Effect of phonons in mixed valence systems

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Effect of phonons in mixed valence systems

M. Avignon, F. Brouers, K. Bennemann

To cite this version:

M. Avignon, F. Brouers, K. Bennemann. Effect of phonons in mixed valence systems. Journal de

Physique Colloques, 1979, 40 (C5), pp.C5-377-C5-379. �10.1051/jphyscol:19795135�. �jpa-00218921�

JOURNAL DE PHYSIQUE Collogue C5, supplement au n° 5, Tome 40, Mai 1979, page C5-377

Effect of phonons in mixed valence systems

M. Avignon

Groupe des Transitions de Phases, C.N.R.S., B.P. 166, 38042 Grenoble Cedex, France F. Brouers and K. H. Bennemann

Institute of Theoretical Physics, Freie Universitat Berlin, 1 Berlin 33, Arnimallee 3, F.R.G.

Résumé. — Nous étudions l'effet du couplage électron-phonon sur les fluctuations de valence dans les systèmes de terres rares. Nous montrons que le couplage électron-phonon affecte à la fois les paramètres électroniques importants et les fréquences de phonons, en particulier les phonons optiques longitudinaux de courte longueur d'onde.

Abstract. — The effect of electron-phonon coupling on valence fluctuations in rare-earth systems is studied.

It is shown that the electron-phonon coupling affects both the important electronic parameters and the phonon- frequencies, in particular the short-wavelength longitudinal optical phonons.

Electronic phase transitions in d- and f- electron systems like transition metals and rare-earth metals are particularly interesting due to strong electron- phonon coupling and strong Coulomb interactions between the electrons. As a result of such interactions, electronic phase transitions involving f «± d electronic transitions occur in many rare-earth systems like SmS, Sm

_

Y

S, etc... These phase transitions involving valence changes and large volume changes occur due to pressure, alloying and temperature, for example [1]. Since large volume changes occur, one expects that the electron-lattice coupling affects strongly the electronic phase transition. In particular, one expects that the f ?± d electronic transitions resulting from the electron-lattice coupling affect the f-d hybridization and in general the electronic f-d interband transitions and thus the valence phase transitions. The indirect electron-electron interactions due to virtual phonon exchange modifies the direct Coulomb interaction between the electrons. Assuming random valence-fluctuations of the rare-earth atoms like Sm in SmS and S n ^ ^ Y J S , one expects for co

t

< 1 that, in particular, the longitudinal short wavelength phonons are softened due to the valence fluctuations Sm

<± Sm

. The valence fluctuation time is denoted by x

. This phonon softening results from the response of the S~ ~ atoms to the fluctuating valence change of the Sm caused by the electron- lattice coupling involving f «± d electronic transitions.

In this communication we want to show how f-d

processes resulting from electron-phonon interactions can influence the mixed valence transition and we shall limit ourselves to a simple atomic local picture.

We want to derive and examine the various possible contributions. A complete discussion of the effect of electron-phonon interactions should take into account the full ^-dependence of the d-electrons and the dispersion relation of optical and acoustical phonons. This is left for a further publication.

In the following we use a simplified tight-binding hamiltonian to describe the effect of the electron- phonon coupling in rare-earth systems with valence transitions. We use for the d- and f- electrons the hamiltonian

where

and the electron-phonon coupling is given by

One includes both d-d, f-f and d-f electron phonon coupling while previously only f-f processes have been considered [2].

Using the usual canonical transformation (i, j = f, d)

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19795135

M. AVIGNON, F. BROUERS A N D K. H. BENNEMANN

then X transforms to

1 aa

Xf = H~ - - 1

- a~: a ^{tim +}

^-

^{+ tic0} c: cj c,+ cm ^-

2

aa+ + a + a )c,+c,c.tc, . (5)

El

-

- t i o

-

+ ^ho I

The two terms in Eq. (5) describe the electron-electron scattering processes resulting from the electron-lattice coupling. These processes are illustrated in figure 1.

