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Progress of Theoretical Physics

Theory of Polaron Mobility

Yukio Ôsaka

Published Apr 1, 1961• 5 citations
OpenAlex

Abstract

The polaron mobility is calculated by making use of the general theory of electrical conductivity. We take the states determined by Feynman's trial action as the unperturbed states and treat the difference between the true action and the trial action as a perturbation. Numerical values of the polaron mobility at very .low temperatures are given and are dis· cussed in comparison with the results obtained by Shultz and .by Morita. § I. Introduction We have investigated the static properties of polaron at finite temperatures in the previous paper1> (hereafter referred to as I). We shall investigB;te the polaron mobility in the present paper. In the same way as in I, we calculate the polaron mobility by using Feynman's path-integral method. In the polaron problem, the interaction between an electron and lattice vibrations is so large that the usual perturbation theoretic treatment fails. Therefore the calculation of the mobility in the present paper will be based on the general theory of electrical conductivity recently developed by many authors,2 > which is applicable to a system which does not allow us to set up the Boltzmann equation. We take the states determined by Feynman's trial action as unperturbed states and treat the difference between the trial action and the true action as a perturba­ tion which yields a decay of electronic current correlation. The following assumptions will be made in the present paper. i) The electrical conductivity is determined by the asymptotic form of a correlation function of electronic current. ii) This asymptotic form has the property of exponential decay in time. These assumptions are valid in the case of weak interaction. It is not sure whether or not they are applicable to the present problem which includes the case of strong interaction. In the present treat­ ment, however, ~he states determined by Feynman's trial action instead of the states of a free electron are chosen as the unperturbed states. It is expected that the perturbation in the former case may be made weaker than in the latter case and that the above mentioned assumptions may be valid. In fact the self­ energy of polaron at 0°K calculated by using these assumptions agrees with that obtained by Feynman3> as will be shown _in § 3. Therefore, the approxi

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The Kadanoff-Baym transport theory leads to an expression for polaron mobility which becomes exact for low, but finite, temperature, ${e}^{\frac{\ensuremath{\Theta}}{T}}\ensuremath{\gg}1$, but not necessarily $\frac{\ensuremath{\Theta}}{T}\ensuremath{\gg}1$. This expression is evaluated exactly in the weak-coupling limit. Comparison with previous variational calculations shows that the latter have not predicted even qualitatively valid corrections to the known asymptotic zero-temperature result. Simple arguments allow the present weak-coupling expression to be extended to finite coupling. These arguments are verified by a quantitative calculation using the Feynman model. It is found that the corrections due to finite temperature are at least as important as those due to a finite coupling constant.

