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Anharmonic effects of phonons with Kohn anomalies
Solid State Communications, Vol. 20, pp. 623—625, 1976.
Pergamon Press.
Printed in Great Britain
ANHARMONIC EFFECTS OF PHONONS WITH KOHN ANOMALIES* L J. Sham Department of Physics, University of California, San Diego, La Jolla, CA 92093, U.S.A. (Received 29 July 1976 by A.A. Maradudin) In a quasi-one-dimensional conductor, the phonons with anomaly at the to soften via the threeFermi diameter, cause at 4kF phonon coupling,2kF, mainly duethe tophonons the phonon modulation of the Coulomb interaction between tight-binding electrons on different sites. The 2kF phonons also cause strong second-order scatterings of X-rays or neutrons. These results may explain the recent observations of phonon anomalies at 4kF in TFF—TCNQ. IN tetrathiofulvalene tetracyanoquinodimethane (TTF—
softest Kohn phonon at wave-vector q
TCNQ), in addition to phonon anomalies at wave-vectors with components 0.295b*, phonon anomalies at wavevectors with components twice as large, 0.59b*, have recently been observed.13 Torrance,4 using the Hubbard model with a large mtrasite electron—electron interaction, suggested that the 0.59b anomaly is the Kohn anomaly and that the 0.295b* anomaly is due to spin waves. More commonly, the 0.295b* anomaly is interpreted as the 2lcF Kohn anomaly (kF being the Fermi radius in the b direction). Emery,5 using the
0 (with 6’7 the cornponent along the chain to be 2kF)2 is given by 2 D(q,w+iO) = 1/(—w +~ —17w). (1) .
q
The damping constant ~yis given in reference 6 and the frequency given approximately by 2 2 + 2 2+ 2 2 2 ~ ~ ~ipi), where ~72 is the electron—phonon coupling times the square of the unrenormalized phonon frequency at qo, p 11 and p1 are the components of q qo parallel and perpendicular to theand chains, ~ and ~ the corresponding coherence lengths, = ~ (TT 3 — —
—
Luttinger model, suggested the correlation 0.59b* anomaly 4lcFthat in the of twois due to the singularity at electron—hole pairs with sufficiently strong repulsive interaction. The anhannonic effects of the 2kF phonons, whose frequencies are lowered by the Kohn effect, could also cause a 4kF anomaly. Consider the self-energy of a 4kF
“
phonon which splits into two phonons, [Fig. 1(a)]. If these two phonons come from the4kF neighborhood ofThis 2kF, phonon soft. they are soft enough to drive the will cause the first-order diffuse X-ray scatterings at 4kF [see Fig. 1(e)] to increase faster than at 2kF on lowering the temperature, and qualitatively explain the fact3 that the 4kF anomaly is observed at a higher temperature range than the 2kF anomaly. For the same reason, the second-order X-ray scattering [Fig. 1(f)] involving two phonons near 2kF is also anomalous at 4kF. Consider a model of a quasi-one-dimensional conductor where the electrons are confmed to move along individual chains of molecules and the phonons are three-dimensional. The calculations are carried out under the mean-field approximation. They should be qualitatively correct above the Peierls transition temperature. Predictions of actual transition temperatures are not qualitatively accurate. The phonon propagator near the *
Work supported by the National Science Foundation.
‘
~°
at a temperature T above the Peierls transition temperature The self-energy of the phonon at wave-vector 2q 0 due to splitting into two Kohn phonons [Fig. 1(a)] is calculated usingwhere equations (1) and (2). Consider rately the case the Kohn phonons are one-sepadimension like (d = 1) at high temperatures3 and the case so close to the transition3 where the Kohn phonons are three-dimensional (d = 3). In the former case (d = 1), the transverse coherence length ~ is negligible and the self-energy term is ll(2q 4e”2e~2(e’~’2+ 0, w + 10) TV1/{2S~ II ~ (4) where S is the cross-sectional area per chain and =
—
i7(~)/2~l2.
