Feynman's variational path-integral model for the Fröhlich polaron. Calculates temperature dependent polaron mobilities, and other polaron observables.
Author jarvist
22 Stars
Updated Last
1 Year Ago
Started In
April 2017


License: MIT made-with-julia DOI docs-latest

Build status codecov.io

PolaronMobility.jl is a Julia package which calculates the temperature-dependent polaron mobility for a material.

This is based on the Feynman variational solution to the Polaron problem. The electron-phonon coupling is treated as an effective α (alpha) Frohlich Hamiltonian parameter. The band structure is treated with an effective mass theory. The variational problem is solved numerically for finite-temperature free energies. (The original 1960s work, and thus textbook solutions, often use asymptotic approximations to the integrals, with a more simple athermal action.)
The mobility is calculated in three ways:

  1. numerically by integrating the polaron self-energy along the imaginary axis (Hellwarth1999)
  2. using Kadanoff's Boltzmann equation approximation (Kadanoff1963)
  3. using the FHIP low-temperature asymptotic solution (FHIP)

These three methods are in approximately descending order of accuracy.

We provide parameters for various metal-halide Perovskites, and other interesting systems.

The motivation for developing these codes was to enable polaron mobility calculations on arbitrary materials. They also provide the only extant implementation of Feynman's variational method.
They offer a convenient basis for writing codes that build on these variational solutions.

More extensive documentation, is perhaps easiest to read and understand alongside the first paper: ArXiv:1704.05404 / Frost2017PRB.


To install, type the following at the Julia (>1.0) REPL:

julia> import Pkg; Pkg.add("PolaronMobility")

Cloud notebook

There is an example notebook which can be run interactively on the (free) MyBinder notebook server. This is the fastest way to calculate a few polaron parameters, if you do not have Julia installed locally.

  1. Click on Binder
  2. That's it!

(Currently plotting does not work, as the Docker image is not built with the (heavy weight) Plots dependency, and I'm not sure how I can do this just for MyBinder, without requiring it generally for PolaronMobility.jl. If this is problematic for you, please open an issue and I'll try to fix it!)


As an example:

using PolaronMobility
MAPIe=polaronmobility(300, 4.5, 24.1, 2.25, 0.12)

Will calculate the polaron mobility for methyl-ammonium lead halide perovskite (f=2.25 THz; ϵoptical=4.5; ϵstatic=24.1; effective-mass=0.12 electron-masses) at 300 K.

An abbreviated output should look like:

T: 300.000000 β: 2.41e+20 βred: 0.36 ħω  = 9.31 meV     Converged? : true
 VariationalParams v= 19.86  w= 16.96   ||   M=0.371407 k=106.835753    
 POLARON SIZE (rf), following Schultz1959. (s.d. of Gaussian polaron ψ )
     Schultz1959(2.4): rf= 0.528075 (int units) = 2.68001e-09 m [SI]
 Polaron Free Energy: A= -6.448815 B= 7.355626 C= 2.911977 F= -3.818788  = -35.534786 meV
Polaron Mobility theories:
    μ(FHIP)= 0.082049 m^2/Vs    = 820.49 cm^2/Vs
     Eqm. Phonon. pop. Nbar: 2.308150 
    μ(Kadanoff1963 [Eqn. 25]) = 0.019689 m^2/Vs      = 196.89 cm^2/Vs
    Tau=1/Gamma0 = 1.15751e-13 = 0.115751 ps
    μ(Hellwarth1999)= 0.013642 m^2/Vs   = 136.42 cm^2/Vs

Further details in the documentation.

Research outputs

The central output of this model are temperature-dependent polaron mobilities:

MAPI Polaron mobility, plotted vs experimental data

From the variational solution, you have characterised the polarons in your system. This gives access to the effective mass renormalisations (phonon drag), polaron binding energies, effective electron-phonon coupling parameters, etc.

Community guidelines

Contributions to the code (extending that which is calculated), or additional physical systems / examples, are very welcome.

If you have questions about the software, scientific questions, or find errors, please create a GitHub issue.


If you find this package (or snippets, such as the entered and tested free-energy expressions) useful for your work, please cite the paper Frost2017PRB.

  doi = {10.1103/physrevb.96.195202},
  url = {https://doi.org/10.1103/physrevb.96.195202},
  year  = {2017},
  month = {nov},
  publisher = {American Physical Society ({APS})},
  volume = {96},
  number = {19},
  author = {Jarvist Moore Frost},
  title = {Calculating polaron mobility in halide perovskites},
  journal = {Physical Review B}

These codes use the Optim.jl optimisation library to do the essential calculation of the Feynman variational theory. DOI

Used By Packages

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