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Journal of the Physical Society of Japan

Theory of Many Wannier Excitons. I

Eiichi Hanamura

Published Dec 1, 1974• 101 citations
Publisher

Abstract

We present the method by which we can describe the many exciton system transparently by making the best use of the boson like character of Wannier excitons. Basing upon this method, the total energy of this system and the number of bose-condensed excitons are calculated as a function of the total concentration of excitons, and we show how the boson like character of Wannier excitons is reflected on the emission spectrum from this system and on the absorption spectrum in the presence of these excitons at finite density.

DOI
10.1143/jpsj.37.1545

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We present the method by which we can describe the many exciton system transparently by making the best use of the boson like character of Wannier excitons. Basing upon this method, the total energy of this system and the number of bose-condensed excitons are calculated as a function of the total concentration of excitons, and we show how the boson like character of Wannier excitons is reflected on the emission spectrum from this system and on the absorption spectrum in the presence of these excitons at finite density.
The present paper has the dual purposes. As the first purpose, it is demonstrated that the boson description of Wannier excitons introduced in the preceding paper is suitable also to discussion of non-linear coherent interaction of Wannier excitons with the radiation field. The second purpose is to discuss the possibility of self-induced transparency due to Wannier excitons on the basis of the boson description by making clear the difference between a two-level atomic system and a system of excitons which can propagate in a crystal as polariton. As a result, stable pulse propagation of the radiation field is shown to be prevented by the polariton effect of excitons.
Chapter 19 addresses bosonic condensation in composite bosons. It first considers the coboson ground-state energy, which is analytically known by now in the case of N Cooper pairs interacting via the reduced BCS potential but not yet known for N Wannier or Frenkel excitons, owing to the complexity of the Coulomb potential. Still, the chapter provides hints on how to derive the density linear term of the N-exciton ground-state energy in the case of Wannier excitons. It also discusses the role of the exciton composite nature in condensate momentum fragmentation, spin fragmentation, and condensate brightness. Indeed, the excitons that have the lowest energy and thus are the ones expected to condense are not coupled to light; so, the exciton condensate must be dark. However, a bright exciton condensate can appear, coherent with the dark exciton condensate, above a density threshold, owing to coupling induced by carrier exchange between dark excitons and bright excitons.
In a first part, a brief review of some excitonic properties is presented, with the accent put on their relevance with respect to the possibility of Bose-Einstein condensation. In a second part, recent experimental evidence is described, which supports the idea that excitonic particles may form a highly quantum fluid. In Cu20, the analysis of the exciton decay spectrum shows a gradual evolution of the gas from a classical regime at low densities up to a strongly degenerate one at high densities, with a chemical potential µ ~ 0. In CuCl excitons are unstable against formation of molecules (biexcitons). It is possible to generate directly a high density gas of biexcitons with momentum K ~ 0 by giant two-photon absorption. At low excitation, the observed molecular emission is in agreement with Maxwell -Boltzmann statistics. At high densities, strong deviations occur. In particular, the appearance of a sharp emission line is attributed to the presence of a Bose-Einstein condensate of excitonic molecules.
The exciton system is described by means of specially constructed coherent states (CS), according to an idea of L.V. Keldysh long ago. The system is analyzed from the geometric point of view and from the relevant symmetry group contained in the displacement operator as generador of the CS system in the Klauder–Perelomov sense. Explicitly important quantities involving the quantum statistics of the system (partition function, mean density n and the Yuen limit) are calculated and briefly discussed in relation to the known case of pure bosons (e.g. photons). The Bogoliubov type transformations for the field operators of the Wannier–Mott type exciton system are presented in an exact way (in sharp contrast with the case of [11] where only a rough approximation was given) showing how these transformations induce nonlinearity (cubic at first order) in the Schrodinger equation of the system. The conditions for a hydrodynamic description of the original Keldysh model, as the possibility of a BEC (Bose–Einstein Condensation), are discussed.
The emission spectrum is given for radiative decay of the excitonic molecules in the Bose-condensed state at finite temperature which is described in the framework of the Bogoliubov approximation. We take into account the final state interaction of an exciton created in the radiative process with the environment of the Bose-condensed excitonic molecules. The emission spectrum is characterized by a sharp line due to the radiative decay of the Bose-condensed excitonic molecules and the side band at lower energy side due to the process associated with the excitation of the Bose-condensed system. The polariton effect on the emission spectrum is also discussed.
A fully quantum mechanical formulation of the electron-hole-exciton-light system is developed following the work of Hanamura. In this formulation 1. The excitons are described by the "bosonisation transformation of Marumori et al. In this method exciton bose operators are introduced to describe electron-hole pairs, and anti symmetrisation effects are taken account of by a unitary transformation of the system operators, which introduces extra nonlinearities into the hamiltonian. This means that exclusion principle effects at high exciton densities are fully accounted for. The theory is formulated for Wannier (non-localised) excitons.