- Email: [email protected]
The dielectric functions of CdI2, CdBr2 and CdCl2
276
THE DIELECTRIC FUNCTIONS OF Cdl2, CdBr2 AND CdCI2 R.D. B R I N G A N S and W,Y. L I A N G
Cavendish Laboratory,Madingley Road, Cambridge CB3 0HE, England The results of electron energy loss measurements on polycrystalline films of CdI2, CdBr2 and CdCI2 are presented and values of E(to)calculated from these results are compared and discussed in terms of a band scheme for these materials. It is found that both valence and conduction bands in the chloride are more separated than those in the iodide with the bromide showing a separation between the upper two valence bands similar to that of the chloride and a conduction band separation similar to Cdl2. Results of angular dependent energy loss measurements for CdCI2 are presented and show effects of the variation in wavevector.
I. Introduction
2. The experiment and calculation of E(k, w)
The compounds CdI2, CdBr2 and CdCI2 are ionic layered materials with the halogen ions being close-packed and the cadmium ions in octahedral sites. Apart from CdI2 for which band structures have recently been calculated [1, 2] there is still little information about the electronic structure of these materials. With the exception again of the iodide [2] no optical determination of their dielectric functions has been made, optical experiments in the main being concerned with the band gap excitons exhibited by all three compounds (see for example, refs. 3 and 4). In the present work both small angle and angular dependent electron energy loss measurements have been made on these materials and the dielectric function E(k, w) determined in an attempt to examine the effect of the different anions on the electronic structure. Transmission electron energy loss spectroscopy in which the probing electrons are analysed in the straight-through direction has been used for some time as an alternative to optical techniques for investigation of vertical interband transitions. In addition, by energyanalysing only those electrons which have been scattered through a particular angle, information about non-vertical transitions can be obtained and the full dielectric function, E(k, to) determined. Section 2 gives some experimental details as well as the method by which e(k, to) can be determined from the energy loss results. Low-k results are given for the three compounds in section 3 and discussed in terms of an electron band scheme and section 4 shows some kdependent results for CdCI2.
The three materials studied have layer type structures, bonding being strong between atoms within a particular layer and weak between layers. Their primitive cells are hexagonal in the case of CdI2 and rhombohedral for CdC12 and CdBr2, however for convenience of comparison the latter case is equivalent to hexagonal structure with three molecules in the unit cell. Polycrystalline films of CdI2, CdBr2 and CdC12 were prepared by evaporation onto 'Formvar' substrates and were found to be highly textured with the basal planes of the polycrystals being parallel to the plane of the substrate. The optimum sample thickness was found to be 400500 .~, thicker films allowing a higher proportion of multiple scattering and much thinner films showing evidence of surface effects such as carbon contamination in the electron beam. Between 200 ~ and 1000 ,~ the shapes of the spectra were independent of thickness. Measurements were carried out as a function of energy loss at fixed scattering angles using 45 keV incident electrons and with energy and wavevector resolutions of 0.6 eV and 0.035 ~-1. The probability that a fast electron loses energy hw and transfers m o m e n t u m hk in a single collision in the bulk of the solid is given by:
Physica 99B (1980) 276-280 © North-Holland
P(k, t o ) = A ~ - ~ I m { ~ } , where A is a constant depending on the particular experimental conditions. In addition to P(k, to), the measured probability may include contributions from surface and (~erenkov radia-
277 tion losses and from electrons which have undergone m o r e than one scattering event. Except at scattering angles much smaller than those used here, surface and radiation losses can be neglected [5] and it is only the contribution from multiple scattering that need be removed. Because P is dominated by a large p e a k corresponding to the plasma resonance of the solid, the relative importance of multiple scattering is indicated by a broad p e a k occurring at about twice the plasmon energy. This p e a k allowed the multiple inelastic contribution to be quantified and removed. Once Im{-1/e(k,w)} was known for each value of k=-lk[, Re{1/e(k, to)} could be calculated using a K r a m e r s - K r 6 n i g technique and el(k, to) and e2(k, to) evaluated. The K r a m e r s Kr6nig integral utilized data from the entire energy range of 0 to 90 eV and values of E are presented for energies up to 30 eV. In order to carry out the K r a m e r s - K r 6 n i g analysis it is necessary that P(k, to) be known absolutely and for low k this can be done most reliably by scaling the measured intensity to an optically determined value of ~ (for which k - 0). For semiconductors and insulators one only requires the optical refractive index n = X/~e~ at a frequency well below the band gap. In principle
1.5
f
....
