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 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 large bipolaron in two and three dimensions is investigated by combining the reasonable asymptotic relative wavefunctions and the Lee–Low–Pines–Huybrechts (LLP-H) variational method. The values of the critical coupling constant above which bipolarons may exist and the critical ratio of dielectric constants below which bipolarons may exist are evaluated, which are very close to the recent results obtained within totally different variational approaches. It is also found that low-dimensional materials are more favorable for the formation of bipolarons.
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.
Feynman path-integral techniques are used to study the large bipolaron system. A new trial action is introduced, which takes into account a nonzero average distance between the electrons. With this trial action, variational expressions are derived for the free energy for arbitrary electron-phonon interactions and spatial dimensions. For large interelectronic repulsions (twice) the Feynman upper bound for a single polaron is reobtained. Therefore it is possible to discuss the single-polaron--bipolaron transition within the same physical picture for both bipolarons and single polarons. Numerical results are presented for the case of LO-phonon interaction and a Coulombic repulsion between the electrons. For this system a scaling relation between the free energies in two (2D) and three dimensions (3D) is obtained. Bipolaron formation is only possible above a critical value for the coupling constant ${\mathrm{\ensuremath{\alpha}}}_{\mathit{c}}$, which is lower in two than in three dimensions (in 2D: ${\mathrm{\ensuremath{\alpha}}}_{\mathit{c}}$\ensuremath{\approxeq}2.9 and in 3D: ${\mathrm{\ensuremath{\alpha}}}_{\mathit{c}}$\ensuremath{\approxeq}6.8). This indicates more favorable conditions for bipolaron formation in two dimensions, which might be of relevance for the bipolaron model of high-
Tc superconductivity. The single-polaron--bipolaron transition behaves much like a first-order phase transition.
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 (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\ensuremath{\approxeq}
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\ensuremath{\approxeq}${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$-${\mathrm{\ensuremath{\alpha}}}^{2}$A(u)
ln2(${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$/${\mathrm{\ensuremath{\alpha}}}^{2}$) [also with an explicit analytical expression for A(u)] whereas in 2D
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.