# Bipolaron Characteristics in a Quasi-0D Spherical Quantum Dot

## Introduction

We examine the bipolaron characteristics in a spherical quantum dot with an all-sided parabolic electronic confinement potential. The confinement potential is selected for technological reasons and aid also for path integration to be performed exactly.

## Model Lagrangian

The model Lagrangian for the bipolaron problem is selected in the same fashion as that of the quasi-ID quantum dot in the one oscillatory form:

## Model Lagrangian

### Equation of Motion and Normal Modes

With the same procedure as for the bipolaron in the quasi-ID quantum wire, we find the equation of motion from

and then set

where, w is the frequency of the normal modes. W then introduce new coordinates with

the centre of mass of the two electrons: the relative distance between the electrons:

the centre of mass of two fictitious particle and

the relative distance between the fictitious particles.

This change of variables transforms the resultant system of equations:

This permits us to have the equations for the normal modes with frequencies: where

The **effective bipolaron mass **may be conveniently obtained from 18.419:

## Diagonalization of the Lagrangian

The variables and *rj _{2}* are the superposition of normal modes with frequencies

*co*and a»,:

_{t}

Substituting these equations into 18.417 then

Similarly, considering 18.420 we represent r and *R* as a superposition of normal modes and the result substituted into 18.418 then we have

Consider,

then from the conservation of kinetic energy

we have

Similarly,

and

In a similar fashion for and

We then have

Substituting all the normal coordinates into the model Lagrangian 18.273 and considering then

## Partition Function

To find the bipolaron energy we start with where

The partition function has the product form:

From the model Lagrangian 18.410, we follow the same procedure as for the bipolaron in a quasi-ID quantum wire to find the partition function:

or

Here,

We write the Lagrangian of the electron-lattice interaction:

The amplitude of the electron-phonon coupling interaction, у*(г) for the *к* moving wave is defined

as:

with

## Full System Influence Phase

After eliminating the phonon variables from the action functional for the full system we obtain:

Here the influence phase of the full system is: with