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Polaron Characteristics in a Cylindrical Quantum DotTable of Contents:
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 zaxis direction to be bounded by an infinite high rectangular potential well while bounded on the oxyplane by a parabolic potential (Figure 18.9). System HamiltonianThe 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 ozaxis. 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 ozaxis bounded by an infinite high rectangular potential while the oxyplane 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 CoordinatesLagrangian DiagonalizationFor 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 FunctionWe follow the same procedure for the evaluation of the energy and effective mass of the polaron via the relation:
The partition function Z_{0}: 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 electronphonon interaction influence phase is The action functional is: / Polaron Generating FunctionWe find now (S — §) with the help of the generating function:
From equation 18.496 then where
then
where,
So,
where Polaron EnergyWe now calculate all quantities in (S  S_{0}^ 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: 
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