In [1]:

    

# JuliaBox-Example.ipynb - live notebook for polarons
# Jarvist Moore Frost

push!(LOAD_PATH,"./src/") # load module from local directory
using PolaronMobility


    

In [2]:

    

# Physical constants
const hbar = const ħ = 1.05457162825e-34;          # kg m2 / s 
const eV = const q = const ElectronVolt = 1.602176487e-19;                         # kg m2 / s2 
const me=MassElectron = 9.10938188e-31;                          # kg
const Boltzmann = const kB =  1.3806504e-23;                  # kg m2 / K s2 
const ε_0 = 8.854E-12 #Units: C2N−1m−2, permittivity of free space
const cm1=2.997e10 # cm-1 to Herz


    

    
Out[2]:





2.997e10

In [3]:

    

?polaronmobility


    

    




search: polaronmobility PolaronMobility







    
Out[3]:





polaronmobility(Trange,
                ε_Inf, ε_S, freq, effectivemass;
                verbose::Bool=false)

Solves the Feynman polaron problem variationally with finite temperature
Osaka energies.  From the resulting v, and w parameters, calculates polaron
structure (wave function size, etc.).  Uses FHIP, Kadanoff (Boltzmann
relaxation time) and Hellwarth direct contour integration to predict
a temperature-dependent mobility for the material system.
Input is a temperature range (e.g. 10:50:1000),
reduced dielectric constants (e.g. 5, 20),
characteristic dielectric phonon frequency (e.g. 2.25E12) - units Hertz
bare-band effective-mass (e.g. 012) - units electron mass.

Returns a structure of type Polaron, containing arrays of useful
information.  Also prints a lot of information to the standard out - which
may be more useful if you're just inquiring as to a particular data point,
rather than plotting a temperature-dependent parameter.

As an example, to calculate the electron polaron in MAPI at 300 K:
polaronmobility(300, 4.5, 24.1, 2.25E12, 0.12)

In [4]:

    

# T=50:25:500 # temperature range
T=300 # Single (room) temperature

# MAPI  4.5, 24.1, 2.25THz - 75 cm^-1 ; α=

MAPIe=polaronmobility(T, 4.5, 24.1, 2.25E12, 0.12)


    

    





Polaron mobility for system ε_Inf=4.5, ε_S=24.1, freq=2.25e12,
                 effectivemass=0.12; with Trange 300 ...
Polaron mobility input parameters: ε_Inf=4.500000 ε_S=24.100000 freq=2.25e+12 α=2.393991 
Derived params in SI: ω =1.41372e+13 mb=1.09313e-31 
T: 300.000000 β: 2.41e+20 βred: 0.36 ħω  = 9.31 meV		Converged? : true
 Polaraon Parameters:  v= 19.8617  w= 16.9603  ||   M=0.371407  k=106.835792	
 Polaron frequency (SI) v= 4.5e+13 Hz  w= 3.8e+13 Hz
 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= -6.448816 B= 7.355626 C= 2.911978 F= -3.818788	 = -35.534786 meV
 Polaron Mobility theories:
	μ(FHIP)= 0.082049 m^2/Vs 	= 820.49 cm^2/Vs
	μ(Kadanoff,via Devreese2016)= 0.019689 m^2/Vs 	= 196.89 cm^2/Vs
		Eqm. Phonon. pop. Nbar: 2.308150 
		Gamma0 = 5.4282e+13 rad/s = 8.63925e+12 /s  
		Tau=1/Gamma0 = 1.15751e-13 = 0.115751 ps
		Energy Loss = 1.288e-08 J/s = 80.3904 meV/ps
	μ(Hellwarth1999)= 0.013642 m^2/Vs 	= 136.42 cm^2/Vs






    
Out[4]:





PolaronMobility.Polaron(Any[300], Any[196.887], Any[136.42], Any[820.494], Any[106.836], Any[0.371407], Any[-6.44882], Any[7.35563], Any[2.91198], Any[-3.81879], Any[0.115751], Any[19.8617], Any[16.9603], Any[0.359943], Any[2.68001e-9], Any[], Any[2.39399], Any[1.09313e-31], Any[1.41372e13])

