- Email: [email protected]
A new interpretation of the fundamental exciton region in LiF
Solid State Communications, Vol. 17, PP. 697—700, 1975.
Pergamon Press.
Printed in Great Britain
A NEW INTERPRETATION OF THE FUNDAMENTAL EXCITON REGION IN LiF* M. Piacentinit Ames Laboratory-ERDA and Department of Physics, Iowa State University, Ames, IA 50010, U.S.A. (Received 2May 1975 byM.F. Collins)
We have reanalyzed LiF optical data between 12 and 15 eV. We show that the conductivity spectrum can be fitted using effective mass theory for excitons with asymmetric Lorentzian lineshapes. We find the band gap is about 14.5 eV. The determination of the effective rydberg is not accurate enough to give the short-range correction to the hydrogenic energy for the n I state.
DETAILED studies of the electronic properties of LiF have been stimulated by considerable recent theoretical and experimental progress. Unfortunately the lack of pronounced sharp structure in the optical spectrum of UF15 makes it difficult to compare energy band calculations with experiments. Often the only data that calculations try to reproduce is the value of the funda-
However, the exciton lineshape is an asymmetric Lorentzian function when a weak exciton-phonon interaction is included in the theory.’°’1’Using this lineshape, Tomiki could fit very well the spin—orbit split n = 1 exciton lines of KCl~and NaC1’3 at several temperatures. Following Tomiki, we tried to fit the LIF conductivity spectrum, o(E) = Ee 2(E)/2, in the first peak region 10 obtained from the experimental 2 of RW, with: I’ +A ‘E—E’ o(E) = a 0 2 ~‘‘ (1) (E E1) + I’1 where E1 is the exciton energy, F1 is approximately its half-value width and A1 is the asymmetry parameter. The intensityfactor a~best-fit is proportional to these the transition oscillator strength. The values for quantities are: E 1 = 12.59 ±0.01 eV, dashed F1 = 0.19 ±0.01 eV,1 2. The curve in Fig. A1 = 0.33, O~ = 13.13 eV corresponds to the conductivity calculated using equation(1), while the solid line is the experimental one. The agreement is excellent and leaves no room for additional structure.
mental band gap energy, E~.In a thorough discussion of the spectrum of the 12imaginary partRoessler of the dielectric function, 2, between and 15eV, and Walker3 (RW) have determined the band gap to be 13.6 eV, a value commonly accepted. We reanalyzed these data and found that the lineshape can be fitted very well within the framework of exciton effective mass theory. We found the band gap at about 69 14.5eV, very close to the value calculated recently.
—
Let at us 12.62 begin our with structure peaking eV. analysis RW fitted thisthe feature withina ~2 symmetric Lorentzian, obtaining a peak energy of 12.62 eV and half-value width of 0.33 eV. They were then left with an extra structure on the high energy side of the peak, having a maximum at 12.85 eV.
Below 12.4 eV the conductivity spectrum deviates from equation (1). This might be due to the fact that around 12.3 eV in LIF at 300K the Lorentzian be-
~Work performed for the U.S. Energy Research and Development Administration under Contract No. W 7405-eng-82.
.
t On leave from Gruppo Nazionale di Struttura della Materia del C.N.R., Sezione di Roma, Istituto di Fisica dell’Universita, Rome, Italy. (Present address). 697
.
14
havior should be replaced by an Urbach tail. Actually, between 11.8 and 12.4 eV, the experimental conductivity increases exponentially according to a(E) ~ exp (sE), with s = 4.22 eV’. This portion of the curve
698
FUNDAMENTAL EXCITON REGION IN LiF ~
a(E)_>~3(EE)2+r2
7°. I I II
60
I I I
4’
~5o
‘‘~~
\
I
*
* + +
EXPERIMENTAL CALCULATED ~ STATE fl~2~STATES
fl
+
N~4UUM
~ I \ ~3O
+ *
c:’.
