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Physical Review

Polaron Mobility at Finite Temperature

David C. Langreth

Published Jul 15, 1967• 79 citations
Publisher

Abstract

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.

DOI
10.1103/physrev.159.717

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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

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Optical Absorption of Polarons in the Feynman-Hellwarth-Iddings-Platzman Approximation

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The optical absorption of polarons at rest at zero temperature is calculated starting from the Feynman-Hellwarth-Iddings-Platzman (FHIP) theory of the impedance. The results are compared with the results of theories whose physical interpretation is clearer [weak-coupling theory of Gurevich, Lang, and Firsov (GLF) and product-ansatz strong-coupling theory of Kartheuser, Evrard, and Devreese (KED)] in order to obtain a better understanding of the FHIP approximation. We are particularly interested in the possible role of lattice relaxation [leading to relaxed excited states (RES)] in the optical absorption process. If the FHIP perturbation method were used to expand the conductivity (this would be the normal procedure), essentially Franck-Condon transitions would be found in the spectrum, and lattice relaxation would be absent. In this case the results do not fit with the product ansatz and provide merely the asymptotic limit $\ensuremath{\alpha}\ensuremath{\rightarrow}0$, where $\ensuremath{\alpha}$ is the electron-phonon coupling constant. If, however, the impedance function rather than the conductivity is expanded (as preferred by FHIP for intuitive reasons, without further justification) more reliable results for the optical absorption appear. For $\ensuremath{\alpha}>5$, intense absorption peaks now occur, which presumably correspond to transitions to RES, and the results are in qualitative agreement with the predictions of the product-ansatz treatment in this coupling region. Also in the limit $\ensuremath{\alpha}\ensuremath{\rightarrow}0$ the correct behavior is found. For $3\ensuremath{\lesssim}\ensuremath{\alpha}\ensuremath{\lesssim}5$, the interpretation of the results is somewhat delicate but the possibility that RES contribute to the oscillator strength as soon as $\ensuremath{\alpha}>3$ should be considered. The results so obtained for the optical absorption seem reliable at all $\ensuremath{\alpha}$. This provides an indirect justification for the expansion of $Z(\ensuremath{\Omega})$ rather than $\frac{1}{Z(\ensuremath{\Omega})}$ in FHIP theory and a confirmation of the qualitative strong-coupling predictions of KED. The present study indicates that optical absorption peaks due to free polarons should be observable experimentally in crystals for which $\ensuremath{\alpha}>1$.

