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Journal of Physics C Solid State Physics

A variational description of the free and localised states of the acoustic deformation potential polaron

A. Erçelebi

Published Mar 30, 1988• 1 citation
Publisher

Abstract

The ground-state property of the acoustic deformation potential polaron is retrieved variationally in the overall range of the coupling constant. The calculation is based on an appropriate extension of the strong-coupling theory and yields a reasonable description of the free and self-localised states of the coupled electron-lattice system.

DOI
10.1088/0022-3719/21/9/015

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The ground-state property of the acoustic deformation potential polaron is retrieved variationally in the overall range of the coupling constant. The calculation is based on an appropriate extension of the strong-coupling theory and yields a reasonable description of the free and self-localised states of the coupled electron-lattice system.
The ground-state characterization of the polaron problem is retrieved within the framework of a variational scheme proposed previously by Devreese et al for the bound polaron. The formulation is based on the standard canonical transformation of the strong coupling ansatz and consists of a variationally determined perturbative extension serving for the theory to interpolate in the overall range of the coupling constant. Specializing our considerations to the bulk and strict two-dimensional polaron models we see that the theory yields significantly improved energy upper bounds in the strong coupling regime and, moreover, extrapolates itself successfully towards the well-established weak coupling limits for all polaron quantities of general interest.
Several variational theories for the energy-momentum (EP\mathrm{EP}EP) relation of the acoustic deformation-potential (ADP) polaron are considered. We show that the trial states of these theories need not be translationally invariant to yield upper bounds to the actual EP\mathrm{EP}EP relation. A theory which interpolates between weak- and strong-coupling theories is thus shown to give an asymptotically linear upper bound to the ADP polaron EP\mathrm{EP}EP relation, as predicted by energy-crossing arguments.
It is well known that two qualitatively different pictures arise in the description of the motion of an electron in an ionic crystal. In the first case, when the electron interacts weakly with the ionic lattice of the crystal, its motion is the same as the motion of a free band electron whose energy is shifted downward with respect to the bottom of the conduction band, and the effective mass is replaced by the renormalized mass (weak-coupling polaron). In the other limiting case, when the electron interacts strongly with the ionic crystal, an entire series of different self-consistent states of the electron and lattice, each of which has its own effective mass and radius (strong-coupling polaron), arises. Although the motion of an electron in an ionic crystal is described in both limiting cases with the help of the Pekar-Frohlich Hamiltonian, each case requires its own special methods of analysis. Thus, for example, in the strong-coupling theory a special form of adiabatic perturbation theory has been developed in which the trans-lational degeneracy is removed before the perturbation expansion is made. It has been possible to relate both descriptions within the framework of Feynman's varia-tional approach, in which a smooth upper bound was obtained for the energy of the ground state of the polaron for all values of the electron-phonon coupling constant a (see Fig. 1). We call attention in this connection to Ref. 6, where the thermodynamic properties are calculated with the help of a special method for performing the calculations and for finding averages of T-products. The approach developed there is more convenient for performing specific calculations than is the use of path integrals. It seems to us that it has definite advantages in the construction of general proofs concerning the choice of the approximating Hamiltonians (this is illustrated in Ref. 6 for the example of the linear Bogolyubov model), in the investigation of the characteristics of the behavior of a polaron in external fields, etc.