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 rj2 are the superposition of normal modes with frequencies cot and a»,:

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

 
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