In [5]:

    

MAPIh=polaronmobility(T, 4.5, 24.1, 2.25E12, 0.15)


    

    





Polaron mobility for system ε_Inf=4.5, ε_S=24.1, freq=2.25e12,
                 effectivemass=0.15; with Trange 300 ...
Polaron mobility input parameters: ε_Inf=4.500000 ε_S=24.100000 freq=2.25e+12 α=2.676564 
Derived params in SI: ω =1.41372e+13 mb=1.36641e-31 
T: 300.000000 β: 2.41e+20 βred: 0.36 ħω  = 9.31 meV		Converged? : true
 Polaraon Parameters:  v= 20.0955  w= 16.8169  ||   M=0.427928  k=121.021906	
 Polaron frequency (SI) v= 4.5e+13 Hz  w= 3.8e+13 Hz
 Polaron size (rf), following Schultz1959. (s.d. of Gaussian polaron ψ )
	 Schultz1959(2.4): rf= 0.499072 (int units) = 2.53282e-09 m [SI]
 Polaron Free Energy: A= -6.847835 B= 8.262481 C= 3.274388 F= -4.689034	 = -43.632641 meV
 Polaron Mobility theories:
	μ(FHIP)= 0.055402 m^2/Vs 	= 554.02 cm^2/Vs
	μ(Kadanoff,via Devreese2016)= 0.013294 m^2/Vs 	= 132.94 cm^2/Vs
		Eqm. Phonon. pop. Nbar: 2.308150 
		Gamma0 = 6.17666e+13 rad/s = 9.83046e+12 /s  
		Tau=1/Gamma0 = 1.01725e-13 = 0.101725 ps
		Energy Loss = 1.46559e-08 J/s = 91.4749 meV/ps
	μ(Hellwarth1999)= 0.009415 m^2/Vs 	= 94.15 cm^2/Vs






    
Out[5]:





PolaronMobility.Polaron(Any[300], Any[132.944], Any[94.1509], Any[554.023], Any[121.022], Any[0.427928], Any[-6.84783], Any[8.26248], Any[3.27439], Any[-4.68903], Any[0.101725], Any[20.0955], Any[16.8169], Any[0.359943], Any[2.53282e-9], Any[], Any[2.67656], Any[1.36641e-31], Any[1.41372e13])

In [6]:

    

# Generate T-dep mobility data; and plot
T=50:25:500
p=polaronmobility(T, 4.5, 24.1, 2.25E12, 0.12)


    

    





