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Physical review. B, Condensed matter

Consistent definition of the effective mass of the polaron

Virulh Sa‐Yakanit, K. Tayanasanti

Published Apr 15, 1998• 5 citations
Publisher

Abstract

The approximated density matrix of the polaron system obtained in our previous paper for deriving the Feynman effective mass ${m}_{F}$ and the Krivoglaz-Pekar mass ${m}_{\mathrm{KP}}$ is reconsidered. It is shown that to obtain a consistent definition of the polaron effective mass one must impose the condition ${m}_{F}{=m}_{\mathrm{KP}}.$ This consistent definition is based on the fact that the wave functions derived from the density matrix corresponding to each excited state must be normalized and orthogonal. This condition leads to a consistent definition of the effective mass and a different variation principle for treating excited states. Numerical results are presented.

DOI
10.1103/physrevb.57.8739

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The approximated density matrix of the polaron system obtained in our previous paper for deriving the Feynman effective mass mF{m}_{F}mF​ and the Krivoglaz-Pekar mass mKP{m}_{\mathrm{KP}}mKP​ is reconsidered. It is shown that to obtain a consistent definition of the polaron effective mass one must impose the condition mF=mKP.{m}_{F}{=m}_{\mathrm{KP}}.mF​=mKP​. This consistent definition is based on the fact that the wave functions derived from the density matrix corresponding to each excited state must be normalized and orthogonal. This condition leads to a consistent definition of the effective mass and a different variation principle for treating excited states. Numerical results are presented.
The polaron effective mass using the path integral approach was first introduced by Feynman. He has shown that from the partition function, he obtained the ground state energy and the effective mass mF of the polaron. By using the partition function, Krivoglaz and Pekar proposed another definition of the effective mass mKP. Sa-yakanit has shown that using the density matrix Feynmam's path integral representation, it is possible to obtain both Feynman's and Krivoglaz and Pekar's results. In this thesis we propose a consistent definition of effective mass that mF and mKP should be equal. We have shown that the wave functions obtained from the density matrix should be normalized. We have also shown that the dynamic mass of the polaron can be obtained from the density matrix. The comparison of our results with recent work by Wang et.al.are given.
We calculate the density matrix elements of the polaron using Feynman's variational method. The density matrix is diagonalized, and the eigenvalues and eigenfunctions are derived. These results and the calculated density matrix are used to evaluate the root-mean-square (rms) displacements ${R}_{\ensuremath{\beta}}$ and the effective mass (EM) m∗{m}^{*}m∗ for various values of coupling strengths between electron and medium and of the temperature. We find that EM is related to the rms displacement. The temperature dependence of the EM is in qualitative agreement with experiment.
Using the exactly soluble Feynman polaron as a point of departure, I discuss and compare the properties of three projected, or momentum dependent, partition functions of a particle coupled to an environment and reconsider the anomalies that appear in some definitions of the effective mass. Although the momentum-dependent free energy associated with the projected partition function at fixed total momentum has exponentially suppressed finite-temperature corrections and yields the correct zero-temperature effective mass, the same is not necessarily true of the free energies associated with the simpler momentum-dependent partition functions derived from either the reduced density matrix of the particle or the density matrixlike quantity obtained by replacing in the particle's nonlocal effective action the phonon Green's function with its zero-temperature form.
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)