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Polaron Characteristics in a Quasi0D Spherical Quantum DotTable of Contents:
IntroductionWe continue to examine the effect of the electronic confinement on the polaron and bipolaron in a quasiOD quantum dot. The problem of the quantum dot will be seen to have reduced variational parameters compared to the case of the quasiID quantum wire. From the Feynman variation principle framework, we examine the polaron energy and effective mass in a quantum dot. For the moment we examine a quantum dot with spherical geometry. In this problem we example the polaron characteristics for low temperatures. Polaron LagrangianThe system of conducting electron interacts with the lattice vibrations in the field of a parabolic confinement potential where the system is described by the Lagrangian:
Here, the electron coordinate is r and m is its effective mass. The model Lagrangian may be selected in the one oscillatory approximation as seen earlier:
where M and a>_{f}are, respectively, the mass of a fictitious particle and the frequency of the elastic coupling that will serve as variational parameters while R is the coordinate of the fictitious particle. The elastic constant describing the force of attraction between the electron and the fictitious particle may also be represented in the form: Normal ModesThe model Lagrangian 18.369 considers the electronic as well as the fictitious particle variables. Path integration will be easily done by moving from real to normal coordinates that will be feasible via frequency eigenmodes. This is possible via the equation of motion in 18.290 and considering 18.369:
Suppose as done earlier for the problems in the quantum wire that
Here w is the frequency of the normal modes of the system. Substituting 18.372 into 18.371 then we have
From the determinant of the system of equations 18.373 we have the following normal frequency eigenmodes: Lagrangian DiagonalizationWith the help of the frequency eigenmodes in 18.374 we can now diagonalize the Lagrangian in 18.369. This will be done via the unified technique applied in 18.315 by the transformation of real to normal coordinates. As observed earlier, such a coordinate transformation has no effect on the particle state as the particle state is an abstract object and makes no reference to a given coordinate system. Transformation to Normal CoordinatesAs seen earlier, to perform Feynman path integration elegantly, we move to normal coordinates via normal modes in 18.374. For the normal mode, a», we have the normal coordinates:
Substituting this into 18.373 then we have
Considering the conservation of the kinetic energy:
Then from 18.375 and 18.376 we have
We do the same change of variables to normal variables via the normal mode m_{2}:
From here considering the kinetic energy
then
The coordinates r and R are not harmonic variables but, however, they represent the superposition of two harmonic oscillations with the frequencies <ы, and w_{2}: So, from 18.375 to 18.383 the model Lagrangian in 18.369 takes the form
The model Lagrangian has two oscillators with Polaron Partition FunctionTo proceed with the polaron energy we find the following quantities:
where
or
and
As done previously for the polaron in a quasiID quantum wire, considering 18.369 then Eliminating the fictitious particle variable R then where
and
So,
Also, from 18.368 we have or
where
is the lattice partition function and the functional of the electronphonon interaction influence phase: Generating FunctionFrom the partition function obtained above, the polaron characteristics can be evaluated via the relation: where,
To evaluate this, it is necessary to evaluate We have to do this via the generating function:
The generating function is obtained in the same manner as done for the polaron in a quasiID quantum wire via the model Lagrangian 18.368:
We then find
All the formulae so far for the quasiOD quantum dot are observed to depend on the quantity r  a . This implies, the formulae express retarded functions that depend on the past histories of the particle. This signifies the interaction with the past where a perturbative motion of the electron (hole) takes “time” to propagate in the crystal lattice. Since the retarded functions have a common argument r , so we again generalize them by the functions G^r сту j and then perform the simplifications of the double integral [1,2]. For p > oo ( T *• 0 ) and considering the equation
and FIGURE 18.7 Depicts the dimensionless polaron ground state energy, —— versus the radius, R of the quantum dot /icOo for different electronphonon interaction coupling constant, a_{e} taken from reference [2]. the polaron variational energy can then be obtained as follows [2]:
Here,
The effective mass is conveniently obtained from the equation of motion [2]:
From Figure 18.7, we observe that the polaron energy increases with reducing radius of the quantum dot and also increasing electronphonon coupling constant. From here we observe also increase polaron energy as a result of enhanced confinement. 
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