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The polaron mobility is calculated by making use of the general theory of electrical conductivity. We take the states determined by Feynman's trial action as the unperturbed states and treat the difference between the true action and the trial action as a perturbation. Numerical values of the polaron mobility at very .low temperatures are given and are dis· cussed in comparison with the results obtained by Shultz and .by Morita. § I. Introduction We have investigated the static properties of polaron at finite temperatures in the previous paper1> (hereafter referred to as I). We shall investigB;te the polaron mobility in the present paper. In the same way as in I, we calculate the polaron mobility by using Feynman's path-integral method. In the polaron problem, the interaction between an electron and lattice vibrations is so large that the usual perturbation theoretic treatment fails. Therefore the calculation of the mobility in the present paper will be based on the general theory of electrical conductivity recently developed by many authors,2 > which is applicable to a system which does not allow us to set up the Boltzmann equation. We take the states determined by Feynman's trial action as unperturbed states and treat the difference between the trial action and the true action as a perturba­ tion which yields a decay of electronic current correlation. The following assumptions will be made in the present paper. i) The electrical conductivity is determined by the asymptotic form of a correlation function of electronic current. ii) This asymptotic form has the property of exponential decay in time. These assumptions are valid in the case of weak interaction. It is not sure whether or not they are applicable to the present problem which includes the case of strong interaction. In the present treat­ ment, however, ~he states determined by Feynman's trial action instead of the states of a free electron are chosen as the unperturbed states. It is expected that the perturbation in the former case may be made weaker than in the latter case and that the above mentioned assumptions may be valid. In fact the self­ energy of polaron at 0°K calculated by using these assumptions agrees with that obtained by Feynman3> as will be shown _in § 3. Therefore, the approxi
It is well known that two qualitatively different pictures arise in the description of the motion of an electron in an ionic crystal. In the first case, when the electron interacts weakly with the ionic lattice of the crystal, its motion is the same as the motion of a free band electron whose energy is shifted downward with respect to the bottom of the conduction band, and the effective mass is replaced by the renormalized mass (weak-coupling polaron). In the other limiting case, when the electron interacts strongly with the ionic crystal, an entire series of different self-consistent states of the electron and lattice, each of which has its own effective mass and radius (strong-coupling polaron), arises. Although the motion of an electron in an ionic crystal is described in both limiting cases with the help of the Pekar-Frohlich Hamiltonian, each case requires its own special methods of analysis. Thus, for example, in the strong-coupling theory a special form of adiabatic perturbation theory has been developed in which the trans-lational degeneracy is removed before the perturbation expansion is made. It has been possible to relate both descriptions within the framework of Feynman's varia-tional approach, in which a smooth upper bound was obtained for the energy of the ground state of the polaron for all values of the electron-phonon coupling constant a (see Fig. 1). We call attention in this connection to Ref. 6, where the thermodynamic properties are calculated with the help of a special method for performing the calculations and for finding averages of T-products. The approach developed there is more convenient for performing specific calculations than is the use of path integrals. It seems to us that it has definite advantages in the construction of general proofs concerning the choice of the approximating Hamiltonians (this is illustrated in Ref. 6 for the example of the linear Bogolyubov model), in the investigation of the characteristics of the behavior of a polaron in external fields, etc.
We have obtained an approximate expression for the impedance function at all frequencies, temperatures, and coupling strengths of an electron coupled to a polar lattice (a system commonly called a polaron). The starting point for the calculation is the quantum mechanical expression for the expected current. The phonon coordinates are eliminated from this expression by well-known field-theory techniques. The resulting exact "influence functional" is then approximated by a corresponding quadratic "influence functional" which, it is hoped, imitates the real polaron. Correction terms are computed to account for the difference between the approximate impedance and the exact polaron impedance in a manner closely analogous to Feynman's treatment of the polaron self-energy. In fact, the analytic evaluation of the expression for the impedance obtained here is carried out using the approximate "influence functional" that was successfully employed in minimizing the binding (and free) energy of the polaron in earlier calculations. However, the accuracy obtained using this approximation, for the impedance calculation, is less satisfactory and its limitations are discussed. Nevertheless, beginning at intermediate coupling strengths, the approximate impedance produces a level structure of increasing complexity and narrowing resonances as the coupling strengthens. This suggests that further refinements may be fruitful. Methods for finding a better quadratic influence functional for use in our impedance expression as well as ways of improving the expression further are suggested. A comparison of our results with those of the Boltzmann equation points up interesting differences which arise from reversing the order of taking limits of zero frequency and coupling.
The free energy of an ideal polaron gas in a static magnetic field is calculated using Feynman's path-integral formalism. The trial action, used in Feynman's polaron theory, is extended to take into account the anisotropy of the effective electron-phonon interaction. This results in a free-energy expression with four variational parameters. According to Feynman's conjecture, the resulting free energy is an upper bound to the exact result. The approximate free energy is expected to provide accurate results for arbitrary electron-phonon coupling strength ($\ensuremath{\alpha}$), temperature (TTT), and magnetic field strength (H\mathcal{H}H). The free energy per polaron is evaluated for limiting values of $\ensuremath{\alpha}$, TTT, and H\mathcal{H}H.
Direct evaluation of Feynman's functional integral for the polaron maps it into a standard type of many-body theory. The perturbation expansion is shown to be well controlled and does not require any renormalization. This perturbation expansion has the pleasing feature that nthn\mathrm{th}nth-order terms in the polaron coupling constant are obtained from nthn\mathrm{th}nth-order perturbation theory, whereas the traditional perturbation formulation in terms of creation and annihilation operators requires a 2nth2n\mathrm{th}2nth-order perturbation calculation to generate the same results. In contrast to previous beliefs, the method is shown to enable the calculation of the polaron effective mass and relaxation time directly from the functional-integral expression. The ground-state energy is computed to second order in the polaron coupling constant to illustrate the use of the method and to establish agreement with previous perturbation treatments. In addition, we provide the first perturbation calculation of the polaron effective mass through second order in the coupling constant. Pad\'e approximants are shown to provide a vehicle for analytically extending the perturbation results into the region of intermediate and strong couplings, and the procedure yields reasonable agreement with variational calculations for values of the polaron coupling constant up to about $\ensuremath{\alpha}=10$. In addition, conditions are derived under which the original polaron action, used in the functional integral, reduces to some of the best variational trial actions utilized in the literature.
We developed a path-integral quantum Monte Carlo-based methodology for calculation of polaron mobility in systems with electron-phonon interaction. Within the method, the current-current correlation function in both the imaginary and real time is calculated in a numerically exact way. The choice of basis for representation of the path integral enabled us to reduce the sign problem and perform real-time calculations for longer times. The DC polaron mobility was extracted by performing analytic continuation that makes use of both the real and imaginary-time data. The method was applied to the Holstein polaron model in one dimension. We obtained reliable results for the temperature dependence of the Holstein polaron mobility for interactions ranging from weak to strong and temperatures that are not too low.
The Kadanoff-Baym transport theory leads to an expression for polaron mobility which becomes exact for low, but finite, temperature, ${e}^{\frac{\ensuremath{\Theta}}{T}}\ensuremath{\gg}1$, but not necessarily $\frac{\ensuremath{\Theta}}{T}\ensuremath{\gg}1$. This expression is evaluated exactly in the weak-coupling limit. Comparison with previous variational calculations shows that the latter have not predicted even qualitatively valid corrections to the known asymptotic zero-temperature result. Simple arguments allow the present weak-coupling expression to be extended to finite coupling. These arguments are verified by a quantitative calculation using the Feynman model. It is found that the corrections due to finite temperature are at least as important as those due to a finite coupling constant.