I
....
I
the higher k m e a s u r e m e n t s can then be normalised to the low-k one. Values of el(k, w = O) determined in this way are very sensitive to uncertainties in the normalization factor and provide a good test of the accuracy of the scaling methods. Values of El(k, to = 0) were c o m p a r e d to those calculated with a simple T h o m a s - F e r m i theory of dielectric screening in semiconductors [6] and showed quite good agreement. Scatter about the theoretical curve indicated that it was m o r e reliable to use the calculated e~(k, to = O) to scale the P(k, to) in an analogous m a n n e r to the use of optical data in the low-k case. 3. Results of measurements at small k and discussion
Fig. 1 shows the normalized single loss functions at the lowest m o m e n t u m transfer k 0.06~-1. The main feature in each spectrum is the broad plasmon which is centred at about 17 eV, 18.5 eV and 20 eV for the iodide, bromide and chloride respectively. Interband transitions are indicated by smaller features in the loss function and the direct band gap by the onset of energy loss. The curves of ~1 and ~2 derived from K r a m e r s - K r 6 n i g analysis of the loss functions of fig. 1 are shown in figs. 2(a) and (b).
....
I
"
'
'
'
]
'
'
'
'_
Xm( )
1-0
0.5
O--
0
10
20
30 1boo (eV)
40
50
Fig. 1. Normalized energy loss functions at near-zero k for the compounds CdI2, CdBr2 and CdCI2.
278
3.1. CdI2
and a shoulder at 6.8eV). They attribute the very sharp peak at 6.0 eV to an exciton, and the fact that the peak at the same energy in our results is much less significant is to be expected with our energy resolution of 0 . 6 e V and the non-zero momentum transfer, BRJ attribute the remaining structure to features in the joint density of states. The band structures calculated by BRJ and previously by McCanny et al. [1] have the same form:
Bordas et al. [2] (hereafter referred to as BRJ) have determined e] and e2 for CdI2 by K r a m e r s Kr6nig analysis of reflectivity measurements using synchrotran radiation. Below 8 eV, BRJ have made use of the results of Greenaway and Nitsche [3]. The features that they find in e2 (peaks at 4.95, 6.02, 6.75, 15.0 and 15.8 eV and a shoulder at ~9.0 eV) agree very well with those in fig. 2(b) (peaks at 5.0, 6.0, 8.25 and 15.25 eV . . . .
I
. . . .
I
. . . .
I
. . . .
I
. . . .
I
. . . .
E1
"/..,"I ~-,
._t \ i",
-
Cd 12
-
- CdBr2
.
.
.
.
.
.,,
v
~
CdCl 2
~
0 10
0
20
30
'hco (eV) (a) . . . .
I
. . . .
I
. . . .
I
. . . .
I
. . . .
I
. . . .
8
6
A
E2
4 2 0
10
20 f~co (eV) (b)
Fig. 2. (a) el(~o) and (b) e2(to) derived from the data in fig. 1.
30
279 (1) two conduction bands, the u p p e r one arising from Cd p states and well separated from it, the lower one, very narrow and formed mainly from Cd s states; (2) three valence bands, the uppermost of which is mainly I Px,y, a band which overlaps slightly with this and consists of mainly I Pz with some Cd s, and then the I s states at a much lower energy; (3) a narrow dispersionless band arising from the Cd 4d level. This band scheme is shown in fig. 3. The two calculated band structures differ mainly in the separation between the two conduction bands, that of [1] having the Cd p band 5-6 eV above the first conduction band whereas in [2] the separation is about 1 eV. B R J calculated a density of states for the u p p e r two valence bands which shows a minimum between the centres of the I p~ band and the u p p e r m o s t one and they attribute the transitions at 6.75 and 9 eV to this shape being combined with the upper conduction band and assign the p e a k at 5.0 eV to a transition between the top of the valence band and the Cd s band. In a p a p e r written independently and showing earlier energy loss results for CdI2 [7], we had attributed both low energy peaks to transitions to the Cd s band and in addition had suggested that the Cd p bands in the band structure of McCanny et al. [1] were too high by about 4 eV to account for the peaks in E2 at 8.25 eV which we attributed to transitions from the upper valence band to the Cd p band and at 15.25 eV which we attributed to transitions from the Cd d level to the same band. Having seen the calculated density of states of B R J it appears that the 6.5 eV p e a k should in fact be assigned to the I Px,y to Cd p transition.
