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Physical Review Letters

Energy-Momentum Relation for Weakly Coupled Optical Polarons

Tianwei Wu, K. Y. Yang

Published Aug 7, 1972• 7 citations
Publisher

Abstract

We have investigated the energy-momentum relationship for optical polarons by taking into consideration the phonon-cross-exchange diagrams. Our result is similar to that of the improved Tamm-Dancoff method, except that the whole curve shifts downward. Therefore, the ground-state energy is lower than the upper bound obtained by the variational method.

DOI
10.1103/physrevlett.29.364

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Journal of Inner Mongolia University for Nationalities•Jan 1, 2003•0 citations
We have investigated the energy-momentum relationship for optical polarons by taking into consideration the phonon-cross-exchange diagrams. Our result is similar to that of the improved Tamm-Dancoff method, except that the whole curve shifts downward. Therefore, the ground-state energy is lower than the upper bound obtained by the variational method.
Using a variational ansatz in which pair correlations between wave vectors of virtually emitted phonons are taken into account, we have obtained better upper bounds to the polaron ground-state energy than have been available heretofore for $\ensuremath{\alpha}\ensuremath{\le}3.5$. The same variational trial functions are also used to obtain lower bounds for $\ensuremath{\alpha}\ensuremath{\le}2.5$. These lower bounds, although not completely rigorous, represent a considerable improvement over the Lieb-Yamazaki values, which are, to the author's knowledge, the only other lower bounds in the literature. The variational ansatz chosen is also suitable for practical calculations of the polaron effective mass.
The optical polaron bounded in a Coulomb potential is investigated within the modified variational scheme of the Lee-Low-Pines theory (1953) which is valid for the whole range of the electron-phonon coupling. The ground state energy, the mean number of phonons in the cloud around the electron and the spatial localisation of the electron for the optical polaron change abruptly at certain values of the electron-phonon coupling constant alpha and of the Coulomb potential beta . The polaron changes its state according to the values of the parameters alpha and beta . Such features are summarised in the phase diagram.
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.
The ground state energy of large polarons is investigated by means of squeezed states. First the authors introduced single-mode squeezed states and used the variational method to calculate this energy. This gives a result that is valid in the weak-coupling regime and only for very small values of the coupling constant. In order to improve the result they have considered two-mode squeezed states in which the correlation between phonons is involved. In this way the ground state energy is found to be lower than the Feynman result in the intermediate-coupling regime for alpha < 3.7.
We present arguments to show that the energy-momentum relation for a polaron is quadratic for small kkk, but bends over and becomes horizontal when the energy approaches the optical-phonon energy above the ground state. We present a calculation for zero temperature based on a thermodynamic Green's-function theory which is especially designed to give a reliable polaron energy-momentum relation.