jarvist/polaronmobility.jl

Feynman's variational path-integral model for the Fröhlich polaron. Calculates temperature dependent polaron mobilities, and other polaron observables.

PolaronMobility.jl

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 dimensionless 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.

Installation

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
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!)

Using

As an example:

julia> using PolaronMobility
julia> MAPIe = material(4.5, 24.1, 0.12, 2.25)
julia> MAPIe_polaron = frohlichpolaron(MAPIe, 300; verbose = true)

Will calculate the polaron mobility for methyl-ammonium lead halide perovskite (ϵ_optical = 4.5, ϵ_static = 24.1, effective_mass = 0.12 electron-masses, f = 2.25 THz) at 300 K.

An abbreviated output should look like:

------------------------------------------
           Material Information           
------------------------------------------
Optic dielectric   | ϵo = 4.5
Static dielectric  | ϵs = 24.1
Ionic dielectric   | ϵi = 19.6
Band mass          | mb = 0.12
Fröhlich coupling  | α = 2.39394
Phonon frequencies | f = 2.25
Eff Phonon freq    | feff = 2.25
IR activities      | ir = 1
Unit cell volume   | V = 1
-------------------------------------------

-----------------------------------------------------------------------
                         Polaron Information:                          
-----------------------------------------------------------------------
Phonon frequencies             | ωeff = 2.25 | ω = 2.25
Fröhlich coupling              | αeff = 2.39394 | α = 2.39394
Number of spatial dimensions   | d = 3
Small α→0 energy               | Fs = -2.46469
Large α→∞ energy               | Fl = -3.43751
Small α→0 fictitious mass      | Ms = 0.542264
Large α→∞ fictitious mass      | Ml = 0.0666023
Small α→0 polaron radius       | Rs = 1.67917
Large α→∞ polaron radius       | Rl = 9.00108
Large α→∞ FC peak freq.        | ΩFC = 0.810764
-----------------------------------------------------------------------
                     Zero Temperature Information:                     
-----------------------------------------------------------------------
Variational parameter          | v0 = 3.30877
Variational parameter          | w0 = 2.66327
Energy                         | E0 = -5.56947
Electron energy                | A0 = -2.17859
Interaction energy             | B0 = 5.78198
Trial energy                   | C0 = 1.96608
Fictitious spring constant     | κ0 = 3.855
Fictitious mass                | M0 = 0.543495
Fictitious mass (asymptotic)   | M0a = 1.24237
Reduced mass                   | M0r = 0.35212
Polaron radius                 | R0 = 0.817278
-----------------------------------------------------------------------
                    Finite Temperature Information:                    
-----------------------------------------------------------------------
Temperatures                   | T = 300
Reduced thermodynamic          | β = 0.159975
Variational parameter          | v = 19.8612
Variational parameter          | w = 16.9599
Free energy                    | F = -8.59191
Electron energy                | A = -14.5096
Interaction energy             | B = 16.5498
Trial energy                   | C = 6.55175
Fictitious spring constant     | κ = 106.831
Fictitious mass                | M = 0.371408
Fictitious mass (asymptotic)   | Ma = 1.17107
Reduced mass                   | Mr = 0.270822
Polaron radius                 | R = 0.0722548
-----------------------------------------------------------------------
                      DC Mobility Information:                         
-----------------------------------------------------------------------
Finite temperature mobility    | μ = 0.487357
FHIP low-temp. mobility        | μFHIP = 2.93118
Devreese low-temp. mobility    | μD = 0.703373
Kadanoff low-temp. mobility    | μK = 0.703373
Hellwarth mobility             | μH = 1.09655
Hellwarth mobility (b=0)       | μH0 = 1.09824
Kadanoff relaxation time       | τ = 0.964611
-----------------------------------------------------------------------

The frohlichpolaron method returns a Frohlich struct type. To access a value we can call the variable from the table above. For example, to get the mobility for MAPIe we do:

julia> MAPIe_polaron.μ
0.48735686547867546

This package makes use of Unitful.jl to convert these values into unitful polaron types. For example, to add units to the above MAPIe polaron:

julia> using Unitful
julia> addunits!(MAPIe_polaron)
julia> MAPIe_polaron.μ
0.48735686547867546 μ₀