2.5
2R J [(E’—E)2
+~2]
Eg
0 12.0
r
,(2)
{l —exp [—2ir.sJR/(E’—Eg)]}
4’I / /
~ 20
Vol. 17, No.6
~
4.5
5.0
FIG. 1. The experimental conductivity of LiF, obtained from the data of reference 3 (solid line), compared to the calculated best fit curve (circles). The calculated contributions from the n = 1 exiton state, then = 2, states and from the continuum are displayed separately.
which is Elliot’s hydrogenic formula,’6 with effective rydbergR. Each exciton state is represented by an asymmetric Lorentzian lineshape with resonance energy Er, = Eg —R/n2 and strength n3. The continuum is broadened by a convolution integral with a symmetric Lorentzian. We used Eg and R as free parameters, rather than constraining one to the other by means of th 1 t E E + R whi h holds in the effe tive e re a ion c c mass approximation. Thus a central cell correction for the n = I state was allowed to appear. All other para. meters entering equation (2) were set constant and equal to the values obtained for the n = 1 line, namely, —
—.
F,, = F = F should not be associated with the intrinsic Urbach tail, since its extrapolation toward higher energies does not meet the cross-over point for the LiFalternative, Urbach tailmore deter4 An mined by reason Tomikifor andthe Miyata.’ plausible, discrepancy in the low energy region depends on the fact that ~2 was calculated from the reflectivity via the Kramers—Kronig relations.3 It is well known that this method does not give good results just around the absorption onset. Let us now analyze the next structure, beginning at approximately 13.7 eV. RW found that it could be represented by the square-root behavior typical of an M 5 thus setting the band gap VaneV, Hove at013.6 thesingularity,’ onset of this structure. But in the presence of the electron—hole interaction such a squareroot behavior is strongly distorted around E~’6and becomes valid again only several exciton rydbergs above Eg, several eV in the case of LiF. In the case of other alkali halides, the onset of a step following the first exciton peak was associated for many years with the 17 However, measurements performed at low band gap. temperatures revealed that the step actually consists of several fine structures,1 l3,l8,~ identified with higher members of the rydberg series and with the exciton ionization threshold. We think that in LiF, too, the structure above 13.7 eV corresponds to the envelope of the n = 2 0~ exiton lines merging into the continuum. We tried to fit this region with equation (2):
1 and A,, = A1. The best fit values are R = 2.09 ±0.02 eV and Eg = 14.53 ±0.02 eV. In Fig. 1 the total calculated conductivity is represented by the circles. calculated one, lineshape agrees almost perfectly with theThe experimental except around 14 eV. where the contribution from the n = 2 exciton state gives a peak resolved from the continuum. We attribute this effect to keeping F,, and A,, all equal to each other and constant. Actually, these parameters depend on the exciton state n.10 Also the lineshape associated with each exciton state might change as n increases, depending on the strength of the coupling between the exciton state and the phonon field.’°” Equation (1) by itself fits the experimental data up to 13.6 eV, as shown in Fig. 1. The onset of the following structure is sharper than that obtained using the asymmetric Lorentzian lineshapes in equation (2). We tried to fit the experimental conductivity above 13.7 eV using different values for F,, and A,,. Gaussian functions instead of Lorentzian functions also were tried. We did not find significant changes in the values of R and Eg, except for the case A,, = 0 and F,, = r = r’ 1. In suchwhile a R case the band gap remained almost constant, was lowered to 1 .83 ±0.02 eV. Upon increasing .1r, and decreasingR, the peak associated with then = 2 exciton tends to disappear and the final lineshape becomes very similar to the experimental one, except for a larger intensity. In Equation (2) we used the same intensity factor Go as in equation (1), which is consistent with Effiott’s theory.’6 However, if central cell
Vol. 17, No.6
FUNDAMENTAL EXCITON REGION IN LiF
corrections to the energy of the n = I state are significant, a change in the oscillator strength should occur too. Considering the above modifications in the model used to fit, the band gap and effective rydberg actually are determined with less precision than in our original fit, our best estimates being 14.5 and 2.0 eV, respect-
ively. Our value of E~is 1 eV larger than that given by RW and is in very close agreement with the value of 14.6 eV determined by Miyakawa,~who used the effective mass approximation with the absorption data of Milgram and Givens.’ Recent band structure calculations on LiF, based on the Hartree—Fock method and including corrections arising from correlation and relaxation effects, obtained a band gap of l4.5,~14.1,8 and 13.3 eV.9 The first two results are indeed in very good agreement with our value. The thermoreflectance spectrum of LiF2’ is better interpreted with a band gap of 14.5 eV than with one of 13.6 eV. The binding energy for the n
=
1 state is Eb
=
699
equation(2), as discussed above. A fmite value of ~ also affects the energies of the n = 2,. 00 states, and the correction A,, = A/n3 24 should be included in the fit. In spite of the small radius for the n = 1 state, A turns out to be only a small correction. (In the context of our analysis A = + 0.1 eV). Hermanson23 has clculated that for rare gas solids, the different terms contributing to ~ tend to cancel each other, with the .
