Four-site crystal model for an intermediate-valence system

Journal

of Magnetism

FOUR-SITE

and Magnetic

CRYSTAL

J.C. PARLEBAS Dtzpurtment

Materials

o/ Phwlcs,

54-57

405

(1986) 405-406

MODEL FOR AN INTERMEDIATE-VALENCE

*, R.H. VICTORA Umrvr.v!,~ of Califrnu,

SYSTEM

and L.M. FALICOV BerXele,v.

C‘A Y4720. tiSA

A model for an intermediate valence four-site tetrahedral crystal. the smallest face-centered consists of one electron and four spin-orhitals per atom and a variety of interactions. Many-body the Hamiltonian are calculated ruuct/v with the aid of the symmetry of the crystal.

The phenomenon of intermediate valence [l-3] in rare-earth compounds arises from the competing effect 3f three difference forces: (A) a strong electron-electron repulsion between electrons in the f-shell: (B) a wide-band one-electron (periodic potential) effect for the s and d conduction electrons; and (C) a sizeable hybridization between f and conduction band states. It is usually modelled by the so-called periodic Anderson Hamiltonian [2,3]. an extension to a periodic lattice of the famous Anderson magnetic impurity problem [4]. Here we present exact calculations of the three-dimensional periodic Anderson Hamiltonian for the smallest face-centered cubic crystal, i.e. a four-site tetrahedral cluster with periodic boundary conditions. The model, an extension to two orbitals of a previously solved Hubbard Hamiltonian 151. consists of (a) One extended orbital per site per spin, with a mean enegy chosen to be zero (the origin of our enerrgy scale) and with an interatomic transfer (hopping) integral t. If t is positive the one-electron band energies are a singlet at -31fl and a triplet at + It 1; if t is negative the lowest one-electron level is a triplet at - 1t 1 and the excited orbital is a singlet at +3lrj. (b) One localized “f” orbital per site per spin, with energy E,,. The value of E,, is arbitrary. positive or negative. (c) A Coulomb repulsion U (always positive) between two electrons in the f orbitals in the same site. (d) A hybridization term between band and localized orbitals in different sites, of strength V (taken to be positive). (e) A number of electrons taken to be one per site: n = 4 in the whole crystal. We exploit the full symmetry of the problem; in particular (i) spin is a good quantum number and the states can be classified as spin singlets, triplets and quintets; (ii) there are 192 space ~ translation, point and mixed ~ operations which yield ten irreducible representations, five of which appear in this problem. The 1820 eigenstates of the n = 4 problem can thus * Permanent address: Strasbourg.

LMSES-ULP.

4 rue B. Pascal.

0 Elsevier

Science

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Eo Fig. 1. The f occupation number per site n, in the four-electron ground state ‘r, in terms of E,, and various hyhridiration energies V for a positive transfer integral t > 0. We show a strong-interaction case (/= 50 in (a) and the non-interacting case L’= 0 m (h). All energies in units of 111.

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be separated into fifteen symmetry-Factorized problems: the largest matrix to diagonalize is of order 50 (of order 33 for the ground state). The most remarkable result of our calculation is that (I) AN rntegrutedproperties, including f-state occupation, are donlinuted not by the electron-electron interaction but I>J the one-electron effects, even when the interaction U tends to infinity. This result is clearly displayed in figs. 1 and 2. The close similarity between (a) and (b) in either case definitely shows that the ground-state properties of the one-electron-per-site case is a sensitive function of E,,. I and V. but very insen-

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Valrncr Inatahilities and Related Narrow-Hnnd Phenomena. ed. R.D. Parks (Plrnum. New York. 1977). PI Valence Fluctuations in Solids, da. L.M. Fal~cov. W. Hanks and M.B. Maple (North-Holland. Amsterdam. IYXI). 131Valence Instabilities. eds. P. Wachtrr and 11. Hoppnrt (North-Holland. Amsterdam. lYX2). [41 P.W. .Ander\on. Phqs. Rev. 124 (lY61) 41. 151 L.M. Fal~cov and R.H. Victora. Phy. Rev. BiO (19X4) lhY5. [61 r-11. LIII and L.M. Falicw. Phvr. Rev. B22 (IYXO) X57. Src in particular figs. I and 2 the&. and K.W.H. Stevena. J. Phvs. (‘12 (1979) [71 R.G. Arnold jOI+7. K.W.H. Steven.\ and R.G. ,Arnold. ihid. 12 (lY7Y) soil.

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