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physica status solidi (a)

Energy of a polaron bound to a Coulomb potential

Ward Huybrechts, G. Remaut

Published Jun 16, 1980• 6 citations
Publisher

Abstract

A variational calculation of the ground state and of the first excited state of a bound polaron is given for all values of the electron-phonon coupling and of the Coulomb potential. The applied unitary transformation formalism is adapted in order to take account of correlation between the longitudinal optical phonons.
DOI
10.1002/pssa.2210590216

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Based on the unitary transformation, a general treatment of the bound polaron is proposed. In distinction from earlier calculations, the present method gives explicit expressions for not only the energy but also the wave functions both in the ground state and in the excited states such as the relaxed excited state and the phonon scattering state. These expressions involve a summation on the complete set of eigenstates of a trial Hamiltonian, and in actual calculations the Coulomb Hamiltonian and the harmonic oscillator Hamiltonian are chosen as trial Hamiltonians. For the ground state, the energy expression is the same as that obtained by Matsuura with the path integral method. The present method is valid in the whole region of the electron-phonon coupling constant α and Coulomb force β except for β→0 and α>6.
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 energy spectrum of a polaron---an electron interacting with the longitudinal-optical phonons of a polar crystal---is studied with particular attention given to the region where the polaron excitation energy closely approaches the energy of a longitudinal-optical phonon. In this region, existing theories of the polaron spectrum are inadequate; in particular, the usual Rayleigh-Schr\"odinger perturbation theory is shown to be inconsistent. A self-consistent weak-coupling theory is developed, and a variational theory of the polaron spectrum which, for small coupling, reduces to this weak-coupling theory is presented. Whitfield and Puff, and Schultz have conjectured that the polaron energy EEpRE\mathrm{E}p\mathrm{R}EEpR, bends over and becomes horizontal as the polaron momentum ppp approaches from below the value at which the polaron excitation energy, $E(p)\ensuremath{-}E(0)$, becomes equal to the optical-phonon energy. Using the weak-coupling method of the present paper, this conjecture has been verified to lowest order and next higher order in the coupling constant.
Abstract The properties of the ground state of the bound large polaron within the Fock approximation are investigated by using a new model spectrum which is known to describe better the polaron at low external fields. The results are compared with those of Matsuura who used the same approximation, obtained from a path integral formulation, with the usual Gaussian and Coulombic trial spectra, Both formulations are variational. It is found that the new spectrum gives better results (lower variational energies) than those of Matsuura for intermediate or large values of electron–phonon coupling and small values of the electrostatic coupling of the electron to the defect. This is related to that fact that the polaron seems to prefer a cylindrical symmetry to a spherical one in the absence of external fields.
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.