Numerical results for the polaron dispersion are presented for an arbitrary number of space dimensions. Upper and lower bounds are calculated for the dispersion curves. They are rather close to each other in the cases of small electron-phonon couplings usual for real polar materials. To describe the dispersion in other materials, we suggest a simple fitting formula which can be applied at intermediate values of the Frohlich electron-phonon coupling constant. Its validity is approved by the comparison with direct calculations and previously obtained results. This makes our results not only reliable and highly accurate but also easy reproducible.

The polaron energy and the effective mass are calculated for an electron confined in an asymmetrical finite quantum well constructed of AlxlGa1-xlAs/GaAs/AlxrGa1-xrAs layers. To simplify the study we suggest a model in which parameters of a medium are averaged over the ground-state wave function. Our model reproduces the correct three-dimensional asymptotics at both small and large widths. We obtained a rather monotonous behavior of the polaron energy as a function of the confining potential width and found a peak of the effective mass. A comparison with theoretical results by other authors is made.

We discuss the (LO)polaron dispersion for arbitrary spatial dimension D. Firstly, we review the existing literature; recent numerical work is critically analyzed. Secondly, we derive novel upper bounds for the dispersion, which incorporate the correct behaviour of the dispersion up to third order of the coupling constant α. A totally analytical evaluation is performed in the case D = 1. We compare the upper bounds with previously published lower bounds. Apart from a surrounding of zero dispersion, the relative deviation is on a few‐percent scale.

We consider the dependence of the MgB_2 superconducting critical temperature on the pressure. Our model exploits the influence of the large polarons on the band structure of the layered MgB_2 superconductor. Namely, the hole Pekar-Froehlich polarons form quasi two-dimensional potential wells in the boron plane which shift the positions of the sigma- and pi-bands. This energy shift depends on the pressure and the Cooper pairing of the correlated sigma-electrons happens inside polaron wells. The results obtained are as follows: dT_c/dp = -\alpha (5.2 \pm 0.9) K/GPa or dT_c/dp = -\alpha (6.9\pm 1.1) K/GPa for a different choice of the Grueneisen parameter. Being compared with known experimental data they give us a resonable interval for the value of the Froehlich electron-phonon coupling constant: \alpha = 0.15 - 0.45.

In this paper we consider the stability of a large bipolaron embedded in a polaron gas. The main conclusion is that an isolated nonstable bipolaron can be stabilized in the presence of a polaron gas exhibiting Fermi statistics. On the other hand, the exchange interaction tends to destabilize the bipolaron. This study has been performed both for bulk three-dimensional materials and for thin two-dimensional films using the Hartree-Fock approximation. The bipolaron has been described by an extension of the Feynman polaron model.

An approximate model to describe a multilayered heterostructure is proposed. The details of the heterostructure are taken into account by the confining potential V(z) generated by the layers. The multilayered GaAs/AlxGa1-xAs heterostructure is considered as an effective medium. Its mean parameters are defined by averaging the subsequent layer-dependent parameters over the ground-state wave function, Only the effective bulk phonon mode inhabits the effective medium with mean characteristics, As a results properties of charge carries in the heterostructure can be described through the Pekar-Frohlich polaron model. The polaron energy and its effective mass are calculated for different quantum wells. We obtained a rather monotonous behavior between the asymptotic values for the polaron energy as a function of the confining potential width. As to the effective polaron mass it exhibits a peak. The comparison is made with theoretical results by other authors.

We study properties of polarons and excitons confined to a potential generated in a planar semiconductor heterostructure of the $Ga_{1-x}Al_{x}As/GaAs/Ga_{1-x}Al_{x}As$ type. In contrast with results of other authors peaks are found for the exciton energy and the polaron effective mass as functions of the potential width while the polaron energy reveals rather monotonous behavior.

The polaron energy and the effective mass are calculated for an electron confined in a finite quantum well constructed of GaAs/AlxGa1-xAs layers. To simplify the study we suggest a model in which parameters of a medium are averaged over the ground-state wave function. The rectangular and the Rosen-Morse potential are used as examples. To describe the confined electron properties explicitly to the second order of perturbations in powers of the electron-phonon coupling constant we use the exact energy-dependent Green's function for the Rosen-Morse confining potential. In the case of the rectangular potential, the sum over all intermediate virtual states is calculated. The comparison is made with the often used leading term approximation when only the ground state is taken into account as a virtual state. It is shown that the results are quite different, so the incorporation of all virtual states and especially those of the continuous spectrum is essential. Our model reproduces the correct three-dimensional asymptotics at both small and large widths. We obtained a rather monotonous behavior of the polaron energy as a function of the confining potential width and found a peak of the effective mass. The comparison is made with theoretical results by other authors. We found that our model gives practically the same (or very close) results as the explicit calculations for potential widths L greater than or equal to 10 Angstrom.

