# 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 Z_{0} 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” **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 e_{2}:
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 *2a _{f}* while each of the electrons interacts with this polarization. This gives the reason why the interaction energy is described as

*a*= 2

_{F}*a*So, the simple doubling of the polaron energy by

_{F}.*a*-> 2

_{F}*a*does not take place.

_{F}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

*m _{c}*

different electron-phonon interaction coupling constant, *a _{F}* taken from reference [2].