Scientific discussion

Overview

These codes solve the Feynman polaron mode [Feynman1955] with Ōsaka'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 Ōsaka'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 / (kBoltzmann * Temperature) ; β=ħω/(kB.T) ) 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.

J.T. Devreese's "Fröhlich Polarons. Lecture course including detailed theoretical derivations" (6th edition, 2016) notes on the ArXiv is a very good place to start & to get an overview of the area. https://arxiv.org/abs/1611.06122


% This introduces two prescriptions for reducing a multi-mode polar lattice to
% a single ~mean-field~ response.
% It contains a modern version of the Ōsaka finite temperature free-energies
% for use in a variational solution of the Feynman temperature problem.
% It also includes how to (numerically) do the contour integration to get the
% DC-response of the polaron developed in Feynman1962.
@article{Hellwarth1999,
  doi = {10.1103/physrevb.60.299},
  url = {https://doi.org/10.1103%2Fphysrevb.60.299},
  year  = {1999},
  month = {jul},
  publisher = {American Physical Society ({APS})},
  volume = {60},
  number = {1},
  pages = {299--307},
  author = {Robert W. Hellwarth and Ivan Biaggio},
  title = {Mobility of an electron in a multimode polar lattice},
  journal = {Physical Review B}
}

% Boltzmann / relaxation time approximation solution of mobility in the Feynman
% polaron problem.
% We extract a relaxation time (+ offer this method of mobility).
@article{Kadanoff1963,
  doi = {10.1103/physrev.130.1364},
  url = {https://doi.org/10.1103%2Fphysrev.130.1364},
  year  = {1963},
  month = {may},
  publisher = {American Physical Society ({APS})},
  volume = {130},
  number = {4},
  pages = {1364--1369},
  author = {Leo P. Kadanoff},
  title = {Boltzmann Equation for Polarons},
  journal = {Physical Review}
}

% A long and very useful article developing response theories for the polaron.
% Mainly known for the FHIP mobility, which is low-temperature only.
@article{Feynman1962,
  doi = {10.1103/physrev.127.1004},
  url = {https://doi.org/10.1103%2Fphysrev.127.1004},
  year  = {1962},
  month = {aug},
  publisher = {American Physical Society ({APS})},
  volume = {127},
  number = {4},
  pages = {1004--1017},
  author = {R. P. Feynman and R. W. Hellwarth and C. K. Iddings and P. M. Platzman},
  title = {Mobility of Slow Electrons in a Polar Crystal},
  journal = {Physical Review}
}

% The original development of Feynman's solution to the polaron problem.
% Zero temperature approximate variational solutions developed (in limits w->0,
% or w=v).
% Perturbative theories of phonon-drag effective-mass renormalisation given.
% (i.e. where the 'me=1+alpha/6' & etc. limits are from. )
@article{Feynman1955,
  doi = {10.1103/physrev.97.660},
  url = {https://doi.org/10.1103%2Fphysrev.97.660},
  year  = {1955},
  month = {feb},
  publisher = {American Physical Society ({APS})},
  volume = {97},
  number = {3},
  pages = {660--665},
  author = {R. P. Feynman},
  title = {Slow Electrons in a Polar Crystal},
  journal = {Physical Review}
}

% Schultz seemed to spend his PhD solving the Feynman polaron problem with
% a digital computer.
% Lots of characterisation of the polaron state, and the introduction of an
% effective polaron size, from considering the variance of the Gaussian
% wavefunction.
% Some work towards polaron mobility, but not as developed as in Feynman et al. 1962.
% Schultz provides units for some of the quantities - which is useful!
@article{Schultz1959,
  doi = {10.1103/physrev.116.526},
  url = {https://doi.org/10.1103%2Fphysrev.116.526},
  year  = {1959},
  month = {nov},
  publisher = {American Physical Society ({APS})},
  volume = {116},
  number = {3},
  pages = {526--543},
  author = {T. D. Schultz},
  title = {Slow Electrons in Polar Crystals: Self-Energy,  Mass,  and Mobility},
  journal = {Physical Review}
}

% Free-energies of the finite interacting Polaron system.
@article{Osaka1961,
  doi = {10.1143/ptp.25.517},
  url = {https://doi.org/10.1143%2Fptp.25.517},
  year  = {1961},
  month = {apr},
  publisher = {Oxford University Press ({OUP})},
  volume = {25},
  number = {4},
  pages = {517--536},
  author = {Yukio \=Osaka},
  title = {Theory of Polaron Mobility},
  journal = {Progress of Theoretical Physics}
}

% Original statement of the Polaron problem + Frohlich Hamiltonian.
@article{Frohlich1952,
  doi = {10.1098/rspa.1952.0212},
  url = {https://doi.org/10.1098%2Frspa.1952.0212},
  year  = {1952},
  month = {dec},
  publisher = {The Royal Society},
  volume = {215},
  number = {1122},
  pages = {291--298},
  author = {H. Frohlich},
  title = {Interaction of Electrons with Lattice Vibrations},
  journal = {Proceedings of the Royal Society A: Mathematical,  Physical and Engineering Sciences}
}

@article{Thornber1970,
  doi = {10.1103/physrevb.1.4099},
  url = {https://doi.org/10.1103%2Fphysrevb.1.4099},
  year  = {1970},
  month = {may},
  publisher = {American Physical Society ({APS})},
  volume = {1},
  number = {10},
  pages = {4099--4114},
  author = {K. K. Thornber and Richard P. Feynman},
  title = {Velocity Acquired by an Electron in a Finite Electric Field in a Polar Crystal},
  journal = {Physical Review B}
}