(5)
The three-phonon coupling is assumed to be a more slowly varying function of the wave-vectors than the phonon propagators and V 3 is the coupling evaluated at q0, q0 and G 2q0 where G is zero or a reciprocal lattice —
vector. The high-temperature approximation, in the 623
624
ANHARMONIC EFFECTS OF PHONONS WITH KOHN ANOMALIES
~v3j~T;:~à
Vol. 20, No.6
~z=~z~
~==g:~’~ ~ (e)
(d)
(f)
~
-~
(g)
PHONON
~
(h)
(i)
ELECTRON
PHOTON OR COULOMB INTERACTION
Fig. 1. Various Feynman diagrams. Explanation in the text. sense that T ~le~ 2, has been used. When the Kohn phonons are three-dimensional (d = 3), the self-energy is approximately ~‘
+~ (6) fl(2q0, w + 10) = TV~/4lr~~11cZ4(ei~’z Since the square of the renormalized phonon frequency is equal to the square of the unrenormalized one minus the real part of the self-energy, at sufficiently low frequencies, 2 = A —BT/e~4~’2, (7) where A and B are constants. This shows that, whether d = 1 or 3, the 2q phonon (with component 4kF) can be driven0 unstable by theparallel Kohn phonons. Its frequency decreases faster than the Kohn phonon ~ cooling above 7’,,. It is strongly 2~3 damped, with the damping constant going like é”4kF phonon then depends on The softening of the the anharinomc constant. The usual contributions, directly from the lattice or via the electron—phonon interaction, are relatively weak. In the tight-binding electron model, there is a strong contribution. When the lattice is distorted, take the zeroth order approximation of the conduction electron wave-function to be made up oflocal orbitals, unchanged in shape from the periodic crystal, but centered at the displaced sites. In this representation, expand the electron Hamiltonian including the Coulomb interaction in powers of the lattice
displacement. The first-order electron—phonon interaction consists of two terms: =
~ 1K
IK(u1+lK_uXc~t1Kc~g+c~t.cc~+iK), (8)
H~2 = ep 1 —
2
~
L(l’,c’, IK)~(u,’~’ U1K)CI’K’Cl’K’CIKCIK,
(9)
where c1~Kdenotes thespin electron at cell 1, site with the indexannihilation understood,operator U 1~the lattice displacement, I~the change in the one-electron energy overlap term, between neighboring sites and L(I’K’, l~)= (a/axl’K.)fdr fdr’ X Ø~’(r xz’K’)12 I r r’ l~OK(r xlK) 2, (10) with the electron orbital 0K at the site position X1K. The first term is the usual electron—phonon term in the tight-binding approximation. The second is the phonon modulation of the Coulomb interaction. This latter provides a strong three-phonon coupling as shown in Fig. 1(b). L is essentially the dipole interaction between sites and is large if the electron orbital is well localized and small if the orbital is wide spread. With —
—
—
Vol. 20, No.6
ANHARMONIC EFFECTS OF PHONONS WITH KOHN ANOMALIES
this coupling, the polarization of the anomalous 4kF phonon is longitudinal, as is the case in TTF—TCNQ.3 By contrast, the anomalous 2k, phonon may be longitudinal or transverse3 since in TFF—TCNQ, the normals to the molecules are nearly in b—c plane, with the consequence that the polarization in either b or c direction can change the perpendicular distance between the parallel molecules and contribute to I. The phonon-modulated electron—electron interaction excites two electron—hole pairs simultaneously. By itself, it does not give a singular second-order Kohn effect. Evaluation of the diagram in Fig. 1(c) shows at most a (q 4kF) hi (q 4kF) singularity analogous to the three-dimension Kohn effect. Even though the electrons are in one dimension, two electron—hole pairs allow sufficient phase space to eliminate the logarithmic singularity of each bubble. Except for the term in (9), the electron-electron interaction does not play an essential role in this theory of2 1c~and 4kF phonon anomalies. However, if the electron—electron interaction is strong enough to cause a singularity in the susceptibility at 2kF, then the sum of all the processes such as the one shown in Fig. 1(d) can cause a singularity at 4kF. This is a somewhat simplified version ofthe effect proposed by Emery.5 The diffuse X-ray scattering is given by the static correlation function of the atomic scattering amplitude at various positions.8 It is evaluated for these soft phonons using the method of Ambegaokar et a19 The neutron scattering is treated in the same way and will be detailed elsewhere. In spite of the heavy dampmg, the one-phonon scattering, Fig. 1(e), is still proportional to —
—
625
the inverse square ofthe frequency for T )~ w~and, therefore, peaks at the soft phonon wave-vectors 2kp. and 4k,. The temperature dependence at 2lcF is proportional to e~and that at 4lcF is proportional to e’ -i for small ‘ where ‘ is e 0, ~,being the value which makes (7) vanish. Thus, the 4kF peak is stronger at high temperatures. The one-phonon scattering has vertex corrections10 given by Fig. 1(g—i), which give correction factors proportional to e((~_4~~2, ~_4~2 and ~ respectively. The two-phonon scattering, Fig. 1(f), will also give a peak at 4kF with the temperature dependence, ~~4)/2, same as the phonon self-energy, Fig. 1(a), since they have the same structure. The temperature dependence is weaker than the one-phonon scattering at 4kF. The polarizations of the phonons in the secondorder scattering could be longitudinal or transverse, whereas the experimental 4kF peak appears to be only longitudinal.3 In the above, we have considered only the soft phonon behaviour just above the Peierls transition. Below the transition, there is a distortion with wavevector qo possessing a transverse component, e.g. in TTF—TCNQ,3 (~a*,2kF, 0). By the anharmonic phonon theory, it is clear that there should be condensation not only at 2q 4kF, 0), coming also at (0,0) and from the combination0 but of (~a*, 2kF, ~a*, 2kF, 0). This is in agreement with the experimental observations.3 —
‘
(—
Acknowledgement I am grateful to Dr. J.B. Torrance for stimulating discussions and unstinting help in bringing me up-to-date on yet unpublished lores. —
REFERENCES 1.