,
consequence that the deviation from the hydrogenic binding energy is small. Mickish et a!.8 have calculated the binding energy of the n = I exciton state of UF
in the one-band, one-site approximation, obtaining Eb = 1.8 eV. This value is comparable to that found by us. Even if ~ is small, the model adopted to describe the n = 1 exciton state in LIF is important for any conclusions one may reach. In the one-site approximation all the conduction band states form the exciton wavefunction.25’26 Mickish eta!.8 found that in LiF most of the contribution comes from L. The n = 1 state lies below the conduction band minimumbecause of the large electron—hole interaction. The other states exist as resonances above the band edge, converging towards
E 1 —E1
=
Eb(eV)
1.9 eV. From this value, using the relations
=
p /1 13.6—
m
2
and r1(A)
=
the shoulder in the density of states at the L point. Our analysis shows that all the exciton states lie below the band gap minimum.
O.54oo~,
with optical dielectric constant e~.= 1.92, we find for the effective mass p the value of 0.52 m and that the effective radius r1 = 1.9 A for LiF. The value of the 2° effective mass is close by to that determined Miyakawa and to that estimated Devreese et al.22by The value of the effective radius is comparable to the nearest neighbor distance of 2.01 A. For this reason the hydrogenic model cannot be applied directly to the n = 1 exciton state. Including short range interactions, the shift of the hydrogenic binding energy can be expressed as = R —Eb.2324 (We neglect here the further correction arising from the self-energy of the exciton in the phonon field, which is small in the weak coupling limit.)1°We shall not discuss the value of ~ in detail, since the value of R depends on the approximations adopted to fit the experimental conductivity with
In conclusion, we have shown that the interpretation of the optical data of LiF between 12 and 15 eV can be modified by using the proper lineshapes in the analysis. The structure peaking at 12.62 eV is single and can be associated with the n = 1 exciton state. The band gap is approximately 14.5 eV, I eV larger than the commonly accepted value. The entire exciton series seems to follow the effective mass approximation, with central-cell corrections included,better than it follows the one-site approximation.
Acknowledgement The author wishes to thank Professor D.W. Lynch for many stimulating discussions and a critical reading of the manuscript. —
REFERENCES 1.
MILGRAM A. & GIVENS M.P.,Phys. Rev. 125, 1506 (1962).
2.
LEMONNIER J., STEPHAN G. & ROBIN S., C.R. hebd. séanc. Sci. Paris 261,2463 (1965). ROESSLER D.M. & WALKER W.C.,J. Phys. Chem. Solids 28, 1507 (1969).
3.
700
FUNDAMENTAL EXCITON REGION IN LiF
Vol. 17, No.6
4.
STEPHAN G., Unpublished thesis, Rennes (1970).
5.
WATANABE M., NISHIDA H. & EJIRI A., Proc. 4th mt. Conf Vacuum Ultraviolet Radiation Physics p. 370. Vieweg, Braunschweig, (1974).
