# Polaron Characteristics in a Quasi-0D Cylindrical Quantum Dot with Asymmetrical Parabolic Potential

We apply path integration in the same fashion as done for other geometries seen earlier. We consider the motion of the electron in the direction of the oz-axis to be bounded by a longitudinal parabolic confinement potential while bounded on the oxy-plane by a transversal parabolic potential (Figure 18.10).

The exact Hamiltonian for the full system is written:

The model Lagrangian of the system is written as:

For normal modes we substitute the following into the equation of motion

FIGURE 18.10 Depicts an asymmetric cylindrical quantum dot where the oz-axis is bounded by a longitudinal parabolic confinement potential and bounded on the oxy-plane by a transversal parabolic potential.

and solving for the frequency eigenmodes we have with

The effective polaron mass is conveniently obtained from 18.591:

We observe from here that there are two transversal oscillatory and two longitudinal oscillatory motions. These frequency eigenmodes may permit us to move to normal mode coordinates as previously done. We do for one variable and apply similarly to all other variables:

Substituting these equations for the normal coordinates into the equation of motion 18.590 and also considering the conservation of kinetic energy in any of the representations then

Considering 18.594 to 18.598 in the model Lagrangian 18.587 then

# Polaron Energy

We follow the same procedure as previously seen above for the evaluation of the energy and effective mass of the polaron via the relation:

For this problem, the partition function Z0: and

Considering the model Lagrangian 18.587 then

from where,

Here,

and

From 18.586 we have or

where lattice partition function is

and the functional of the electron-phonon interaction influence phase is The action functional is:

We find now{S — with the help of the generating function:

Here, for example, where

The above polaron formulae have the dependence on the quantity |r - a, which is a property of retarded functions. We again generalize the functions with G^|r —

that after the change of variables then

We do again the change of variable p = py, where 0 < 1. Ify = 1 then p = /?. We do another subsequent change of variable

This renders all our integrals convergent and consequently, or

where

and

For the symmetric case when

Then we have the result, as seen earlier, for the spherical quantum dot: where the dimensionless variational polaron energy is obtained as: