Bipolaron Characteristics in 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 and bounded on the oxy-plane by a transversal parabolic potential.
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 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 Lagrangian 18.534 by the wave function 18.535 we have
ч /
Here,
The model Lagrangian of the transversal motion:
Model System Action Functional
From the model Lagrangian 18.538 we have the model action functional
Following the procedure of path integration seen earlier then
Equation of Motion / Normal Modes
The equations of motion for the transversal motion are
For normal modes we substitute the following
into the equation of motion and letting,
then
Solving for the eigenmodes we have
with
The bipolaron effective mass is conveniently obtained from the eigenmode equations:
Lagrangian Diagonalization
To diagonalize the model Lagrangian, we move to normal coordinates:
Substituting these equations into the equation of motion and also considering the conservation of the kinetic energy in any representation then
and the model diagonalized Lagrangian in normal coordinates:
It shows that in the motion of the bipolaron we have four oscillators indicating four internal motions.
Bipolaron Partition Function
The bipolaron partition function can be obtained from the equation:
or
The Coulomb interaction can be written in the form:
Bipolaron Generating Function
We find now (Svia the generating function:
Letting,
then
where
Bipolaron Energy
We again observe all the formulae in the bipolaron problem have the same dependence on the quantity |r -
where,
and
with
For Л0» 1 then
18.19 Polaron Characteristics in a Quasi-OD Cylindrical