6.
KUNZ A.B., MIYAKAWA T. & OYAJIIA S.,Phys. Status Solidi 34,581(1969).
7.
PERROT F.,Phys. Status Solidi B52, 163 (1972).
8.
MICKISH D.J., KUNZ A.B. & COLLINS T.C., Phys. Rev. B9, 4461 (1974).
9.
BRENERN.E.,Phys.Rev. BIl, 1600 (1975).
10.
11.
TOYOZAWA Y.,Prog. Theor. Phys. 20,53 (1958). SUMI H.,J. Phys. Soc. Japan 32, 616 (1972).
12. 13.
TOMIKI T.,J. Phys. Soc. Japan 22,463 (1967). MIYATA T. & TOMIKI T.,J. Phys. Soc. Japan 24, 1286 (1968).
14.
TOMIKI T. & MIYATA T.,J. Phys. Soc. Japan 27, 658 (1969).
15. 16.
VANHOVEL.,Phys.Rev. 89, 1189 (1953). ELLIOTT R.J.,Phys. Rev. 108, 1384 (1957).
17.
PHILLIPS J.C., Solid State Phys. 18,55 (1966).
18. 19.
ROESSLER D.M. & WALKER W.C., Phys. Rev. 166, 599 (1968). BALDINI G. & BOSACCHI B., Phys. Rev. 166, 863 (1968).
20.
MIYAKAWA T.,J. Phys. Soc. Japan 17, 1898 (1962).
21.
PIACENTINI M., LYNCH D.W. & OLSON C.G. (to be published).
22.
23.
DEVREESE J.T., KUNZ A.B. & COLLINS T.C.,Solid State C’ommun. 11,673(1972). HERMANSON J., Phys. Rev. 150, 660 (1966).
24.
PASTOR! PARRAVICINI A. & RESCA L.,Phys. Lett. A45, 73(1973).
25.
PHILLIPS J.C. & HERMANSON J., Phys. Rev. 150, 652 (1966).
26.
ALTARELLI M.& BASSANI F.,J. Phys. C4, L328 (1971).
Pergamon Press.
Printed in Great Britain
A NEW INTERPRETATION OF THE FUNDAMENTAL EXCITON REGION IN LiF* M. Piacentinit Ames Laboratory-ERDA and Department of Physics, Iowa State University, Ames, IA 50010, U.S.A. (Received 2May 1975 byM.F. Collins)
We have reanalyzed LiF optical data between 12 and 15 eV. We show that the conductivity spectrum can be fitted using effective mass theory for excitons with asymmetric Lorentzian lineshapes. We find the band gap is about 14.5 eV. The determination of the effective rydberg is not accurate enough to give the short-range correction to the hydrogenic energy for the n I state.
DETAILED studies of the electronic properties of LiF have been stimulated by considerable recent theoretical and experimental progress. Unfortunately the lack of pronounced sharp structure in the optical spectrum of UF15 makes it difficult to compare energy band calculations with experiments. Often the only data that calculations try to reproduce is the value of the funda-
However, the exciton lineshape is an asymmetric Lorentzian function when a weak exciton-phonon interaction is included in the theory.’°’1’Using this lineshape, Tomiki could fit very well the spin—orbit split n = 1 exciton lines of KCl~and NaC1’3 at several temperatures. Following Tomiki, we tried to fit the LIF conductivity spectrum, o(E) = Ee 2(E)/2, in the first peak region 10 obtained from the experimental 2 of RW, with: I’ +A ‘E—E’ o(E) = a 0 2 ~‘‘ (1) (E E1) + I’1 where E1 is the exciton energy, F1 is approximately its half-value width and A1 is the asymmetry parameter. The intensityfactor a~best-fit is proportional to these the transition oscillator strength. The values for quantities are: E 1 = 12.59 ±0.01 eV, dashed F1 = 0.19 ±0.01 eV,1 2. The curve in Fig. A1 = 0.33, O~ = 13.13 eV corresponds to the conductivity calculated using equation(1), while the solid line is the experimental one. The agreement is excellent and leaves no room for additional structure.
mental band gap energy, E~.In a thorough discussion of the spectrum of the 12imaginary partRoessler of the dielectric function, 2, between and 15eV, and Walker3 (RW) have determined the band gap to be 13.6 eV, a value commonly accepted. We reanalyzed these data and found that the lineshape can be fitted very well within the framework of exciton effective mass theory. We found the band gap at about 69 14.5eV, very close to the value calculated recently.
