- Email: [email protected]
Calculation of the photoelectric effect in germanium
J. Phys. Chem. Solids
Pergamon Press 1971. Vol. 32, pp. 154 I-1552.
Printed in G r e a t Britain.
CALCULATION OF THE PHOTOELECTRIC EFFECT IN GERMANIUM L. R. SARAVIA and L. CASAMAYOU
Instituto de Fisica, Facultad de lngenierla y Agrimensura, Universidad de la Reptlblica, Montevideo, Uruguay
( Received 7 August 1970; in revised form 16 October 1970) A b s t r a c t - T h e contribution from direct interband transitions to the photoelectric effect in Germanium is calculated for phg~'ton energies up to 12 eV. Refined computational techniques for the calculation of the energy distribut!on curves allow a detailed comparison of the theoretical and experimental curves. The agreement is good and the structure is interpreted in terms of the details of the energy bands. A new interpretation is given for some of the peaks. 1. INTRODUCTION
THE POWERof the photoemission experiments for determining important features of the electron energy band structure of semiconductors, especially for the higher electron energies, has been emphasized by Spicer and Eden [1]. From a theoretical point of view, the calculations of the energy distribution curves, necessary for a quantitative analysis of the experimental structure, has been limited to Si. Brust[2] has performed a calculation for Si in the lower photon energy range. Recently, Saravia and Casamayou [3] have extended the calculation in Si up to a photon energy of 10 eV, and used numerical techniques that improved considerably the resolution of the calculation. It was shown that the agreement between the experimental and theoretical distribution curves is excellent for the higher electron energies, allowing a detailed interpretation of the structure in terms of the energy bands. In the present work a similar calculation is performed in Ge, for which there exist experimental results [4, 5] for photon energies covering the range up to 12eV that show very interesting structure. The procedure to compute the photoemission energy distribution curves is described in Section 2. A direct transition model is adopted. The contributions for the different
energy bands are calculated, but no detailed study of the scattering of the electrons by the crystal is performed. T h e model adopted for the energy bands of Ge 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 results are given and discussed in Sections 4 and 5. The agreement with the experimerrtal results is good for the higher electron energies, and a detailed analysis is made in terms of the contributions from the different interband transitions. The regions of the Brillouin Zone (BZ) producing the most important structure are indicated. Particularly, the interpretation of the experimental structure corresponding to higher electron energies, previously discussed by Donovan and Spicer [5], is clarified. 2. CALCULATION OF THE ENERGY DISTRIBUTION CURVES We are dealing with photoelectrons produced in the volume of a semiconductor by direct optical excitation. If is the frequency of the incident photon and E the final conduction band energy of the excited electrons, the energy distribution curve N(E, hco) is given by
N(E, hoJ) = A , ~ f Ip,~(k)I2 P(E, k) 8(oJ~(k)--co) BZ
8(En(k)--E)dak.
(1)
The sum is over all conduction bands n and valence bands s. The integral is performed over all the BZ, p,~
1541
1542
L.R.
S A R A V I A and L. C A S A M A Y O U
(k) is the momentum matrix element between bands n and s, A is a normalization constant, and hto,,~(k) is equal to E,,(k)--Es(k). P ( E , k) is the probability that the electron produced with energy'E and wave vector k will escape. We perform this integration using a method developed elsewhere[3]. Briefly, the integration is reduced to one along a line defined by the surfaces E,,s(k) = E and to,s = to, which is performed analytically using a linear interpolation for the functions E,(k) and co,,,(k) inside each cube in the mesh defined by the points of the B Z where the bands are calculated numerically. 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. in (l) 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[2]. We assume that an electron produced with an energy E.(k) has a momentum 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: h2kT Constant, if E,(k) -- Eva,. > 2m P( E, k) =
h2kT 0, if E,(k) - E,,,,~ <
(2)
2m
where kr is the component of k parallel to the surface of the crystal and E,,,,. is the vacuum level. This factor is obtained in 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 ofk.
3. CALCULATION OF THE ENERGY BANDS We need to compt,te the energy bands and the dipole matrix elements over a cubic mesh in the B Z 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[6]. Briefly, one computes eigenvalues and eigenvectors at a small number of points in a coarse cubic mesh by solving the secular equation generated by the pseudopotential approach. Then, a refined mesh of points is generated from the previous one by a k. p extrapolation procedure. The combination has the advantage of greatly expediting the computational work. The spin-orbit terms have also been considered. We use the formalism developed elsewhere [7]. The pseudopotential parameters used for Ge are the same as those used in a previous w0rk[7]: V(l, l, 0 ) = --0'282 Ry, V(2, 2, 0) = 0.058 Ry, V(3, l, l) = 0'018 Ry.
The spin-orbit parameter was selected to fit the experimental value of the spin-orbit splitting at the F25, level [8]. The comparison with other energy bands, calculated by Brust[9], by Cohen and Bergstresser[10], by Herman et al.[l l], by Cardona and Pollak[12], and by Dresselhaus and Dresselhaus[13] are performed in the mentioned work[7]. The selected pseudopotential formfactors are somewhat different from those used by Brust[9] and by Cohen and Bergstresser [ 10]. The changes were performed in order to bring down the position of the I't~ level. The spin-orbit splittings in Ge, about 0"3 eV, are of the order of the broadening constants present in the photoemission experiments, what makes them not very important from the point of view of the interpretation of the experimental structure. The resulting energy bands along several directions in the B Z are shown in Fig. I. The double group notation is not used. In the discussion that follows the energy values at a general point in the B Z are denoted by Ei (i = I, 2, ...), not taking into account Kramer's degeneracy, i.e., each value of i corresponds to two eigenvalues. All the electron energies are referred to the top of the valence band. 4. RESULTS
(a) Escape probability Donovan and Spicer[5] have measured energy distribution curves obtained from Ge surfaces which have received different treatement in order to change their vacuum level. We have calculated the escape probability curves for two values of the vacuum level: 1.65 and 4.80 eV. They are shown in Fig. 2. (b) Energy distribution curves They have been calculated for different values of fifo up to 12 eV and include interband transitions involving valence bands 2, 3, 4 and conduction bands 5 to 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 two-dimensional saddle critical points is logarithmic, and the discontinuities related to maxima and minima .critical points can take large values, even approaching infinite. It is not possible to make a direct interpretation of the experimental peaks, since they are fewer and look much broader than the calculated ones. This is expected because effects such as electron-phonon and electron-electron scat-
PHOTOELECTRIC EFFECT IN GERMANIUM
1543
I0
LJ
2
-2
-4
K
W
L
]"
X
UK
Fig. 1. Energy bands of Ge along, some principal symmetry lines. They were computed using the pseudopotential method together with a k.p extrapolation procedure. The energy bands are labeled E~ (i = I, 2. . . . ), with i increasing with energy. The single group notation is given for some of the points of the BZ.
Escape
Probability ~
f
.3
.2
.I .0
/
/Vacuum
2
/
level
oev
/
Vacuum level
6
a
E,
8 eV
Fig. 2. Shows the two different escape probability functions used in the present calculation. The differences are produced by the position assumed for the vacuum level in each case.
t e r i n g , i n t e r a c t i o n o f t h e e l e c t r o n s with the c e s i u m film a n d b a n d b e n d i n g h a v e n o t b e e n considered. I n o r d e r to p e r f o r m a c o m p a r i s o n , w e i n t r o -
duce a p h e n o m e n o lo g ic a l b r o a d e n i n g of the form N ' ( E , ho~) ---
f •N ( E ' , ho.)) L ( E ' , E) dE'
(2)
1544
L.R.