3.
POUGET J.P., KANNA Sic., DENOYER F., COM1~SR., GARITO A.F. & HEEGER AJ., Phys. Rev. Lett. 37, 437 (1976). ELLENSON W.D., COMES R., SHAPIRO S.M., SHIRANE G., GARITO A.F. & HEEGER AJ. (quoted in reference 3). KAGOSHIMA S., ISHIGURO T. & ANZAI H. (to be published).
4.
TORRANCE J.B., MOOK H.A. & WATSON C.R. (to be published).
5.
EMERY VJ.,Phys. Rev. Lett. 37,107(1976).
6.
PATFON B.R. & SHAM LJ.,Phys. Rev. Letr. 31,631(1973); SHAM U. & PATTON B.R.,Phys. Rev. Lett. 36, 733 (1976).
7.
Frequencies and temperatures are expressed in energy units.
8.
MARADUDIN A.A., MONTROLL E.W., WEISS G.H. & IPATOVA I.P., Theory ofLattice Dynamics in the Harmonic Approximation, Cli. VII. Academic Press, New York (1971).
9.
AMBEGAOKAR V., CONWAY J.M. & BAYM G., LatticeDynamics (Edited by WALLIS R.F.) p.261. Pergamon Press, New York (1964).
2.
10.
MARADUDIN A.A.
AMBEGAOKAR V., Phys. Rev. 135, A1071 (1964).
Pergamon Press.
Printed in Great Britain
ANHARMONIC EFFECTS OF PHONONS WITH KOHN ANOMALIES* L J. Sham Department of Physics, University of California, San Diego, La Jolla, CA 92093, U.S.A. (Received 29 July 1976 by A.A. Maradudin) In a quasi-one-dimensional conductor, the phonons with anomaly at the to soften via the threeFermi diameter, cause at 4kF phonon coupling,2kF, mainly duethe tophonons the phonon modulation of the Coulomb interaction between tight-binding electrons on different sites. The 2kF phonons also cause strong second-order scatterings of X-rays or neutrons. These results may explain the recent observations of phonon anomalies at 4kF in TFF—TCNQ. IN tetrathiofulvalene tetracyanoquinodimethane (TTF—
softest Kohn phonon at wave-vector q
TCNQ), in addition to phonon anomalies at wave-vectors with components 0.295b*, phonon anomalies at wavevectors with components twice as large, 0.59b*, have recently been observed.13 Torrance,4 using the Hubbard model with a large mtrasite electron—electron interaction, suggested that the 0.59b anomaly is the Kohn anomaly and that the 0.295b* anomaly is due to spin waves. More commonly, the 0.295b* anomaly is interpreted as the 2lcF Kohn anomaly (kF being the Fermi radius in the b direction). Emery,5 using the
0 (with 6’7 the cornponent along the chain to be 2kF)2 is given by 2 D(q,w+iO) = 1/(—w +~ —17w). (1) .