All the values above are given in "polaron" units. We can convert to a specific unit using Unitful. For example, to obtain the MAPIe polaron ground-state energy (meV), room-temperature free energy (meV) and mobility (cm^2/V/s):

julia> MAPIe_polaron.F0 |> u"meV"
-23.03349215713356 meV

julia> MAPIe_polaron.F |> u"meV"
-35.5332946810453 meV

julia> MAPIe_polaron.μ |> u"cm^2/V/s"
136.42332082324646 cm² s⁻¹ V⁻¹

The frohlichpolaron method can generally accept a range of α electron-phonon coupling parameters, a range of temperatures for finite temperature properties such as free energy and mobility, and range of Electric Field frequencies for calculating dynamical properties such as the complex conductivity. For example:

julia> frohlich_polaron = frohlichpolaron(1:12, 1:400, 0:0.1:10)

will calculate all properties for α's 1 to 12, temperature 1K to 400K and Electric field frequencies 0 to 10 THz.

It is also possible to change the number of spatial dimensions for the polaron using dims = # or as an array/range dims = [2, 3]. For example:

julia> frohlich_polaron = frohlichpolaron(1:12, 1:400, 0:0.1:10; dims = 2)
julia> frohlich_polaron = frohlichpolaron(1:12, 1:400, 0:0.1:10; dims = [2, 3])

Note that the Frohlich polaron does not exist in 1D.

TIP: If the frohlichpolaron errors or you interrupt the calculation whilst you have verbose = true, use print("\e[?25h") to return your cursor.

Multiple phonon modes

To calculate polaron properties for a material with multiple phonon modes the code requires a vector of phonon frequencies (at the gamma point), a vector of corresponding infrared activities for each phonon mode, and the unitcell volume of the material. The code uses this additional information to calculate the ionic dielectric contributions to the static dielectric function. For example, for MAPIe:

julia> volume = (6.29e-10)^3
julia> m_eff = 0.12
julia> ϵ_static = 24.1
julia> ϵ_optic = 4.5
julia> phonon_freqs = [4.016471586720514, 3.887605410774121, 3.5313112232401513, 2.755392921480459, 2.4380741812443247, 2.2490917637719408, 2.079632190634424, 2.0336707697261187, 1.5673011873879714, 1.0188379384951798, 1.0022960504442775, 0.9970130778462072, 0.9201781906386209, 0.800604081794174, 0.5738689505255512]
julia> ir_activities = [0.08168931020200264, 0.006311654262282101, 0.05353548710183397, 0.021303020776321225, 0.23162784335484837, 0.2622203718355982, 0.23382298607799906, 0.0623239656843172, 0.0367465760261409, 0.0126328938653956, 0.006817361620021601, 0.0103757951973341, 0.01095811116040592, 0.0016830270365341532, 0.00646428491253749]

julia> MAPIe = material(ϵ_optic, ϵ_static, m_eff, phonon_freqs, ir_activities, volume)

Hellwarth B Scheme... (athermal)
Hellwarth (58) summation: 0.03803673767058733
Hellwarth (59) summation (total ir activity ^2): 0.19283002835623678
Hellwarth (59) W_e (total ir activity ): 0.4391241605243747
Hellwarth (61) Omega (freq): 2.251571287857919
------------------------------------------
           Material Information           
------------------------------------------
Optic dielectric   | ϵo = 4.5
Static dielectric  | ϵs = 24.1
Ionic dielectric   | ϵi = [0.299962, 0.0247382, 0.254308, 0.166213, 2.30827, 3.07073, 3.20261, 0.892655, 0.886139, 0.720913, 0.401989, 0.618315, 0.766623, 0.155541, 1.16275]
Band mass          | mb = 0.12
Fröhlich coupling  | α = [0.0340093, 0.0028509, 0.03075, 0.0227523, 0.335905, 0.465256, 0.50462, 0.142232, 0.160835, 0.162287, 0.0912368, 0.140706, 0.181593, 0.0394994, 0.348765]
Phonon frequencies | f = [4.01647, 3.88761, 3.53131, 2.75539, 2.43807, 2.24909, 2.07963, 2.03367, 1.5673, 1.01884, 1.0023, 0.997013, 0.920178, 0.800604, 0.573869]
Eff Phonon freq    | feff = 2.25157
IR activities      | ir = [0.0816893, 0.00631165, 0.0535355, 0.021303, 0.231628, 0.26222, 0.233823, 0.062324, 0.0367466, 0.0126329, 0.00681736, 0.0103758, 0.0109581, 0.00168303, 0.00646428]
Unit cell volume   | V = 2.48858e-28
-------------------------------------------