Polaron mobility for system ε_Inf=4.5, ε_S=24.1, freq=2.25e12,
                 effectivemass=0.12; with Trange 50:25:500 ...
Polaron mobility input parameters: ε_Inf=4.500000 ε_S=24.100000 freq=2.25e+12 α=2.393991 
Derived params in SI: ω =1.41372e+13 mb=1.09313e-31 
T: 50.000000 β: 1.45e+21 βred: 2.16 ħω  = 9.31 meV		Converged? : true
 Polaraon Parameters:  v= 4.4204  w= 3.1294  ||   M=0.995232  k=9.746668	
 Polaron frequency (SI) v= 9.9e+12 Hz  w= 7e+12 Hz
 Polaron size (rf), following Schultz1959. (s.d. of Gaussian polaron ψ )
	 Schultz1959(2.4): rf= 0.8248 (int units) = 4.18591e-09 m [SI]
 Polaron Free Energy: A= -3.269485 B= 3.425492 C= 1.307481 F= -1.463488	 = -13.618128 meV
 Polaron Mobility theories:
	μ(FHIP)= 0.057936 m^2/Vs 	= 579.36 cm^2/Vs
	μ(Kadanoff,via Devreese2016)= 0.083414 m^2/Vs 	= 834.14 cm^2/Vs
		Eqm. Phonon. pop. Nbar: 0.130409 
		Gamma0 = 8.80655e+12 rad/s = 1.40161e+12 /s  
		Tau=1/Gamma0 = 7.13468e-13 = 0.713468 ps
		Energy Loss = 2.0896e-09 J/s = 13.0423 meV/ps
	μ(Hellwarth1999)= 0.052358 m^2/Vs 	= 523.58 cm^2/Vs
T: 75.000000 β: 9.66e+20 βred: 1.44 ħω  = 9.31 meV		Converged? : true
 Polaraon Parameters:  v= 5.8852  w= 4.3313  ||   M=0.846249  k=15.875837	
 Polaron frequency (SI) v= 1.3e+13 Hz  w= 9.7e+12 Hz
 Polaron size (rf), following Schultz1959. (s.d. of Gaussian polaron ψ )
	 Schultz1959(2.4): rf= 0.745692 (int units) = 3.78443e-09 m [SI]
 Polaron Free Energy: A= -3.990232 B= 3.964199 C= 1.546485 F= -1.520452	 = -14.148190 meV
 Polaron Mobility theories:
	μ(FHIP)= 0.043815 m^2/Vs 	= 438.15 cm^2/Vs
	μ(Kadanoff,via Devreese2016)= 0.042055 m^2/Vs 	= 420.55 cm^2/Vs
		Eqm. Phonon. pop. Nbar: 0.310585 
		Gamma0 = 1.88768e+13 rad/s = 3.00433e+12 /s  
		Tau=1/Gamma0 = 3.32853e-13 = 0.332853 ps
		Energy Loss = 4.47905e-09 J/s = 27.956 meV/ps
	μ(Hellwarth1999)= 0.031456 m^2/Vs 	= 314.56 cm^2/Vs
T: 100.000000 β: 7.24e+20 βred: 1.08 ħω  = 9.31 meV		Converged? : true
 Polaraon Parameters:  v= 7.4101  w= 5.6419  ||   M=0.725005  k=23.077985	
 Polaron frequency (SI) v= 1.7e+13 Hz  w= 1.3e+13 Hz
 Polaron size (rf), following Schultz1959. (s.d. of Gaussian polaron ψ )
	 Schultz1959(2.4): rf= 0.693998 (int units) = 3.52208e-09 m [SI]
 Polaron Free Energy: A= -4.559900 B= 4.458681 C= 1.753534 F= -1.652314	 = -15.375200 meV
 Polaron Mobility theories:
	μ(FHIP)= 0.043105 m^2/Vs 	= 431.05 cm^2/Vs
	μ(Kadanoff,via Devreese2016)= 0.031031 m^2/Vs 	= 310.31 cm^2/Vs
		Eqm. Phonon. pop. Nbar: 0.514357 
		Gamma0 = 2.73814e+13 rad/s = 4.35788e+12 /s  
		Tau=1/Gamma0 = 2.29469e-13 = 0.229469 ps
		Energy Loss = 6.49701e-09 J/s = 40.5512 meV/ps