—
Let at us 12.62 begin our with structure peaking eV. analysis RW fitted thisthe feature withina ~2 symmetric Lorentzian, obtaining a peak energy of 12.62 eV and half-value width of 0.33 eV. They were then left with an extra structure on the high energy side of the peak, having a maximum at 12.85 eV.
Below 12.4 eV the conductivity spectrum deviates from equation (1). This might be due to the fact that around 12.3 eV in LIF at 300K the Lorentzian be-
~Work performed for the U.S. Energy Research and Development Administration under Contract No. W 7405-eng-82.
.
t On leave from Gruppo Nazionale di Struttura della Materia del C.N.R., Sezione di Roma, Istituto di Fisica dell’Universita, Rome, Italy. (Present address). 697
.
14
havior should be replaced by an Urbach tail. Actually, between 11.8 and 12.4 eV, the experimental conductivity increases exponentially according to a(E) ~ exp (sE), with s = 4.22 eV’. This portion of the curve
698
FUNDAMENTAL EXCITON REGION IN LiF ~
a(E)_>~3(EE)2+r2
7°. I I II
60
I I I
4’
~5o
‘‘~~
\
I
*
* + +
EXPERIMENTAL CALCULATED ~ STATE fl~2~STATES
fl
+
N~4UUM
~ I \ ~3O
+ *
c:’.
2.5
2R J [(E’—E)2
+~2]
Eg
0 12.0
r
,(2)
{l —exp [—2ir.sJR/(E’—Eg)]}
4’I / /
~ 20
Vol. 17, No.6
~
4.5
5.0
FIG. 1. The experimental conductivity of LiF, obtained from the data of reference 3 (solid line), compared to the calculated best fit curve (circles). The calculated contributions from the n = 1 exiton state, then = 2, states and from the continuum are displayed separately.
which is Elliot’s hydrogenic formula,’6 with effective rydbergR. Each exciton state is represented by an asymmetric Lorentzian lineshape with resonance energy Er, = Eg —R/n2 and strength n3. The continuum is broadened by a convolution integral with a symmetric Lorentzian. We used Eg and R as free parameters, rather than constraining one to the other by means of th 1 t E E + R whi h holds in the effe tive e re a ion c c mass approximation. Thus a central cell correction for the n = I state was allowed to appear. All other para. meters entering equation (2) were set constant and equal to the values obtained for the n = 1 line, namely, —
—.