We can then put this into the frohlichpolaron function as before. For example here we would get:

julia> MAPIe_polaron = frohlichpolaron(MAPIe, 300; verbose = true)

-----------------------------------------------------------------------
                         Polaron Information:                          
-----------------------------------------------------------------------
Phonon frequencies             | ωeff = 2.25157 | ω = [4.01647, 3.88761, 3.53131, 2.75539, 2.43807, 2.24909, 2.07963, 2.03367, 1.5673, 1.01884, 1.0023, 0.997013, 0.920178, 0.800604, 0.573869]
Fröhlich coupling              | αeff = 2.6633 | α = [0.0340093, 0.0028509, 0.03075, 0.0227523, 0.335905, 0.465256, 0.50462, 0.142232, 0.160835, 0.162287, 0.0912368, 0.140706, 0.181593, 0.0394994, 0.348765]
Number of spatial dimensions   | d = 3
Small α→0 energy               | Fs = -2.75087
Large α→∞ energy               | Fl = -3.58205
Small α→0 fictitious mass      | Ms = 0.621212
Large α→∞ fictitious mass      | Ml = 0.102027
Small α→0 polaron radius       | Rs = 1.592
Large α→∞ polaron radius       | Rl = 10.0138
Large α→∞ FC peak freq.        | ΩFC = 1.00348
-----------------------------------------------------------------------
                     Zero Temperature Information:                     
-----------------------------------------------------------------------
Variational parameter          | v0 = 3.29227
Variational parameter          | w0 = 2.67919
Energy                         | E0 = -4.71904
Electron energy                | A0 = -1.83134
Interaction energy             | B0 = 4.88955
Trial energy                   | C0 = 1.66083
Fictitious spring constant     | κ0 = 3.66096
Fictitious mass                | M0 = 0.510019
Fictitious mass (asymptotic)   | M0a = 1.22883
Reduced mass                   | M0r = 0.337757
Polaron radius                 | R0 = 0.858447
-----------------------------------------------------------------------
                    Finite Temperature Information:                    
-----------------------------------------------------------------------
Temperatures                   | T = 300
Reduced thermodynamic          | β = [0.159975, 0.159975, 0.159975, 0.159975, 0.159975, 0.159975, 0.159975, 0.159975, 0.159975, 0.159975, 0.159975, 0.159975, 0.159975, 0.159975, 0.159975]
Variational parameter          | v = 35.1921
Variational parameter          | w = 32.4542
Free energy                    | F = -10.3578
Electron energy                | A = -11.5987
Interaction energy             | B = 15.4728
Trial energy                   | C = 6.48363
Fictitious spring constant     | κ = 185.207
Fictitious mass                | M = 0.175839
Fictitious mass (asymptotic)   | Ma = 1.08436
Reduced mass                   | Mr = 0.149544
Polaron radius                 | R = 0.0554786
-----------------------------------------------------------------------
                      DC Mobility Information:                         
-----------------------------------------------------------------------
Finite temperature mobility    | μ = 0.572962
FHIP low-temp. mobility        | μFHIP = 4.71425
Devreese low-temp. mobility    | μD = 0.0673372
Kadanoff low-temp. mobility    | μK = 0.0673372
Hellwarth mobility             | μH = 0.0360092
Hellwarth mobility (b=0)       | μH0 = 0.0359907
Kadanoff relaxation time       | τ = 0.0572674
-----------------------------------------------------------------------

Using Unitful we then get the following energies and mobility:

julia> MAPIe_polaron.F0 |> u"meV"
-19.516365044323575 meV

julia> MAPIe_polaron.F |> u"meV"
-42.83622939936868 meV

julia> MAPIe_polaron.μ |> u"cm^2/V/s"
160.38638000423998 cm² s⁻¹ V⁻¹

Multiple variational parameters

The base trial action used in the Feynman-Jensen variational path integral approximation has only two variational parameters called v and w. This can be generalised to more variational parameters v1, w1, v2, w2 etc where v1 > w1 > v2 > w2 > .... Including more variational paramaters can produce better variational results, improving the upper-bound on the polaron free energy.