	μ(Hellwarth1999)= 0.024562 m^2/Vs 	= 245.62 cm^2/Vs
T: 125.000000 β: 5.79e+20 βred: 0.864 ħω  = 9.31 meV		Converged? : true
 Polaraon Parameters:  v= 8.9558  w= 7.0024  ||   M=0.635764  k=31.173527	
 Polaron frequency (SI) v= 2e+13 Hz  w= 1.6e+13 Hz
 Polaron size (rf), following Schultz1959. (s.d. of Gaussian polaron ψ )
	 Schultz1959(2.4): rf= 0.656455 (int units) = 3.33155e-09 m [SI]
 Polaron Free Energy: A= -5.019532 B= 4.910741 C= 1.938017 F= -1.829227	 = -17.021414 meV
 Polaron Mobility theories:
	μ(FHIP)= 0.045771 m^2/Vs 	= 457.71 cm^2/Vs
	μ(Kadanoff,via Devreese2016)= 0.026360 m^2/Vs 	= 263.60 cm^2/Vs
		Eqm. Phonon. pop. Nbar: 0.728700 
		Gamma0 = 3.39918e+13 rad/s = 5.40997e+12 /s  
		Tau=1/Gamma0 = 1.84844e-13 = 0.184844 ps
		Energy Loss = 8.06553e-09 J/s = 50.3411 meV/ps
	μ(Hellwarth1999)= 0.021111 m^2/Vs 	= 211.11 cm^2/Vs
T: 150.000000 β: 4.83e+20 βred: 0.72 ħω  = 9.31 meV		Converged? : true
 Polaraon Parameters:  v= 10.5708  w= 8.4524  ||   M=0.564063  k=40.298827	
 Polaron frequency (SI) v= 2.4e+13 Hz  w= 1.9e+13 Hz
 Polaron size (rf), following Schultz1959. (s.d. of Gaussian polaron ψ )
	 Schultz1959(2.4): rf= 0.62727 (int units) = 3.18343e-09 m [SI]
 Polaron Free Energy: A= -5.397726 B= 5.327466 C= 2.110581 F= -2.040320	 = -18.985692 meV
 Polaron Mobility theories:
	μ(FHIP)= 0.049979 m^2/Vs 	= 499.79 cm^2/Vs
	μ(Kadanoff,via Devreese2016)= 0.023986 m^2/Vs 	= 239.86 cm^2/Vs
		Eqm. Phonon. pop. Nbar: 0.948588 
		Gamma0 = 3.90684e+13 rad/s = 6.21793e+12 /s  
		Tau=1/Gamma0 = 1.60825e-13 = 0.160825 ps
		Energy Loss = 9.2701e-09 J/s = 57.8594 meV/ps
	μ(Hellwarth1999)= 0.019003 m^2/Vs 	= 190.03 cm^2/Vs
T: 175.000000 β: 4.14e+20 βred: 0.617 ħω  = 9.31 meV		Converged? : true
 Polaraon Parameters:  v= 12.0700  w= 9.7987  ||   M=0.517324  k=49.670680	
 Polaron frequency (SI) v= 2.7e+13 Hz  w= 2.2e+13 Hz
 Polaron size (rf), following Schultz1959. (s.d. of Gaussian polaron ψ )
	 Schultz1959(2.4): rf= 0.603739 (int units) = 3.06401e-09 m [SI]
 Polaron Free Energy: A= -5.696157 B= 5.715127 C= 2.261191 F= -2.280161	 = -21.217469 meV
 Polaron Mobility theories:
	μ(FHIP)= 0.054485 m^2/Vs 	= 544.85 cm^2/Vs
	μ(Kadanoff,via Devreese2016)= 0.022413 m^2/Vs 	= 224.13 cm^2/Vs
		Eqm. Phonon. pop. Nbar: 1.171725 
		Gamma0 = 4.30984e+13 rad/s = 6.85932e+12 /s  
		Tau=1/Gamma0 = 1.45787e-13 = 0.145787 ps
		Energy Loss = 1.02263e-08 J/s = 63.8277 meV/ps
	μ(Hellwarth1999)= 0.017502 m^2/Vs 	= 175.02 cm^2/Vs
T: 200.000000 β: 3.62e+20 βred: 0.54 ħω  = 9.31 meV		Converged? : true
 Polaraon Parameters:  v= 13.6291  w= 11.2168  ||   M=0.476367  k=59.935186	
 Polaron frequency (SI) v= 3.1e+13 Hz  w= 2.5e+13 Hz