q
The damping constant ~yis given in reference 6 and the frequency given approximately by 2 2 + 2 2+ 2 2 2 ~ ~ ~ipi), where ~72 is the electron—phonon coupling times the square of the unrenormalized phonon frequency at qo, p 11 and p1 are the components of q qo parallel and perpendicular to theand chains, ~ and ~ the corresponding coherence lengths, = ~ (TT 3 — —
—
Luttinger model, suggested the correlation 0.59b* anomaly 4lcFthat in the of twois due to the singularity at electron—hole pairs with sufficiently strong repulsive interaction. The anhannonic effects of the 2kF phonons, whose frequencies are lowered by the Kohn effect, could also cause a 4kF anomaly. Consider the self-energy of a 4kF
“
phonon which splits into two phonons, [Fig. 1(a)]. If these two phonons come from the4kF neighborhood ofThis 2kF, phonon soft. they are soft enough to drive the will cause the first-order diffuse X-ray scatterings at 4kF [see Fig. 1(e)] to increase faster than at 2kF on lowering the temperature, and qualitatively explain the fact3 that the 4kF anomaly is observed at a higher temperature range than the 2kF anomaly. For the same reason, the second-order X-ray scattering [Fig. 1(f)] involving two phonons near 2kF is also anomalous at 4kF. Consider a model of a quasi-one-dimensional conductor where the electrons are confmed to move along individual chains of molecules and the phonons are three-dimensional. The calculations are carried out under the mean-field approximation. They should be qualitatively correct above the Peierls transition temperature. Predictions of actual transition temperatures are not qualitatively accurate. The phonon propagator near the *
Work supported by the National Science Foundation.
‘
~°
at a temperature T above the Peierls transition temperature The self-energy of the phonon at wave-vector 2q 0 due to splitting into two Kohn phonons [Fig. 1(a)] is calculated usingwhere equations (1) and (2). Consider rately the case the Kohn phonons are one-sepadimension like (d = 1) at high temperatures3 and the case so close to the transition3 where the Kohn phonons are three-dimensional (d = 3). In the former case (d = 1), the transverse coherence length ~ is negligible and the self-energy term is ll(2q 4e”2e~2(e’~’2+ 0, w + 10) TV1/{2S~ II ~ (4) where S is the cross-sectional area per chain and =
—
i7(~)/2~l2.
(5)
The three-phonon coupling is assumed to be a more slowly varying function of the wave-vectors than the phonon propagators and V 3 is the coupling evaluated at q0, q0 and G 2q0 where G is zero or a reciprocal lattice —
vector. The high-temperature approximation, in the 623
624
ANHARMONIC EFFECTS OF PHONONS WITH KOHN ANOMALIES
~v3j~T;:~à
Vol. 20, No.6
~z=~z~
~==g:~’~ ~ (e)
(d)
(f)
~
-~
(g)
PHONON
~
(h)
(i)
ELECTRON
PHOTON OR COULOMB INTERACTION
Fig. 1. Various Feynman diagrams. Explanation in the text. sense that T ~le~ 2, has been used. When the Kohn phonons are three-dimensional (d = 3), the self-energy is approximately ~‘
+~ (6) fl(2q0, w + 10) = TV~/4lr~~11cZ4(ei~’z Since the square of the renormalized phonon frequency is equal to the square of the unrenormalized one minus the real part of the self-energy, at sufficiently low frequencies, 2 = A —BT/e~4~’2, (7) where A and B are constants. This shows that, whether d = 1 or 3, the 2q phonon (with component 4kF) can be driven0 unstable by theparallel Kohn phonons. Its frequency decreases faster than the Kohn phonon ~ cooling above 7’,,. It is strongly 2~3 damped, with the damping constant going like é”4kF phonon then depends on The softening of the the anharinomc constant. The usual contributions, directly from the lattice or via the electron—phonon interaction, are relatively weak. In the tight-binding electron model, there is a strong contribution. When the lattice is distorted, take the zeroth order approximation of the conduction electron wave-function to be made up oflocal orbitals, unchanged in shape from the periodic crystal, but centered at the displaced sites. In this representation, expand the electron Hamiltonian including the Coulomb interaction in powers of the lattice
displacement. The first-order electron—phonon interaction consists of two terms: =
~ 1K
IK(u1+lK_uXc~t1Kc~g+c~t.cc~+iK), (8)
H~2 = ep 1 —
2
~
L(l’,c’, IK)~(u,’~’ U1K)CI’K’Cl’K’CIKCIK,
(9)