S A R A V I A and L. C A S A M A Y O U
where 'hw= 4.17 eV
1
r
L ( E ' , E ) -- r r 2 Jr ( E
-
-
E') 2
(3) i.d Z
F is the broadening constant. We adopt different values of F according to the surface treatement: 0.15 and 0.10eV for the surfaces with vacuum levels at 1.65 eV (cesiated surface) and 4.80 eV (uncesiated surface). The calculated energy distribution curves for different photon energies, including the effect of the escape probability factor and broadening, are given in Figs. 3 to 6. The experimental ones, as measured by Donovan [5], are shown in Figs. 7 to 10.
,
, 2,
4.
4-6
hw= 5.15 eV
3-6 :3-7 7
5. DISCUSSION
A general remark can be made about the calculated results shown in Figs. 3 to 6. The calculated structure is clearly observed in the experiments, Figs. 7 to 10, but it appears superposed in a contribution not reproduced at all by the present calculation. This contribution is particularly large for the case of cesiated surfaces and low photon energies. It shows a large shoulder and peak which appear for all hto at electron energies equal to 2-2 and 3-0 eV. For the uncesiated surface the effect is not so important, but all the same there is a shoulder beginning at 5.0 eV for all hoJ which probably results from the action of the escape probability function on the mentioned contribution. This structure disappears after the surface is cesiated, indicating that it is not related to direct transitions. A similar situation was found in the case of Si[3]. Several authors have mentioned the possibility for this type of structure to be produced by physical effects not mentioned here, such as scattering mechanisms[I, 14, 15, 16] or contributions from indirect transitions [17]. On the following we study the interpretation of the different peaks in terms of the direct interband transition contributions. The position of the structure in the calculated distribution curves is given in Figs. 11 and 12 a s
2.
\
]
I
l
I
4.
6.
4.
6.
hw =6.18 eV
I
I
I 2.
E,
eV
Fig. 3. Shows the calculated energy distribution curves in Ge for several values of photon energy corresponding to a sample with a cesiated surface. The calculation includes an escape probability factor with t h e vacuum level at 1-65 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 ET). For the sake of clarity only partial contributions important from the point of view of the interpretation of the peaks were included.
an E - - h o ~ plot. The experimental results[5] have been superposed in the same diagrams and the structure has been denoted by capital letters, from A to J. We find that each piece of
PHOTOELECTRIC
7/
~
EFFECT
IN G E R M A N I U M
1545
hw=7.0eV
flw=7.0 eV
z
[
11 #"
I
4.
i~
o ~l 6.
I
I 8. 6.
~,~ ~ /
V 4.
I
I B.
E
I
io.
1~w=8.0eV
41- 3-10 8.
~6.
~w=9.0eV
6.
I
8.
io.
/~w- 9.0eV "
I
3,~(74_8%~/3_10 ~ ~4 - 1 0
4.
"hw=I0.0eV
l
6.
~
/
l
""~-6/r~
4.
6,
E,
I'
8,
~~
I
6.
B.
"'1\ 6.
"T-'-"~~ .. BE,
I
I
io.
-I0
4 - 7 ~ s~k'C ~ 4 i
lc
8.
eV
eV
~
I
I
Io.
Fig. 4. Shows the calculated energy distribution curves in G e for several values of photon energy corresponding to a sample with a cesiated surface. The calculation includes an escape probability factor with the vacuum level at I '65 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 ET). For the sake of clarity only partial contributions important from the point of view of the interpretation of the peaks were included.
Fig. 5. Shows the calculated distribution curves in G e for several values of photon energy corresponding to a sample with no cesium on its surface. The calculation includes an escape probability factor with the vacuum level at 4.80 eV and a b r o a d e ~ n g factor equal to 0- I0 eV. Partial results indicating the c~'ntribution from some of the interband transitions are also shown. They are identified by the numbers labeling the bands involved in the transition. For the sake of clarity only partial contributions important from the point of view of the interpretation of the peaks were included.
structure is mainly produced by contributions from a few interband transitions. They are indicated in Table 1. The region of the B Z producing the structure is also given in Table 1. The region of the B Z producing the struc-
ture is also given in Table I. Usually, the region of interest in the B Z is related to a symme;try line. Plots E - - hw for the calculated symmetry lines are given in Figs. 13 and 14, together with the position of the calculated
1546
L.R. SARAVIA and L. CASAMAYOU i
hw: 10.OeV
/'lw=10.5eV
i
I
I
I
6e
W~O.4
Uncesiotod surfoce Experimentt
LJ 4-9
3-10
I
~
I 9
I I0
//~E,
I II
eV l"lW=11.0eV
I 8
9
IO
,
Ii
E, eV ~hw =12.0eV
I I0
9
hw=ll. SeV
1 3 "lO~i'~r-"~ "~'" 9
10
E,
~~w-e.a
ev
t
6.6
o
7.7
0.0 '4
,-9 \ ~ - I O
e
8
"6
I
II
eV
E,
~
3-1~4_10 (E e
'~0,3 -
3-10
-t0 e
Oonovon Ref. 5
4-9
I
II
~0.6
eV
t
I
I Ge
I
Uncesiated
hw =12.5 eV
I
I I "hw =10.0
~ 0 . 5 F Surface
\ I
31101 N 9
E,
~ . I0
eV
f -II II
I 8
I3 9
E,
~ I0
I II
b_o~ I "8 o.~
eV
Fig. 6. Shows the calculated distribution curves in Ge for several values of photon energy and high electron energies corresponding to a sample with no cesium on its surface. The calculation includes an escape probability factor with the vacuum level at 4"80 eV and a broadening factor equal to 0"10eV. 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. For the sake of clarity only partial contributions important from the point of view of the interpretation of the peaks were included.
Fig. 7. S h o w s the experimental normalized energy distribution curves for a p - t y p e crystal of Ge with no cesium on its surface, as measured by Donovan [5].
structure. This data establishes the connection between the experimental structure and the energy bands. The structure labeled B is well reproduced by the present calculation. The main contribution comes from interband transitions Es-~0 and E4-9. Transition Es-~0 determines the position of the peaks for the higher photon energies. There is a region of the BZ, in the FKL plane, where the surfaces of constant E and constant hto are quite parallel. This region begins near the F point and moves mainly along the FK line a s the photon energy increases.
The contribution from transitions E4-9 are clearly seen at lower photon energies. An extense region in the FKX plane is responsible for the structure. Band E9 shows a saddle point which is not at the r point, but somewhat displaced, in the FXK plane, as can be seen in Fig. 1. It has an energy value below the one corresponding to Flz and for this reason interband E4-9 produces peaks in the region labeled F in Fig. 12, where peaks from Ea-10 have disappeared. This interpretation clearly indicates that level FI~, that is the double degenerate level E9 and El0, is the one related to structure B, but its position in the E -
~ ~ F
~.6'
0.0 /
4
5
6
7 E,
8 9 eV
I0
PHOTOELECTRIC
EFFECT
Table 1. Interpretation of the structure present in the photoemission experiments in terms of the different interband transitions. Regions o f the B Z involved in the t?ansitions as well as energy levels corresponding to the related symmetry points are indicated Interband transition
Structure
L o c a t i o n in B Z
Related energylevel
IN G E R M A N I U M I
1547 I
I
I
I
Ge Cesioted surface 5w=3.6 V O X g o J~ Q.
