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arXiv:1203.5836 · Physical Review B

Trapping of three-dimensional Holstein polarons by various impurities

Hadi Ebrahimnejad, Mona Berciu

Published Apr 11, 2012• 34 citations
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Abstract

We study the bound states of a three-dimensional Holstein polaron near various kinds of single impurities, using the momentum-average approximation. We show that the electron-phonon coupling is responsible for a strong renormalization of the impurity potential, resulting in an effective potential with significant retardation effects, which describes essential physics ignored by ``instantaneous'' approximations. The accuracy of our approximation is gauged by comparison with results from diagrammatic Monte Carlo for the case of an impurity that modifies the on-site energy of the electron. We also discuss impurities that modify the local strength of the electron-phonon coupling as well as isotope substitutions that change both the electron-phonon coupling and the phonon frequency, and contrast and highlight the difference between these cases.

DOI
10.1103/physrevb.85.165117

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The Physics of Metals and Metallography•Feb 1, 2011•0 citations
We study the bound states of a three-dimensional Holstein polaron near various kinds of single impurities, using the momentum-average approximation. We show that the electron-phonon coupling is responsible for a strong renormalization of the impurity potential, resulting in an effective potential with significant retardation effects, which describes essential physics ignored by ``instantaneous'' approximations. The accuracy of our approximation is gauged by comparison with results from diagrammatic Monte Carlo for the case of an impurity that modifies the on-site energy of the electron. We also discuss impurities that modify the local strength of the electron-phonon coupling as well as isotope substitutions that change both the electron-phonon coupling and the phonon frequency, and contrast and highlight the difference between these cases.
Non-local, inhomogeneous and retarded response similar to that observed in experiments is studied theoretically by introducing the Inhomogeneous Momentum Average (IMA) approximation for single-polaron problems with disorder in the on-site potential and/or spatial variations of the electron-phonon couplings and/or phonon frequencies. We show that the electron-phonon coupling gives rise to an additional inhomogeneous, strongly retarded potential. This potential describes essential physics ignored by "instanteneous" approximations. The accuracy of IMA is demonstrated by comparison with single-impurity results from the approximation-free Diagrammatic Monte Carlo (DMC) method. Its simplicity allows for easy study of many problems that were previously unaccessible.
We solve the problem of polaron localization on an attractive impurity by means of direct-space diagrammatic Monte Carlo implemented for the system in the thermodynamic limit. In particular we determine the ground-state phase diagram in dependence on the electron-phonon coupling and impurity potential strength for the whole phonon frequency range. Including the quantum phonon effects we find and characterize a phase where self-trapped polarons are not localized at shallow impurities, which is missing in the adiabatic approximation. We show that near the localization transition a region with a mixture of weak- and strong-coupling spectral responses is realized.
Using the momentum average (MA) approximation, we find an analytical expression for the disorder-averaged Green's function of a Holstein polaron in a three-dimensional simple cubic lattice with random on-site energies. The on-site disorder is assumed to be weak compared to the kinetic energy of the polaron, and is treated perturbationally. Within this scheme, the states at the bottom of the polaron band are found to have an infinite lifetime, signaling a failure of perturbation theory at these energies. The higher-energy polaron states have a finite lifetime. We study this lifetime and the disorder-induced energy shift of these eigenstates for various strengths of disorder and electron-phonon coupling. We compare our findings to the predictions of Fermi's golden rule and the average TTT-matrix method, and find a significant quantitative discrepancy at strong electron-phonon coupling, where the polaron lifetime is much shorter than Fermi's golden rule prediction. We attribute this to the renormalization of the on-site potential by the electron-phonon coupling.
We consider the effects of impurities on polarons in three dimensions using a continuous time quantum Monte Carlo algorithm. An exact treatment of the phonon degrees of freedom leads to a very efficient algorithm, and we are able to compute the polaron dynamics on an infinite lattice by developing an auxiliary weighting scheme. The magnitude of the impurity potential, the electron-phonon coupling, and the phonon frequency are varied. We determine the magnitude of the impurity potential required for polaron trapping. For small electron-phonon coupling the number of phonons increases dramatically on trapping combined with a sharp decrease in kinetic energy. The polaron binding diagram is computed, showing that intermediate-coupling low-phonon-frequency polarons are localized by exceptionally small impurities.
In most cases, as the strength of electron-phonon coupling increases, the effective mass of polarons typically increases. However, in this paper, we uncover a fascinating phenomenon: the presence of light polarons even within the strong coupling regime, where electron-phonon coupling includes both Holstein and bond types (electron-phonon coupling occurring on the hopping bonds). To investigate this, we employ a diagrammatic Monte Carlo method based on the path-integral formulation of the particle sector and the Fock-state representation of the phonon sector. Our study centers on analyzing the impact of bond electron-phonon coupling on the Holstein polaron's essential properties, including its effective mass, ground-state energy, and the average number of phonons. We examine two distinct scenarios: one where the phonon frequency of both Holstein and bond types is the same, and another where the phonon frequency of the Holstein type is twice as large as that of the bond type. In both of these cases, when the Holstein coupling falls within the light mass regime, we observe minimal changes in the effective mass as a function of bond coupling, compared to the case of the bare bond polaron. However, the behavior of the effective mass undergoes a significant change in the heavy mass regime. Most intriguingly, we discover a nonmonotonic dependence of the effective mass on the bond coupling gB{g}_{B}gB​ when the Holstein coupling gH{g}_{H}gH​ falls within the heavy mass regime. This finding holds promise, particularly in the context of bipolarons, where achieving a light effective mass and a large binding energy to get a compact size is crucial.
The Hubbard model augmented by the Holstein interaction of localized electrons and optical dispersionless phonons is reduced (with the help of a canonical transformation) to a model, in which there occur jumps of polarons and scattering on “impurity” scatterers, which are accompanied by jumps and scatterings of an unlimited number of phonons. This circumstance and the fact that there is a strong on-site interaction of electrons inherent in the Hubbard model, lead to the need for the formulation of a new theory of the phenomenon and the use of a diagrammatic technique. A theory of perturbations in disorder for electrons moving in the random field of “impurity” scatterers has been developed. A collective mode of the phonon cloud surrounding a polaron has been found and the influence of the processes of scattering of this mode on the energy spectrum of both localized and delocalized polarons has been established. The motion of the delocalized polarons related to their scattering is described in the approximation of chain diagrams, in which the manyparticle irreducible Green’s functions of the polaron are ignored. The electron-energy spectrum obtained consists of four subbands.