To use this functionality, you have to provide an initial vector of guesses for the variational parameters. For example for the single effective phonon mode MAPIe:

julia> MAPIe_polaron = frohlichpolaron(MAPIe; v_guesses = [4, 3], w_guesses = [3.5, 2.5], verbose = true)

-----------------------------------------------------------------------
               Polaron Information: [1 / 1 (100.0 %)]
-----------------------------------------------------------------------
Phonon frequencies             | ωeff = 2.25 | ω = 2.25
Fröhlich coupling              | αeff = 2.39394 | α = 2.3939410167951287
Small α→0 energy               | Fs = -2.46469
Large α→∞ energy               | Fl = -3.43751
Small α→0 fictitious mass      | Ms = 0.542264
Large α→∞ fictitious mass      | Ml = 0.0666023
Small α→0 polaron radius       | Rs = 1.67917
Large α→∞ polaron radius       | Rl = 9.00108
Large α→∞ FC peak freq.        | ΩFC = 0.810764
Number of dimensions [1 / 1]   | d = 3
-----------------------------------------------------------------------
                   Zero Temperature Information:                       
-----------------------------------------------------------------------
Variational parameter          | v0 = [2.28596, 12.4981]
Variational parameter          | w0 = [1.86037, 12.1284]
Energy                         | E0 = -5.57292
Electron energy                | A0 = -2.684
Interaction energy             | B0 = 5.78926
Trial energy                   | C0 = 2.46766
Fictitious spring constant     | κ0 = [1.76463, 9.1037]
Fictitious mass                | M0 = [0.509867, 0.0618885]
Fictitious mass (asymptotic)   | M0a = [1.22877, 1.03048]
Reduced mass                   | M0r = [0.33769, 0.0582815]
Polaron radius                 | R0 = [1.48402, 0.672612]
-----------------------------------------------------------------------

which gives a (polaron units) ground-state energy of -5.57292 compared to the previous prediction -5.56947.

Note that whilst the code is capable of finite temperatures, more variational parameters and multiple modes all at once - the resultant energy landscape is difficult to optimise on and find converged results and requires well informed initial guesses to arrive at a correct solution.

The Holstein polaron model

To calculate properties for a small Holstein-like polaron, use the holstenpolaron() function. It accepts all the same arguements as for the previous case with Frohlich, but will use the Holstein model instead and store the information in the Holstein type.

For example:

julia> holstein_polaron = holsteinpolaron(1:3, [0, 300], [0, 1]; dims = [1, 2, 3], verbose = true)