Physical review. B, Solid state•Mar 15, 1972•138 citations
The Kadanoff-Baym transport theory leads to an expression for polaron mobility which becomes exact for low, but finite, temperature, e\ensuremathΘT\ensuremath≫1{e}^{\frac{\ensuremath{\Theta}}{T}}\ensuremath{\gg}1eT\ensuremathΘ​\ensuremath≫1, but not necessarily \ensuremathΘT\ensuremath≫1\frac{\ensuremath{\Theta}}{T}\ensuremath{\gg}1T\ensuremathΘ​\ensuremath≫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.
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
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 Feynman path-integral variational approach to the polaron problem, along with the associated Feynman-Hellwarth-Iddings-Platzman (FHIP) linear-response mobility theory, provides a computationally amenable method to predict the frequency-resolved temperature-dependent charge-carrier mobility, and other experimental observables in polar semiconductors. We show that the FHIP mobility theory predicts non-Drude transport behavior, and shows remarkably good agreement with the recent diagrammatic Monte Carlo mobility simulations of Mishchenko et al. [Phys. Rev. Lett. 123, 076601 (2019)] for the abstract Fr\"ohlich Hamiltonian. We extend this method to multiple phonon modes in the Fr\"ohlich model action. This enables a slightly better variational solution, as inferred from the resulting energy. We carry forward this extra complexity into the mobility theory, which shows a richer structure in the frequency and temperature-dependent mobility, due to the different phonon modes activating at different energies. The method provides a computationally efficient and fully quantitative method of predicting polaron mobility and response in real materials.
How a polaron loses its quasiparticle coherence with increasing temperature is a long-standing open problem. Holstein addressed it in 1959 only in the extreme antiadiabatic, strong-coupling limit, while more recent numerically exact approaches are largely restricted to one dimension. Here we solve this problem on square and simple cubic lattices across the weak-, intermediate-, and strong-coupling regimes. We show that the polaron effective mass m∗m^*m∗ and inverse lifetime 1/τ1/τ1/τ increase monotonically with temperature until, at T∼0.5 ΩT \sim 0.5\,ΩT∼0.5Ω, the quasiparticle peak dissolves into a broad incoherent thermal continuum. The phonon frequency ΩΩΩ therefore defines a universal coherence scale, independent of dimensionality and coupling strength, validating Holstein's prediction far beyond the regime in which it was derived. These results follow from a finite-temperature generalization of the Momentum Average (MA) approximation, yielding a closed-form, diagrammatically derived self-energy that is asymptotically exact in the strong-coupling limit at all temperatures. Benchmark comparisons demonstrate excellent quantitative agreement of the resulting 1D spectral functions with the numerically exact Variational Exact Diagonalization-Finite-Temperature Lanczos Method (VED-FTLM) and finite-TTT Density Matrix Renormalization Group (DMRG).
A method is presented to obtain the linear response coefficients of a system coupled to a bath. The method is based on a systematic truncation of the Liouville equation for the reduced distribution function. The first-order truncation results are expected to be accurate in the low-temperature and weak-coupling regime. Explicit expressions for the conductivity of the Fröhlich polaron are obtained, and the discrepancy between the Kadanoff and the Feynman-Hellwarth-Iddings-Platzman mobility is elucidated.
The optical absorption of polarons at rest at zero temperature is calculated starting from the Feynman-Hellwarth-Iddings-Platzman (FHIP) theory of the impedance. The results are compared with the results of theories whose physical interpretation is clearer [weak-coupling theory of Gurevich, Lang, and Firsov (GLF) and product-ansatz strong-coupling theory of Kartheuser, Evrard, and Devreese (KED)] in order to obtain a better understanding of the FHIP approximation. We are particularly interested in the possible role of lattice relaxation [leading to relaxed excited states (RES)] in the optical absorption process. If the FHIP perturbation method were used to expand the conductivity (this would be the normal procedure), essentially Franck-Condon transitions would be found in the spectrum, and lattice relaxation would be absent. In this case the results do not fit with the product ansatz and provide merely the asymptotic limit $\ensuremath{\alpha}\ensuremath{\rightarrow}0$, where $\ensuremath{\alpha}$ is the electron-phonon coupling constant. If, however, the impedance function rather than the conductivity is expanded (as preferred by FHIP for intuitive reasons, without further justification) more reliable results for the optical absorption appear. For $\ensuremath{\alpha}>5$, intense absorption peaks now occur, which presumably correspond to transitions to RES, and the results are in qualitative agreement with the predictions of the product-ansatz treatment in this coupling region. Also in the limit $\ensuremath{\alpha}\ensuremath{\rightarrow}0$ the correct behavior is found. For $3\ensuremath{\lesssim}\ensuremath{\alpha}\ensuremath{\lesssim}5$, the interpretation of the results is somewhat delicate but the possibility that RES contribute to the oscillator strength as soon as $\ensuremath{\alpha}>3$ should be considered. The results so obtained for the optical absorption seem reliable at all $\ensuremath{\alpha}$. This provides an indirect justification for the expansion of $Z(\ensuremath{\Omega})$ rather than $\frac{1}{Z(\ensuremath{\Omega})}$ in FHIP theory and a confirmation of the qualitative strong-coupling predictions of KED. The present study indicates that optical absorption peaks due to free polarons should be observable experimentally in crystals for which $\ensuremath{\alpha}>1$.