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Journal of the Physical Society of Japan

The Optical Absorption by Free Piezoelectric and Anisotropic Optical Polarons

Kazuo Hattori

Published Feb 1, 1975• 4 citations
Publisher

Abstract

The absorption coefficients of free piezoelectric and anisotropic optical polarons are calculated to the first order of the electron-phonon coupling constant, in the framework of the Lee, Low and Pines variational method. In the case of the anisotropic optical polaron, a bare electron is assumed to be on a spheroidal energy surface.
DOI
10.1143/jpsj.38.360

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Physical review. B, Condensed matter•Mar 15, 1985•122 citations
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 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.
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
The optical absorption of a many (continuum) polaron gas is derived in the framework of a variational approach at zero temperature and weak electron-phonon coupling strength. We derive a compact formula for the optical conductivity of the many-polaron system, taking into account many-body effects in the electron or hole system. Within the method presented here, these effects are contained completely in the dynamical structure factor of the electron or hole system. This allows one to build on well-established studies of the interacting electron gas. Based on this approach a feature in the absorption spectrum of the many-polaron gas, related to the emission of a plasmon together with a phonon, is identified. As an application and illustration of the technique, we compare the theoretical many-polaron optical absorption spectrum as derived in the present work with the ‘‘d``d‘‘d-band'' absorption feature in ${\mathrm{Nd}}_{2}{\mathrm{CuO}}_{4\ensuremath{-}\ensuremath{\delta}} (\ensuremath{\delta}<0.004).$ Similarities are shown between the theoretically and the experimentally derived first frequency moment of the optical absorption of a family of differently doped ${\mathrm{Nd}}_{2\ensuremath{-}x}{\mathrm{Ce}}_{x}{\mathrm{CuO}}_{4}$ materials.
The ground-state energy of a two-dimensional (2D) Fr\"ohlich polaron is calculated to second order in the coupling constant (\ensuremath{\alpha}) and gives E/\ensuremath{\Elzxh}${\ensuremath{\omega}}_{s}$=-(\ensuremath{\pi}/2)\ensuremath{\alpha}-0.063 97${\ensuremath{\alpha}}^{2}$ with \ensuremath{\Elzxh}${\ensuremath{\omega}}_{s}$ the surface optical-phonon energy. In the strong-coupling limit the adiabatic approximation is used and E/\ensuremath{\Elzxh}${\ensuremath{\omega}}_{s}$=-0.4047${\ensuremath{\alpha}}^{2}$ is found to leading order in \ensuremath{\alpha}. The Feynman path-integral approximation, the Gaussian approximation, and the modified Lee-Low-Pines unitary transformation approximation to the polaron ground-state energy all satisfy the scaling relation E2D{E}_{2\mathrm{D}}E2D​(\ensuremath{\alpha})=(2/3)E3D{E}_{3\mathrm{D}}E3D​(( 3\ensuremath{\pi}/4)\ensuremath{\alpha}), where E2D{E}_{2\mathrm{D}}E2D​ (E3D{E}_{3\mathrm{D}}E3D​) is the ground-state energy of the 2D (3D) polaron.