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Effect of electron-phonon coupling on Raman spectrum of C8K
Physica 105B (1981) 381-385 North-Holland Publishing Company
E F F E C T OF E L E C T R O N - P H O N O N COUPLING ON RAMAN SPECTRUM OF CsK H. M I Y A Z A K I , T. H A T A N O , G. K U S U N O K I , T. W A T A N A B E and C. H O R I E Department of Applied Physics, Tohoku University, Sendai, Japan
A theoretical analysis is presented of an asymmetricbroad line shape at ~1500 cm-t in the observed Raman spectrum of CaK. The anomalous line-shape is attributable to the coupled excitations of symmetry-allowedphonons and electrons. The electron-phonon coupling strength relevant to the Raman scattering under consideration is also estimated.
1. Introduction The Raman spectra of the first stage graphite compounds intercalated with alkali metals exhibit a feature markedly different from those of higher stage compounds [1-3]. Characteristic of the observed spectra for CsK, for example, is a broad and asymmetric line near 1500 cm -1 and a set of sharp lines near 560 cm -~, both of which are superimposed on an intense continuum background. On the other hand, theoretical phonon dispersion curves for CsK based on a force constant mode [4] predict that there are at least 13 Raman-active modes allowed for the backscattering configuration, in which the Raman measurements have been performed, as shown in table I. The set of lines near 560 cm -1 has been interpreted as being associated with out-of-plane modes produced by the folding of L and M phonons of pristine graphite [3, 4]. However, a recent experiment indicates a contribution from disorder-induced scattering [5]. The broad line near 1500 cm -1, on the other hand, exhibits no distinct features of the in-plane modes predicted theoretically for to -> 800 cm -1. Instead, the asymmetric line shape has been interpreted as being due to the Fano resonance between symmetry-allowed phonon modes and the continuum background, though there are conflicting interpretations of the origin of the background [1,2,5,61. In the present paper we focus our attention on the anomalous line shape near 1500 cm -1 and
present a theoretical analysis based on the assumption that the coupled excitations of electrons and phonons play a dominant role. In the analysis, a simplified band structure is assumed for electrons by referring to the recently calculated band structures [7-9]. Moreover, we consider electron-phonon interactions involving not only the mode at 1580 cm -1, but also those at 1280 and 858 cm-L It is shown that the asymmetric shape of the line is attributable to the coupled excitations of electrons and phonons, which can be expressed in terms of the Fano formula, and that the electronic excitations also contribute to the broad background. From the result of the present analysis the electron-phonon coupling strength relevant to the Rarnan scattering under consideration is estimated to be of the order of 0.08 eV.
2. Raman scattering due to coupled electronphonon excitations According to the recent calculation of the electronic band structure of CsK [8, 9], the Itband of pristine graphite suffers a slight lifting in its degeneracy when it crosses or comes close to the 4s band originating from potassium atom. This results in a small gap between filled and empty bands near the Fermi surface. IS this small gap is comparable w i t h frequencies of the Raman-active modes, then the electronic excitations across the small gap are expected to modify significantly, by resonant interactions, the intrin-
0378-4363/81/0000-0000/$2.50 O North-Holland Publishing Company and Yamada Science Foundation
382
H. Miyazaki et al./Electron-phonon coupling in CsK
Table I Raman-active modes of CsK (back-scattering configuration)
v (cm -l)
Symmetry"
Polarization direction
Relevant atoms
1580 1280 858 592 583 582 203
2A~ + 2 B i g Ag +Btg Ag + BIB As + B~B Big A~ B~s
in-plane in-plane in-plane out-of-plane out-of-plane out-of-plane out-of-plane
C C C C C C K
• Ai is allowedfor Z(X, X)Z or Z(Y, Y)2,; Bls is allowedfor Z(X, Y)Z or Z(Y, X)Z.