Fig.

- Electron scattering processes resulting from the electron-lattice coupling

(a), (b) and

contribute to the renormalization of electron-electron interactions

(d) and (e) give contributions to 'the hybridization.

Evaluating in detail the various terms in the sums in Eq. (5), one obtains where

with the renormalised energies

and where

Eq. (10) can be rewritten as

1 + a + a [ 8 d - E f 1 AH ⁼ - g:,

Ed

-

+ ^{f i o} + ^ho +

^-

The renormalised phonon energies are given by the functional derivative

Thus, from (1 1) one obtains approximately

EFFECT OF PHONONS IN MIXED VALENCE SYSTEMS C5-379

Assuming intra-atomic electron-hole excitations only, thus the phonons frequencies get only modified due to f-electrons d-hole and f-hole d-electrons excitations. We expect this to be the case for semi- conducting SmS. For Sm,-,Y,S and metallic SmS further renormalization of the phonon frequencies may result from electron-hole excitations involving d-states only. Note, for example, along the (1, 1, 1)- direction for the phonons wavevector q, one obtains for optical phonons g,, cc sin (qa), where a denotes the distance between Sm and S along the (111)- crystal direction.

Assuming that the f-d hybridization is essential for the valence phase-transition, thus evidently the renormalization of V,, plays an important role.

The detailed study of the different contributions will be published in another paper. However, note the renormalization due to the electron-phonon coupling should become ineffective at temperatures larger than the maximal phonon frequencies. Due to the f e d transitions resulting from the electron-lattice coupling, the f-states remain now partially filled in metallic SmS, etc.

In Eq. (1) the electronic hopping between different atomic sites has been neglected for simplicity. How- ever, it is straightforward to include a term tij Ci+ Cj.

This will change the discussed effects due to the electron-lattice coupling involving intra-atomic elec- tronic transitions only quantitatively. For discussing inter-atomic Coulomb interactions and electronic transitions caused by lattice vibrations, one must consider simultaneously this hopping tij Ci' Cj.

Note, on general physical grounds, one expects that the transverse optical phonons are much weakly affected by the valence transitions than the longitudi- nal optical phonons. In the case of transverse phonons the motion of the S-- atoms in the long wavelength limit is approximately perpendicular to the change in the Coulomb force acting on the S-- atoms due to Sm2+ a Sm3+. Consequently g,,(long) b g,,(trans) and one might observe a t the zone boundary a crossing of the optical phonon branches. Furthermore, the acoustic phonons are mainly softened in the long wavelength limit where o, - ^{o:(1 -} ctN(O)), N(0) being the electronic density of states a t the Fermi energy E,. Since w:(a) increases when the lattice constant a decreases, the change of the phonon fre- quencies should be observable most clearly in semi- conducting SmS, etc. close to the phase transitions.

For the phase diagram, one expects that the depen- dence of the transition temperature T,(p, x) on pressure p or alloy composition x may be significantly affected a t temperatures smaller than the maximal phonon energies.

Note added in proof: - A local picture similar to the one developped in this paper including d-f and f-f processes has also been considered by P. Entel and H. J. Leder (Proceedings of this conference).

However, they treated it in a mean-field approxima- tion and the renormalization of the parameters is different from what we obtained with the canonical transformation.

References

[I] JAYARAMAN, A., DERNIER, P. D . and LONGINOTTI, L. D., in High temp., High press. 7 (1975) 1 .

VARMA, C. M., Rev. Mod. Phys. 48 (1976) 219; for a recent review see also

Proc. Int. Conf. on Valence Instabilities and Related Narrow Band Phenomena, Ed. R. D. Parks (Plenum Press, N.Y.) 1977.

[2] SHERRINGTON, D. and RISEBOROUGH, S., J. Physique Colloq.

37 (1976) C4-255 and references therein.

HALDANE, F. D. M., Phys. Rev. B 15 (1977) 281.