3.:
2 -
Experimen~"
-
oo
8
3-10
Plane K L
4--9
Plane K X
C hoJ ~< 8.2 eV
4-7
L W line in the
hoJ /> 8.2 eV
4-8
D
3-7
E ho~ ~< 9.0 eV
3-6
B
F12
hoJ >/9.0 eV G hto/> 5-5 eV hto ~< 5.5 eV
"~ 0.6 ,c
I
h e x a g o n a l face Plane FKX, n e a r line r x . Line L W, n e a r the L point.
2 I
3-7
3-6
L W line, n e a r the
4-6
L point L W line, n e a r the L point
I
I
Ge
I
I
W ( E3)
E,
eV
I
I
,.E
Experiment ] ~
d
D~
L3,
I
I 5 I
I
Ge Cesiated surfoce
/_ \\
z
"hw= 5.8 eV
O D x
.5.0
g
o D LLI
|
J
1\
3
L3,
I 4
3
X4
H e x a g o n a l face, n e a r the W point. H e x a g o n a l face, n e a r the W point
I
0.7
taJ
2 I
0
I I
I 2
1 4
5
6
h w = 11.4 eV
-7222'"
E~
eV
Fig. 9, S h o w s the e x p e r i m e n t a l n o r m a l i z e d e n e r g y distribution c u r v e s for a p - t y p e crystal o f G e with a cesiated surface, as m e a s u r e d by D o n o v a n [5].
uS Z 0-5
g o 0.4
o 0.3 hJO'2 o. o.o
Experiment, Donoven, R e f . 5 I 6
I
E
I I 8 eV
I0
Fig. 8. Shows the experimental normalized energy distribution curves f o r a p-type crystal o f Ge with no cesium on its surface, as measured by D o n o v a n [5].
JPCS Vol. 32, No. 7 - J
hto diagram is not directly given by the intersection of the line defined by structure B and the line E = h o J . Its position can be determined taking into account the fact that a shift of the conduction bands E9 and Elo in order to put the F12 level in the right place, essentially shifts structure B in the diagram E--hoJ along a direction parallel to the line E - - hie. In order to superpose the calculated and experimental structure, the calculated F12 level has to be shifted in 0.8 eV, what places this level at 8.1 eV approximately. Level F1 has been men-
1548
L . R . SARAVIA and L. CASAMAYOU I
I
I
i
I
Be
+e
~:
/~/
i ,N~
I/"
J
Cesioted surface Vacuum level =1.65 eV
-
Experiment,
/ S ,,,/ (B)_ ~ c-,S$ ppP SS
Donovan Ref.5
I~
Cesioted
/ ~ / ~
surf~
Experiment
J Brood Shoulder
The~
"~
s
/~S S
7 6
~(C)~7
S~p
p
5
///[
~ e
4
=10
/
5
pSp ~ } "---ompPPPPPPP P PPP P PPPpp pF*. . . . . PP, . . . . PPPPSS SPP P PPPPPPPPPPPP~.
.
.
.
.
.
.
.
.
.
.
.
.
2 C
I
I
2
4
6 E,
r
+
W
5
8
I
I0
I
eV
r~ "hw=6.2ev ,~'hw=6.4 ev
oe 0 A
Cesia re
Experiment, %~ ]Oonovon \\\ 2
4 E,
6
-
8
eV
Fig. I0. Shows the experimental normalized energy distribution curves for a p-type crystal of Ge with a cesiated surface, as measured by Donovan [5].
tioned in the p a s t [ I , 5] as a possible source for this structure. W e h a v e found that interb a n d transitions related to l't, here d e n o t e d as En, give no i m p o r t a n t contribution. T h i s is due to the fact that the dipole matrix e l e m e n t s for these transitions are v e r y small n e a r the 171 point, and transition r2s,-l', is indeed forbidden. Structure labeled A is not well r e p r o d u c e d . W e p e r f o r m e d a detailed calculation o f the N ( E , hoJ) c u r v e s in the region o f higher photon
I 3
I 4
I 5
I 6 hw,
I 7 ev
P B
I 9
I I0
HI
Fig. I 1. E--hoJ curves for the most prominent calculated structure and for the experimental structure as given by Donovan[5] for a crystal of Ge with a cesiated surface. The experimental peaks are labeled by capital letters from A to J. The calculated peaks are indicated by the numbers labeling the bands involved in the transition mainly responsible for the structure. energies in o r d e r to detect these shoulders. T h e results are s h o w n in Fig. 6. T h e r e a p p e a r s s o m e small s t r u c t u r e p r o d u c e d by E4-9 and E4-,0 transitions f r o m regions in the BZ near the h e x a g o n a l face. T h e a g r e e m e n t in position is not very g o o d and we are not sure w h e t h e r they c o r r e s p o n d to s t r u c t u r e A or they are shoulders in the p h o t o n region w h e r e m e a s u r e m e n t s w e r e not p e r f o r m e d . T h e e x p e r i m e n t a l s t r u c t u r e is indeed v e r y w e a k , as can be seen in Fig. 8 and 9, and small shifts+ in the calculated contribution to p e a k s B could e v e n t u a l l y p r o d u c e the shoulders. S t r u c t u r e labeled C is mainly related to interbands E4-T and E4-8. E4-7 is r e s p o n s i b l e for the p e a k s c o r r e s p o n d i n g to the l o w e r photon energies. T h e contributions are produced in regions near the L W line. F o r this r e a s o n they a p p e a r w h e n hco b e c o m e s larger than the values c o r r e s p o n d i n g to the L point. F o r the case of the uncesiated surface, the probability e s c a p e function highly distorts the contribution f r o m the E4-7 transition. E4-8
P H O T O E L E C T R I C E F F E C T IN G E R M A N I U M
1549
Uncesioted sur fete Vocuum level : 4 . 8 0 eV
. IC
.,
Exp . . . . . . .
+
V
Volley
S
Shoulder
P P /
[ Broo~+hoo,0er
S
/
O~p P
s o P /
_
P
~
u/
//
E
For
/,
s s
I 5
~
c~/~.
V v
S
~ p
V v
].j..~.
/,[~.~' /'+ ~
.~/-'vv
S
~
/
P
"~
(e) _ '
oj -
.~ _,~
':'4 / + + + ~ + + " + +
I 6
I 7
I 8
I 9 "["1W.
P I0
I II
I f2
eV
Fig. 12. E--/i(o curves for the most prominent calculated structure and rot the experimental structure as given by Donovan [5] for a crystal of Ge with no cesium on its surface. The experimental peaks are labeled by capital letters from A to J. The calculated peaks are indicated by numbers labeling the bands involved in the transitions mainly responsible for the structure.
9
Ge
o Calculated peeks Symmetry lines
8
--
7
~
6
u~ 5
,re W
4
~ I
.~o '
T 3
I 4
LW(4-6)
] 5
1 6 "hw,
\
I 7
I 8
p 9
I I0
I II
eV
Fig. 13. E--hoJ curves for the most prominent calculated structure (open circles) for a crystal o r G e with a cesiated surface, and for the points along the symmetry lines of the B Z related to the structure (solid lines). The interband transition plotted for a given point are indicated by the numbers labeling the bands involved in the transition. The symmetry lines are identified through the symmetry points in the B Z that determine the line.
1550
L.R.