 Polaron size (rf), following Schultz1959. (s.d. of Gaussian polaron ψ )
	 Schultz1959(2.4): rf= 0.584034 (int units) = 2.964e-09 m [SI]
 Polaron Free Energy: A= -5.939239 B= 6.078636 C= 2.405972 F= -2.545369	 = -23.685300 meV
 Polaron Mobility theories:
	μ(FHIP)= 0.059588 m^2/Vs 	= 595.88 cm^2/Vs
	μ(Kadanoff,via Devreese2016)= 0.021448 m^2/Vs 	= 214.48 cm^2/Vs
		Eqm. Phonon. pop. Nbar: 1.396922 
		Gamma0 = 4.62861e+13 rad/s = 7.36667e+12 /s  
		Tau=1/Gamma0 = 1.35747e-13 = 0.135747 ps
		Energy Loss = 1.09827e-08 J/s = 68.5487 meV/ps
	μ(Hellwarth1999)= 0.016389 m^2/Vs 	= 163.89 cm^2/Vs
T: 225.000000 β: 3.22e+20 βred: 0.48 ħω  = 9.31 meV		Converged? : true
 Polaraon Parameters:  v= 15.1885  w= 12.6441  ||   M=0.442967  k=70.818500	
 Polaron frequency (SI) v= 3.4e+13 Hz  w= 2.8e+13 Hz
 Polaron size (rf), following Schultz1959. (s.d. of Gaussian polaron ψ )
	 Schultz1959(2.4): rf= 0.567192 (int units) = 2.87853e-09 m [SI]
 Polaron Free Energy: A= -6.130523 B= 6.421787 C= 2.542274 F= -2.833539	 = -26.366790 meV
 Polaron Mobility theories:
	μ(FHIP)= 0.064961 m^2/Vs 	= 649.61 cm^2/Vs
	μ(Kadanoff,via Devreese2016)= 0.020784 m^2/Vs 	= 207.84 cm^2/Vs
		Eqm. Phonon. pop. Nbar: 1.623505 
		Gamma0 = 4.88707e+13 rad/s = 7.77802e+12 /s  
		Tau=1/Gamma0 = 1.28567e-13 = 0.128567 ps
		Energy Loss = 1.1596e-08 J/s = 72.3764 meV/ps
	μ(Hellwarth1999)= 0.015507 m^2/Vs 	= 155.07 cm^2/Vs
T: 250.000000 β: 2.9e+20 βred: 0.432 ħω  = 9.31 meV		Converged? : true
 Polaraon Parameters:  v= 16.7478  w= 14.0786  ||   M=0.415137  k=82.283005	
 Polaron frequency (SI) v= 3.8e+13 Hz  w= 3.2e+13 Hz
 Polaron size (rf), following Schultz1959. (s.d. of Gaussian polaron ψ )
	 Schultz1959(2.4): rf= 0.552548 (int units) = 2.80421e-09 m [SI]
 Polaron Free Energy: A= -6.276074 B= 6.747520 C= 2.671359 F= -3.142805	 = -29.244590 meV
 Polaron Mobility theories:
	μ(FHIP)= 0.070527 m^2/Vs 	= 705.27 cm^2/Vs
	μ(Kadanoff,via Devreese2016)= 0.020309 m^2/Vs 	= 203.09 cm^2/Vs
		Eqm. Phonon. pop. Nbar: 1.851066 
		Gamma0 = 5.09992e+13 rad/s = 8.11678e+12 /s  
		Tau=1/Gamma0 = 1.23202e-13 = 0.123202 ps
		Energy Loss = 1.2101e-08 J/s = 75.5287 meV/ps
	μ(Hellwarth1999)= 0.014783 m^2/Vs 	= 147.83 cm^2/Vs
T: 275.000000 β: 2.63e+20 βred: 0.393 ħω  = 9.31 meV		Converged? : true
 Polaraon Parameters:  v= 18.3053  w= 15.5173  ||   M=0.391621  k=94.297308	
 Polaron frequency (SI) v= 4.1e+13 Hz  w= 3.5e+13 Hz
 Polaron size (rf), following Schultz1959. (s.d. of Gaussian polaron ψ )
	 Schultz1959(2.4): rf= 0.539616 (int units) = 2.73858e-09 m [SI]
 Polaron Free Energy: A= -6.380886 B= 7.058182 C= 2.794352 F= -3.471648	 = -32.304558 meV
