calculation. For the higher photon energies most of the contribution comes from interbands E3-~0 and E H 0 . The main contribution comes from a region near the F point. Since the bands Ea and E4 have very similar

h= = 7.21 eV 'hr =7.56 eV

3 il

E4-9

~---'h~ T n ~ =7.38 eV =8.92eV

E3-~

2 I

0 4

5

6

8

z

- - h m = 7'21 eV

14

5

7

I

0

8

4

E4-7

2

6

2

i ii,'

,,Q

~ 3

h(~= 7.21 eV . . . . hoJ=756eV

.-~ 3

3

4-

E3-6

~

I

'I

I,

5

6

ii II

i- -

Ii ) I

I

~4

4

7

8

E ,eV

Fig. 8. S h o w s the contributions from different interband transitions in Si to the energy distribution curves corresponding to several values of p h o t o n energy. E s c a p e probability and broadening factors are not included.

0

7

li

t

3 '

.... -hoJ = 8.92eV

!

j, I I

i

i i

6

- - "h~ =7"?-I eV - - - h~ =7-56 eV

ii i i 2

5

!, t,

4

' ~ E,eV

6

7

Fig. 9. Shows the contributions from different interband transitions in Si to the energy distribution curves corresponding to several values of photon energy. Escape probability and broadening factors are not included.

1082

L.R.

S A R A V I A and I. C A S A M A Y O U

9 --

E4-5

Theory

. . . II

..... Experiment

"h= =4-41 e V

.** I0 s

- - - hw=5.39 eV

~ -

8

I01

,! /"

(4 - 7 ) / / / 7

(3-8L... 9~'L ;-.~; -/-~/y(3-7) 3 w Z

o

" '

j

,

.

.

.

.

.

.

-

. . . . . . . . . . . . . . . . . . .

2

9 /i 2

o

I~

,~

~

E4-6

hco=4.41eV hw=5.39e~

IO I

:.. o

i

[--~-

I

2

i

3

i

'....~ ,

Ii

;

4

i

5

hw ,eV

Fig. 11. E-hoJ curves for the most promiment calculated peaks (solid lines) and for the experimental peaks as given by Callcott (Ref.[2]), (broken lines). The curve corresponding to transition 3-7 is shown twice. The second one corresponds to the peak shifted by the inclusion of the escape probability factor. X

U

E ,eV Fig. 10. Shows the contributions from different interband transitions in Si to the energy distribution curves corresponding to several values of photon energy. Escape probability and broadening factors are not included.

values at F, it is reasonable to expect them to contribute to the same peak. The photon energy corresponding to the F25,- F12, transition, about 7.7 eV, indicates the limit below which these bands will give no contribution to the peak. Below 7-7 eV interbands E3-9 and E4-9 are mainly responsible for peak No. 10. The analysis of their energy distribution curves is complicated somewhat by the fact that level E9 is really composed of different crossing bands. For example, for ho~ = 7.56 eV there are two different surfaces in the BZ contributing to the energy distribution curves. Peak No. 9 appears in a small range of values of photon energy. Interband Ea-s is mainly responsible for it. The energy distribution function shows two peaks. The upper

E=o .

.

.

.

(~3-10

['\ 7.7

8-2~ 8.7

Fig. 12. Contours of constant electron energy (solid lines) for band E~0 and constant photon.energy (broken lines) for interband transition E3-~0 for some of the principal planes of the BZ (in eV).

CALCULATION x

OF THE PHOTOELECTRIC

u

E F F E C T IN S I L I C O N

x

1083

u

E8

E9 .

.

.

.

.

Od4_9

.

.

.

.

.

(03_ 8

ii6', Ft 4

7.7

5.

;,~ 8.92

5.i

Fig. 13. Contours of constant electron energy (solid lines) for band E,~ and constant photon energy (broken lines) for interband transition E4-., in some of the principal planes of the BZ (in eV).

Fig. 14. Contours of constant electron energy (solid lines) for band E8 and constant photon energy (broken lines) for interband transition E3-s in some of the principal planes of the BZ (in eV).

Table 2. Interpretation of experimental structure in terms o f interband transition contributions. The region of the Brillouin Zone responsible for the peak is indicated. Photon (hoJ) and electron (E) energies in eV for the most prominent theoretical peaks are given for each interband transition Peak No. 10 hco > 7-7

Interband transition

Location in* Brillouin Zone

3-10

Region along A line near F point Similar

4-10 hco < 7.7

4-9

Near E line

3-9 3-8

Similar A l o n g E line

4-7

Plane L K W U

3-7

Plane L K W U

3-6

Plane FLK

3-6

Plane X W U near W U line

* For details of location see Figs. 12-19.

Data for prominent peak h co- upper value (eV) E - l o w e r value (eV)

8-16 8.55 8-92 9-31 7-50 7.70 7.86 8-06 7.21 7-38 7.56 6"90 7.00 7.12 7.21 6.20 7-56 5.65 7-21 4-76 7.21 4.40 8-55 4.36

7.38 6.28 8.16 6-10 7-56 4.98 7.38 4.46 8-92 4-62

7.56 6.40 8.92 6.60 8.16 8.92 5.32 5.80 7.56 6.56 9.31 4-98

L. R. S A R A V I A and L. C A S A M A Y O U

1084 •

X

1.2 2'2 324'2 U

t~

Fig. 15. Contours of constant electron energy (solid lines) for band E7 and constant photon energy (broken lines) for interband transition E4-7 in some of the principal symmetry planes of the BZ (in eV). X

X

U

v

Fig. 16. Contours of constant electron energy (solid lines) for band E7 and constant photon energy (broken lines) for interband transition E3-r in some of the principal symmetry planes of the BZ (in eV).

v

Fig. 17. Contours of constant electron energy (solid lines) for band E6 and constant photon energy (broken lines) for interband transition E3-6 in some of the principal symmetry planes of the BZ (in eV). 1,2

O

v

Fig. 18. Contours of constant electron energy (solid lines) for band Es and constant photon energy (broken lines) for interband transition E4-s in some of the principal symmetry planes of the BZ (in eV).

