.~ Solid State Communications, Vol.53,No.10, pp.827-830, 1985. 0038-1098/85 $3.00 + .00 Printed in Great Britain. Pergamon Press Ltd. SHORT RANGE ORDER EFFECTS AND THE FALICOV-KIMBALL MODEL Gast6n Martfnez Departmento de Ffsica, Universidad de Tarapac~, Arica, Chile and Jaime Rgssler* Facultad de Ciencias B~sicas y Farmac~uticas, Universidad de Chile, Casilla 653, Santiago, Chile and Miguel Kiwi *t Facultad de Fisica, Universidad Cat61ica de Chile, Casilla II4-D, Santiago, Chile (Received 15 November 1984 by H. Suhl) The model of Falicov and Kimball is investigated incorporating short range order effects. This modifies both the electronic density of states and the configurational entropy, relative to the fully disordered case. The way this affects the conditions under which phase transitions occur is evaluated and discussed. ] . Introduction The model of Falicov and Kimball I (FK), has been extensively used to obtain an under- standing of a wide variety of phenomena such as first- and second-order phase transitions, 2 the electronic and magnetic properties of the rare- earths 3-5 and of excitonic phases. 3 Here we incorporate short range order ef- fects, and the consequent modification of the configuration entropy, in the formulation and H = [li>£o<il + [ li>tij<J I - ~' i <i,j> i solution of the FK model. These effects were treated qualitatively by Sakurai and Schlottmann,Swho suggested the formation of a short range ordered superstructure as a mechanism to stabilize a valence of 2/3 of Sm in SmS. Also Lazo et al 7 gave some attention to these effects in previous work. In this paper we first formulate the prob- lem writing an appropiate Hamiltonian. Next, we outline the techniques of R~ssler and Lazo 8 and of Kikuchi, 9 which allow to handle the ef- fects due to short-range-order on the electronic density of states and to evaluate the configu- rational entropy, respectively. Finally, the scheme used to obtain numerical results, as well as the results themselves are presented, its implications are discussed and some relevant conclusions are drawn. * Supported in part by CONICYT under Project # 1098 + Supported in part by the Organization of American 827 II. Model Hamiltonian and its Helmholtz Free Energy Falicov and Kimball (FK) introduced I their Hamiltonian as a simple theoretical tool to study metal-insulator phase transitions. Sub- sequently, Ramlrez, Falicov and Kimball dis- cussed it in detail 2 using the mean-field a~- proximation. Later on RSssler and Ramfrez, ~ and also Trias, Ramirez and Kiwi I0 reformulated the FK model as an alloy problem and obtained ad- ditional information of its features. We follow the notation of the latter work I0 and write H as li>G<ij, c2.1) where li> denotes a localized Wannier state on lattice site ~i, G>O describes the attractive interaction between an electron and a hole at site Ri; actually, the prime in the last summa- tion in (2.1) implies that it is limited to ionized sites only. The hopping parameter t ij= -t connects two first neighbor atoms denoted by <i,j>. In this way an equivalence is estab- lished between the fn+l rare-earth atoms and say tile A-type species of ~ binary alloy, and be- tween the fully ionized 4f n atomic states and species B. (For example, in the fluctuating valence compound SmS we identify the Sm ++ (4f 8) and the Sm 4-~+ (4f 5) with species A and B, respectively). Using the notation {RI, R 2 .... R N } = {R.} to denote the set of lattlce sites B with B-#ype atoms, we can write the total Helmholtz free energy as and by the DIB of the U. of Chile. States (OAS).