Linear and Nonlinear Optical Characteristics of the Falicov-Kimball Model

Abstract

We calculate the linear and nonlinear optical properties of the Falicov-Kimball model for a mixedvalent system within the self-consistent mean-field approximation. Second-harmonic generation can only occur if the mixed-valent state has a built-in coherence between the itinerant d-electrons and the localized f -holes. By contrast, second-harmonic generation cannot occur for solutions of the model with f -site occupation as a good quantum number. As an experimental test of coherence in mixed-valent compounds we propose a measurement of the dynamic second-order susceptibility.The Falicov-Kimball We report here the nonlinear optical responses of these two classes of solutions. Solutions of the model with a built-in coherence can sustain second-harmonic generation. Solutions with classical f -electron site distributions cannot. Therefore, we propose the measurement of the second-order susceptibility of a mixed-valent compound as a test to distinguish between these theories. The existence of such second-harmonic generation due to coherence in the ground state would, of course, be of interest in its own right as a manifestation of strong electron correlation.Four-Wave-Mixing (FWM) spectroscopy has become a powerful tool for studying coherence in semiconductor systems [6]. In a three-beam FWM experiment, two incoming beams of wavevectors k 1 and k 2 set up a transient polarization grating. The third incoming beam of wavevector k 3 diffracts off the grating to produce the outgoing signal in the direction k 4 = k 3 + k 2 − k 1 . Being a third-order process, FWM is allowed in media with or without inversion symmetry. We pose the question: what happens if the state being probed already has a polarization built into it instead of being created artificially by optical pumping? An example of such a system is the self-consistent mean-field (SCMF) solution of the FK model resulting in the Bose-Einstein condensation of d-f excitons.As shown below, the built-in polarization leads to a nonlinear optical response to second order in the exter-1

Bound state formation and the nature of the excitonic insulator phase in the extended Falicov-Kimball model

“…13 and 14 in contrast to the findings in Ref. 12. Afterwards it has been demonstrated that spontaneous electronic ferroelectricity also exists in the EFKM, provided that the c-and f-bands involved have different parity.…”

“…To be a good model of the mixed-valence state, however, one should build in a coherence between c and f particles. 12 This can be achieved by a c-f hybridization term V . Alternatively, a finite f-bandwidth, which is certainly more realistic than entirely localized f-electrons, can also induce c-f coherence.…”

“…12 The origin is a non-vanishing c † f expectation value, causing a finite electrical polarization. In the limit V → 0 there is no fer-roelectric ground state, as was shown in Refs.…”

Existence of excitonic insulator phase in the extended Falicov-Kimball model: SO(2)-invariant slave-boson approach

“…An onsite Coulomb interaction U of both species determines the distribution of the electrons between the f and c sub-systems, and therefore may drive a valence transition, provided there is a way to establish f -c coherence. At first glance this can only be achieved by including a hybridization of the f and c bands 27,28 . It has been shown, however, that a finite f bandwidth, being certainly more realistic than entirely localized f electrons, will also induce f -c transitions 29,30 .…”

“…In the past few years, both the Falicov-Kimball model with hybridization 27,28,[32][33][34] and the EFKM [29][30][31]35 have been studied in connection with the exciting idea of electronic ferroelectricity. The origin of electronic ferroelectricity is a spontaneously broken symmetry due to a non-vanishing c † f average, which causes finite electrical polarizability without an external, interbandtransition driving field.…”