A bipolaron in a spherical quantum dot with parabolic confinement

Abstract

A theory of bipolaron states in a spherical 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, and the bipolaron radius) are studied as functions of the radius R of the potential well. Analytical expressions for bipolaron parameters are obtained at large and small sizes of the quantum well. It is shown that at R>>1 (where R is expressed in units of the polaron radius), the influence of confinement on the bipolaron binding energy W(R) is described by the function ~1/R2, while at small sizes this influence is more complicated: W(R) passes through a maximum in the region R<1.

On the stability of Fröhlich bipolarons in spherical quantum dots

“…Relying on the stability criterion based essentially on this inequality, Mukhopadhyay and Chatterjee [27] and the present authors [30] have reached the conclusion that for strong-coupling polarons the role of the geometric confinement is to disfavour the bipolaron stability in small quantum dots. An outcome along the same lines has been reported by Pokatilov et al [29] under the Feynman variational principle. In their plots of the electron-phonon coupling constant versus the dot size we note that, even for fairly weak Coulomb coefficients, the value of the coupling constant below which the bipolaron phase is unfavourable increases inevitably when the dot radius is made smaller than the size of a polaron.…”

“…Of particular relevance to the content of the present article are the recent solutions for the bipolaron state in quantum dots [27][28][29][30][31]. In most of the preceding works, including the study of bipolarons in quantum dots, the phase boundary of the bipolaron stability region is determined by the inequality…”

Stability of low-dimensionally-confined bipolarons

“…We should remark that the oscillator-oscillator type waveform (15,16) for φ(R) × ϕ(r) has proved to be the most efficient approximation yielding the largest lower bound on η c among the possible sets of different combinations of Pekar, Coulomb and oscillator type trialwavefunctions tackled earlier by Verbist et al [8] for the three-and two-dimensional bipolarons. We hope that the results derived here will provide us a revision of the phase stability of bipolarons within a broader context beyond the already existing literature concerning the bulk and strict 2D cases.…”

“…An enormous amount of literature published within this context leads to the evidence that bipolarons can exist under certain circumstances defined critically by the Coulomb repulsion coefficient and the electron-phonon coupling strength [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. Of particular relevance to the content of the present article are the recent solutions of this problem in strict two dimensions (2D) [6][7][8][9]12] where it has been found that bipolaron formation should be easier in a space of low dimensions.…”

Polaron in Cylindrical and Spherical Quantum Dots

“…When investigating the polaron in nanocrystals, we should consider both the electron and the phonon confinements. The electron confinement is described in [1,[22][23][24][25][26][27].…”