S A R A V I A and L. C A S A M A Y O U
gives contribution for the higher photon energies. An extended region in the r K X plane, related to the r X symmetry line, is responsible for the contribution. Donovan[5] gives the same interpretation, based on the work of Spicer and Eden for GaAs[1]. It is interesting to notice that the peaks obtained from E4-8 define a line in the E--ho~ diagram which becomes parallel to E---hoJ for the higher energies. This means that for the region of the B Z near the X point, the valence band E4(X4) is very flat and the energy, value corresponding to )(4 can be determined from the experiment. A value between --2.7 and - 2 . 9 eV is obtained [5].
Ge o --
C a l c u l a t e d peaks S y m m e t r y lines o
o
.i
5
6
3"
I 8 hw,
I 9
I I0
I II
I 12
eV
Fig. 14. E--hco curves for the most prominent calculated structure (open circles) for a crystal of Ge-with no cesium on its surface, and for the interband transitions corresponding to points along the symmetry lines of the B Z related to the structure (solid lines). The interband transitions are indicated by the numbers labeling the bands involved in the transition. The symmetry lines are identified through the symmetry points in the B Z that determine the line.
Structure labeled D is partially produced by contributions from interband transition E3-~, highly distorted by its superposition on the shoulder produced by contributions not due to direct transitions. The main contribution comes from a region in the B Z related to the L W line near the L point. The structure is not very pronounced and it i,,J practically lost between the structure C and E when a cesiated surface is used. For hoJ larger than 9.0 eV, E 3 _ 7 intersects
the hexagonal face near the W point, and its contribution becomes significant producing the structure labeled E. As hoJ increases, the region of interest moves from the W to the K point. A similar interpretation has been given by Donovan[5], who noticed that constant values of E3 near the W point produce a structure moving like E in the E--hoJ diagram. The peaks determine the experimental value of E3, --4.2 eV approximately. The calculated value is off by 0.6 eV and for this reason the calculated peaks have to be shifted 0 . 6 e V in a direction parallel to the hoJ axis in the E--ho~ diagrams, Figs. 11 and 12, to reproduce the position of the experimental structure E. When a cesiated surface is used, the contributions to structure E from interband E3-6 become important. They are also produced in a region of the hexagonal face related to the W point. Structure labeled G is also produced in part by Ez-6. For the lower photon energy range the surfaces of constant hoJ find a region in the hexagonal face near the L point where the electron energies are practically constant. We get peaks which are locked at a certain value of E for hco up to 7.5 eV. Interband E4-6 also contributes to structure G and the structure is also produced in a region near the L point. Structure G determines the position of level E6 (L~), giving a value of 4.2 eV approximately [5]. Our calculated bands give a value for La, offby 0.5 eV from the experimental results. This can be clearly seen in the E--hoJ diagram, Fig. 1 1, where the calculated structure G is about 0.5 eV below the experimental one. Finally, we have not attempted a study of the structure labeled H , since it is superposed to a large contribution not predicted at all by our calculation not allowing any detailed comparison. Donovan[5] has performed a detailed experimental study and relates the onset of this structure to the 1"15level. Unfortunately, the spin orbit splittings are not resolved. A value of 3.13 eV is obtained for the center of mass transition energy of the r25,--rl~ transition.
P H O T O E L E C T R I C E F F E C T IN G E R M A N I U M 6. CONCLUSIONS
The present calculation has allowed a detailed interpretation of the structure present in the photoemission experiments in terms of the energy bands of the semiconductor. In general, this interpretation is similar to the one given by Donovan[5], based on a direct inspection of the energy bands calculated by Herman[11], and by Brust and Kane [18]. The most important difference is the change in the identification of structure B, which is now related to level F~2, with an assigned energy of 8-1eV, and not l'v It is interesting to notice that the same problem in the interpretation of the experimental structure is present in other semiconductors[l]. Actually, we found the same interpretation for Si [3]. These results add a considerable amount of information about the value of the energy levels of some of the symmetry points to that obtained in the past from other types of experiments. The available information has been included in Table 2, where a comparison is performed with some of the energy band calculations. The values indicated in Table 2 for
1551
1"2,, L1, and L3, are obtained from optical experiments[19, 20]. For L3, we used a value of the L3,--LI transition energy which is 0-1 eV smaller than the one measured for the A3--A~ transition in reflectivity experiments [20]. The position of L3, is determined since L1 is known. The L3,--L3 transition, also measured in reflectivity experiments[20], can be used as a check of the position assigned to L3 through the photoemission experiment. It is important to remark that most of the values included as results of experiments are really obtained through processes of interpretation which could be eventually revised. Table 2 shows that the calculation performed by H e r m a n [ I l l is in excellent agreement with experiment for the valence bands, but the discrepancies become larger for the higher conduction bands. Actually, we see from the pseudopotential calculations shown in Table 2 that the conduction levels 1"15,L3, and 1"12are quite sensible to small changes in the parameters. A better overall agreement can be expected if the new data is taken into account in the process of adjustment of the parameters.
Table 2. Shows experimental and theoretical energy level locations in eV. The first experimental value is that obtained from experiment. The second value, between parenthesis, represents the experimental level reduced to. the center o f mass value without spin-orbit splitting. The theoretical values do not include spin-orbit splittings Level F2, F15 Ft2 La, Lt L3 X4 W(E3)
Exp. 0-8 [b] (0"9) (3" 1) [a] 8" 1[a] (8-2) (--l'4)[b] 0"7 [b] (0"8) 4"2 [a] (4.3) --2-8 [a] --3.0[a] (--2.7, --2.9) --4.2[a] (--4-1)
Herman et al. [c]
Brust [d]
Present calc.
New model
0.9 2.9 9'5 --1"3 0"8 4.1
0"7 3-5 8-5 --1"I 0"7 4.3
0.9 3.0 7-3 --1"2 0"9 3.7
0-8 3.2 7.9 --1'2 0"8 4-0
--2.8 --4-1
--2.5 --3"4
--2.8 --3-5
--2.7 --3-5
[a] Obtained from photoemission experiments. [b] Obtained from optical experiments, Ref. [19-21]. [c] Ref. [11]. [d] Ref. [9].
1552
L.R.
S A R A V 1 A and L. C A S A M A Y O U
The pseudopotential model used in the present calculation was obtained in a previous work [7] from a careful adjustment to the experimental energy levels near the band gap, but now it shows large discrepancies for the other levels. We have tried a general adjustment of the pseudopotential parameters at posteriori of the photoemission calculation performed here. The energy levels corresponding to the new model, calculated with the following parameters: V(I, 1, 1 ) = - - 0 . 2 5 1 Ry, V(2, 2, 0) = 0.028 Ry, V(3. 1. 1) = 0.039 Ry, are given in the last column of Table 2. They show an average error of 0-2 eV respect to experiment. Although a better general agreement is obtained in the conduction levels, it was not possible to improve the position of the W level. It appears as practically unchanged in all the available pseudopotential calculations and contrasting with the value obtained by Herman [l l], which is the same as the experimental one. Acknowledgements-We are indebted to Dr. T. M. Donovan who made available to us the experimental results of Ge prior to publication, as well as to Dr. D. Brust and Professor W. E. Spicer for bringing some references to our attention. The calculation was performed at the 'Centro de Computaci6n de la Universidad de la Rcpfiblica'. We acknowledge its staff for technical assistante. This research was supported in part by the "Fondo de Investigaci6n Cient[fica de la Universidad de la Repfblica, Montevideo, Uruguay'.