where c1~Kdenotes thespin electron at cell 1, site with the indexannihilation understood,operator U 1~the lattice displacement, I~the change in the one-electron energy overlap term, between neighboring sites and L(I’K’, l~)= (a/axl’K.)fdr fdr’ X Ø~’(r xz’K’)12 I r r’ l~OK(r xlK) 2, (10) with the electron orbital 0K at the site position X1K. The first term is the usual electron—phonon term in the tight-binding approximation. The second is the phonon modulation of the Coulomb interaction. This latter provides a strong three-phonon coupling as shown in Fig. 1(b). L is essentially the dipole interaction between sites and is large if the electron orbital is well localized and small if the orbital is wide spread. With —
—
—
Vol. 20, No.6
ANHARMONIC EFFECTS OF PHONONS WITH KOHN ANOMALIES
this coupling, the polarization of the anomalous 4kF phonon is longitudinal, as is the case in TTF—TCNQ.3 By contrast, the anomalous 2k, phonon may be longitudinal or transverse3 since in TFF—TCNQ, the normals to the molecules are nearly in b—c plane, with the consequence that the polarization in either b or c direction can change the perpendicular distance between the parallel molecules and contribute to I. The phonon-modulated electron—electron interaction excites two electron—hole pairs simultaneously. By itself, it does not give a singular second-order Kohn effect. Evaluation of the diagram in Fig. 1(c) shows at most a (q 4kF) hi (q 4kF) singularity analogous to the three-dimension Kohn effect. Even though the electrons are in one dimension, two electron—hole pairs allow sufficient phase space to eliminate the logarithmic singularity of each bubble. Except for the term in (9), the electron-electron interaction does not play an essential role in this theory of2 1c~and 4kF phonon anomalies. However, if the electron—electron interaction is strong enough to cause a singularity in the susceptibility at 2kF, then the sum of all the processes such as the one shown in Fig. 1(d) can cause a singularity at 4kF. This is a somewhat simplified version ofthe effect proposed by Emery.5 The diffuse X-ray scattering is given by the static correlation function of the atomic scattering amplitude at various positions.8 It is evaluated for these soft phonons using the method of Ambegaokar et a19 The neutron scattering is treated in the same way and will be detailed elsewhere. In spite of the heavy dampmg, the one-phonon scattering, Fig. 1(e), is still proportional to —
—
625
the inverse square ofthe frequency for T )~ w~and, therefore, peaks at the soft phonon wave-vectors 2kp. and 4k,. The temperature dependence at 2lcF is proportional to e~and that at 4lcF is proportional to e’ -i for small ‘ where ‘ is e 0, ~,being the value which makes (7) vanish. Thus, the 4kF peak is stronger at high temperatures. The one-phonon scattering has vertex corrections10 given by Fig. 1(g—i), which give correction factors proportional to e((~_4~~2, ~_4~2 and ~ respectively. The two-phonon scattering, Fig. 1(f), will also give a peak at 4kF with the temperature dependence, ~~4)/2, same as the phonon self-energy, Fig. 1(a), since they have the same structure. The temperature dependence is weaker than the one-phonon scattering at 4kF. The polarizations of the phonons in the secondorder scattering could be longitudinal or transverse, whereas the experimental 4kF peak appears to be only longitudinal.3 In the above, we have considered only the soft phonon behaviour just above the Peierls transition. Below the transition, there is a distortion with wavevector qo possessing a transverse component, e.g. in TTF—TCNQ,3 (~a*,2kF, 0). By the anharmonic phonon theory, it is clear that there should be condensation not only at 2q 4kF, 0), coming also at (0,0) and from the combination0 but of (~a*, 2kF, ~a*, 2kF, 0). This is in agreement with the experimental observations.3 —
‘
(—
Acknowledgement I am grateful to Dr. J.B. Torrance for stimulating discussions and unstinting help in bringing me up-to-date on yet unpublished lores. —
REFERENCES 1.
3.
POUGET J.P., KANNA Sic., DENOYER F., COM1~SR., GARITO A.F. & HEEGER AJ., Phys. Rev. Lett. 37, 437 (1976). ELLENSON W.D., COMES R., SHAPIRO S.M., SHIRANE G., GARITO A.F. & HEEGER AJ. (quoted in reference 3). KAGOSHIMA S., ISHIGURO T. & ANZAI H. (to be published).
4.
TORRANCE J.B., MOOK H.A. & WATSON C.R. (to be published).
5.
EMERY VJ.,Phys. Rev. Lett. 37,107(1976).
6.
PATFON B.R. & SHAM LJ.,Phys. Rev. Letr. 31,631(1973); SHAM U. & PATTON B.R.,Phys. Rev. Lett. 36, 733 (1976).
7.
Frequencies and temperatures are expressed in energy units.
8.
MARADUDIN A.A., MONTROLL E.W., WEISS G.H. & IPATOVA I.P., Theory ofLattice Dynamics in the Harmonic Approximation, Cli. VII. Academic Press, New York (1971).
9.
AMBEGAOKAR V., CONWAY J.M. & BAYM G., LatticeDynamics (Edited by WALLIS R.F.) p.261. Pergamon Press, New York (1964).
2.
10.
MARADUDIN A.A.
AMBEGAOKAR V., Phys. Rev. 135, A1071 (1964).