# Bipolaron Energy

## Generating Function

In a similar manner as done previously, we find now(S-§)^ where it is necessary to evaluate

with the help of the generating function: or

where

## Bipolaron Characteristics

From here, we now evaluate the quantities in *(S* - :

Again as seen previously, all the formulae in the bipolaron has the dependence on the quantity |r - *a *confirming the fact that the quantities (retarded functions) depend on the past with the significance of interaction with the past being the perturbation due to the moving electrons (holes) that take **“time” **propagating in the crystal lattice.

We again generalize the function in the integrand of the bipolaron to be G^|r -oj --y j to aid in the simplification of the evaluation of the integral [1,2]:

We do the change of variables then

We consider again the change of variable *p = /5y,* where 0 < *у* < 1. Ify = 1 then *p* = *fi.* Again, if we do the change of variable

then from *[)* -> с» ( *T -*■* 0), we evaluate the asymptotic expression of Feynman bipolaron dimensionless energy [1,2]:

Here,

where,

The effective mass is conveniently obtained from the equation of motion:

For the transition to the polaron problem for the quasi-OD spherical quantum dot we set:

w

FIGURE 18.8 Depicts the dimensionless bipolaron coupling energy,-versus the radius, *R* of the quantum dot

/ico_{0}

for different electron-phonon interaction coupling constant, a, and fixed *t)* = 0.01 taken from reference [2].

We observe after substituting this into the bipolaron energy that

From here

This shows the bipolaron to constitute a stable pair of two polarons.

In [2], from the analytical results of the bipolaron coupling energy as well as Figure 18.8, it is observed from the curve of the bipolaron coupling energy versus the radius of the quantum dot, that for different coupling constants, *a _{t}* there exist a non-monotonic dependence. The bipolaron coupling energy faster decreases through a minimum at the neighborhood of the confinement radius

*Rx*0, and then starts increasing. This effect may be explained by the fact that, when electrons are forcefully brought closer together, there arises enhanced polarization as well as increased Coulomb repulsion. For small

*R*, when the bipolaron radius becomes greater than the radius of the quantum dot, this obviously leads to bringing the electrons very near to each other and obviously mutual repulsion is enhanced and, consequently, there is an increase of the bipolaron energy. This phenomenon is observed in references [2,56].