F,, = F = F should not be associated with the intrinsic Urbach tail, since its extrapolation toward higher energies does not meet the cross-over point for the LiFalternative, Urbach tailmore deter4 An mined by reason Tomikifor andthe Miyata.’ plausible, discrepancy in the low energy region depends on the fact that ~2 was calculated from the reflectivity via the Kramers—Kronig relations.3 It is well known that this method does not give good results just around the absorption onset. Let us now analyze the next structure, beginning at approximately 13.7 eV. RW found that it could be represented by the square-root behavior typical of an M 5 thus setting the band gap VaneV, Hove at013.6 thesingularity,’ onset of this structure. But in the presence of the electron—hole interaction such a squareroot behavior is strongly distorted around E~’6and becomes valid again only several exciton rydbergs above Eg, several eV in the case of LiF. In the case of other alkali halides, the onset of a step following the first exciton peak was associated for many years with the 17 However, measurements performed at low band gap. temperatures revealed that the step actually consists of several fine structures,1 l3,l8,~ identified with higher members of the rydberg series and with the exciton ionization threshold. We think that in LiF, too, the structure above 13.7 eV corresponds to the envelope of the n = 2 0~ exiton lines merging into the continuum. We tried to fit this region with equation (2):
1 and A,, = A1. The best fit values are R = 2.09 ±0.02 eV and Eg = 14.53 ±0.02 eV. In Fig. 1 the total calculated conductivity is represented by the circles. calculated one, lineshape agrees almost perfectly with theThe experimental except around 14 eV. where the contribution from the n = 2 exciton state gives a peak resolved from the continuum. We attribute this effect to keeping F,, and A,, all equal to each other and constant. Actually, these parameters depend on the exciton state n.10 Also the lineshape associated with each exciton state might change as n increases, depending on the strength of the coupling between the exciton state and the phonon field.’°” Equation (1) by itself fits the experimental data up to 13.6 eV, as shown in Fig. 1. The onset of the following structure is sharper than that obtained using the asymmetric Lorentzian lineshapes in equation (2). We tried to fit the experimental conductivity above 13.7 eV using different values for F,, and A,,. Gaussian functions instead of Lorentzian functions also were tried. We did not find significant changes in the values of R and Eg, except for the case A,, = 0 and F,, = r = r’ 1. In suchwhile a R case the band gap remained almost constant, was lowered to 1 .83 ±0.02 eV. Upon increasing .1r, and decreasingR, the peak associated with then = 2 exciton tends to disappear and the final lineshape becomes very similar to the experimental one, except for a larger intensity. In Equation (2) we used the same intensity factor Go as in equation (1), which is consistent with Effiott’s theory.’6 However, if central cell
Vol. 17, No.6
FUNDAMENTAL EXCITON REGION IN LiF
corrections to the energy of the n = I state are significant, a change in the oscillator strength should occur too. Considering the above modifications in the model used to fit, the band gap and effective rydberg actually are determined with less precision than in our original fit, our best estimates being 14.5 and 2.0 eV, respect-
ively. Our value of E~is 1 eV larger than that given by RW and is in very close agreement with the value of 14.6 eV determined by Miyakawa,~who used the effective mass approximation with the absorption data of Milgram and Givens.’ Recent band structure calculations on LiF, based on the Hartree—Fock method and including corrections arising from correlation and relaxation effects, obtained a band gap of l4.5,~14.1,8 and 13.3 eV.9 The first two results are indeed in very good agreement with our value. The thermoreflectance spectrum of LiF2’ is better interpreted with a band gap of 14.5 eV than with one of 13.6 eV. The binding energy for the n
=
1 state is Eb
=
699
equation(2), as discussed above. A fmite value of ~ also affects the energies of the n = 2,. 00 states, and the correction A,, = A/n3 24 should be included in the fit. In spite of the small radius for the n = 1 state, A turns out to be only a small correction. (In the context of our analysis A = + 0.1 eV). Hermanson23 has clculated that for rare gas solids, the different terms contributing to ~ tend to cancel each other, with the .
,
consequence that the deviation from the hydrogenic binding energy is small. Mickish et a!.8 have calculated the binding energy of the n = I exciton state of UF
in the one-band, one-site approximation, obtaining Eb = 1.8 eV. This value is comparable to that found by us. Even if ~ is small, the model adopted to describe the n = 1 exciton state in LIF is important for any conclusions one may reach. In the one-site approximation all the conduction band states form the exciton wavefunction.25’26 Mickish eta!.8 found that in LiF most of the contribution comes from L. The n = 1 state lies below the conduction band minimumbecause of the large electron—hole interaction. The other states exist as resonances above the band edge, converging towards
E 1 —E1
=
Eb(eV)
1.9 eV. From this value, using the relations
=
p /1 13.6—
m
2
and r1(A)
=
the shoulder in the density of states at the L point. Our analysis shows that all the exciton states lie below the band gap minimum.