61
I
5 t
.~
...............
4 3
i...........
1 [
Cd
S
Halide Holide
2
Cd J
Interpretation of the results for CdBr2 and CdCI2 will also use the band scheme of fig. 3, there having been no band structures calculated yet for these compounds. Dealing with the chloride first, the most likely assignment is to identify the four peaks A, B, C and D in fig. 2(b) (which occur at 6.2, 8.0, 11.6 and 16.2 eV respectively) with the transitions 4 ~ 5, 4--> 6, 3 ~ 6 and 2-->6 where the numbers of the bands are as shown in fig. 3. The peaks are much m o r e distinct and better separated than in CdI2 and indicate that (1) the "centres of gravity" of the two conduction bands in fig. 3 are now separated by 1.8eV c o m p a r e d with 1.3eV for CdI2, these centres occurring at about 6 and 8 eV above the valence band; (2) the upper two valence bands are also much m o r e clearly separated than in CdI2 and their centres are 3.5 eV apart (cf. 1.9 eV in CdI2); (3) the Cd d level is now 8 eV below the top of the valence band c o m p a r e d with 9 e V in the iodide. If the same assignments of peaks A - D are m a d e for the bromide then it shows that the two conduction bands have a similar separation to the iodide but that the two u p p e r valence bands resemble those of the chloride. It should be pointed out that XPS m e a s u r e m e n t s on CdI2, CdBr2 and CdCI2 [8] all show little separation between the two upper valence bands. In summary, the overall effect of an increase in ionicity is a sharpening and increased separation of features in the E2(~0) spectra indicating a narrowing and moving apart of the electron energy bands.
4. k-dependent results for CdCI2
EF
J
3.2. CdBre and CdCle
Px,y Pz
d
Halide
s
Fig. 3. Band scheme for the cadmium halides and 2.
based on
refs. 1
Curves of E2(k, oJ) determined for six different values of k are shown in fig. 4. Because the c axes of the polycrystals are parallel to the incident electron b e a m the m o m e n t u m transfer k lies in the plane of the Brillouin Z o n e containing F, M and K. The m o m e n t u m transfers are shown in fig. 4 as a fraction of the dimension /'M (0.94 •-1 for CdC12) and vary from near zero up to 94% of the F to M separation. The following effects of varying k can be seen in the e2(k, ~) spectra:
280
'
.
.
.
.
.
.
.
i
'
'
'
'
I
. . . .
I
. . . .
I
. . . .
CdCI2
E ~ / r M =0.06 0 0.24
.
.
.
.
.
10
.
~co(eV)
20
30
Fig. 4. Curves of e2(k, w) for CdCI2, for values of k from 6% to 94% of the F M dimension.
(1) the p e a k at 6 . 2 e V at low k, which was assigned to transitions from the top of the valence band to the Cd s band, moves steadily from 6.3 eV at k / F M = 0.06 to 6.8 eV at k / F M = 0.94; (2) the broad p e a k at 8.0eV at low k steadily loses intensity relative to the others and moves up in energy to 9.2 eV by k / F M = 0.94, (3) the peaks at l l . 6 e V (bands 3 ~ 6 ) and 16.2 eV (bands 2--->6) show no change with k. T h e effect on E(k, w) of varying k can yield information about the shapes of the energy bands involved in transitions and is the subject of further study at present.
References [1] J.V. McCanny, R.H. Williams, R.B. Murray and P.C. Kemeny, J. Phys. C.: Solid State Physics 10 (1977) 4255. [2] J. Bordas, J. Robertson and A. Jakobsson, J. Phys. C.: Solid State Physics 11 (1978)2607. [31 D.L. Greenaway and R. Nitsche, J. Phys. Chem. Solids 26 (1%5) 1445. [4] S. Kondo and H. Matsumoto, Solid State Comm. 24 (1977) 695. [51 J. Daniels, C. v. Festenberg, H. Raether and K. Zeppenreid, Springer Tracts in Modern Physics 54 (1970) 77. [6] R. Resta, Phys. Rev. B 16 (1977) 2717. [7] R D . Bringans and W.Y. Liang in Proceedings of the 14th Int. Conf. on the Physics of Semiconductors, B.L.H. Wilson, ed. (Inst. of Physics, London, 1978) p. 891. [8] T, Matsukawa and T. lshii, J. Phys. Soc. Jap. 41 (1976) 1285.
