# Bipolaron Partition Function

As done previously for the polaron problem, the energy and effective mass of the bipolaron can be found from the relation:

Considering 18.321, the partition function Z0 is obtained:

So, the associated partition function has therefore the product of the contributions from one free motion and three oscillatory motions:

where /. is the length of the wire,

is the thermal de Broglie wavelength and frequency describing the relative oscillations of the electronic and fictitious particle subsystems:

We write the Lagrangian corresponding to the Hamiltonian 18.284:

The amplitude of the electron-phonon coupling interaction у* (r) for the ic moving wave is defined: with

From equation 18.328, the action functional for the full system after eliminating the phonon variables is obtained similarly as for the polaron problem:

Here the influence phase of the exact system is:

with

From 18.289, following the same procedure like for the polaron when finding the partition function then considering 18.324 we have:

or

Here, the model bipolaron action functional is obtained:

# Bipolaron Generating Function

We find now^S — with the help of the generating function:

From change of variables and considering

Feynman Path Integrals in Quantum Mechanics and Statistical Physics

then

where

# Bipolaron Asymptotic Characteristics

From 18.323, we now evaluate the quantities in (S - to permit the evaluation of the polaron characteristics. In order to do this, we first find the following expectation values:

All the formulae in the bipolaron have the same dependence on the quantity |r - “time” propagating in the crystal. The exact action functional shows again that the potential of the electrons (holes) feel at any “time” the dependence on its position at previous times. This confirms the effect of the electron (hole) on the crystal as it propagates at a finite velocity and can make itself felt by the crystal at a later time.

We again generalize the function in the integrand of the bipolaron to be G^|r — cr| — у j to help in the

simplification of the evaluation of the integral. So, we look again at the following function that yields the

(. . p)

value of the integral for a given G г -

V 2,

We do the change of variables then

We consider again the change of variable p = py, where 0 < у < 1. Ify = 1 then p = /?. Again, if we also do the change of variable

then from 18.323 considering/? -> oo (T -> 0), we evaluate the asymptotic expansions of Feynman bipolaron dimensionless energy and consider strong coupling for bipolarons where the fictitious particle is more massive than the electron (hole) [2]:

For brevity, we exclude the Coulomb interaction by letting: and

where

Extremizing 18.359 over the energy of the oscillatory motion e2: then

The solution of 18.362 can be obtained via the iteration method as done for the polaron problem:

The bipolaron coupling energy without the Coulomb contribution:

We introduce the effective electron-phonon coupling constant

then, the bipolaron coupling energy without the Coulomb contribution:

The polaron counterpart is

From these two results we see the bipolaron energy appears to be twice the polaron energy [2]. This is an indication that the induced polarization due to the two electrons in the bipolaron is proportional to 2af while each of the electrons interacts with this polarization. This gives the reason why the interaction energy is described as aF = 2aF. So, the simple doubling of the polaron energy by aF -> 2aF does not take place.

From Figures 18.5 and 18.6 we observe that the bipolaron coupling ground state energy has a monotonic decrease with an increase in the radius of the quantum wire. This implies the bipolaron coupling energy increases with confinement enhancement. In a similar fashion, the bipolaron effective mass has a monotonic decrease with an increased radius of the quantum wire and implies an increase of the bipolaron effective mass with confinement enhancement.

w

FIGURE 18.5 Depicts the dimensionless bipolaron coupling energy,-versus the radius, R of the quantum wire

for different electron-phonon interaction coupling constant, a, taken from reference [2].

FIGURE 18.6 Depicts the dimensionless bipolaron effective mass, —— versus the radius, R of the quantum wire for

mc

different electron-phonon interaction coupling constant, aF taken from reference [2].