 Polaron Mobility theories:
	μ(FHIP)= 0.076232 m^2/Vs 	= 762.32 cm^2/Vs
	μ(Kadanoff,via Devreese2016)= 0.019956 m^2/Vs 	= 199.56 cm^2/Vs
		Eqm. Phonon. pop. Nbar: 2.079339 
		Gamma0 = 5.27779e+13 rad/s = 8.39986e+12 /s  
		Tau=1/Gamma0 = 1.1905e-13 = 0.119050 ps
		Energy Loss = 1.25231e-08 J/s = 78.1628 meV/ps
	μ(Hellwarth1999)= 0.014170 m^2/Vs 	= 141.70 cm^2/Vs
T: 300.000000 β: 2.41e+20 βred: 0.36 ħω  = 9.31 meV		Converged? : true
 Polaraon Parameters:  v= 19.8617  w= 16.9604  ||   M=0.371398  k=106.834219	
 Polaron frequency (SI) v= 4.5e+13 Hz  w= 3.8e+13 Hz
 Polaron size (rf), following Schultz1959. (s.d. of Gaussian polaron ψ )
	 Schultz1959(2.4): rf= 0.528079 (int units) = 2.68003e-09 m [SI]
 Polaron Free Energy: A= -6.448766 B= 7.355622 C= 2.911932 F= -3.818788	 = -35.534786 meV
 Polaron Mobility theories:
	μ(FHIP)= 0.082050 m^2/Vs 	= 820.50 cm^2/Vs
	μ(Kadanoff,via Devreese2016)= 0.019689 m^2/Vs 	= 196.89 cm^2/Vs
		Eqm. Phonon. pop. Nbar: 2.308150 
		Gamma0 = 5.42818e+13 rad/s = 8.63922e+12 /s  
		Tau=1/Gamma0 = 1.15751e-13 = 0.115751 ps
		Energy Loss = 1.28799e-08 J/s = 80.3902 meV/ps
	μ(Hellwarth1999)= 0.013642 m^2/Vs 	= 136.42 cm^2/Vs
T: 325.000000 β: 2.23e+20 βred: 0.332 ħω  = 9.31 meV		Converged? : true
 Polaraon Parameters:  v= 21.4166  w= 18.4065  ||   M=0.353819  k=119.873156	
 Polaron frequency (SI) v= 4.8e+13 Hz  w= 4.1e+13 Hz
 Polaron size (rf), following Schultz1959. (s.d. of Gaussian polaron ψ )
	 Schultz1959(2.4): rf= 0.517678 (int units) = 2.62724e-09 m [SI]
 Polaron Free Energy: A= -6.483091 B= 7.641372 C= 3.024842 F= -4.183124	 = -38.925023 meV
 Polaron Mobility theories:
	μ(FHIP)= 0.087958 m^2/Vs 	= 879.58 cm^2/Vs
	μ(Kadanoff,via Devreese2016)= 0.019483 m^2/Vs 	= 194.83 cm^2/Vs
		Eqm. Phonon. pop. Nbar: 2.537375 
		Gamma0 = 5.55679e+13 rad/s = 8.84391e+12 /s  
		Tau=1/Gamma0 = 1.13072e-13 = 0.113072 ps
		Energy Loss = 1.31851e-08 J/s = 82.2948 meV/ps
	μ(Hellwarth1999)= 0.013179 m^2/Vs 	= 131.79 cm^2/Vs
T: 350.000000 β: 2.07e+20 βred: 0.309 ħω  = 9.31 meV		Converged? : true
 Polaraon Parameters:  v= 22.9707  w= 19.8559  ||   M=0.338343  k=133.394306	
 Polaron frequency (SI) v= 5.2e+13 Hz  w= 4.5e+13 Hz
 Polaron size (rf), following Schultz1959. (s.d. of Gaussian polaron ψ )
	 Schultz1959(2.4): rf= 0.508234 (int units) = 2.57932e-09 m [SI]
 Polaron Free Energy: A= -6.486522 B= 7.916679 C= 3.133536 F= -4.563692	 = -42.466303 meV
 Polaron Mobility theories:
	μ(FHIP)= 0.093943 m^2/Vs 	= 939.43 cm^2/Vs
	μ(Kadanoff,via Devreese2016)= 0.019322 m^2/Vs 	= 193.22 cm^2/Vs
		Eqm. Phonon. pop. Nbar: 2.766925 
		Gamma0 = 5.66777e+13 rad/s = 9.02054e+12 /s  
