Polaron Characteristics in a Cylindrical Quantum Dot

We apply path integration in the same fashion as done for the geometries above but now for a cylindrical quantum dot. We consider the motion of the electron in the z-axis direction to be bounded by an infinite high rectangular potential well while bounded on the oxy-plane by a parabolic potential (Figure 18.9).

System Hamiltonian

The Hamiltonian of the system is written in the form:

Here, V(z) is the confinement potential (infinite high rectangular potential well) in the direction of the oz-axis. The fourth summand of the Hamiltonian is the transversal parabolic confinement potential.

The state of the electron is described by the variational wave function that has a large spread compared to the ground state wave function:

Averaging the Hamiltonian 18.486 by the wave function 18.487 we have Here,

FIGURE 18.9 Depicts a cylindrical quantum dot with the oz-axis bounded by an infinite high rectangular potential while the oxy-plane is bounded by a parabolic potential.

From the Feynman variational principle, the interaction of the electron with the polarized vibrations of the crystal is modelled by an elastic coupling of the electron and a fictitious particle that attracts the electron to itself. So, the effect of the polarized crystal lattice on the electron is approximated to an elastic attraction of the second particle. From these analyses, the model Lagrangian may be selected in the one oscillatory approximation:

Here M and a>, are, respectively, the mass of the fictitious particle and the frequency of the elastic coupling serving as variational parameters; R is the coordinate of the fictitious particle.

Transformation to Normal Coordinates

Lagrangian Diagonalization

For normal modes we substitute the following harmonic coordinates: into the equation of motion

and solving for the frequency eigenmodes we have with

The effective polaron mass is conveniently obtained from the frequency eigenmodes in 18.493:

The frequency eigenmodes in 18.493 permit us to move to normal mode coordinates as previously done:

Inserting these equations for the normal coordinates into the equation of motion 18.492 and also considering the conservation of kinetic energy in any of the representations then

Substituting 18.497 and 18.496 into the model Lagrangian 18.490 is then diagonalized:

Polaron Energy/Partition Function

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

The partition function Z₀: and

or

From the model Lagrangian 18.490 then from where,

Here,

where,

From 18.488 we have or

where lattice partition function is

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

/

Polaron Generating Function

We find now (S — §) with the help of the generating function:

From equation 18.496 then

where

then

where,

So,

where

Polaron Energy

We now calculate all quantities in (S - S₀^ with the generating function and, in particular,

We observe again all the formulae in the polaron have the same dependence on the quantity |г - о|, 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 functions to be |r - oj - -y j that aid in the

evaluation of the twofold integral [1,2]:

that after the change of variables then

and again, considering the change of variable p = py, where 0 < у < 1. If y= 1 then p = p. Subsequently, we also do a the change of variable

This renders all our integrals convergent and, consequently,

where

and

The polaron energy: