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Physical review. B, Condensed matter

Linear energy-momentum relations for acoustic deformation-potential polarons

George B. Norris, George Whitfield

Published Apr 15, 1980• 6 citations
Publisher

Abstract

Several variational theories for the energy-momentum ($\mathrm{EP}$) relation of the acoustic deformation-potential (ADP) polaron are considered. We show that the trial states of these theories need not be translationally invariant to yield upper bounds to the actual $\mathrm{EP}$ relation. A theory which interpolates between weak- and strong-coupling theories is thus shown to give an asymptotically linear upper bound to the ADP polaron $\mathrm{EP}$ relation, as predicted by energy-crossing arguments.

DOI
10.1103/physrevb.21.3658

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The present work is a reply to the paper [1]. It is proven that the argumentation of Refs. [1] is inconsistent. The variational functional for the polaron ground state energy considered in Ref. [1] contains an incomplete recoil energy. Since the variational functional of Ref. [1] is incomplete, it is not proven to provide a variational upper bound for the polaron ground-state energy. The same conclusion follows also for the bipolaron ground-state energy.

arXiv:1208.1166•Solid State Communications•Oct 16, 2012•10 citations
Several variational theories for the energy-momentum (EP\mathrm{EP}EP) relation of the acoustic deformation-potential (ADP) polaron are considered. We show that the trial states of these theories need not be translationally invariant to yield upper bounds to the actual EP\mathrm{EP}EP relation. A theory which interpolates between weak- and strong-coupling theories is thus shown to give an asymptotically linear upper bound to the ADP polaron EP\mathrm{EP}EP relation, as predicted by energy-crossing arguments.
A general class of acoustical-phonon mode induced-polaron systems is considered. The electron-phonon interaction for these systems is described by a Fr\"ohlich-interaction Hamiltonian involving Fourier components proportional to ${q}^{\ensuremath{\gamma}}$, where $\ensuremath{\gamma}$ is an arbitrary real number. The energy versus total momentum ($E\ensuremath{-}P$) relations for these polarons are investigated. At weak coupling, in the context of the intermediate coupling theory, it is found that when $\ensuremath{-}1<\ensuremath{\gamma}\ensuremath{\le}\ensuremath{-}\frac{1}{2}$ a linear structure is present in the $E\ensuremath{-}P$ relations. As $\ensuremath{\gamma}$ decreases from -1/2 to -1, this linear structure becomes apparent only for larger and larger values of PPP. For $\ensuremath{\gamma}>\ensuremath{-}\frac{1}{2}$, this linear structure is not present, but rather the $E\ensuremath{-}P$ relations cut off at a critical value of PPP. When $\ensuremath{-}1<\ensuremath{\gamma}\ensuremath{\le}0$, the polaron mass for these systems grows large as the polaron velocity vvv approaches the speed of sound sss, but for $\ensuremath{\gamma}>0$ the polaron mass has a finite value as $v\ensuremath{\rightarrow}s$. As a function of $\ensuremath{\gamma}$, the self-energy reaches a maximum value for $\ensuremath{\gamma}\ensuremath{\approx}\ensuremath{-}0.73$ and the $v\ensuremath{\rightarrow}0$ polaron mass reaches a minimum value for $\ensuremath{\gamma}\ensuremath{\approx}\ensuremath{-}0.25$, independent of coupling strength. From a strong-coupling theory appropriate for moving polarons it is found that a linear structure in the $E\ensuremath{-}P$ relations is obtained for all $\ensuremath{\gamma}>\ensuremath{-}1$. At both weak and strong coupling it is unclear that the $E\ensuremath{-}P$ relations are meaningful when $\ensuremath{\gamma}\ensuremath{\le}\ensuremath{-}1$.
The ground-state characterization of the polaron problem is retrieved within the framework of a variational scheme proposed previously by Devreese et al for the bound polaron. The formulation is based on the standard canonical transformation of the strong coupling ansatz and consists of a variationally determined perturbative extension serving for the theory to interpolate in the overall range of the coupling constant. Specializing our considerations to the bulk and strict two-dimensional polaron models we see that the theory yields significantly improved energy upper bounds in the strong coupling regime and, moreover, extrapolates itself successfully towards the well-established weak coupling limits for all polaron quantities of general interest.