O.54oo~,
with optical dielectric constant e~.= 1.92, we find for the effective mass p the value of 0.52 m and that the effective radius r1 = 1.9 A for LiF. The value of the 2° effective mass is close by to that determined Miyakawa and to that estimated Devreese et al.22by The value of the effective radius is comparable to the nearest neighbor distance of 2.01 A. For this reason the hydrogenic model cannot be applied directly to the n = 1 exciton state. Including short range interactions, the shift of the hydrogenic binding energy can be expressed as = R —Eb.2324 (We neglect here the further correction arising from the self-energy of the exciton in the phonon field, which is small in the weak coupling limit.)1°We shall not discuss the value of ~ in detail, since the value of R depends on the approximations adopted to fit the experimental conductivity with
In conclusion, we have shown that the interpretation of the optical data of LiF between 12 and 15 eV can be modified by using the proper lineshapes in the analysis. The structure peaking at 12.62 eV is single and can be associated with the n = 1 exciton state. The band gap is approximately 14.5 eV, I eV larger than the commonly accepted value. The entire exciton series seems to follow the effective mass approximation, with central-cell corrections included,better than it follows the one-site approximation.
Acknowledgement The author wishes to thank Professor D.W. Lynch for many stimulating discussions and a critical reading of the manuscript. —
REFERENCES 1.
MILGRAM A. & GIVENS M.P.,Phys. Rev. 125, 1506 (1962).
2.
LEMONNIER J., STEPHAN G. & ROBIN S., C.R. hebd. séanc. Sci. Paris 261,2463 (1965). ROESSLER D.M. & WALKER W.C.,J. Phys. Chem. Solids 28, 1507 (1969).
3.
700
FUNDAMENTAL EXCITON REGION IN LiF
Vol. 17, No.6
4.
STEPHAN G., Unpublished thesis, Rennes (1970).
5.
WATANABE M., NISHIDA H. & EJIRI A., Proc. 4th mt. Conf Vacuum Ultraviolet Radiation Physics p. 370. Vieweg, Braunschweig, (1974).
6.
KUNZ A.B., MIYAKAWA T. & OYAJIIA S.,Phys. Status Solidi 34,581(1969).
7.
PERROT F.,Phys. Status Solidi B52, 163 (1972).
8.
MICKISH D.J., KUNZ A.B. & COLLINS T.C., Phys. Rev. B9, 4461 (1974).
9.
BRENERN.E.,Phys.Rev. BIl, 1600 (1975).
10.
11.
TOYOZAWA Y.,Prog. Theor. Phys. 20,53 (1958). SUMI H.,J. Phys. Soc. Japan 32, 616 (1972).
12. 13.
TOMIKI T.,J. Phys. Soc. Japan 22,463 (1967). MIYATA T. & TOMIKI T.,J. Phys. Soc. Japan 24, 1286 (1968).
14.
TOMIKI T. & MIYATA T.,J. Phys. Soc. Japan 27, 658 (1969).
15. 16.
VANHOVEL.,Phys.Rev. 89, 1189 (1953). ELLIOTT R.J.,Phys. Rev. 108, 1384 (1957).
17.
PHILLIPS J.C., Solid State Phys. 18,55 (1966).
18. 19.
ROESSLER D.M. & WALKER W.C., Phys. Rev. 166, 599 (1968). BALDINI G. & BOSACCHI B., Phys. Rev. 166, 863 (1968).
20.
MIYAKAWA T.,J. Phys. Soc. Japan 17, 1898 (1962).
21.
PIACENTINI M., LYNCH D.W. & OLSON C.G. (to be published).
22.
23.
DEVREESE J.T., KUNZ A.B. & COLLINS T.C.,Solid State C’ommun. 11,673(1972). HERMANSON J., Phys. Rev. 150, 660 (1966).
24.
PASTOR! PARRAVICINI A. & RESCA L.,Phys. Lett. A45, 73(1973).
25.
PHILLIPS J.C. & HERMANSON J., Phys. Rev. 150, 652 (1966).
26.
ALTARELLI M.& BASSANI F.,J. Phys. C4, L328 (1971).