C A L C U L A T I O N O F T H E P H O T O E L E C T R I C E F F E C T IN S I L I C O N X 1.2 2'2 3-2 4 , 2

Fig. 19. Contours of constant electron energy (solid lines) for band E~ and constant photon energy (broken lines) for interband transition E.,-0 in some of the principal symmetry planes of the BZ (in eV).

one, produced by regions near the ~ line, is responsible for the experimental structure. Interband E4-r gives the main contribution to peak No. 8. The region responsible for it is related to the plane L K W U . It approaches the L point as the photon energy decreases. Peaks marked as No. 6 and 7 in the experiment are treated together because they are mostly produced in the theoretical calculation by the same interband transition E3-r, although there is some contribution from E3-8. Their position are determined through experiments with Cesium films of different thicknesses and differences in the position of the structure are produced by changes in the escape probability factors. The peak produced by Ea-7 is shifted because P(E) is changing at very different rates in both cases. The E3-8 interband, at a slightly larger energy, enhances the effect. E3-0 produces peaks No. 4 and 5. There are two regions in the BZ giving important

1085

contributions to E3-6. One, very near the W point, is important at higher photon energies and produces peak No. 4. The other becomes important at lower photon energies and it is related to a flat zone in electron energies near the L point. We have not been able to reproduce clearly peaks No. 1 and 2, as well as dip No. 3. Interband transition E4-~ produces a structure near peak No. 1 but the agreement is not very good. Up to photon energies about 4.41 eV the theoretical peak follows the experimental one, although its position is at slightly higher electron energies. For higher photon energies the experimental peak becomes very large, while the contribution from E4-s practically disappears. The peak produced by E4-.5 reaches a maximum near 4.41 eV, when the surface in the BZ contributing to N(E, co) is near a three dimensional optical critical point in the FXL plane. Contribution from interband E4-6 appears as related to peak No. 2. Again, for the lower photon energies the agreement is reasonable, but for the higher energies the experimental structure becomes very large and it is difficult to perform any comparison. Peak No. 11 was not reproduced by our calculation. A possible source for this structure are transitions from band E~, which were not included in our calculation. 6. CONCLUSIONS

We have shown that a direct transition model for the photoelectric effect provides a good prediction of the experiments for higher electron energies, giving a possibility to study the details of conduction band energies no accessible to optical experiments. Through the use of refined computational techniques we have been able to obtain in detail the critical point structure present on the theoretical spectrum. This has been useful to correlate the most important peaks and the energy bands. We have shown that many details are erased from the experimental spectra by a large broadening. If it were possible to decrease it, the information to be extracted from

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the energy distribution curves could be increased considerably. At lower energies the agreement is worse but we expect to improve it by the consideration of other physical effects. Acknowledgements-One

of the authors (L.R.S.) is grateful to Dr. David Brust for many discussions on the subject. We thank G. Lesino for her help on the preparation of some of the computer programs used in this work. The calculations were performed at the 'Centro de Computaci6n de la Universidad de la Repfiblica,. Montevideo, Uruguay'. We acknowledge its staff for technical assistance. This research was supported bythe 'Fondo de Investigaci6n Cientifica de la Universidad de la Repfiblica, Montevideo, Uruguay'. REFERENCES 1. ALLEN F. G. and GOBELLI G. W., Phys. Rev. 144, 558 (1966), and previous works by the same authors, cited in the mentioned reference. 2. CALLCOTT T. A., Phys. Rev. 161, 146 (1967). 3. SPICER W. E. and SIMON R. E., Phys. Rev. Lett. 9, 385 (1962).

4. SPICER W. E. and EDEN R. C., Proc. lntrnL Conf. Phys. Semiconductors, Vol. 1, p. 65 Moscow, (1968). 5. KANE E. O., Phys. Rev. 175, 1039 (1968). 6. BRUST D., Phys. Reo. 139, A489 (1965). 7. G I L A T G. and R A U B E N H E I M E R L. J., Phys. Rev. 144, 390 (1966). 8. SARAVIA L. R. and BRUST D., Phys. Reo. 171. 916(1968). 9. KANE E. 0., J. phys. Soc. Japan Suppl. 21, 37 (1966). 10. KANE E. O., Phys. Rev. 147,335 (1966). 11. K A N E E . O.,Phys. Rev. 159, 624(1967). 12. SPICER W. E. and EDEN R. C., Ball. Am. phys. Soc. 10, 1198 (1965). 13. SPICER W. E., d. phys. Soc. Japan SuppL 21, 42 (1966). 14. SARAV1A L. R. and BRUST D., Phys. Rev. 176, 915 (1968). 15. DRESSELHAU~ G. G., KIP A. F. and KITTEL C., Phys. Rev. 98, 368 (1955). 16. HERMAN F., KORTUM R. L., K U G L I N C. D. and SHORT R. A., Quantum Theory of Atoms, Molecules, and the Solid State p. 381. Academic Press, New York (I966). 17. BRUST D., Phys. Rev. 134, A1337 (1964).