-----------------------------------------------------------------------
                         Polaron Information:                          
-----------------------------------------------------------------------
Phonon frequencies             | ωeff = 1 | ω = 1
Holstein coupling              | αeff = [1, 2, 3] | α = [1, 2, 3]
Number of spatial dimensions   | d = 3
-----------------------------------------------------------------------
                     Zero Temperature Information:                     
-----------------------------------------------------------------------
Variational parameter          | v0 = [4.06018, 4.80557, 7.07573]
Variational parameter          | w0 = [3.30811, 1.51507, 1.16092]
Total energy                   | F0 = [-7.3982, -9.21291, -12.1558]
Electron energy                | A0 = [-1.1281, -4.93576, -8.87222]
Interaction energy             | B0 = [1.50268, 4.90273, 9.86404]
Trial energy                   | C0 = [1.02362, 3.24593, 5.16394]
Fictitious spring constant     | κ0 = [5.54143, 20.7981, 48.7182]
Fictitious mass                | M0 = [0.506361, 9.06064, 36.1484]
Fictitious mass (asymptotic)   | M0a = [1.22734, 3.17185, 6.09495]
Reduced mass                   | M0r = [0.336149, 0.900603, 0.973081]
Polaron radius                 | R0 = [0.629813, 0.182562, 0.0945703]
-----------------------------------------------------------------------
                    Finite Temperature Information:                    
-----------------------------------------------------------------------
Temperatures                   | T = [0, 300]
Reduced thermodynamic          | β = [Inf, 0.00333333]
Variational parameter          | v = [4.06018 4.80557 7.07573; 6.76201 8.98156 10.7774]
Variational parameter          | w = [3.30811 1.51507 1.16092; 3.75187 3.22804 2.85246]
Free energy                    | F = [-7.3982 -9.21291 -12.1558; -1751.64 -1757.63 -1763.61]
Electron energy                | A = [-1.1281 -4.93576 -8.87222; 1739.64 1739.63 1739.61]
Interaction energy             | B = [1.50268 4.90273 9.86404; 5.98483 11.9697 17.9545]
Trial energy                   | C = [1.02362 3.24593 5.16394; 0.0131867 0.0292696 0.0450057]
Fictitious spring constant     | κ = [5.54143 20.7981 48.7182; 31.6483 70.2482 108.016]
Fictitious mass                | M = [0.506361 9.06064 36.1484; 2.2483 6.7415 13.2754]
Fictitious mass (asymptotic)   | Ma = [1.22734 3.17185 6.09495; 1.80231 2.78236 3.77828]
Reduced mass                   | Mr = [0.336149 0.900603 0.973081; 0.692147 0.870826 0.929949]
Polaron radius                 | R = [0.629813 0.182562 0.0945703; 0.142314 0.0738927 0.0526417]
-----------------------------------------------------------------------
                      DC Mobility Information:                         
-----------------------------------------------------------------------
Finite temperature mobility    | μ = [Inf Inf Inf; 440.224 199.198 95.5733]
-----------------------------------------------------------------------
                  Frequency Response Information:                      
-----------------------------------------------------------------------
Electric field frequency       | Ω = [0, 1]
Memory function                | χ = [Inf+0.0im Inf+0.0im; -2.90365+5.53403im -3.85738e-5+0.00226794im;;; Inf+0.0im Inf+0.0im; -185.241+292.499im -0.000229113+0.00496805im;;; Inf+0.0im Inf+0.0im; -767.896+2281.86im -0.00127186+0.0101249im]
Complex impedance              | z = [0.0+Inf*im 0.0+Inf*im; 5.53403+1.90365im 0.00226794-0.999961im;;; 0.0+Inf*im 0.0+Inf*im; 292.499+184.241im 0.00496805-0.999771im;;; 0.0+Inf*im 0.0+Inf*im; 2281.86+766.896im 0.0101249-0.998728im]
Complex conductivity           | σ = [0.0+0.0im 0.0+0.0im; 0.16158-0.0555821im 0.0022681+1.00003im;;; 0.0+0.0im 0.0+0.0im; 0.00244768-0.00154176im 0.00497021+1.0002im;;; 0.0+0.0im 0.0+0.0im; 0.000393763-0.000132337im 0.0101496+1.00117im]
-----------------------------------------------------------------------

Using addunits!(holstein_polaron) will then add the specific Holstien polaron units (energy in terms of the electron hopping energy instead of phonon energy as for the Frohlich model) which can then be transformed into other SI units using Unitful.

General electron-phonon matrices and k-space integration

TBC

Saving and loading polaron data

The Frohlich and Holstein polaron types can be saved as .jld files and loaded back into Julia. For example:

julia> save_frohlich_polaron(MAPIe_polaron, "MAPIe_polaron")
Saving polaron data to MAPIe_polaron.jld ...
... Polaron data saved.

