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arXiv:1204.4142 · Solid State Communications

Comments on “Translation-invariant bipolarons and the problem of high-temperature superconductivity”

Serghei N. Klimin, J. T. Devreese

Published May 21, 2012• 9 citations
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Abstract

We comment on the recent results of Refs. [1], [2] on the bipolaron problem derived using an approximation of Gross–Tulub. It is proved that, contrary to the claim made in Refs. [1], [2], the bipolaron ground-state energy calculated there in the strong-coupling approximation has not been shown to constitute a variational upper bound.
DOI
10.1016/j.ssc.2012.05.013

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Comments are given on the application of the Bogoliubov-Tyablikov approach to the bipolaron problem in a recent paper by Lakhno [Phys. Rev. B 51, 3512 (1995)]. This author believes that his model (1) is the translation-invariant adiabatic theory of bipolarons and (2) gives asymptotically exact solutions in the adiabatic limit while the other approaches are considered as either phenomenological or variational in nature. Numerical results by Lakhno are in contradiction with all other papers published on the subject because his model leads to much lower energies. Thus, the author concludes that bipolarons ``are more stable than was considered before.'' We prove that both the analytical and the numerical results presented by Lakhno are wrong. \textcopyright{} 1996 The American Physical Society.
A detailed consideration is given to the translation-invariant theory of Tulub polaron constructed without the use of Pekar ansatz.A fundamental result of the theory is that the value of the polaron energy is lower than that obtained on the basis of Pekar ansatz which was considered as an asymptotically exact solution in the strong coupling limit.In the case of bipolarons the theory yields the best values of the coupling energy and critical parameters of their stability.Numerous physical consequences of the existence of translation-invariant polarons and bipolarons are discussed.
A detailed consideration is given to the translation-invariant theory of Tulub polaron constructed without the use of Pekar ansatz. A fundamental result of the theory is that the value of the polaron energy is lower than that obtained on the basis of Pekar ansatz which was considered as an asymptotically exact solution in the strong coupling limit. In the case of bipolarons the theory yields the best values of the coupling energy and critical parameters of their stability. Numerous physical consequences of the existence of translation-invariant polarons and bipolarons are discussed.
The energies of bipolaron states in three- and two-dimensional systems are calculated variationally, treating the electron-phonon interaction in the Fr\"ohlich approximation, and separating the relative motion from the center-of-mass motion. The bipolaron is bound if the electron-phonon coupling constant \ensuremath{\alpha} is larger than 6 in three dimensions and 2 in two dimensions, provided the ratio \ensuremath{\eta}=${\mathrm{\ensuremath{\epsilon}}}_{\mathrm{\ensuremath{\infty}}}$/${\mathrm{\ensuremath{\epsilon}}}_{0}$ is smaller than a critical value which depends on \ensuremath{\alpha}. The theory yields the free-polaron energies as given in the Lee, Low, and Pines approximation for arbitrary values of the electron-phonon coupling constant \ensuremath{\alpha}. For \ensuremath{\alpha} larger than \ensuremath{\sim}10 and \ensuremath{\sim}4 in the three- and two-dimensional cases, respectively, the lowest free-polaron self-energies are obtained in the strong-coupling approach; it is shown that a trial harmonic envelope wave function, localized both in the center-of-mass and relative coordinates, with a coherent phonon state gives a bound localized bipolaron state of lower energy. It is interesting for application to high-Tc{\mathit{T}}_{\mathit{c}}Tc​ superconductivity that the bipolaron bounds more easily in two dimensions than in three, and that the mean value of the pair radius is a few angstroms. Furthermore, bipolaron states obtained when the linear total momentum is conserved have intrinsically high mobility, which is also an important condition to make the bipolaron mechanism consistent with high-temperature superconductivity.