REFERENCES
I. S P I C E R W. E. and E D E N R. C., Proc. IX Int. Con./~ Phys. Semicond., Moscow, 65 (1968). 2. B R U S T D., Phys. Rev. 139, A 489 (1965). 3. S A R A V I A L. R. and C A S A M A Y O U L., J. Phys. Chem. Solids 32, 1075 ( 1971 ). 4. A L L E N F. G and G O B E L L I G. W., Phys. Rev. 144, 558 (1966). 5. D O N O V A N T. M., private communication. D O N O V A N T. M. and S P I C E R W. E., Bttll. Am. Phys. Sot., 113, 476 (1968). 6. S A R A V I A L. R. and B R U S T D., Phys. Rev. 171, 916(1968). 7. S A R A V I A L. R. and B R U S T D.. Phys. Rev. 176, 915 (1968). 8. C A R D O N A , M. and S O M M E R S H. S., Phys. Rev. 122, 1382(1961). 9. B R U S T D.,Phys. Rev. 134. A 1337 (1964). 10. C O H E N M. L. and B E R G S T R E S S E R T. K., Phys. Rev. 141,789 (1966). I1. H E R M A N F., K O R T U M R. k., K U G L E M A N D , C. D. and S H O R T , R. D. Quantum Theort of Atoms, Molecules, and the Solid State, Per-Olov L6wdin ed., (Academic Press, N.Y.). 12. C A R D O N A M. and P O L L A K F. H., Phys. Reo. 142.530 (1966). 13. D R E S S E L H A U S G. and D R E S S E L H A U S M. S., Phys. Rev. 160,649 (1967). 14. K A N E E. 0., J. phys. Soc. Japan 21, Suppl., 37 (1966). 15. KAN E E. O., Phys. Rev. 147,335 (1966). 16. KAN E E. O., Phys. Reu. 159,624 (1967). 17. S P I C E R W. E., Phys. Rev. 154,385 (1967). 18. B R U S T D. and K A N E E. O., Phys. Rev. 176, 894 (1968). 19. M A C F A R L A N E G. G., M c L E A N J. P., Q U A R R I N G T O N J. E. and R O B E R T S V., Proc. Phys. Soc. London 71,863 (1958). 20. G HOSH A. K.,Phys. Rev. 165,888(1968).
Pergamon Press 1971. Vol. 32, pp. 154 I-1552.
Printed in G r e a t Britain.
CALCULATION OF THE PHOTOELECTRIC EFFECT IN GERMANIUM L. R. SARAVIA and L. CASAMAYOU
Instituto de Fisica, Facultad de lngenierla y Agrimensura, Universidad de la Reptlblica, Montevideo, Uruguay
( Received 7 August 1970; in revised form 16 October 1970) A b s t r a c t - T h e contribution from direct interband transitions to the photoelectric effect in Germanium is calculated for phg~'ton energies up to 12 eV. Refined computational techniques for the calculation of the energy distribut!on curves allow a detailed comparison of the theoretical and experimental curves. The agreement is good and the structure is interpreted in terms of the details of the energy bands. A new interpretation is given for some of the peaks. 1. INTRODUCTION
THE POWERof the photoemission experiments for determining important features of the electron energy band structure of semiconductors, especially for the higher electron energies, has been emphasized by Spicer and Eden [1]. From a theoretical point of view, the calculations of the energy distribution curves, necessary for a quantitative analysis of the experimental structure, has been limited to Si. Brust[2] has performed a calculation for Si in the lower photon energy range. Recently, Saravia and Casamayou [3] have extended the calculation in Si up to a photon energy of 10 eV, and used numerical techniques that improved considerably the resolution of the calculation. It was shown that the agreement between the experimental and theoretical distribution curves is excellent for the higher electron energies, allowing a detailed interpretation of the structure in terms of the energy bands. In the present work a similar calculation is performed in Ge, for which there exist experimental results [4, 5] for photon energies covering the range up to 12eV that show very interesting structure. The procedure to compute the photoemission energy distribution curves is described in Section 2. A direct transition model is adopted. The contributions for the different
energy bands are calculated, but no detailed study of the scattering of the electrons by the crystal is performed. T h e model adopted for the energy bands of Ge 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 results are given and discussed in Sections 4 and 5. The agreement with the experimerrtal results is good for the higher electron energies, and a detailed analysis is made in terms of the contributions from the different interband transitions. The regions of the Brillouin Zone (BZ) producing the most important structure are indicated. Particularly, the interpretation of the experimental structure corresponding to higher electron energies, previously discussed by Donovan and Spicer [5], is clarified. 2. CALCULATION OF THE ENERGY DISTRIBUTION CURVES We are dealing with photoelectrons produced in the volume of a semiconductor by direct optical excitation. If is the frequency of the incident photon and E the final conduction band energy of the excited electrons, the energy distribution curve N(E, hco) is given by
N(E, hoJ) = A , ~ f Ip,~(k)I2 P(E, k) 8(oJ~(k)--co) BZ
8(En(k)--E)dak.
(1)
The sum is over all conduction bands n and valence bands s. The integral is performed over all the BZ, p,~
1541
1542
L.R.
S A R A V I A and L. C A S A M A Y O U
(k) is the momentum matrix element between bands n and s, A is a normalization constant, and hto,,~(k) is equal to E,,(k)--Es(k). P ( E , k) is the probability that the electron produced with energy'E and wave vector k will escape. We perform this integration using a method developed elsewhere[3]. Briefly, the integration is reduced to one along a line defined by the surfaces E,,s(k) = E and to,s = to, which is performed analytically using a linear interpolation for the functions E,(k) and co,,,(k) inside each cube in the mesh defined by the points of the B Z where the bands are calculated numerically. 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. in (l) 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[2]. We assume that an electron produced with an energy E.(k) has a momentum 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: h2kT Constant, if E,(k) -- Eva,. > 2m P( E, k) =
h2kT 0, if E,(k) - E,,,,~ <
(2)
2m
where kr is the component of k parallel to the surface of the crystal and E,,,,. is the vacuum level. This factor is obtained in 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 ofk.
3. CALCULATION OF THE ENERGY BANDS We need to compt,te the energy bands and the dipole matrix elements over a cubic mesh in the B Z 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[6]. Briefly, one computes eigenvalues and eigenvectors at a small number of points in a coarse cubic mesh by solving the secular equation generated by the pseudopotential approach. Then, a refined mesh of points is generated from the previous one by a k. p extrapolation procedure. The combination has the advantage of greatly expediting the computational work. The spin-orbit terms have also been considered. We use the formalism developed elsewhere [7]. The pseudopotential parameters used for Ge are the same as those used in a previous w0rk[7]: V(l, l, 0 ) = --0'282 Ry, V(2, 2, 0) = 0.058 Ry, V(3, l, l) = 0'018 Ry.