julia> MAPIe_polaron = load_frohlich_polaron("MAPIe_polaron.jld")
Loading polaron data from MAPIe_polaron.jld ...
... Polaron loaded.
-----------------------------------------------------------------------
                         Polaron Information:                          
-----------------------------------------------------------------------
Phonon frequencies             | ωeff = 2.25 | ω = 2.25
Fröhlich coupling              | αeff = 2.39394 | α = 2.39394
Number of spatial dimensions   | d = 3
Small α→0 energy               | Fs = -2.46469
Large α→∞ energy               | Fl = -3.43751
Small α→0 fictitious mass      | Ms = 0.542264
Large α→∞ fictitious mass      | Ml = 0.0666023
Small α→0 polaron radius       | Rs = 1.67917
Large α→∞ polaron radius       | Rl = 9.00108
Large α→∞ FC peak freq.        | ΩFC = 0.810764
-----------------------------------------------------------------------
                     Zero Temperature Information:                     
-----------------------------------------------------------------------
Variational parameter          | v0 = 3.30877
Variational parameter          | w0 = 2.66327
Energy                         | E0 = -5.56947
Electron energy                | A0 = -2.17859
Interaction energy             | B0 = 5.78198
Trial energy                   | C0 = 1.96608
Fictitious spring constant     | κ0 = 3.855
Fictitious mass                | M0 = 0.543495
Fictitious mass (asymptotic)   | M0a = 1.24237
Reduced mass                   | M0r = 0.35212
Polaron radius                 | R0 = 0.817278
-----------------------------------------------------------------------
                    Finite Temperature Information:                    
-----------------------------------------------------------------------
Temperatures                   | T = 300
Reduced thermodynamic          | β = 0.159975
Variational parameter          | v = 19.8612
Variational parameter          | w = 16.9599
Free energy                    | F = -8.59191
Electron energy                | A = -14.5096
Interaction energy             | B = 16.5498
Trial energy                   | C = 6.55175
Fictitious spring constant     | κ = 106.831
Fictitious mass                | M = 0.371408
Fictitious mass (asymptotic)   | Ma = 1.17107
Reduced mass                   | Mr = 0.270822
Polaron radius                 | R = 0.0722548
-----------------------------------------------------------------------
                      DC Mobility Information:                         
-----------------------------------------------------------------------
Finite temperature mobility    | μ = 0.487357
FHIP low-temp. mobility        | μFHIP = 2.93118
Devreese low-temp. mobility    | μD = 0.703373
Kadanoff low-temp. mobility    | μK = 0.703373
Hellwarth mobility             | μH = 1.09655
Hellwarth mobility (b=0)       | μH0 = 1.09824
Kadanoff relaxation time       | τ = 0.964611
-----------------------------------------------------------------------
                  Frequency Response Information:                      
-----------------------------------------------------------------------
Electric field frequency       | Ω = 0
Memory function                | χ = Inf+0.0im
Complex impedance              | z = 0.0+Inf*im
Complex conductivity           | σ = 0.0+0.0im
-----------------------------------------------------------------------

To do the same for a Holstein polaron use save_holstein_polaron and load_holstein_polaron.

Further details in the documentation.

Research outputs

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

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.

Reference

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.

@article{Frost2017,
  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.

polaronmobility.jl's Issues

Type instability in the Polaron type

This is very messy, and should be refactored. I think it also ends up as type Any which is not great for efficiency.

PolaronMobility.jl/src/types.jl

Lines 11 to 36 in 51aa09a

	# Structure to store data of polaron solution + other parameters, for each temperature
	struct Polaron
	T
	# Mobilities
	Kμ; Hμ; FHIPμ
	# Spring constant and renormalised (phonon-drag) mass
	k; M
	# Osaka free energy components (A,B,C) and total (F). See Hellwarth et al. 1999 PRB Part IV
	A; B; C; F
	# Relaxation time from Kadanoff Boltzmann transport equation
	Tau
	# Raw variational parameters
	v; w
	# Reduced thermodynamic beta
	βred
	# Feynman polaron radius (Schultz), in SI units. Then also the small-alpha asymptotic approx
	rfsi; rfsmallalpha
	# Setup of simulation. These parameters are sent to the function.
	# Alpha = Frohlich alpha
	α
	# Band effective mass
	mb
	# Effective dielectric frequency
	ω
	end
	Polaron()=Polaron([],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[],[]) # structure initialisation

Multiple phonon branches

You have a load of half interpreted versions of this in various lab books. So you should actually implement it.
The basic idea is to put multiple phonon branches in the 'true' Lagrangian, against which the model v and w Lagrangian is varied. This should give more temperature dependent structure in the e.g. mobility curves, as individual phonon modes go into resonance.

