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

The exact Hamiltonian for the system is written:

The model Lagrangian of the system is written: where,

The equations of motion for the transversal motion are and for the longitudinal motion:

For normal modes we substitute the following

into the equation of motion and letting, then

Solving for the frequency eigenmodes we have with

Hie bipolaron effective mass is conveniently obtained from the frequency eigenmode equations:

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

The model Lagrangian in normal coordinates is obtained:

This shows that in the motion of the bipolaron we have four oscillators showing four internal motions. The Coulomb interaction can be written in the form:

We find now^Svia the generating function:




The resultant formulae for the bipolaron are observed to have the same dependence on the quantity |r- rr. This is the property of retarded functions as earlier indicated.. We generalize the resultant bipolaron functions G^|r -<т| - у j which helps in the evaluation of the twofold integral [1,2] • This procedure renders all our integrals convergent and, consequently, the variational bipolaron energy:




Theoretical studies show that the electronic confinement is observed to imitate the magnetic field subjected to the system [22,57]. In this case, similar effects of the cyclotron frequency [9] and confinement frequencies on the electronic and polaron as well as the bipolaron states are expected.

Polaron in a Magnetic Field

We have so far examined a lot of problems on the harmonic oscillator. We examine another problem when a charged particle, for example say an electron, is subjected to a homogenous magnetic field in a polar medium and interacts with lattice vibrations. This will be calculated using the same Feynman variational technique where, in this case, Feynman employed a special approach for the evaluation of the polaron in a magnetic field by introducing a transition amplitude permitting the action to be written in quadratic form over the electronic coordinates:

We express now the transition amplitude К via the temperature dependence by using the analytic prolongation:

So, for the action functional or

we employ a special approach via

We consider first the kinetic energy term:

and consider for the Dirac delta function the property

Apply twice integration by parts in 18.679 then





Here, the differential operator F(r,cr):

We have just seen that the functional is independent of the velocity but dependent only on the coordinates. For the excursion of the particle from the new path we select a fluctuating path

about the classical one:

We write now the actional functional considering the introduction of a perturbation y(r):

This should be the action functional of a driven harmonic oscillator. We find the equation of motion: or

From this equation of motion, we find first the homogenous solution. From definition of the inverse operator F (cr,r):


This gives us the solution of some partial solution for the classical path.

We write now the action in terms of the fluctuation/(r) to the classical pathq(r):


The fluctuation у may be found from the equation of motion after varying the action functional S[q]. Considering the problem of the oscillators in the previous chapters as well as the quasi-classical approximation then the transition amplitude is:

Here, we introduce the following function:

Considering 18.674, then

where now,

This first factor is due to the classical solution while the second is due to the fluctuation of the classical path. Expression 18.698 permit to find the partition function of a driven harmonic oscillator.

Let us find the properties of the operator F(r,cr):

Here, Ф„(г) is the eigenfunction of the operator F(r,cr) and is the eigenvalue. From the above we have

If this relation is satisfied then Ф„(г) is obviously the eigenfunction and eigenvalue of the operator

^(reconsidering F(r,o-) to be a symmetrical operator we examine the following properties:




This follows that the eigenfunctions Ф„(г) and Ф„-(г') satisfy, respectively, the orthogonality and completeness relations:

We will treat with operators of the form:

In order to find the inverse operator F (cr,r), it is important to find a„ and a~':

Multiplying 18.709 and 18.707 and then taking the integral of the resultant expression we have

Comparing the coefficients of the left-hand and right-hand sides then we have:

Considering the operator F(r,cr) in 18.709 and the definition of the Dirac delta function in 18.708 then




Thus, we find From,


where the function Fa,(|r - ш[г] for the full system has been found this time via the operator method:

Hie transition amplitude for this problem is thus:

We now apply the above procedure for the case of an electron interacting with lattice vibrations in a polar medium and subjected to a magnetic field which is described by the following Lagrangian:

Here the vector potential A is related to the magnetic field strength H expected to point in the oz-axis direction:

The Feynman approach can also be applicable for the case of a particle in a magnetic field where the full partition function:

This is the product of the partition functions over all possible oscillators:

Here is the lattice partition function:

and the electron-phonon influence phase:


is the full functional of the electron-phonon interaction (influence phase). So, the coupling to the environment leads to an additional, non-local term to the action where the original bonafide many- body problem is now reduced to a one-body problem and we rewrite the full action of the full system via its influence phase ф[г]:

The action functional 18.730 describes a driven harmonic oscillator in a magnetic field. In the full action functional 18.730 the first term is the contribution of the kinetic energy of the electron, the second term that due to the magnetic interaction, and the last, the full functional of the electron-phonon interaction described by ф[г] and is observed to depend on the function F(l,(|r - o) with the time difference |r - a indicative of a retarded function depending on the past thereby signifying interaction with the past where the perturbative motion of the electron takes “time” to propagate in the crystal lattice. This function Fw(|r -

As mentioned earlier, a classical potential for the polaron problem might well be expected to be a good approximation with tight binding and so, to imitate the polaron problem, a good model is one where instead of the electron being coupled to the lattice, it is coupled by some “spring” to another particle (fictitious particle) and the pair of particles are free to wander about the crystal where the model Lagrangian may be selected in the one oscillatory approximation:

The first term describes the translational motion; the second term due to the magnetic field acting on the particle, the third and fourth describe the oscillatory motion. It is an approximation of the full Lagrangian of the system, L. Here M and ay are, respectively, the mass of a fictitious particle and the frequency of the elastic coupling that will serve as variational parameters; R is the coordinate of the fictitious particle. The model system conserves the translational symmetry of the system. The judicious choice of L0 is to simulate a physical situation that may give a better upper bound for the energy, as mentioned earlier.

The partition function of the model system can be written:

We find the action functional of the model system in terms of only the path r by eliminating the fictitious particle subsystem variables R in 18.732:

Hie partition function due to the fictitious lattice is

and the influence phase of the interaction of the electron with the fictitious particle: where

The action functional of the interaction of the electron (hole) with the fictitious particle via its influence phase:



Apply the Feynman inequality then from

we have



In the limit of zero temperature T -> 0, the free energy becomes the ground state energy F = E: or

We evaluate Z0 via the model action functional that considers the magnetic field:

The second summand in 18.747 considers the interaction of the particle with the magnetic field and the first summand, the action functional that considers the interaction of the particle with the fictitious particle subsystem of oscillators:

So, after path integration, then: where,

and the influence phase Фга/ [r] of the interaction of the electron with the fictitious particle is defined in 18.735. The first three terms in 18.750 can be rewritten via the following kernel:


Then from here and 18.749 we have or


In the absence of the magnetic field term, then we have the normal polaron integral:

This integral can be sequentially transformed to Here,

To find the magneto-polaron characteristics we move to normal coordinates that permit the finding of the magneto-polaron ground state energy:


The model Lagrangian may be conveniently represented in quadratic form by doing a change of variables:



The quantities px and p2 are, respectively, the coordinates of the relative motion and of the center of mass. The parameter p is a reduced mass and v is a scaled frequency.


The coordinate p2 describes the free motion of the particle with mass M + m, while the coordinate pv the harmonic oscillator with frequency v. Hence the partition function in 18.762 now becomes


So, from equation 18.754 then

Here, the quantity lz is the length in the oz-direction, le(hf)) is the thermal de Broglie wavelength.

The role of the driving force is represented by

From this denotation relation 18.769 can be rewritten:

Here, the additional factor lz is due to integration over

Introducing the operator then

We observe that the integral overly has a quadratic form and may be evaluated easily. The most convenient approach may be by introducing normal variables that will permit the integral to be transformed into the sum of independent quadratic parts. There is the possibility of writing the Lagrangian corresponding to 18.774 in the form:


Taking the form of the Lagrangian in 18.775 and doing path integration over, , у, and y2 then the result imitates the initial path integral overy:

The expression for Z0 is rather convenient than Z0 due to the fact that if, from the very beginning, we took 18.775 before path integration over , y( and y2 then we could have obtained the normal terms.

Let us select

and substitute into the equation of motion:


Substituting 18.778 into 18.780 then the resulting system has non-trivial solutions when

From here,



This is when v2 is at the neighborhood of d). Also, for at the neighborhood of v we have:

So, the four components of the frequency can permit us to move to normal coordinates. If we transform у via normal coordinates and substitute into 18.775, then we have four independent terms. In this case, the integral is easily calculated and for the partition function we have:


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