The spin-orbit parameter was selected to fit the experimental value of the spin-orbit splitting at the F25, level [8]. The comparison with other energy bands, calculated by Brust[9], by Cohen and Bergstresser[10], by Herman et al.[l l], by Cardona and Pollak[12], and by Dresselhaus and Dresselhaus[13] are performed in the mentioned work[7]. The selected pseudopotential formfactors are somewhat different from those used by Brust[9] and by Cohen and Bergstresser [ 10]. The changes were performed in order to bring down the position of the I't~ level. The spin-orbit splittings in Ge, about 0"3 eV, are of the order of the broadening constants present in the photoemission experiments, what makes them not very important from the point of view of the interpretation of the experimental structure. The resulting energy bands along several directions in the B Z are shown in Fig. I. The double group notation is not used. In the discussion that follows the energy values at a general point in the B Z are denoted by Ei (i = I, 2, ...), not taking into account Kramer's degeneracy, i.e., each value of i corresponds to two eigenvalues. All the electron energies are referred to the top of the valence band. 4. RESULTS
(a) Escape probability Donovan and Spicer[5] have measured energy distribution curves obtained from Ge surfaces which have received different treatement in order to change their vacuum level. We have calculated the escape probability curves for two values of the vacuum level: 1.65 and 4.80 eV. They are shown in Fig. 2. (b) Energy distribution curves They have been calculated for different values of fifo up to 12 eV and include interband transitions involving valence bands 2, 3, 4 and conduction bands 5 to 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 two-dimensional saddle critical points is logarithmic, and the discontinuities related to maxima and minima .critical points can take large values, even approaching infinite. It is not possible to make a direct interpretation of the experimental peaks, since they are fewer and look much broader than the calculated ones. This is expected because effects such as electron-phonon and electron-electron scat-
PHOTOELECTRIC EFFECT IN GERMANIUM
1543
I0
LJ
2
-2
-4
K
W
L
]"
X
UK
Fig. 1. Energy bands of Ge along, some principal symmetry lines. They were computed using the pseudopotential method together with a k.p extrapolation procedure. The energy bands are labeled E~ (i = I, 2. . . . ), with i increasing with energy. The single group notation is given for some of the points of the BZ.
Escape
Probability ~
f
.3
.2
.I .0
/
/Vacuum
2
/
level
oev
/
Vacuum level
6
a
E,
8 eV
Fig. 2. Shows the two different escape probability functions used in the present calculation. The differences are produced by the position assumed for the vacuum level in each case.
t e r i n g , i n t e r a c t i o n o f t h e e l e c t r o n s with the c e s i u m film a n d b a n d b e n d i n g h a v e n o t b e e n considered. I n o r d e r to p e r f o r m a c o m p a r i s o n , w e i n t r o -
duce a p h e n o m e n o lo g ic a l b r o a d e n i n g of the form N ' ( E , ho~) ---
f •N ( E ' , ho.)) L ( E ' , E) dE'
(2)
1544
L.R.
S A R A V I A and L. C A S A M A Y O U
where 'hw= 4.17 eV
1
r
L ( E ' , E ) -- r r 2 Jr ( E
-
-
E') 2
(3) i.d Z
F is the broadening constant. We adopt different values of F according to the surface treatement: 0.15 and 0.10eV for the surfaces with vacuum levels at 1.65 eV (cesiated surface) and 4.80 eV (uncesiated surface). The calculated energy distribution curves for different photon energies, including the effect of the escape probability factor and broadening, are given in Figs. 3 to 6. The experimental ones, as measured by Donovan [5], are shown in Figs. 7 to 10.
,
, 2,
4.
4-6
hw= 5.15 eV
3-6 :3-7 7
5. DISCUSSION
A general remark can be made about the calculated results shown in Figs. 3 to 6. The calculated structure is clearly observed in the experiments, Figs. 7 to 10, but it appears superposed in a contribution not reproduced at all by the present calculation. This contribution is particularly large for the case of cesiated surfaces and low photon energies. It shows a large shoulder and peak which appear for all hto at electron energies equal to 2-2 and 3-0 eV. For the uncesiated surface the effect is not so important, but all the same there is a shoulder beginning at 5.0 eV for all hoJ which probably results from the action of the escape probability function on the mentioned contribution. This structure disappears after the surface is cesiated, indicating that it is not related to direct transitions. A similar situation was found in the case of Si[3]. Several authors have mentioned the possibility for this type of structure to be produced by physical effects not mentioned here, such as scattering mechanisms[I, 14, 15, 16] or contributions from indirect transitions [17]. On the following we study the interpretation of the different peaks in terms of the direct interband transition contributions. The position of the structure in the calculated distribution curves is given in Figs. 11 and 12 a s
2.
\
]
I
l
I
4.
6.
4.
6.
hw =6.18 eV
I
I
I 2.
E,
eV
Fig. 3. Shows the calculated energy distribution curves in Ge for several values of photon energy corresponding to a sample with a cesiated surface. The calculation includes an escape probability factor with t h e vacuum level at 1-65 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 ET). For the sake of clarity only partial contributions important from the point of view of the interpretation of the peaks were included.
an E - - h o ~ plot. The experimental results[5] have been superposed in the same diagrams and the structure has been denoted by capital letters, from A to J. We find that each piece of
PHOTOELECTRIC
7/
~
EFFECT
IN G E R M A N I U M
1545
hw=7.0eV
flw=7.0 eV
z
[
11 #"
I
4.
i~
o ~l 6.
I
I 8. 6.
~,~ ~ /
V 4.
I
I B.
E
I
io.
1~w=8.0eV
41- 3-10 8.
~6.
~w=9.0eV
6.
I
8.
io.
/~w- 9.0eV "
I
3,~(74_8%~/3_10 ~ ~4 - 1 0
4.
"hw=I0.0eV
l
6.
~
/
l
""~-6/r~
4.
6,
E,
I'
8,
~~
I
6.
B.
"'1\ 6.
"T-'-"~~ .. BE,
I
I
io.
-I0
4 - 7 ~ s~k'C ~ 4 i
lc
8.
eV
eV
~
I
I
Io.
Fig. 4. Shows the calculated energy distribution curves in G e for several values of photon energy corresponding to a sample with a cesiated surface. The calculation includes an escape probability factor with the vacuum level at I '65 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 ET). For the sake of clarity only partial contributions important from the point of view of the interpretation of the peaks were included.
Fig. 5. Shows the calculated distribution curves in G e for several values of photon energy corresponding to a sample with no cesium on its surface. The calculation includes an escape probability factor with the vacuum level at 4.80 eV and a b r o a d e ~ n g factor equal to 0- I0 eV. Partial results indicating the c~'ntribution from some of the interband transitions are also shown. They are identified by the numbers labeling the bands involved in the transition. For the sake of clarity only partial contributions important from the point of view of the interpretation of the peaks were included.
structure is mainly produced by contributions from a few interband transitions. They are indicated in Table 1. The region of the B Z producing the structure is also given in Table 1. The region of the B Z producing the struc-
ture is also given in Table I. Usually, the region of interest in the B Z is related to a symme;try line. Plots E - - hw for the calculated symmetry lines are given in Figs. 13 and 14, together with the position of the calculated
1546
L.R. SARAVIA and L. CASAMAYOU i
hw: 10.OeV
/'lw=10.5eV
i
I
I
I
6e
W~O.4
Uncesiotod surfoce Experimentt
LJ 4-9
3-10
I
~
I 9
I I0
//~E,
I II
eV l"lW=11.0eV
I 8
9
IO
,
Ii
E, eV ~hw =12.0eV
I I0
9
hw=ll. SeV
1 3 "lO~i'~r-"~ "~'" 9
10
E,
~~w-e.a
ev
t
6.6
o
7.7
0.0 '4
,-9 \ ~ - I O
e
8
"6
I
II
eV
E,
~
3-1~4_10 (E e
'~0,3 -
3-10
-t0 e
Oonovon Ref. 5
4-9
I
II
~0.6
eV
t
I
I Ge
I
Uncesiated
hw =12.5 eV
I
I I "hw =10.0
~ 0 . 5 F Surface
\ I
31101 N 9
E,
~ . I0
eV
f -II II
I 8
I3 9
E,
~ I0
I II
b_o~ I "8 o.~
eV
Fig. 6. Shows the calculated distribution curves in Ge for several values of photon energy and high electron energies corresponding to a sample with no cesium on its surface. The calculation includes an escape probability factor with the vacuum level at 4"80 eV and a broadening factor equal to 0"10eV. 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. For the sake of clarity only partial contributions important from the point of view of the interpretation of the peaks were included.