frequencies, respectively, and ImO~,vs(vi, v,) describes the Raman tensor. For the first order Raman process, the Raman tensor consists of the terms represented by the diagrams shown in fig. 1, where a solid line represents the thermal Green function for an electron, a wavy line stands for a Raman-active phonon, and a dotted line refers to the incident or scattered light. The Green functions for electrons are calculated so as to include the interactions with acoustic phonons into the self-energy part. The Green functions for the Raman-active phonons are approximated by /9,(0, v ) = g240
sic Raman lines due to phonons. Moreover, it is also noticed that the symmetry-allowed phonon modes with frequencies 1580, 1280 and 858 cm -~ are in-plane modes, so that they can easily interact with electrons movable along the carbon basal plane. Therefore, we propose a model in which the coupled electron-phonon excitations play a dominant role in determining the line shape observed near 1500 cm -~. In terms of the annihilation and creation operators an,. and a,~, + for electrons and ba q and
b~,q for phonons, the interaction Hamiltonian between electrons and phonons is generally written as
I"Ii,,t = ~
ga..,,(p, q)a.,,p+qa,,,t,(b~,q + b4,-q). +
1
iv+O
-H,-, {i,,-'124 + iF4
1
+ir4}' (3) (4)
/22 = 12~0+ F~ + 2n40 R e / / 4 (124),
r4 =
n40 tm n,(a4),
(5)
where 040 are either 1580, 1280, or 858 cm -1, and the self-energy parts Ha are calculated by taking into account virtual excitations between the filled and empty bands near the Fermi surface. It should be noted that the contribution to the Raman spectrum from diagram (c) in fig. 1 can be written as tr to) = ~rlc) + o'~~),
(6)
where
ii,m
(1) Here, the phonons refer to not only the Ramanactive modes but also to the longitudinal acoustic phonons. The coupling parameter g~. is assumed to be constant for interactions with the Ramanactive modes and to be of the deformation potential type for interactions with the acoustic modes. In accordance with Ipatova and Subashiev [10], we start with the differential cross-section for Raman scattering d2-y _ v~
trt¢) = A v~ ~2 x~ 12,o (Im R ) 2 ~ c-i ~° ~ N4 12, F4
1 e4+l "',
"'"
"%
"""
,'"
(b) ,,/
"',
~" =dto d---~- -~ e.(k )e;(k )
x Im G.o,va(Vi, v,)e*(k')es(k'),
(2)
where vi and v~ are incident and scattered light
"""
(c)
"'"
Fig. 1. Diagrams contributing to the first order Raman tensor.
H. Miyazaki et al./Electron-phonon coupling in CsK
e, = (v~- v , - O~)IF~,
(8)
O = - ( R e R)/(Im R),
(9)
383
1.5
R = -~-l ~ ~ ~,fGi(IJ, O).)Grm(P,to. + lYi) x G~(p, to, + v i - v,),
(10)
//,\
and
-o.:/2 -
1.12
0
2
4
k
WAVE VECTOR
3. Spectral shape near 1500 cm -t As has been mentioned in the previous section, the key features of the band structure near the Fermi surface consistent with the recently calculated band structures [8, 9] can be represented by the following expressions:
el(k)
=
h2 E 1 - 2-m l ( k l 2 + k 2 + k 2) + h2kl (k 2 + k~) ~r2, ml
(11)
h~
e~(k) = E2--~22(k22+ k~ + k~) + h2k2 (k~ + k2y)~,
(12)
m2
h2 e3(k) = E3 + ~m-~m3(k2 + k 2 + k 2) h2k3 (k~ + k2) 'r2.