THE DIELECTRIC FUNCTIONS OF Cdl2, CdBr2 AND CdCI2 R.D. B R I N G A N S and W,Y. L I A N G
Cavendish Laboratory,Madingley Road, Cambridge CB3 0HE, England The results of electron energy loss measurements on polycrystalline films of CdI2, CdBr2 and CdCI2 are presented and values of E(to)calculated from these results are compared and discussed in terms of a band scheme for these materials. It is found that both valence and conduction bands in the chloride are more separated than those in the iodide with the bromide showing a separation between the upper two valence bands similar to that of the chloride and a conduction band separation similar to Cdl2. Results of angular dependent energy loss measurements for CdCI2 are presented and show effects of the variation in wavevector.
I. Introduction
2. The experiment and calculation of E(k, w)
The compounds CdI2, CdBr2 and CdCI2 are ionic layered materials with the halogen ions being close-packed and the cadmium ions in octahedral sites. Apart from CdI2 for which band structures have recently been calculated [1, 2] there is still little information about the electronic structure of these materials. With the exception again of the iodide [2] no optical determination of their dielectric functions has been made, optical experiments in the main being concerned with the band gap excitons exhibited by all three compounds (see for example, refs. 3 and 4). In the present work both small angle and angular dependent electron energy loss measurements have been made on these materials and the dielectric function E(k, w) determined in an attempt to examine the effect of the different anions on the electronic structure. Transmission electron energy loss spectroscopy in which the probing electrons are analysed in the straight-through direction has been used for some time as an alternative to optical techniques for investigation of vertical interband transitions. In addition, by energyanalysing only those electrons which have been scattered through a particular angle, information about non-vertical transitions can be obtained and the full dielectric function, E(k, to) determined. Section 2 gives some experimental details as well as the method by which e(k, to) can be determined from the energy loss results. Low-k results are given for the three compounds in section 3 and discussed in terms of an electron band scheme and section 4 shows some kdependent results for CdCI2.
The three materials studied have layer type structures, bonding being strong between atoms within a particular layer and weak between layers. Their primitive cells are hexagonal in the case of CdI2 and rhombohedral for CdC12 and CdBr2, however for convenience of comparison the latter case is equivalent to hexagonal structure with three molecules in the unit cell. Polycrystalline films of CdI2, CdBr2 and CdC12 were prepared by evaporation onto 'Formvar' substrates and were found to be highly textured with the basal planes of the polycrystals being parallel to the plane of the substrate. The optimum sample thickness was found to be 400500 .~, thicker films allowing a higher proportion of multiple scattering and much thinner films showing evidence of surface effects such as carbon contamination in the electron beam. Between 200 ~ and 1000 ,~ the shapes of the spectra were independent of thickness. Measurements were carried out as a function of energy loss at fixed scattering angles using 45 keV incident electrons and with energy and wavevector resolutions of 0.6 eV and 0.035 ~-1. The probability that a fast electron loses energy hw and transfers m o m e n t u m hk in a single collision in the bulk of the solid is given by:
Physica 99B (1980) 276-280 © North-Holland
P(k, t o ) = A ~ - ~ I m { ~ } , where A is a constant depending on the particular experimental conditions. In addition to P(k, to), the measured probability may include contributions from surface and (~erenkov radia-
277 tion losses and from electrons which have undergone m o r e than one scattering event. Except at scattering angles much smaller than those used here, surface and radiation losses can be neglected [5] and it is only the contribution from multiple scattering that need be removed. Because P is dominated by a large p e a k corresponding to the plasma resonance of the solid, the relative importance of multiple scattering is indicated by a broad p e a k occurring at about twice the plasmon energy. This p e a k allowed the multiple inelastic contribution to be quantified and removed. Once Im{-1/e(k,w)} was known for each value of k=-lk[, Re{1/e(k, to)} could be calculated using a K r a m e r s - K r 6 n i g technique and el(k, to) and e2(k, to) evaluated. The K r a m e r s Kr6nig integral utilized data from the entire energy range of 0 to 90 eV and values of E are presented for energies up to 30 eV. In order to carry out the K r a m e r s - K r 6 n i g analysis it is necessary that P(k, to) be known absolutely and for low k this can be done most reliably by scaling the measured intensity to an optically determined value of ~ (for which k - 0). For semiconductors and insulators one only requires the optical refractive index n = X/~e~ at a frequency well below the band gap. In principle
1.5
f
....
