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Physical review. B, Condensed matter

Stability of bipolarons in the presence of a magnetic field

F. Brosens, J. T. Devreese

Published Oct 1, 1996• 13 citations
Publisher

Abstract

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 ${\mathit{U}}_{\mathit{C}}$ (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 ${\mathit{E}}_{\mathrm{estim}}$\ensuremath{\approxeq}${\mathit{E}}_{\mathrm{estim}}$(${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$=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 ${\mathit{E}}_{\mathrm{estim}}$\ensuremath{\approxeq}${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$-${\mathrm{\ensuremath{\alpha}}}^{2}$A(u)${\mathrm{ln}}^{2}$(${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$/${\mathrm{\ensuremath{\alpha}}}^{2}$) [also with an explicit analytical expression for A(u)] whereas in 2D ${\mathit{E}}_{\mathrm{estim}}^{2\mathrm{D}}$\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.

DOI
10.1103/physrevb.54.9792

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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 \ensuremathωc{\mathrm{\ensuremath{\omega}}}_{\mathit{c}}\ensuremathωc​) favors bipolaron formation: (i) The critical electron-phonon coupling parameter \ensuremathαc{\mathrm{\ensuremath{\alpha}}}_{\mathit{c}}\ensuremathαc​ (above which the bipolaron is stable) decreases with increasing \ensuremathωc{\mathrm{\ensuremath{\omega}}}_{\mathit{c}}\ensuremathωc​ and (ii) the critical Coulomb repulsion strength UC{\mathit{U}}_{\mathit{C}}UC​ (below which the bipolaron is stable) increases with increasing \ensuremathωc{\mathrm{\ensuremath{\omega}}}_{\mathit{c}}\ensuremathωc​. The binding energy and the corresponding variational parameters are calculated as a function of \ensuremath{\alpha}, U, and \ensuremathωc{\mathrm{\ensuremath{\omega}}}_{\mathit{c}}\ensuremathωc​. Analytical results are obtained in various limiting cases. In the limit of strong electron-phonon coupling (\ensuremath{\alpha}\ensuremath{\gg}1) we obtain for \ensuremathωc{\mathrm{\ensuremath{\omega}}}_{\mathit{c}}\ensuremathωc​\ensuremath{\ll}1 that Eestim{\mathit{E}}_{\mathrm{estim}}Eestim​\ensuremath{\approxeq}Eestim{\mathit{E}}_{\mathrm{estim}}Eestim​(\ensuremathωc{\mathrm{\ensuremath{\omega}}}_{\mathit{c}}\ensuremathωc​=0)+c(u)\ensuremathωc{\mathrm{\ensuremath{\omega}}}_{\mathit{c}}\ensuremathωc​/\ensuremathα4{\mathrm{\ensuremath{\alpha}}}^{4}\ensuremathα4 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 \ensuremathωc{\mathrm{\ensuremath{\omega}}}_{\mathit{c}}\ensuremathωc​\ensuremath{\gg}\ensuremathα2{\mathrm{\ensuremath{\alpha}}}^{2}\ensuremathα2\ensuremath{\gg}1 we find in 3D Eestim{\mathit{E}}_{\mathrm{estim}}Eestim​\ensuremath{\approxeq}\ensuremathωc{\mathrm{\ensuremath{\omega}}}_{\mathit{c}}\ensuremathωc​-\ensuremathα2{\mathrm{\ensuremath{\alpha}}}^{2}\ensuremathα2A(u)ln2{\mathrm{ln}}^{2}ln2(\ensuremathωc{\mathrm{\ensuremath{\omega}}}_{\mathit{c}}\ensuremathωc​/\ensuremathα2{\mathrm{\ensuremath{\alpha}}}^{2}\ensuremathα2) [also with an explicit analytical expression for A(u)] whereas in 2D Eestim2D{\mathit{E}}_{\mathrm{estim}}^{2\mathrm{D}}Eestim2D​\ensuremath{\approxeq}\ensuremathωc{\mathrm{\ensuremath{\omega}}}_{\mathit{c}}\ensuremathωc​-\ensuremath{\alpha}\ensuremath{\surd}\ensuremathωc{\mathrm{\ensuremath{\omega}}}_{\mathit{c}}\ensuremathωc​\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.
A large polaron is a quasiparticle that consists of a nearly free electron interacting with the phonons of a material, whose lattice parameters are much smaller than the polaron scale. The electron-phonon interaction also leads to an attractive interaction between electrons, which can allow two polarons to pair up and form a bipolaron. It has been shown that large bipolarons can form in theory due to strong one-electron--one-phonon coupling, but they have not been seen in real materials because the critical value of the required electron-phonon interaction is too large. Here, we investigate the effect of one-electron--two-phonon coupling on the large bipolaron problem. Starting from a generalization of the Fr\"ohlich Hamiltonian that includes both the standard one-electron--one-phonon interaction as well as an anharmonic one-electron--two-phonon interaction, we use the path-integral method to find a semianalytical upper bound for the bipolaron energy that is valid at all values of the Fr\"ohlich coupling strength $\ensuremath{\alpha}$. We find the bipolaron phase diagram and conditions for the bipolaron stability by comparing the bipolaron energy to the energy of two free polarons. The critical value of the Fr\"ohlich coupling strength ${\ensuremath{\alpha}}_{\text{crit}}$ is calculated as a function of the strength of the one-electron--two-phonon interaction. The results suggest that large bipolaron formation is more likely in materials with significant one-electron--two-phonon interaction as well as strong one-electron--one-phonon interaction, such as strontium titanate.
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.
A variational calculation is presented for large bipolarons in the strong-coupling limit. By means of the phonon coherent state, two coupled self-consistent integrodifferential equations for the center-of-mass and relative motions have been derived. Compared with previous works, lower binding energies for large bipolarons in three dimensions (3D) and two dimensions (2D) have been calculated. As a result, the values of lower bounds to critical ratios between dielectric constants below which bipolarons can exist are obtained as ${\mathrm{\ensuremath{\eta}}}_{\mathit{c}}$=1.23 in 3D and ${\mathrm{\ensuremath{\eta}}}_{\mathit{c}}$=1.64 in 2D.
The large bipolaron in two and three dimensions is investigated by combining anapproximate wave functions for the relative motion and the modified Lee-Low-Pines variational method. The values of the critical coupling constant αc above which bipolarons may exist and the critical ratio of dielectric constants ηc below which bipolarons may exist are evaluated, which are very close to the recent results obtained within a totally different variational approach. In addition, the stability region of bipolarons in the 1/α,η plane is clearly plotted. It is also found that low dimensional materials are more favorable for the formation of bipolarons.
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.