(13)
m3
Here the origin of the wave vector is taken to be at the peripheral corner of the Brillouin zone, and the electron mass in the z-direction is taken as an infinity. Because of the quantitative disagreement among the calculated band structures [8,9], we have chosen the appropriate set of band parameters so as to reproduce the essential feature of the band splitting near the Fermi surface under the following conditions. (1) Frequencies of the Raman-active modes under
\ -
,
6
8
"
F
( xIOTcm'' ]
Fig. 2. Model of the band structure. The curves in the range k m = 2.785 ~ k ~ kFl = 3.034 are given by eqs. (11)-(13) with Et = 0.99, E2 = 0.93, E3 = 0.97 (eV), kl = 8.53, k2 = 5.00, k3 = 7.00 (x107 cm -I) and rnl = 1.16, m2 = 0.2, m3 = 2.0 (in units of electron mass).
consideration fall within the energy gap between band 1 and 2 in the region of momentum, kF2--< k ~
[1
~, (cm -1)
O
a
b
a
b
1580 1280 858
-1.00 0.076 0.448
46.0 99.3 92.6
131.7 298.5 271.2
1557.0 1280.3 881.5
1523.0 1300.6 944.4
384
H. Miyazaki et al./Electron-phonon coupling in CsK
I
1700
I
I
=
I
I
I
1500 1300 I100 RAMAN SHIFT (cm -I)
J
I
900
Fig. 3. A plot of o-(c) 1 vs the Raman shift with the choice of parameters g0=0.0875eV and me/mo=0.200. The dots represent the observed spectrum [1]. The calculated curve for tr~c) is normalized to fit the observed height at the maximum and minimum.
In the numerical calculation we tentatively take gA = 0.75eV for the coupling p a r a m e t e r with acoustic phonons and vary the coupling p a r a m e t e r with Raman-active modes go from 0.01 to 0.1eV. A m o n g contributions from diagrams in fig. 1, the line shape due to tr~c) is found to be most sensitive to the value of go, as is seen from table II. Fig. 3 shows a plot of o'tc) vs. the R a m a n shift ~'i-~'s for go = 0.0875 eV. This value of go seems most plausible in the sense that the curve o-~~) demonstrates the a s y m m e t r y of Fano-type resonance near 1500cm -~, and that the structure indicative of other modes at 1280 and 858 cm -1 is smeared out as a result of their large damping p a r a m e t e r F~. For a smaller value of go, there appears a structure with peaks corresponding to these two modes, which may account for the observed structure below 1500 c m -1 in the R a m a n spectra of CsCs [5]. The contribution to the spectral shape from o'~~) and the diagrams (a) and (b) in fig. 1 will be discussed in the next section. 4. D i s c u s s i o n
We have shown in the previous section that o'I ~) of diagram (c) in fig. 1 gives rise to the
Fano-type resonance. This explains the observed asymmetric structure at ~ 1 5 0 0 c m -1. The fact that the observed spectrum has no clear sign of corresponding structure for the modes at 1280 and 858 cm -1 can be also explained by their large Fa. The spectrum due to the diagram (b) in fig. 1 turns out to be a broad band in the range from 800 to 1600cm -~, with no distinct peaks corresponding to the three modes under consideration. Moreover, this contributes to diminishing the apparent difference between the calculated curve due to o-~c) and the observed data in the frequency region below 1500cm -~. The diagram (a) of fig. 1 corresponds to the R a m a n scattering due to electronic excitations and yields a continuum spectrum over the frequency region extending from kw and kFx. Hence, the diagram (a) yields no anomalous lineshape near 1500 cm -1 but serves as a part of the intense background observed in the experiment in the frequency range under consideration. T h e total contribution from all the diagrams is shown in fig. 4 by a solid line, which is normalized to fit the observed height at the m a x i m u m and minimum. The almost constant contribution from o'~c) is neglected because of its smallness. From these facts we conclude that the anomalous line-shape near 1500cm -~ is due to the resonance between excitations of electrons
b
e '
x
/
I
1700
~
I
,
I,,500 R,~YlAN
I
~
t
1300 II00 SHIFT (cm-*)
~
I
900
Fig. 4. Comparison with the experimental data. The solid line represents the total contribution from all the diagrams which is normalized to fit the observed height at the maximum and minimum. The dashed line represents o-~c). The dots are the experimental data [1].