I
....
I
the higher k m e a s u r e m e n t s can then be normalised to the low-k one. Values of el(k, w = O) determined in this way are very sensitive to uncertainties in the normalization factor and provide a good test of the accuracy of the scaling methods. Values of El(k, to = 0) were c o m p a r e d to those calculated with a simple T h o m a s - F e r m i theory of dielectric screening in semiconductors [6] and showed quite good agreement. Scatter about the theoretical curve indicated that it was m o r e reliable to use the calculated e~(k, to = O) to scale the P(k, to) in an analogous m a n n e r to the use of optical data in the low-k case. 3. Results of measurements at small k and discussion
Fig. 1 shows the normalized single loss functions at the lowest m o m e n t u m transfer k 0.06~-1. The main feature in each spectrum is the broad plasmon which is centred at about 17 eV, 18.5 eV and 20 eV for the iodide, bromide and chloride respectively. Interband transitions are indicated by smaller features in the loss function and the direct band gap by the onset of energy loss. The curves of ~1 and ~2 derived from K r a m e r s - K r 6 n i g analysis of the loss functions of fig. 1 are shown in figs. 2(a) and (b).
....
I
"
'
'
'
]
'
'
'
'_
Xm( )
1-0
0.5
O--
0
10
20
30 1boo (eV)
40
50
Fig. 1. Normalized energy loss functions at near-zero k for the compounds CdI2, CdBr2 and CdCI2.
278
3.1. CdI2
and a shoulder at 6.8eV). They attribute the very sharp peak at 6.0 eV to an exciton, and the fact that the peak at the same energy in our results is much less significant is to be expected with our energy resolution of 0 . 6 e V and the non-zero momentum transfer, BRJ attribute the remaining structure to features in the joint density of states. The band structures calculated by BRJ and previously by McCanny et al. [1] have the same form:
Bordas et al. [2] (hereafter referred to as BRJ) have determined e] and e2 for CdI2 by K r a m e r s Kr6nig analysis of reflectivity measurements using synchrotran radiation. Below 8 eV, BRJ have made use of the results of Greenaway and Nitsche [3]. The features that they find in e2 (peaks at 4.95, 6.02, 6.75, 15.0 and 15.8 eV and a shoulder at ~9.0 eV) agree very well with those in fig. 2(b) (peaks at 5.0, 6.0, 8.25 and 15.25 eV . . . .
I
. . . .
I
. . . .
I
. . . .
I
. . . .
I
. . . .
E1
"/..,"I ~-,
._t \ i",
-
Cd 12
-
- CdBr2
.
.
.
.
.
.,,
v
~
CdCl 2
~
0 10
0
20
30
'hco (eV) (a) . . . .
I
. . . .
I
. . . .
I
. . . .
I
. . . .
I
. . . .
8
6
A
E2
4 2 0
10
20 f~co (eV) (b)
Fig. 2. (a) el(~o) and (b) e2(to) derived from the data in fig. 1.
30
279 (1) two conduction bands, the u p p e r one arising from Cd p states and well separated from it, the lower one, very narrow and formed mainly from Cd s states; (2) three valence bands, the uppermost of which is mainly I Px,y, a band which overlaps slightly with this and consists of mainly I Pz with some Cd s, and then the I s states at a much lower energy; (3) a narrow dispersionless band arising from the Cd 4d level. This band scheme is shown in fig. 3. The two calculated band structures differ mainly in the separation between the two conduction bands, that of [1] having the Cd p band 5-6 eV above the first conduction band whereas in [2] the separation is about 1 eV. B R J calculated a density of states for the u p p e r two valence bands which shows a minimum between the centres of the I p~ band and the u p p e r m o s t one and they attribute the transitions at 6.75 and 9 eV to this shape being combined with the upper conduction band and assign the p e a k at 5.0 eV to a transition between the top of the valence band and the Cd s band. In a p a p e r written independently and showing earlier energy loss results for CdI2 [7], we had attributed both low energy peaks to transitions to the Cd s band and in addition had suggested that the Cd p bands in the band structure of McCanny et al. [1] were too high by about 4 eV to account for the peaks in E2 at 8.25 eV which we attributed to transitions from the upper valence band to the Cd p band and at 15.25 eV which we attributed to transitions from the Cd d level to the same band. Having seen the calculated density of states of B R J it appears that the 6.5 eV p e a k should in fact be assigned to the I Px,y to Cd p transition.