• figure out a nice way to go from i.r. activity + frequency to dielectric constant contribution
• units? interface with a VASP script? Intermediate .dat file to keep interface cleanly separate?
• implement this simply as a set of contributions to the Frohlich Hamiltonian, used in the calculation of a single v and w polaron at finite T.
• if this is too expensive, consider some variational approximation where you first project the complex spectra onto a representative set of modes
• stare at the results
• compare and contrast to the Hellwarth1999 averaging techniques
• sort out the Hellwarth1999 A scheme (finite temperature). Half implemented in a Jupyter notebook somewhere?
• see whether you can treat mode lifetime (i.e. from Phonopy3) at the same time, by adding a Gaussian width to the resonance

Fix MyBinder

Error at build currently:

Waiting for build to start...
Picked Git content provider.
Cloning into '/tmp/repo2dockeruxnjeh13'...
HEAD is now at 51aa09a Julia 1.0 style Project.toml via gen_project.jl
Building conda environment for python=3.7Error during build:  0.6.0 is not valid SemVer string

Should be on 1.x ?

JOSS Review: Documentation

I think there are a number of steps that can be taken in the documentation to help users get up and running more easily. This is connected to the JOSS review at openjournals/joss-reviews#566

• First off, the documentation is built as a single, large page, if this could be split into multiple pages, that would help ease navigation
• Installation instructions: these are in the README, but should also be included in the documentation
• Example usage: in addition to the plots shown in the readme, it would be helpful to include examples which show both the code usage and the resulting plot

@yxqd may also have some suggestions here.

JOSS review: readme

These are just a few minor suggestions accompanying the review, they don't need to hold up the publication.

• it is common to put all of the badges at the top of the readme (where you have the build and coverage), I would suggest including the badge for the docs there as well.
• do you have an example that runs on JuliaBox? https://www.juliabox.com/. I tried getting the package to build there, but the build failed. This might be a helpful way to get people up and running, but again, it is not required for the JOSS submission

Update documentation for v2

Documentation has been touched for years!
Probably this will also be in concert with a code-clean.

• get Documenter.jl working under CI, so it builds
• Fix bug where the sidebar has blank titles
• Update README on /docs subfolder
• Update docs ^)_(^

Optical absorption / Re χ

Optical absorption: https://github.com/jarvist/PolaronMobility.jl/blob/51aa09add590bd6b68406bbf16eb872d2e502a69/src/OedipusRex.jl

• Finish implementing it
• possibly by sorting out the integral
• reproduce the data in https://doi.org/10.1103/PhysRevB.5.2367
• think of some science to do with this?

JOSS Review: README.md

@jarvist Please consider simplify the README.md. Some of the information there is now available at the documentation site https://jarvist.github.io/PolaronMobility.jl/ and looks duplicate. It would be nice that README.md is a succinct summary and defer more elaborate discussions to the documentation site. This might be a good example.

See also: #1, openjournals/joss-reviews#566

Example Pluto notebook on Binder for v.2

This little webapp will build a working environment / URL to kick off a Binder run: http://pluto-on-binder.glitch.me/

BUT, currently all the Pluto notebooks won't run as they assume you can find PolaronMobility in '../'.
Now that v2 is registered, I suppose just including PolaronMobility as a dependency within the Pluto notebook should be quite consistent & reproducible.

• Figure out best way to run PolaronMobility on Binder
• Put together an exemplary notebook showing off some useful features of the package
• Cross link from the docs

TagBot trigger issue

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Register v.2.0.0

@JuliaRegistrator register

Massive performance regression at #1b7f414

Massive performance regression introduced with 1b7f414 . The tests almost suddenly hang, as they start running so slowly! The old 2017 era tests that run first seem to run fine though.
I couldn't see immediately where the issue was, as the diff on GitHub looked pretty confusing - lots of formatting updates at the same time as something mechanistic?

Susceptibility / Im χ

Get this code: https://github.com/jarvist/PolaronMobility.jl/blob/51aa09add590bd6b68406bbf16eb872d2e502a69/src/Susceptibility.jl

• working
• with the oscillatory integral understood
• some kind of substitution by parts (?) / rescaling to fix the numerics
• doing some useful science (i.e. like by like comparison to the Drude model, to show where the disagreement becomes stark)

Sort out / delete PlotPolaron

Sort out PlotPolaron (https://github.com/jarvist/PolaronMobility.jl/blob/51aa09add590bd6b68406bbf16eb872d2e502a69/src/PlotPolaron.jl) - this was a temporary fix to keep the Plots dependency away. Either it should be refactored into a Plots recipe, or perhaps just discarded.