This paper presents a variational study of the ground-state energy of an exciton-(LO) phonon system, which is spatially confined to a quantum well. The exciton-phonon interaction is of Frohlich type, the confinement potentials are assumed to be parabolic functions of the coordinates. Making use of functional integral techniques, the phonon part of the problem can be eliminated exactly, leading us to an effective two-particle system, which has the same spectral properties as the original one. Subsequently, Jensen's inequality is applied to obtain an upper bound on the ground-state energy. The main intention of this paper is to analyze the influence of the quantum-well-induced localization of the exciton on its ground-state energy (or its binding energy, respectively). To do so, we neglect any mismatch of the masses or the dielectric constants, but admit an arbitrary strength of the confinement potentials. Our approach allows for a smooth interpolation of the ultimate limits of vanishing and infinite confinement, corresponding to the cases of a free three-dimensional and a free two-dimensional exciton-phonon system. The interpolation formula for the ground-state energy bound corresponds to similar formulas for the free polaron or the free exciton-phonon system. These bounds in turn are known to compare favorably with all previous ones, which we are aware of. [S0163-1829(99)01844-5].

We consider a model describing the one-dimensional confinement of an exciton in a symmetrical, rectangular quantum-well structure and derive upper and lower bounds for the binding energy $E_b$ of the exciton. Based on these bounds, we study the dependence of $E_b$ on the width of the confining potential with a higher accuracy than previous reports. For an infinitely deep potential the binding energy varies as expected from $1 Ry$ at large widths to $4 Ry$ at small widths. For a finite potential, but without consideration of a mass mismatch or a dielectric mismatch, we substantiate earlier results that the binding energy approaches the value $1 Ry$ for both small and large widths, having a characteristic peak for some intermediate size of the slab. Taking the mismatch into account, this result will in general no longer be true. For the specific case of a $Ga_{1-x}Al_{x}As/GaAs/Ga_{1-x}Al_{x}As$ quantum-well structure, however, and in contrast to previous findings, the peak structure is shown to survive.

A criterion, proposed by the present authors, is used to derive numerical results for the stability of a large bipolaron embedded in a polaron gas. The main conclusion is that an isolated metastable bipolaron can be stabilized by the polaron gas surroundings because of the Fermi statistics of polarons. On the other hand, it is found that the exchange interaction tends to destabilize the bipolaron. The study is performed both for bulk (3D) materials and for thin (2D) films within the Hartree-Fock approximation. The bipolaron is described by an extension of the Feynman

A condition for the bipolaron stability has to be reformulated to take into account the fact that a bipolaron is a part of the charge carrier system. Possible consequences of this condition are considered in the special case when a single bipolaron is placed into the polaron gas. A possibility for the large bipolaron with zero spin to be stabilized in three-space dimensions by the polaron environment is discussed.

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.

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.

The nonlinear Schrodinger equation is solved on an infinitesimal thin ring or circle. We obtained the exact real wave functions with their corresponding energies for the ground state and the excited states. Critical values of the circle perimeter are found at which the ground state changes its structure and additional higher excited states appear. Also, the complex wave functions that correspond to energy levels with finite angular momentum are studied.

The strong coupling limit is studied for a Pekar-Fröhlich polaron confined to a one-dimensional (1D) structure. The non-linear effective Schrödinger equation is solved exactly in the case of two different external potentials which imitate a finite size 1D sample: an infinite and a finite deep rectangular well. The ground state and excited states are calculated. We found that taking the limit of a finite size box to an infinitely large box leads to additional solutions which are not found in a treatment on an infinite axis. The additional solutions, which have a 1/n 2 discrete spectrum, correspond to polaron states in which the wave function is split up in identical parts which are infinitely apart from each other.

The one-dimensional (1D) large bipolaron is investigated in the limit of strong electron-phonon coupling. The nonlinear integro-differential equation for the bipolaron wave function is solved numerically, from which we obtained estimates for the main characteristics. An enlargement of the stability region for the bipolaron ground state is found in 1D as compared to the stability regions in 2D and 3D. The energy of the first relaxed excited state (RES) equals the energy of two single polarons and the ground state in the potential generated by the first RES has a slightly lower energy than this RES and is therefore stable. The nonlinearity causes the feature that the combination of the ground state and an excited state of one-particle wave functions could lead to a higher bipolaron energy than the combination of two excited states.

Widely reported broadening of a bipolaron formation region in two dimensions should be revised in view of a concrete mechanism of electron confinement to a two-dimensional layer.

Equations for the bipolaron wavefunction, ground-state energy and effective mass are derived which are exact in the strong-coupling limit. The results are obtained for large bipolarons in an arbitrary number of spatial dimensions (D). We apply our results to the cases D = 1, 2, 3.