<commit_before>## Overview 

These codes solve Osaka's [Osaka1961] finite-temperature parameters for the Feynman
polaron model [Feynman1955]. 
We copy the form used in [Hellwarth1999].

For each temperature, the total free energy of the Feynman coupled
phonon-electron system is minimised by optimising coefficients which control
the spring-coupling coefficient and effective-mass for the phonon cloud. 
You have integrated through the infinite quantum field of the harmonic
oscillators which make up the lattice, to collapse the problem abck to
a quasi-particle. 

The codes then calculate the polaron mobility, with both the original low-temperature
FHIP approximation [Feynman1962]; Kadanoff's [Kadanoff1964] Boltzmann equation
motivated phonon-emission correction to the FHIP; and Hellwarth et al.'s
[Hellwarth1999] method. 
This last method (Hellwarth) is probably the most accurate. 
It uses a more general (non low-temperature-approximation) result in
[Feynman1962], then directly calculates the contour integral for the polaron
self-energy with numerical integration.

Underlying all this is the simplified Frohlich Hamiltonian [Frohlich1952] for
a single electron interacting with a phonon cloud of non-interacting (harmonic)
phonons.
The electron-phonon interaction for a polar system is treated at the simple
level of being the dielectric response. 
These are the infrared-active modes present at the Gamma point in the Brillouin
zone. 
Along with an effective mode frequency, the dielectric constants are used to
calculate the dimensionless 'α' parameter describing the electron-phonon
coupling.  
(In a simple covalent semiconductor system, the only dielectric active mode is
the linear-optical mode.)

The Feynman model offers a direct solution of this most simple quantum field
problem. 
The the infinite phonon (quantum) field is 'integrated out' by path
integration. 
The soluble system is one in which you have an electron interacting by
a (harmonic) spring constant with a mass representing the phonon drag. 
The variational method allows you to find a set of parameters for this
simplified system which produces the smallest free-energy. 

This is now a (renormalised) single particle system, a quasi-particle.

These Julia codes use Hellwarth's [Hellwarth1999] presentation of Osaka's variational
free-energies for the Feynman model.
We optimise the `v` and `w` parameters for these finite-temperature free energies.
These can be alternatively restated the mass 'M' and spring-constant 'k' of the
coupled phonon-electron Feynman model.

Here we apply these methods to the case of hybrid halide perovskites.
The method provides the temperature dependent polaron-mobility without any free parameters.
No arbitrary relaxation time is needed or used. The scattering processes are
treated directly, by including an effective electron-phonon coupling in the
specification of the Frohlich 'alpha' parameter, and then all other features
come from solving the model.
The original Feynman model is correct to all orders in alpha, and the Hellwarth
direct contour-integration of the general Feynman mobility statement is
suitable for high temperature.

It was necessary to return to these (rather old!) papers and resolve the
models, as hybrid halide perovskites are soft materials with low energy
phonons. Therefore the effective temperature in terms of a reduced
thermodynamic beta (Beta=hbar omega / (k_Boltzmann * Temperature) ) is much
smaller than previously considered.

A finaly note that in [Hellwarth1999], there is a mistake in the formula for 'b',
which is also present in their prior PRL [Biaggio1997].
It is correct in [Feynman1962], where there is no factor of b on the right-hand
side.
It doesn't matter too much, as setting it to zero makes ~0.1% difference in the
eventual mobility.
However, since we're integrating numerically anyway, we may as well calculate
it explicitly.


## Bibliography

This bibliography is listed in vague order of utility; I recommend reading the
first ones first!

Feynman also describes his Polaron model in more detail in both 'Statistical
Mechanics' [Feynman1972] and 'Quantum Mechanics and Path Integrals'
[FeynmanHibbs1965]. Note that the differing presentations of Feynman do not always agree in detail.

Schulman's 'Techniques and applications of path integration' has a 10-page
chapter on the Polaron problem. It tries to unify the Feynman prescriptions.
<commit_msg>Edits / corrections. Part of the JOSS documentation push.<commit_after>## Overview 

These codes solve the Feynman polaron mode [Feynman1955] with Osaka's
[Osaka1961] finite-temperature energies. 
We use the form of these free energies, as presented in [Hellwarth1999].

