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arXiv:cond-mat/0004336 · Physical review. B, Condensed matter

Bipolaron binding in quantum wires

Evghenii P. Pokatilov, V. M. Fomin, J. T. Devreese, S. N. Balaban, Serghei N. Klimin

Published Jan 15, 2000• 28 citations
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Abstract

A theory of bipolaron states in quantum wires with a parabolic potential well is developed applying the Feynman variational principle. The basic parameters of the bipolaron ground state (the binding energy, the number of phonons in the bipolaron cloud, the effective mass, and the bipolaron radius) are studied as a function of sizes of the potential well. Two cases are considered in detail: a cylindrical quantum wire and a planar quantum wire. Analytical expressions for the bipolaron parameters are obtained at large and small sizes of the quantum well. It is shown that at $R\ensuremath{\gg}1$ [where $R$ means the radius (half width) of a cylindrical (planar) quantum wire, expressed in Feynman units], the influence of confinement on the bipolaron binding energy is described by the function $\ensuremath{\sim}{1/R}^{2}$ for both cases, while at small sizes this influence is different in each case. In quantum wires, the bipolaron binding energy $W(R)$ increases logarithmically with decreasing radius. The shapes and the sizes of a nanostructure, which are favorable for observation of stable bipolaron states, are determined.

DOI
10.1103/physrevb.61.2721

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Physical review. B, Condensed matter•Oct 1, 1996•13 citations
A theory of bipolaron states in quantum wires with a parabolic potential well is developed applying the Feynman variational principle. The basic parameters of the bipolaron ground state (the binding energy, the number of phonons in the bipolaron cloud, the effective mass, and the bipolaron radius) are studied as a function of sizes of the potential well. Two cases are considered in detail: a cylindrical quantum wire and a planar quantum wire. Analytical expressions for the bipolaron parameters are obtained at large and small sizes of the quantum well. It is shown that at R\ensuremath≫1R\ensuremath{\gg}1R\ensuremath≫1 [where RRR means the radius (half width) of a cylindrical (planar) quantum wire, expressed in Feynman units], the influence of confinement on the bipolaron binding energy is described by the function \ensuremath∼1/R2\ensuremath{\sim}{1/R}^{2}\ensuremath∼1/R2 for both cases, while at small sizes this influence is different in each case. In quantum wires, the bipolaron binding energy W(R)W(R)W(R) increases logarithmically with decreasing radius. The shapes and the sizes of a nanostructure, which are favorable for observation of stable bipolaron states, are determined.
The bipolaron ground state binding energy and the effective masses are calculated self-consistently in a scheme where the electron-phonon interaction is described by the Fröhlich interaction. We explicitly use the total linear momentum conservation and both two-and three-dimensional systems are considered. We review results for binding energies and show that the bipolaron effective mass increases with the electron-phonon coupling constant α more rapidly than two free polaron masses. As expected, the increase is greater in two than in three dimensions. We estimate the screening effects due to an electronic or hole density n in a range of values such that n R 2 b ≪ 1 (here R b is the bipolaron radius). We find that the bipolaron binding energy decreases with n and eventually becomes positive indicating the existence of a metastable bipolaron state. Finally we discuss the possible connections between our results and high T c superconductivity.
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.
The stability of large Fr\"ohlich bipolarons in the presence of a static magnetic field is investigated with the path integral formalism. We find that the application of a magnetic field (characterized by the cyclotron frequency ${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$) favors bipolaron formation: (i) The critical electron-phonon coupling parameter ${\mathrm{\ensuremath{\alpha}}}_{\mathit{c}}$ (above which the bipolaron is stable) decreases with increasing ${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$ and (ii) the critical Coulomb repulsion strength UC{\mathit{U}}_{\mathit{C}}UC​ (below which the bipolaron is stable) increases with increasing ${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$. The binding energy and the corresponding variational parameters are calculated as a function of \ensuremath{\alpha}, U, and ${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$. Analytical results are obtained in various limiting cases. In the limit of strong electron-phonon coupling (\ensuremath{\alpha}\ensuremath{\gg}1) we obtain for ${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$\ensuremath{\ll}1 that Eestim{\mathit{E}}_{\mathrm{estim}}Eestim​\ensuremath{\approxeq}Eestim{\mathit{E}}_{\mathrm{estim}}Eestim​(${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$=0)+c(u)${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$/${\mathrm{\ensuremath{\alpha}}}^{4}$ with c(u) an explicitly calculated constant, dependent on the ratio u=U/\ensuremath{\alpha}, where U is the strength of the Coulomb repulsion. This relation applies both in two dimensions (2D) and in 3D, but with a different expression for c(u). For ${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$\ensuremath{\gg}${\mathrm{\ensuremath{\alpha}}}^{2}$\ensuremath{\gg}1 we find in 3D Eestim{\mathit{E}}_{\mathrm{estim}}Eestim​\ensuremath{\approxeq}${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$-${\mathrm{\ensuremath{\alpha}}}^{2}$A(u)ln2{\mathrm{ln}}^{2}ln2(${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$/${\mathrm{\ensuremath{\alpha}}}^{2}$) [also with an explicit analytical expression for A(u)] whereas in 2D Eestim2D{\mathit{E}}_{\mathrm{estim}}^{2\mathrm{D}}Eestim2D​\ensuremath{\approxeq}${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$-\ensuremath{\alpha}\ensuremath{\surd}${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$\ensuremath{\pi}(u-2-\ensuremath{\surd}2)/2. The validity region of the Feynman-Jensen inequality for the present problem, bipolarons in a magnetic field, remains to be examined. \textcopyright{} 1996 The American Physical Society.