H. Miyazaki et al./Eiectron-phonon coupling in CsK
and phonons, in agreement with the interpretation by Nemanich et al. [1], who attributed the entire continuum to electronic scattering. However, Caswell and Solin have shown that the low frequency continuum ~o < 600 cm -1 is due to single phonon scattering [5]. It is remarkable that we can almost reproduce the observed spectrum by varying only one parameter, go, if we choose the proper values of band parameters near the Fermi energy. Within the reasonable choice of band parameters, we find that the most plausible value for the coupling constant between electrons and Raman-active modes is about 0.08 eV which is of the order of one-tenth the deformation potential constant.
Acknowledgement The authors wish to thank Dr. S.A. Solin for helpful discussions.
385
References [1] R.J. Nemanich, S.A. Solin and D. Gu6rard, Phys. Rev. B16 (1977) 2965. [2] P.C. Eklund, G. Dresselhaus, M.S. Dresselhaus and J.E. Fischer, Phys. Rev. B16 (1977) 3330. [3] S.A. Solin, Physica 99B (1980) 443. [4] C. Horie, M. Maeda and Y. Kuramoto, Physica 99B (1980) 430. [5] N. Caswell and S.A. Solin, Phys. Rev. B20 (1979) 2551. [6] P.C. Eklund and K.R. Subbaswamy, Phys. Rev. B20 (1979) 5157. [7] T. Inoshita, K. Nakao and H. Kamimura, J. Phys. Soc. Japan 43 (1977) 1237. [8] T. Ohno, K. Nakao and H. Kamimura, J. Phys. Soc. Japan 47 (1979) 1125. [9] D.P. DiVincenzo, N.A.W. Holzwarth and S. Rabii, Physica 99B (1980) 406. [10] I.P. Ipatova and A.V. Subashiev, in: Theory of Light Scattering in Condensed Matter, B. Bendow, J.L. Birman and V.M. Agranovich, eds. (Plenum, N e w York, 1975) p. 201. [II] H. Suematsu, K. Higuchi and S. Tanuma, Tech. Rep. I.S.S.P.,Ser. A. no. 927 (1978). [12] U. Mizutani, I. Kondow and T.B. Massalski, Phys. Rev. B17 (1978) 3165.
E F F E C T OF E L E C T R O N - P H O N O N COUPLING ON RAMAN SPECTRUM OF CsK H. M I Y A Z A K I , T. H A T A N O , G. K U S U N O K I , T. W A T A N A B E and C. H O R I E Department of Applied Physics, Tohoku University, Sendai, Japan
A theoretical analysis is presented of an asymmetricbroad line shape at ~1500 cm-t in the observed Raman spectrum of CaK. The anomalous line-shape is attributable to the coupled excitations of symmetry-allowedphonons and electrons. The electron-phonon coupling strength relevant to the Raman scattering under consideration is also estimated.
1. Introduction The Raman spectra of the first stage graphite compounds intercalated with alkali metals exhibit a feature markedly different from those of higher stage compounds [1-3]. Characteristic of the observed spectra for CsK, for example, is a broad and asymmetric line near 1500 cm -1 and a set of sharp lines near 560 cm -~, both of which are superimposed on an intense continuum background. On the other hand, theoretical phonon dispersion curves for CsK based on a force constant mode [4] predict that there are at least 13 Raman-active modes allowed for the backscattering configuration, in which the Raman measurements have been performed, as shown in table I. The set of lines near 560 cm -1 has been interpreted as being associated with out-of-plane modes produced by the folding of L and M phonons of pristine graphite [3, 4]. However, a recent experiment indicates a contribution from disorder-induced scattering [5]. The broad line near 1500 cm -1, on the other hand, exhibits no distinct features of the in-plane modes predicted theoretically for to -> 800 cm -1. Instead, the asymmetric line shape has been interpreted as being due to the Fano resonance between symmetry-allowed phonon modes and the continuum background, though there are conflicting interpretations of the origin of the background [1,2,5,61. In the present paper we focus our attention on the anomalous line shape near 1500 cm -1 and
present a theoretical analysis based on the assumption that the coupled excitations of electrons and phonons play a dominant role. In the analysis, a simplified band structure is assumed for electrons by referring to the recently calculated band structures [7-9]. Moreover, we consider electron-phonon interactions involving not only the mode at 1580 cm -1, but also those at 1280 and 858 cm-L It is shown that the asymmetric shape of the line is attributable to the coupled excitations of electrons and phonons, which can be expressed in terms of the Fano formula, and that the electronic excitations also contribute to the broad background. From the result of the present analysis the electron-phonon coupling strength relevant to the Rarnan scattering under consideration is estimated to be of the order of 0.08 eV.
