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Physical review. B, Solid state

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

J. T. Devreese, J. DE SITTER, Marc J. Goovaerts

Published Mar 15, 1972• 138 citations
Publisher

Abstract

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 $\ensuremath{\alpha}>1$.

DOI
10.1103/physrevb.5.2367

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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.

Physical Review•Aug 15, 1962•401 citations
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 \ensuremathα\ensuremath→0\ensuremath{\alpha}\ensuremath{\rightarrow}0\ensuremathα\ensuremath→0, where \ensuremathα\ensuremath{\alpha}\ensuremathα 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 \ensuremathα>5\ensuremath{\alpha}>5\ensuremathα>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 \ensuremathα\ensuremath→0\ensuremath{\alpha}\ensuremath{\rightarrow}0\ensuremathα\ensuremath→0 the correct behavior is found. For 3\ensuremath≲\ensuremathα\ensuremath≲53\ensuremath{\lesssim}\ensuremath{\alpha}\ensuremath{\lesssim}53\ensuremath≲\ensuremathα\ensuremath≲5, the interpretation of the results is somewhat delicate but the possibility that RES contribute to the oscillator strength as soon as \ensuremathα>3\ensuremath{\alpha}>3\ensuremathα>3 should be considered. The results so obtained for the optical absorption seem reliable at all \ensuremathα\ensuremath{\alpha}\ensuremathα. This provides an indirect justification for the expansion of Z(\ensuremathΩ)Z(\ensuremath{\Omega})Z(\ensuremathΩ) rather than 1Z(\ensuremathΩ)\frac{1}{Z(\ensuremath{\Omega})}Z(\ensuremathΩ)1​ 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 \ensuremathα>1\ensuremath{\alpha}>1\ensuremathα>1.
In the present work, the problem of an all-coupling analytic description for the optical conductivity of the Fr\"ohlich polaron is treated, with the goal being to bridge the gap in the validity range that exists between two complementary methods: on the one hand, the memory-function formalism and, on the other hand, the strong-coupling expansion based on the Franck-Condon picture for the polaron response. At intermediate coupling, both methods were found to fail as they do not reproduce diagrammatic quantum Monte Carlo results. To resolve this, we modify the memory-function formalism with respect to the Feynman-Hellwarth-Iddings-Platzman approach in order to take into account a nonquadratic interaction in a model system for the polaron. The strong-coupling expansion is extended beyond the adiabatic approximation by including in the treatment nonadiabatic transitions between excited polaron states. The polaron optical conductivity that we obtain at T=0T=0T=0 by combining the two extended methods agrees well, both qualitatively and quantitatively, with the diagrammatic quantum Monte Carlo results in the whole available range of the electron-phonon coupling strength.
The optical absorption of the Fröhlich polaron model is obtained by an approximation-free diagrammatic Monte Carlo method and compared with two new approximate approaches that treat lattice relaxation effects in different ways. We show that: (i) a strong coupling expansion, based on the Franck-Condon principle, well describes the optical conductivity for large coupling strengths (alpha > 10); (ii) a memory function formalism with phonon broadened levels reproduces the optical response for weak coupling strengths (alpha < 6) taking the dynamic lattice relaxation into account. In the coupling regime 6 < alpha < 10, the optical conductivity is a rapidly changing superposition of both Franck-Condon and dynamic contributions.
A free-energy theorem for polarons is derived as an extension to arbitrary lattice temperature of the ground-state theorem derived by Lemmens, deSitter, and and Devreese for zero lattice temperature. These theorems relate the exact free energy of the polaron to the exact absorption spectrum integrated over frequency. With approximate solutions in general, and with approximate solutions to many-body problems in particular, it is of interest to know how closely such sum rules are satisfied. We prove that the free-energy theorem is valid for any Feynman-approximation solution to the polaron problem. Using this analytic result, we are able to determine how sensitively satisfying the free-energy theorem depends upon the details of the absorption spectrum. Considering examples of two quite different spectra at zero temperature, we conclude that the ground-state theorem is of little value in discriminating between different calculated absorption spectra. A more sensitive alternative is discussed which makes use of the fluctuation-dissipation theorem for comparing the calculated absorption with the fluctuation energy at each frequency. Since this alternative method requires equality between two functions for each value of their argument (frequency), rather than merely between two numbers as with sum rules, the method easily distinguishes between spectra, each of which may satisfy the ground-state rule very closely. This method is closely related to a self-consistent procedure developed previously. The latter procedure removed the ambiguity of which approximate influence functional to use in path-integral treatments of polaronlike problems.
We present accurate results for optical conductivity of the three dimensional Fröhlich polaron in all coupling regimes. The systematic-error free diagrammatic quantum Monte Carlo method is employed where the Feynman graphs for the momentum-momentum correlation function in imaginary time are summed up. The real-frequency optical conductivity is obtained by the analytic continuation with stochastic optimization. We compare numerical data with available perturbative and nonperturbative approaches to the optical conductivity and show that the picture of sharp resonances due to relaxed excited states in the strong-coupling regime is "washed out" by large broadening of these states. As a result, the spectrum contains only a single-maximum broad peak with peculiar shape and a shoulder.
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.