For each temperature, the total free energy of the Feynman coupled
phonon-electron system is minimised by optimising coefficients (`v` and `w`)
which, equivalently, describe the spring-coupling coefficient (`k`) and
effective-mass (`M`) of the phonon cloud (i.e. the phonon drag / phonon surfing
contribution). 
This integrates through the infinite quantum field of the harmonic oscillators
which make up the dynamic response of the lattice, to simplify the problem back
to a quasi-particle (the polaron). 

The codes calculate the polaron mobility, with both the original
low-temperature FHIP asymptotic approximation [Feynman1962]; Kadanoff's
[Kadanoff1964] Boltzmann equation motivated phonon-emission correction to the
FHIP; and Hellwarth et al.'s [Hellwarth1999] method. 
This last method (Hellwarth) is probably the most accurate. 
This uses a more general result (Eqn. 44-47) in [Feynman1962], directly
evaluating the contour integral for the polaron self-energy numerically. 

Underlying all this is the simplified Fröhlich Hamiltonian [Frohlich1952] for
a single electron interacting with a phonon cloud of non-interacting (harmonic)
phonons.
The electron-phonon interaction for a polar system is treated at the simple
level of being the dielectric response. 
These are the infrared-active modes present at the Gamma point in the Brillouin
zone. 
Along with an effective mode frequency, the dielectric constants are used to
calculate the dimensionless 'α' parameter describing the electron-phonon
coupling.  
(In a simple covalent semiconductor system, the only dielectric active mode is
the linear-optical mode.)

The Feynman model offers a direct solution of this most simple quantum field
problem. 
The infinite phonon (quantum) field is 'integrated out' by path
integration. 
The soluble system is one in which you have an electron interacting by
a (harmonic) spring constant with a mass representing the phonon drag. 
The variational method allows you to find a set of parameters for this
simplified system which produces the smallest free-energy. 

This is now a (renormalised) single particle system, a quasi-particle.

These Julia codes use Hellwarth's [Hellwarth1999] presentation of Osaka's variational
free-energies for the Feynman model.
We optimise the `v` and `w` parameters for these finite-temperature free energies.
These can be alternatively restated the mass 'M' and spring-constant 'k' of the
coupled phonon-electron Feynman model.

Here we apply these methods to the case of hybrid halide perovskites.
The method provides the temperature dependent polaron-mobility without any free parameters.
No arbitrary relaxation time is needed or used. The scattering processes are
treated directly, by including an effective electron-phonon coupling in the
specification of the Fröhlich α/'alpha' parameter, and then all other features
come from solving the model.
The original Feynman model is correct to all orders in alpha, and the Hellwarth
direct contour-integration of the general Feynman mobility statement is
suitable for high temperature.

It was necessary to return to these (rather old!) papers and resolve the
models, as hybrid halide perovskites are soft materials with low energy
phonons. Therefore the effective temperature in terms of a reduced
thermodynamic beta (Beta=hbar omega / (k_Boltzmann * Temperature) ) is much
smaller than previously considered.

A final note that: in [Hellwarth1999], there is a mistake in the formula for 'b',
which is also present in their prior PRL [Biaggio1997].
It is correct in [Feynman1962], where there is no factor of b on the right-hand
side.
They probably didn't notice this, as they set it to zero. 
This doesn't make too much difference (~0.1%, for the hybrid halide
perovskites) to the calculated mobility.  
Since we're integrating numerically anyway, we may as well calculate it
explicitly.


## Bibliography

This bibliography is listed in vague order of utility; I recommend reading the
first ones first!

Feynman also describes his Polaron model in more detail in both 'Statistical
Mechanics' [Feynman1972] and 'Quantum Mechanics and Path Integrals'
[FeynmanHibbs1965]. Note that the differing presentations of Feynman do not always agree in detail.

Schulman's 'Techniques and applications of path integration' has a 10-page
chapter on the Polaron problem. It tries to unify the Feynman prescriptions.

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