2. Raman scattering due to coupled electronphonon excitations According to the recent calculation of the electronic band structure of CsK [8, 9], the Itband of pristine graphite suffers a slight lifting in its degeneracy when it crosses or comes close to the 4s band originating from potassium atom. This results in a small gap between filled and empty bands near the Fermi surface. IS this small gap is comparable w i t h frequencies of the Raman-active modes, then the electronic excitations across the small gap are expected to modify significantly, by resonant interactions, the intrin-
0378-4363/81/0000-0000/$2.50 O North-Holland Publishing Company and Yamada Science Foundation
382
H. Miyazaki et al./Electron-phonon coupling in CsK
Table I Raman-active modes of CsK (back-scattering configuration)
v (cm -l)
Symmetry"
Polarization direction
Relevant atoms
1580 1280 858 592 583 582 203
2A~ + 2 B i g Ag +Btg Ag + BIB As + B~B Big A~ B~s
in-plane in-plane in-plane out-of-plane out-of-plane out-of-plane out-of-plane
C C C C C C K
• Ai is allowedfor Z(X, X)Z or Z(Y, Y)2,; Bls is allowedfor Z(X, Y)Z or Z(Y, X)Z.
frequencies, respectively, and ImO~,vs(vi, v,) describes the Raman tensor. For the first order Raman process, the Raman tensor consists of the terms represented by the diagrams shown in fig. 1, where a solid line represents the thermal Green function for an electron, a wavy line stands for a Raman-active phonon, and a dotted line refers to the incident or scattered light. The Green functions for electrons are calculated so as to include the interactions with acoustic phonons into the self-energy part. The Green functions for the Raman-active phonons are approximated by /9,(0, v ) = g240
sic Raman lines due to phonons. Moreover, it is also noticed that the symmetry-allowed phonon modes with frequencies 1580, 1280 and 858 cm -~ are in-plane modes, so that they can easily interact with electrons movable along the carbon basal plane. Therefore, we propose a model in which the coupled electron-phonon excitations play a dominant role in determining the line shape observed near 1500 cm -~. In terms of the annihilation and creation operators an,. and a,~, + for electrons and ba q and
b~,q for phonons, the interaction Hamiltonian between electrons and phonons is generally written as
I"Ii,,t = ~
ga..,,(p, q)a.,,p+qa,,,t,(b~,q + b4,-q). +
1
iv+O
-H,-, {i,,-'124 + iF4
1
+ir4}' (3) (4)
/22 = 12~0+ F~ + 2n40 R e / / 4 (124),
r4 =
n40 tm n,(a4),
(5)
where 040 are either 1580, 1280, or 858 cm -1, and the self-energy parts Ha are calculated by taking into account virtual excitations between the filled and empty bands near the Fermi surface. It should be noted that the contribution to the Raman spectrum from diagram (c) in fig. 1 can be written as tr to) = ~rlc) + o'~~),
(6)
where
ii,m
(1) Here, the phonons refer to not only the Ramanactive modes but also to the longitudinal acoustic phonons. The coupling parameter g~. is assumed to be constant for interactions with the Ramanactive modes and to be of the deformation potential type for interactions with the acoustic modes. In accordance with Ipatova and Subashiev [10], we start with the differential cross-section for Raman scattering d2-y _ v~
trt¢) = A v~ ~2 x~ 12,o (Im R ) 2 ~ c-i ~° ~ N4 12, F4
1 e4+l "',
"'"
"%
"""
,'"
(b) ,,/
"',
~" =dto d---~- -~ e.(k )e;(k )
x Im G.o,va(Vi, v,)e*(k')es(k'),
(2)
where vi and v~ are incident and scattered light
"""
(c)
"'"
Fig. 1. Diagrams contributing to the first order Raman tensor.
