Theory of the Contribution of Excitons to the Complex Dielectric Constant of Crystals*t

completely satisfactory (in contrast to the case of absorption due to interband transitions). A more complete theory is developed. It is shown that excitons are approximate bosons, and, in interaction with the electromagnetic field, the exciton field plays the role ef the classical polarization field. The eigenstates of the system of crystal and radiation field are mixtures of photons and excitons. The ordinary one-quantum optical lifetime of an excitation is infinite. Absorption occurs only when "three-body" processes are introduced. The theory includes "local field" effects, leading to the Lorentz local field correction when it is applicable. A Smakula equation for the oscillator strength in terms of the integrated absorption constant is derived. for which the wavelength of light is much greater than a lattice constant. ' In Sec. III the quantum theory of a classical dielectric is developed. This problem reduces to the interaction of the radiation field with a second boson field, the polarization field. Some possible generalizations of the classical dielectric theory are brieQy discussed. Section IV is devoted to constructing an approximate exciton Hamiltonian and to finding the interaction between excitons and photons for this approximate Hamiltonian. It is shown that excitons are approximate bosons and, in interactions with the electromagnetic 6eld, play the role of the quantized polarization field of Sec. III. In Sec. V the interactions which cause true absorption in crystals are introduced. These interactions result in finite lifetimes for the mixed eigenstates constructed in Sec. IV, and can be treated in terms of a complex dielectric constant. The final section is a summary of the theory, with emphasis on the application to experiments. I. INTRODUCTION 'HE lowest energy excited states of insulating crystals are usually states of an electron in the conduction band bound to a hole in the valence band. These bound states, called excitons, were first introduced by Frenkel' in 1931. The existence of such nonconducting excited states has been tentatively verified in both extremes of exciton models, the Frenkel (tight-bonding) model and the Wannier (weak-bonding) model, by means of optical absorption experiments in the visible and near ultraviolet regions. ' The usual method of calculating the optical properties due to exciton states is by use of the semiclassical theory of radiation. This method is satisfactory for the calculation of the dielectric constant in frequency regions of no absorption. The use of this method to treat optical absorption by exciton states raises difhculties perculiar to sets of energy states for which there is but one crystal state having a given wave number k in a 6nite energy interval. The problem of the description of the fundamental absorption process is the subject of Sec. II. This problem was the motivation for the investigation of the exciton-photon system. The purpose of the present work is to formulate the problem of the optical properties of excitons in a more rigorous manner than the semiclassical theory through the use of a quantum-electrodynamical formalism and to present a more complete view of the absorption process. The theory, as developed here, is applicable only to crystals exhibiting optical isotropy. The frequency region considered is limited to frequencies