Fig. 7. S h o w s the experimental normalized energy distribution curves for a p - t y p e crystal of Ge with no cesium on its surface, as measured by Donovan [5].
structure. This data establishes the connection between the experimental structure and the energy bands. The structure labeled B is well reproduced by the present calculation. The main contribution comes from interband transitions Es-~0 and E4-9. Transition Es-~0 determines the position of the peaks for the higher photon energies. There is a region of the BZ, in the FKL plane, where the surfaces of constant E and constant hto are quite parallel. This region begins near the F point and moves mainly along the FK line a s the photon energy increases.
The contribution from transitions E4-9 are clearly seen at lower photon energies. An extense region in the FKX plane is responsible for the structure. Band E9 shows a saddle point which is not at the r point, but somewhat displaced, in the FXK plane, as can be seen in Fig. 1. It has an energy value below the one corresponding to Flz and for this reason interband E4-9 produces peaks in the region labeled F in Fig. 12, where peaks from Ea-10 have disappeared. This interpretation clearly indicates that level FI~, that is the double degenerate level E9 and El0, is the one related to structure B, but its position in the E -
~ ~ F
~.6'
0.0 /
4
5
6
7 E,
8 9 eV
I0
PHOTOELECTRIC
EFFECT
Table 1. Interpretation of the structure present in the photoemission experiments in terms of the different interband transitions. Regions o f the B Z involved in the t?ansitions as well as energy levels corresponding to the related symmetry points are indicated Interband transition
Structure
L o c a t i o n in B Z
Related energylevel
IN G E R M A N I U M I
1547 I
I
I
I
Ge Cesioted surface 5w=3.6 V O X g o J~ Q.
3.:
2 -
Experimen~"
-
oo
8
3-10
Plane K L
4--9
Plane K X
C hoJ ~< 8.2 eV
4-7
L W line in the
hoJ /> 8.2 eV
4-8
D
3-7
E ho~ ~< 9.0 eV
3-6
B
F12
hoJ >/9.0 eV G hto/> 5-5 eV hto ~< 5.5 eV
"~ 0.6 ,c
I
h e x a g o n a l face Plane FKX, n e a r line r x . Line L W, n e a r the L point.
2 I
3-7
3-6
L W line, n e a r the
4-6
L point L W line, n e a r the L point
I
I
Ge
I
I
W ( E3)
E,
eV
I
I
,.E
Experiment ] ~
d
D~
L3,
I
I 5 I
I
Ge Cesiated surfoce
/_ \\
z
"hw= 5.8 eV
O D x
.5.0
g
o D LLI
|
J
1\
3
L3,
I 4
3
X4
H e x a g o n a l face, n e a r the W point. H e x a g o n a l face, n e a r the W point
I
0.7
taJ
2 I
0
I I
I 2
1 4
5
6
h w = 11.4 eV
-7222'"
E~
eV
Fig. 9, S h o w s the e x p e r i m e n t a l n o r m a l i z e d e n e r g y distribution c u r v e s for a p - t y p e crystal o f G e with a cesiated surface, as m e a s u r e d by D o n o v a n [5].
uS Z 0-5
g o 0.4
o 0.3 hJO'2 o. o.o
Experiment, Donoven, R e f . 5 I 6
I
E
I I 8 eV
I0
Fig. 8. Shows the experimental normalized energy distribution curves f o r a p-type crystal o f Ge with no cesium on its surface, as measured by D o n o v a n [5].
JPCS Vol. 32, No. 7 - J
hto diagram is not directly given by the intersection of the line defined by structure B and the line E = h o J . Its position can be determined taking into account the fact that a shift of the conduction bands E9 and Elo in order to put the F12 level in the right place, essentially shifts structure B in the diagram E--hoJ along a direction parallel to the line E - - hie. In order to superpose the calculated and experimental structure, the calculated F12 level has to be shifted in 0.8 eV, what places this level at 8.1 eV approximately. Level F1 has been men-
1548
L . R . SARAVIA and L. CASAMAYOU I
I
I
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Cesioted surface Vacuum level =1.65 eV
-
Experiment,
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Donovan Ref.5
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Experiment
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.
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.
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.
.
.
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Fig. I0. Shows the experimental normalized energy distribution curves for a p-type crystal of Ge with a cesiated surface, as measured by Donovan [5].
tioned in the p a s t [ I , 5] as a possible source for this structure. W e h a v e found that interb a n d transitions related to l't, here d e n o t e d as En, give no i m p o r t a n t contribution. T h i s is due to the fact that the dipole matrix e l e m e n t s for these transitions are v e r y small n e a r the 171 point, and transition r2s,-l', is indeed forbidden. Structure labeled A is not well r e p r o d u c e d . W e p e r f o r m e d a detailed calculation o f the N ( E , hoJ) c u r v e s in the region o f higher photon
I 3
I 4
I 5
I 6 hw,
I 7 ev
P B
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Fig. I 1. E--hoJ curves for the most prominent calculated structure and for the experimental structure as given by Donovan[5] for a crystal of Ge with a cesiated surface. The experimental peaks are labeled by capital letters from A to J. The calculated peaks are indicated by the numbers labeling the bands involved in the transition mainly responsible for the structure. energies in o r d e r to detect these shoulders. T h e results are s h o w n in Fig. 6. T h e r e a p p e a r s s o m e small s t r u c t u r e p r o d u c e d by E4-9 and E4-,0 transitions f r o m regions in the BZ near the h e x a g o n a l face. T h e a g r e e m e n t in position is not very g o o d and we are not sure w h e t h e r they c o r r e s p o n d to s t r u c t u r e A or they are shoulders in the p h o t o n region w h e r e m e a s u r e m e n t s w e r e not p e r f o r m e d . T h e e x p e r i m e n t a l s t r u c t u r e is indeed v e r y w e a k , as can be seen in Fig. 8 and 9, and small shifts+ in the calculated contribution to p e a k s B could e v e n t u a l l y p r o d u c e the shoulders. S t r u c t u r e labeled C is mainly related to interbands E4-T and E4-8. E4-7 is r e s p o n s i b l e for the p e a k s c o r r e s p o n d i n g to the l o w e r photon energies. T h e contributions are produced in regions near the L W line. F o r this r e a s o n they a p p e a r w h e n hco b e c o m e s larger than the values c o r r e s p o n d i n g to the L point. F o r the case of the uncesiated surface, the probability e s c a p e function highly distorts the contribution f r o m the E4-7 transition. E4-8
P H O T O E L E C T R I C E F F E C T IN G E R M A N I U M
1549
Uncesioted sur fete Vocuum level : 4 . 8 0 eV
. IC
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Fig. 12. E--/i(o curves for the most prominent calculated structure and rot the experimental structure as given by Donovan [5] for a crystal of Ge with no cesium on its surface. The experimental peaks are labeled by capital letters from A to J. The calculated peaks are indicated by numbers labeling the bands involved in the transitions mainly responsible for the structure.
9
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o Calculated peeks Symmetry lines
8
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Fig. 13. E--hoJ curves for the most prominent calculated structure (open circles) for a crystal o r G e with a cesiated surface, and for the points along the symmetry lines of the B Z related to the structure (solid lines). The interband transition plotted for a given point are indicated by the numbers labeling the bands involved in the transition. The symmetry lines are identified through the symmetry points in the B Z that determine the line.
1550
L.R.