H. Miyazaki et al./Electron-phonon coupling in CsK
e, = (v~- v , - O~)IF~,
(8)
O = - ( R e R)/(Im R),
(9)
383
1.5
R = -~-l ~ ~ ~,fGi(IJ, O).)Grm(P,to. + lYi) x G~(p, to, + v i - v,),
(10)
//,\
and
-o.:/2 -
1.12
0
2
4
k
WAVE VECTOR
3. Spectral shape near 1500 cm -t As has been mentioned in the previous section, the key features of the band structure near the Fermi surface consistent with the recently calculated band structures [8, 9] can be represented by the following expressions:
el(k)
=
h2 E 1 - 2-m l ( k l 2 + k 2 + k 2) + h2kl (k 2 + k~) ~r2, ml
(11)
h~
e~(k) = E2--~22(k22+ k~ + k~) + h2k2 (k~ + k2y)~,
(12)
m2
h2 e3(k) = E3 + ~m-~m3(k2 + k 2 + k 2) h2k3 (k~ + k2) 'r2.
(13)
m3
Here the origin of the wave vector is taken to be at the peripheral corner of the Brillouin zone, and the electron mass in the z-direction is taken as an infinity. Because of the quantitative disagreement among the calculated band structures [8,9], we have chosen the appropriate set of band parameters so as to reproduce the essential feature of the band splitting near the Fermi surface under the following conditions. (1) Frequencies of the Raman-active modes under
\ -
,
6
8
"
F
( xIOTcm'' ]
Fig. 2. Model of the band structure. The curves in the range k m = 2.785 ~ k ~ kFl = 3.034 are given by eqs. (11)-(13) with Et = 0.99, E2 = 0.93, E3 = 0.97 (eV), kl = 8.53, k2 = 5.00, k3 = 7.00 (x107 cm -I) and rnl = 1.16, m2 = 0.2, m3 = 2.0 (in units of electron mass).
consideration fall within the energy gap between band 1 and 2 in the region of momentum, kF2--< k ~
[1
~, (cm -1)
O
a
b
a
b
1580 1280 858
-1.00 0.076 0.448
46.0 99.3 92.6
131.7 298.5 271.2
1557.0 1280.3 881.5
1523.0 1300.6 944.4
384
H. Miyazaki et al./Electron-phonon coupling in CsK
I
1700
I
I
=
I
I
I
1500 1300 I100 RAMAN SHIFT (cm -I)
J
I
900
Fig. 3. A plot of o-(c) 1 vs the Raman shift with the choice of parameters g0=0.0875eV and me/mo=0.200. The dots represent the observed spectrum [1]. The calculated curve for tr~c) is normalized to fit the observed height at the maximum and minimum.
