STUDY OF EXTENDED FALICOV-KIMBALL MODEL ON A TRIANGULAR LATTICE FOR CORRELATED SYSTEMS

Abstract

The study presented in this thesis is mainly concerned with the electronic and mag-netic properties of certain correlated electronic systems such as transition metal dichalcogenides, cobaltates, Gd12 and its doped variant Cd12Hx, NaTiO2 and NaV02 etc. These are layered systems with a triangular lattice and can often be character-ized by the presence of two types of electrons, one is itinerant and other localized, confined to the two-dimensional triangular lattice. In these systems by tuning the U/t ratio (where U is the strength of on-site Coulomb repulsion between itinerant and localized electrons and t is the width of itinerant electron band) and position of localized electron energy level Ef , several interesting cooperative phenomena such as valence and metal-insulator transitions, charge and magnetic order, excitonic in-stability and possible non-Fermi liquids have been observed experimentally. There-fore we believe that these systems can very well be modeled by the two-dimensional spin-independent and spin-dependent Falicov-Kimball model (FKM) on a triangular lattice. This thesis studies the ground and excited state properties of above models with various extended interaction terms on a non-bipartite (triangular) lattice This thesis consists of six chapters which are discussed one by one in the following. Chapter 1: This chapter carries general introduction related to strongly corre-lated electron systems like transition metal, rare earth and their compounds. We 11 have reviewed available experimental results for the systems of recent interest like transition metal dichalcogenides, cobaltates, Gd/2 and its doped variant GdIglx, NaTiO2 and NaVO2 etc. A review of different theoretical approaches such as Mott hypothesis, Hubbard model and Falicov-Kimball model to study the correlated elec-tron systems is presented. After that different analytical and exact calculation results using FKM on different lattices (bipartite/non-bipartite) are discussed. An explicit description of ground state energy calculation of Falicov-Kimball Hamiltonian on a triangular lattice using numerical diagonalization technique with periodic boundary conditions (PBC) and calculation of average value of a physical quantity at a finite temperature is given in methodology section of this chapter. Chapter 2: In this chapter, we have discussed the spinless Falicov-Kimball model (FKM) on the triangular lattice with non-degenerate localized f—electron level with correlated hopping (t'). Unusual correlated hopping (t') term includes the effect of occupation of nearest neighboring sites of localized (f —) electrons on hopping of itin-erant (d—) electrons. The different interesting ground state configurations e.g. reg-ular, bounded, hexagonal and phase segregation of localized (f) electrons are found. The first order (discontinuous) metal-insulator transition corresponding to the differ-ent values of the parameters like correlated hopping (t'), on-site Coulomb repulsion (U) between d— and f —electrons, f —electron energy level position (E f)and filling of the f —electrons (Nf) is seen. Phase segregation with correlated hopping (t') even at small U—values, is one of the important results of our analysis which is recently observed in Gd/2 system, experimentally. Earlier analysis without the correlated hopping term shows that phase segregation can find only at the larger U—values. Chapter 3: In this chapter we have studied the spinless Falicov-Kimball model on the triangular lattice with doubly degenerate localized f—electron levels. The model considers on-site Coulomb repulsion (U) between the d— and f—electrons and on-site Coulomb repulsion (Uf) between f —electrons of doubly degenerate orbitals (say fi and f2). We have calculated the effect of U and Uf on the ground state con-figurations at different commensurate fillings of f1 and 12 electrons. We have found 111 metal-insulator transition by calculating the energy gap (around the Fermi energy) and density of states of d-electrons. We have seen different types of long range orbital ordered phases at different fillings of fi and f2 electrons and different values of U and Uf. We have found that there is a competition between U and Uf. At a fixed U, first order metal to insulator transition is observed with increasing Uf. When U is larger than Uf, doubly occupied sites by fi and f2 electrons with disordered phase and with metallic nature, is observed. In opposite limit well long range orbital ordered phase with insulting nature is observed. At a fixed U1, 're-entrant' metallic phase is observed with increasing U. The disordered to ordered phase transition in ground state configuration of f —electrons is accompanied by a metal to insulator transition. Changing the parameters U and Uf, we can easily switch between metal and insulator phases. In addition we have discussed the spontaneous symmetry breaking conjec-ture (SSBC) on triangular lattice which was observed on square lattice by adding hybridization (V) term between d— and f —electrons in the Hamiltonian. We have reported that there is no spontaneous symmetry breaking when the ground state is inhomogeneous. These results are quite relevant for the Gd12 and NaTiO2 systems. Chapter 4: This chapter is divided into two parts. In first part of the chapter we have studied thermodynamic properties of the spinless FKM on the triangular lat-tice. We have calculated metal-insulator (i.e. of — Ef) transition, specific heat (C„) at different values of on-site Coulomb repulsion (U) between d— and f —electrons, f —electron energy level (E1) and different values of temperature (T). We have ob-served that the temperature is able to smear the discontinuous (nf — E1) transition on triangular lattice opposite to the case of square lattice where it has been reported that the discontinuous metal-insulator transition observed in ground state, smear out with temperature. Single and two-peak patterns are observed in Cu with tem-perature at different U and Ef values. These peaks are related to energy gaps in the many body spectrum of d—electrons. In second part of the chapter we have discussed the type of phase transitions at finite temperatures in a spinless, extended Falicov-Kimball model on the triangular lattice. Study of phase transitions from low iv temperature (ordered) phase to high temperature (disordered) phase with correlated hopping (t') is presented. We have calculated temperature dependence of specific heat (Cr), charge susceptibility (x), probability energy distribution P(E) and average f—electron occupation on each site (c.,), (T)) at different values of t', U and at differ-ent fillings of f —electrons. It is found that the low temperature (regular, bounded, striped and segregated) phases persist up to a finite critical temperature (71). In addition, we observe that the critical temperature decreases in regular, striped and bounded phases and sharply increases in the segregated phase with increasing the correlated hopping. The critical temperature depends upon filling of the localized electrons. Chapter 5: In this chapter, the ground state properties of spin-dependent FKM on a triangular lattice are studied as a function of Coulomb correlation (U), exchange correlation (J) between itinerant and localized electrons and Coulomb correlations between localized f—electrons (Uf). The results indicate various charge and spin or-dering of the localized electrons in the ground state. The ground state configuration is found to be long range ordered,either antiferromagnetic or ferromagnetic depend-ing on values of the above mentioned parameters. We have also studied variation of d— and f—electrons magnetic moments as a function of U, Uf and J. Chapter 6: This chapter contains an overall summary and conclusions along with interesting future plans.