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Physical review. B, Solid state

Self-consistent equation-of-motion approach for polarons

J. T. Devreese, R. Evrard, E. Kartheuser

Published Oct 15, 1975• 22 citations
Publisher

Abstract

The polaron problem is treated in a self-consistent manner. The treatment is based on Heisenberg's equations of motion starting from a trial expression for the electron position which includes only one real phonon but any number of virtual phonons. Numerical results for the polaron effective mass and the optical-absorption coefficient are given for electron-LO-phonon coupling strengths ranging from 0 to 4.5. The results are discussed and compared with those obtained by Feynman and by Lee, Low, and Pines.

DOI
10.1103/physrevb.12.3353

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北海道大學工學部研究報告 = Bulletin of the Faculty of Engineering, Hokkaido University•Mar 25, 1986•0 citations
The polaron problem is treated in a self-consistent manner. The treatment is based on Heisenberg's equations of motion starting from a trial expression for the electron position which includes only one real phonon but any number of virtual phonons. Numerical results for the polaron effective mass and the optical-absorption coefficient are given for electron-LO-phonon coupling strengths ranging from 0 to 4.5. The results are discussed and compared with those obtained by Feynman and by Lee, Low, and Pines.
We have calculated the polaron effective mass m∗{m}^{*}m∗ by extending the variational ansatz of a previous paper, where, in the Lee-Low-Pines representation, pair correlations between wave vectors of virtually emitted phonons are taken into account. Our results suggest that (1) effective masses calculated by fourth-order perturbation theory lie below the true values but are quite accurate for $\ensuremath{\alpha}<~3$ and (2) the Feynman theory gives effective masses which are somewhat too high in the intermediate-coupling region. For $\ensuremath{\alpha}\stackrel{\ensuremath{\sim}}{<}4.5$ we propose the estimate $\frac{{m}^{*}}{m}=1+\frac{\ensuremath{\alpha}}{6}+0.02363{\ensuremath{\alpha}}^{2}+0.0014{\ensuremath{\alpha}}^{3}$.
A concise approach for static polarons is extended to study the dynamic polaron system having nonspherical symmetry. The general properties of moving polarons, such as the ground-state energy, the phonon number, the effective mass, and the density distribution of phonon momentum---studied as functions of the total polaron momentum and electron-phonon coupling strength---are calculated numerically in a wide range of intermediate coupling strengths. In particular, a modified Fr\"ohlich Hamiltonian is derived for the problem of polaron momentum Q\ensuremath{\geqslant}1. \textcopyright{} 1996 The American Physical Society.
An approximate calculation is made of the polaron self-energy and effective mass as defined by Frohlich.The approach taken is a variational one based on an ex tended version of Hohler's ansatz in which the trial state function employed is an exact eigenfunction of the total wave vector of the polaron.The effective mass obtained simulates the Feynmann-Schultz result while the corresponding polaron self-energy is fairly accurate but inferior to the result of Feynman and Schultz.CONTENTS Chapters « -« T -| > £ ( l ^' (1'2) n in which n is the optical refractive index, k is the static dielectric constant, e is the electronic charge and 2 t\& is Plank's constant.The limit S->°° is to be taken with lim I = -~ S-*oo v 8tt d 3v .(1,3) A natural unit system is employed in which 2^=o)=2m=l.The first term in the Hamiltonian is the quantum mechanical operator corresponding to the kinetic energy of the electron.The second term represents the energy of the polarization field resulting from the long wave length longitudinal optical crystal modes in the absence of interaction with the electron.The last term represents the energy of interaction between the electron and the phonons.The name "polaron" has been given to the entity consisting of the Bloch conduction electron together with its accompanying nonradiative polarization field.Accordingly Frohlich defines the polaron self-energy, EQ (a), and the (ground-state) polaron effective mass, y(a), by the expansion EQ (a,ic) = Ec (a) + k 2/y(a) + 0(k4 ), (1,4)
The polaron band theory of Lee, Low and Pines is extended beyond the effective mass approximation to a general form of the electron–phonon interaction within two different self-consistent schemes. In the first one a variational Bloch state depending on the polaron crystalline momentum is used to study polaron features taking into account umklapp processes. In particular, the self-energy and the effective mass at the Γ point of the Brillouin zone are calculated as functions of the electron–phonon coupling constant α. It is found that a sudden transition of these quantities from a large band to a narrow band polaron behaviour occurs at a critical value αc. Such value increases indefinitely with the bare electron band width; moreover the transition becomes sharper. The theory recovers, as particular cases, a) the Lee-Low-Pines regime valid for small α and large electron band width Δ˜(Δ˜ ≫ ħω and b) the Holstein limit for large α and small band width (Δ˜ ≫ ħω). This occurs as result of the competing role of α and Δ˜. In the second scheme, the electron–phonon interaction is considered only in a single elementary cell and the self-trapped polaron state is calculated self-consistently. It is shown that, if only one electronic band is taken into account, such state becomes a Δ function with infinitely negative energy when α is larger than a critical value depending on the electron band width. To avoid this fact, electronic band states of higher energy must be considered. Finally, through the tight-binding method, the polaron band is set up. It is shown that at the Γ point of the Brillouin zone, the contribution of the higher electronic bands prevents the self-energy and the polaron mass to have a sharp increase from the large to the narrow band regime.
The problem of an electron interacting wlth acouseic as well as with optical modes of lattice vibrations is investlgated wlthin the modjfjed variatioRal scheme of the Lee-Low -Pines theory． lt is shown that the po｝aron changes its state according to the values of the electroR-optical phonon and ￡he electron-acoustic phonoR coupling constants by examining the ground-state energy， the mean numbers of the optical and acoustic phonons iR the cloud around the e｝ectron and the spatlal localization of the electron as fuRctions of the coupling