61
I
5 t
.~
...............
4 3
i...........
1 [
Cd
S
Halide Holide
2
Cd J
Interpretation of the results for CdBr2 and CdCI2 will also use the band scheme of fig. 3, there having been no band structures calculated yet for these compounds. Dealing with the chloride first, the most likely assignment is to identify the four peaks A, B, C and D in fig. 2(b) (which occur at 6.2, 8.0, 11.6 and 16.2 eV respectively) with the transitions 4 ~ 5, 4--> 6, 3 ~ 6 and 2-->6 where the numbers of the bands are as shown in fig. 3. The peaks are much m o r e distinct and better separated than in CdI2 and indicate that (1) the "centres of gravity" of the two conduction bands in fig. 3 are now separated by 1.8eV c o m p a r e d with 1.3eV for CdI2, these centres occurring at about 6 and 8 eV above the valence band; (2) the upper two valence bands are also much m o r e clearly separated than in CdI2 and their centres are 3.5 eV apart (cf. 1.9 eV in CdI2); (3) the Cd d level is now 8 eV below the top of the valence band c o m p a r e d with 9 e V in the iodide. If the same assignments of peaks A - D are m a d e for the bromide then it shows that the two conduction bands have a similar separation to the iodide but that the two u p p e r valence bands resemble those of the chloride. It should be pointed out that XPS m e a s u r e m e n t s on CdI2, CdBr2 and CdCI2 [8] all show little separation between the two upper valence bands. In summary, the overall effect of an increase in ionicity is a sharpening and increased separation of features in the E2(~0) spectra indicating a narrowing and moving apart of the electron energy bands.
4. k-dependent results for CdCI2
EF
J
3.2. CdBre and CdCle
Px,y Pz
d
Halide
s
Fig. 3. Band scheme for the cadmium halides and 2.
based on
refs. 1
Curves of E2(k, oJ) determined for six different values of k are shown in fig. 4. Because the c axes of the polycrystals are parallel to the incident electron b e a m the m o m e n t u m transfer k lies in the plane of the Brillouin Z o n e containing F, M and K. The m o m e n t u m transfers are shown in fig. 4 as a fraction of the dimension /'M (0.94 •-1 for CdC12) and vary from near zero up to 94% of the F to M separation. The following effects of varying k can be seen in the e2(k, ~) spectra:
280
'
.
.
.
.
.
.
.
i
'
'
'
'
I
. . . .
I
. . . .
I
. . . .
CdCI2
E ~ / r M =0.06 0 0.24
.
.
.
.
.
10
.
~co(eV)
20
30
Fig. 4. Curves of e2(k, w) for CdCI2, for values of k from 6% to 94% of the F M dimension.
(1) the p e a k at 6 . 2 e V at low k, which was assigned to transitions from the top of the valence band to the Cd s band, moves steadily from 6.3 eV at k / F M = 0.06 to 6.8 eV at k / F M = 0.94; (2) the broad p e a k at 8.0eV at low k steadily loses intensity relative to the others and moves up in energy to 9.2 eV by k / F M = 0.94, (3) the peaks at l l . 6 e V (bands 3 ~ 6 ) and 16.2 eV (bands 2--->6) show no change with k. T h e effect on E(k, w) of varying k can yield information about the shapes of the energy bands involved in transitions and is the subject of further study at present.
References [1] J.V. McCanny, R.H. Williams, R.B. Murray and P.C. Kemeny, J. Phys. C.: Solid State Physics 10 (1977) 4255. [2] J. Bordas, J. Robertson and A. Jakobsson, J. Phys. C.: Solid State Physics 11 (1978)2607. [31 D.L. Greenaway and R. Nitsche, J. Phys. Chem. Solids 26 (1%5) 1445. [4] S. Kondo and H. Matsumoto, Solid State Comm. 24 (1977) 695. [51 J. Daniels, C. v. Festenberg, H. Raether and K. Zeppenreid, Springer Tracts in Modern Physics 54 (1970) 77. [6] R. Resta, Phys. Rev. B 16 (1977) 2717. [7] R D . Bringans and W.Y. Liang in Proceedings of the 14th Int. Conf. on the Physics of Semiconductors, B.L.H. Wilson, ed. (Inst. of Physics, London, 1978) p. 891. [8] T, Matsukawa and T. lshii, J. Phys. Soc. Jap. 41 (1976) 1285.