In the numerical calculation we tentatively take gA = 0.75eV for the coupling p a r a m e t e r with acoustic phonons and vary the coupling p a r a m e t e r with Raman-active modes go from 0.01 to 0.1eV. A m o n g contributions from diagrams in fig. 1, the line shape due to tr~c) is found to be most sensitive to the value of go, as is seen from table II. Fig. 3 shows a plot of o'tc) vs. the R a m a n shift ~'i-~'s for go = 0.0875 eV. This value of go seems most plausible in the sense that the curve o-~~) demonstrates the a s y m m e t r y of Fano-type resonance near 1500cm -~, and that the structure indicative of other modes at 1280 and 858 cm -1 is smeared out as a result of their large damping p a r a m e t e r F~. For a smaller value of go, there appears a structure with peaks corresponding to these two modes, which may account for the observed structure below 1500 c m -1 in the R a m a n spectra of CsCs [5]. The contribution to the spectral shape from o'~~) and the diagrams (a) and (b) in fig. 1 will be discussed in the next section. 4. D i s c u s s i o n
We have shown in the previous section that o'I ~) of diagram (c) in fig. 1 gives rise to the
Fano-type resonance. This explains the observed asymmetric structure at ~ 1 5 0 0 c m -1. The fact that the observed spectrum has no clear sign of corresponding structure for the modes at 1280 and 858 cm -1 can be also explained by their large Fa. The spectrum due to the diagram (b) in fig. 1 turns out to be a broad band in the range from 800 to 1600cm -~, with no distinct peaks corresponding to the three modes under consideration. Moreover, this contributes to diminishing the apparent difference between the calculated curve due to o-~c) and the observed data in the frequency region below 1500cm -~. The diagram (a) of fig. 1 corresponds to the R a m a n scattering due to electronic excitations and yields a continuum spectrum over the frequency region extending from kw and kFx. Hence, the diagram (a) yields no anomalous lineshape near 1500 cm -1 but serves as a part of the intense background observed in the experiment in the frequency range under consideration. T h e total contribution from all the diagrams is shown in fig. 4 by a solid line, which is normalized to fit the observed height at the m a x i m u m and minimum. The almost constant contribution from o'~c) is neglected because of its smallness. From these facts we conclude that the anomalous line-shape near 1500cm -~ is due to the resonance between excitations of electrons
b
e '
x
/
I
1700
~
I
,
I,,500 R,~YlAN
I
~
t
1300 II00 SHIFT (cm-*)
~
I
900
Fig. 4. Comparison with the experimental data. The solid line represents the total contribution from all the diagrams which is normalized to fit the observed height at the maximum and minimum. The dashed line represents o-~c). The dots are the experimental data [1].
H. Miyazaki et al./Eiectron-phonon coupling in CsK
and phonons, in agreement with the interpretation by Nemanich et al. [1], who attributed the entire continuum to electronic scattering. However, Caswell and Solin have shown that the low frequency continuum ~o < 600 cm -1 is due to single phonon scattering [5]. It is remarkable that we can almost reproduce the observed spectrum by varying only one parameter, go, if we choose the proper values of band parameters near the Fermi energy. Within the reasonable choice of band parameters, we find that the most plausible value for the coupling constant between electrons and Raman-active modes is about 0.08 eV which is of the order of one-tenth the deformation potential constant.
Acknowledgement The authors wish to thank Dr. S.A. Solin for helpful discussions.
385
References [1] R.J. Nemanich, S.A. Solin and D. Gu6rard, Phys. Rev. B16 (1977) 2965. [2] P.C. Eklund, G. Dresselhaus, M.S. Dresselhaus and J.E. Fischer, Phys. Rev. B16 (1977) 3330. [3] S.A. Solin, Physica 99B (1980) 443. [4] C. Horie, M. Maeda and Y. Kuramoto, Physica 99B (1980) 430. [5] N. Caswell and S.A. Solin, Phys. Rev. B20 (1979) 2551. [6] P.C. Eklund and K.R. Subbaswamy, Phys. Rev. B20 (1979) 5157. [7] T. Inoshita, K. Nakao and H. Kamimura, J. Phys. Soc. Japan 43 (1977) 1237. [8] T. Ohno, K. Nakao and H. Kamimura, J. Phys. Soc. Japan 47 (1979) 1125. [9] D.P. DiVincenzo, N.A.W. Holzwarth and S. Rabii, Physica 99B (1980) 406. [10] I.P. Ipatova and A.V. Subashiev, in: Theory of Light Scattering in Condensed Matter, B. Bendow, J.L. Birman and V.M. Agranovich, eds. (Plenum, N e w York, 1975) p. 201. [II] H. Suematsu, K. Higuchi and S. Tanuma, Tech. Rep. I.S.S.P.,Ser. A. no. 927 (1978). [12] U. Mizutani, I. Kondow and T.B. Massalski, Phys. Rev. B17 (1978) 3165.