S A R A V I A and L. C A S A M A Y O U
gives contribution for the higher photon energies. An extended region in the r K X plane, related to the r X symmetry line, is responsible for the contribution. Donovan[5] gives the same interpretation, based on the work of Spicer and Eden for GaAs[1]. It is interesting to notice that the peaks obtained from E4-8 define a line in the E--ho~ diagram which becomes parallel to E---hoJ for the higher energies. This means that for the region of the B Z near the X point, the valence band E4(X4) is very flat and the energy, value corresponding to )(4 can be determined from the experiment. A value between --2.7 and - 2 . 9 eV is obtained [5].
Ge o --
C a l c u l a t e d peaks S y m m e t r y lines o
o
.i
5
6
3"
I 8 hw,
I 9
I I0
I II
I 12
eV
Fig. 14. E--hco curves for the most prominent calculated structure (open circles) for a crystal of Ge-with no cesium on its surface, and for the interband transitions corresponding to points along the symmetry lines of the B Z related to the structure (solid lines). The interband transitions are indicated by the numbers labeling the bands involved in the transition. The symmetry lines are identified through the symmetry points in the B Z that determine the line.
Structure labeled D is partially produced by contributions from interband transition E3-~, highly distorted by its superposition on the shoulder produced by contributions not due to direct transitions. The main contribution comes from a region in the B Z related to the L W line near the L point. The structure is not very pronounced and it i,,J practically lost between the structure C and E when a cesiated surface is used. For hoJ larger than 9.0 eV, E 3 _ 7 intersects
the hexagonal face near the W point, and its contribution becomes significant producing the structure labeled E. As hoJ increases, the region of interest moves from the W to the K point. A similar interpretation has been given by Donovan[5], who noticed that constant values of E3 near the W point produce a structure moving like E in the E--hoJ diagram. The peaks determine the experimental value of E3, --4.2 eV approximately. The calculated value is off by 0.6 eV and for this reason the calculated peaks have to be shifted 0 . 6 e V in a direction parallel to the hoJ axis in the E--ho~ diagrams, Figs. 11 and 12, to reproduce the position of the experimental structure E. When a cesiated surface is used, the contributions to structure E from interband E3-6 become important. They are also produced in a region of the hexagonal face related to the W point. Structure labeled G is also produced in part by Ez-6. For the lower photon energy range the surfaces of constant hoJ find a region in the hexagonal face near the L point where the electron energies are practically constant. We get peaks which are locked at a certain value of E for hco up to 7.5 eV. Interband E4-6 also contributes to structure G and the structure is also produced in a region near the L point. Structure G determines the position of level E6 (L~), giving a value of 4.2 eV approximately [5]. Our calculated bands give a value for La, offby 0.5 eV from the experimental results. This can be clearly seen in the E--hoJ diagram, Fig. 1 1, where the calculated structure G is about 0.5 eV below the experimental one. Finally, we have not attempted a study of the structure labeled H , since it is superposed to a large contribution not predicted at all by our calculation not allowing any detailed comparison. Donovan[5] has performed a detailed experimental study and relates the onset of this structure to the 1"15level. Unfortunately, the spin orbit splittings are not resolved. A value of 3.13 eV is obtained for the center of mass transition energy of the r25,--rl~ transition.
P H O T O E L E C T R I C E F F E C T IN G E R M A N I U M 6. CONCLUSIONS
The present calculation has allowed a detailed interpretation of the structure present in the photoemission experiments in terms of the energy bands of the semiconductor. In general, this interpretation is similar to the one given by Donovan[5], based on a direct inspection of the energy bands calculated by Herman[11], and by Brust and Kane [18]. The most important difference is the change in the identification of structure B, which is now related to level F~2, with an assigned energy of 8-1eV, and not l'v It is interesting to notice that the same problem in the interpretation of the experimental structure is present in other semiconductors[l]. Actually, we found the same interpretation for Si [3]. These results add a considerable amount of information about the value of the energy levels of some of the symmetry points to that obtained in the past from other types of experiments. The available information has been included in Table 2, where a comparison is performed with some of the energy band calculations. The values indicated in Table 2 for
1551
1"2,, L1, and L3, are obtained from optical experiments[19, 20]. For L3, we used a value of the L3,--LI transition energy which is 0-1 eV smaller than the one measured for the A3--A~ transition in reflectivity experiments [20]. The position of L3, is determined since L1 is known. The L3,--L3 transition, also measured in reflectivity experiments[20], can be used as a check of the position assigned to L3 through the photoemission experiment. It is important to remark that most of the values included as results of experiments are really obtained through processes of interpretation which could be eventually revised. Table 2 shows that the calculation performed by H e r m a n [ I l l is in excellent agreement with experiment for the valence bands, but the discrepancies become larger for the higher conduction bands. Actually, we see from the pseudopotential calculations shown in Table 2 that the conduction levels 1"15,L3, and 1"12are quite sensible to small changes in the parameters. A better overall agreement can be expected if the new data is taken into account in the process of adjustment of the parameters.
Table 2. Shows experimental and theoretical energy level locations in eV. The first experimental value is that obtained from experiment. The second value, between parenthesis, represents the experimental level reduced to. the center o f mass value without spin-orbit splitting. The theoretical values do not include spin-orbit splittings Level F2, F15 Ft2 La, Lt L3 X4 W(E3)
Exp. 0-8 [b] (0"9) (3" 1) [a] 8" 1[a] (8-2) (--l'4)[b] 0"7 [b] (0"8) 4"2 [a] (4.3) --2-8 [a] --3.0[a] (--2.7, --2.9) --4.2[a] (--4-1)
Herman et al. [c]
Brust [d]
Present calc.
New model
0.9 2.9 9'5 --1"3 0"8 4.1
0"7 3-5 8-5 --1"I 0"7 4.3
0.9 3.0 7-3 --1"2 0"9 3.7
0-8 3.2 7.9 --1'2 0"8 4-0
--2.8 --4-1
--2.5 --3"4
--2.8 --3-5
--2.7 --3-5
[a] Obtained from photoemission experiments. [b] Obtained from optical experiments, Ref. [19-21]. [c] Ref. [11]. [d] Ref. [9].
1552
L.R.
S A R A V 1 A and L. C A S A M A Y O U
The pseudopotential model used in the present calculation was obtained in a previous work [7] from a careful adjustment to the experimental energy levels near the band gap, but now it shows large discrepancies for the other levels. We have tried a general adjustment of the pseudopotential parameters at posteriori of the photoemission calculation performed here. The energy levels corresponding to the new model, calculated with the following parameters: V(I, 1, 1 ) = - - 0 . 2 5 1 Ry, V(2, 2, 0) = 0.028 Ry, V(3. 1. 1) = 0.039 Ry, are given in the last column of Table 2. They show an average error of 0-2 eV respect to experiment. Although a better general agreement is obtained in the conduction levels, it was not possible to improve the position of the W level. It appears as practically unchanged in all the available pseudopotential calculations and contrasting with the value obtained by Herman [l l], which is the same as the experimental one. Acknowledgements-We are indebted to Dr. T. M. Donovan who made available to us the experimental results of Ge prior to publication, as well as to Dr. D. Brust and Professor W. E. Spicer for bringing some references to our attention. The calculation was performed at the 'Centro de Computaci6n de la Universidad de la Rcpfiblica'. We acknowledge its staff for technical assistante. This research was supported in part by the "Fondo de Investigaci6n Cient[fica de la Universidad de la Repfblica, Montevideo, Uruguay'.
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