		Tau=1/Gamma0 = 1.10858e-13 = 0.110858 ps
		Energy Loss = 1.34484e-08 J/s = 83.9384 meV/ps
	μ(Hellwarth1999)= 0.012768 m^2/Vs 	= 127.68 cm^2/Vs
T: 375.000000 β: 1.93e+20 βred: 0.288 ħω  = 9.31 meV		Converged? : true
 Polaraon Parameters:  v= 24.5237  w= 21.3080  ||   M=0.324597  k=147.377665	
 Polaron frequency (SI) v= 5.5e+13 Hz  w= 4.8e+13 Hz
 Polaron size (rf), following Schultz1959. (s.d. of Gaussian polaron ψ )
	 Schultz1959(2.4): rf= 0.4996 (int units) = 2.5355e-09 m [SI]
 Polaron Free Energy: A= -6.461371 B= 8.182590 C= 3.238420 F= -4.959639	 = -46.150693 meV
 Polaron Mobility theories:
	μ(FHIP)= 0.099994 m^2/Vs 	= 999.94 cm^2/Vs
	μ(Kadanoff,via Devreese2016)= 0.019196 m^2/Vs 	= 191.96 cm^2/Vs
		Eqm. Phonon. pop. Nbar: 2.996737 
		Gamma0 = 5.76437e+13 rad/s = 9.17428e+12 /s  
		Tau=1/Gamma0 = 1.09e-13 = 0.109000 ps
		Energy Loss = 1.36776e-08 J/s = 85.3689 meV/ps
	μ(Hellwarth1999)= 0.012399 m^2/Vs 	= 123.99 cm^2/Vs
T: 400.000000 β: 1.81e+20 βred: 0.27 ħω  = 9.31 meV		Converged? : true
 Polaraon Parameters:  v= 26.0752  w= 22.7619  ||   M=0.312311  k=161.810132	
 Polaron frequency (SI) v= 5.9e+13 Hz  w= 5.1e+13 Hz
 Polaron size (rf), following Schultz1959. (s.d. of Gaussian polaron ψ )
	 Schultz1959(2.4): rf= 0.491651 (int units) = 2.49515e-09 m [SI]
 Polaron Free Energy: A= -6.409757 B= 8.440005 C= 3.339957 F= -5.370205	 = -49.971105 meV
 Polaron Mobility theories:
	μ(FHIP)= 0.106098 m^2/Vs 	= 1060.98 cm^2/Vs
	μ(Kadanoff,via Devreese2016)= 0.019095 m^2/Vs 	= 190.95 cm^2/Vs
		Eqm. Phonon. pop. Nbar: 3.226761 
		Gamma0 = 5.84912e+13 rad/s = 9.30917e+12 /s  
		Tau=1/Gamma0 = 1.07421e-13 = 0.107421 ps
		Energy Loss = 1.38787e-08 J/s = 86.6242 meV/ps
	μ(Hellwarth1999)= 0.012065 m^2/Vs 	= 120.65 cm^2/Vs
T: 425.000000 β: 1.7e+20 βred: 0.254 ħω  = 9.31 meV		Converged? : true
 Polaraon Parameters:  v= 27.6255  w= 24.2175  ||   M=0.301245  k=176.676956	
 Polaron frequency (SI) v= 6.2e+13 Hz  w= 5.4e+13 Hz
 Polaron size (rf), following Schultz1959. (s.d. of Gaussian polaron ψ )
	 Schultz1959(2.4): rf= 0.484295 (int units) = 2.45783e-09 m [SI]
 Polaron Free Energy: A= -6.333403 B= 8.689676 C= 3.438430 F= -5.794702	 = -53.921162 meV
 Polaron Mobility theories:
	μ(FHIP)= 0.112251 m^2/Vs 	= 1122.51 cm^2/Vs
	μ(Kadanoff,via Devreese2016)= 0.019014 m^2/Vs 	= 190.14 cm^2/Vs
		Eqm. Phonon. pop. Nbar: 3.456961 
		Gamma0 = 5.92399e+13 rad/s = 9.42832e+12 /s  
		Tau=1/Gamma0 = 1.06063e-13 = 0.106063 ps
		Energy Loss = 1.40564e-08 J/s = 87.7329 meV/ps
	μ(Hellwarth1999)= 0.011760 m^2/Vs 	= 117.60 cm^2/Vs
T: 450.000000 β: 1.61e+20 βred: 0.24 ħω  = 9.31 meV		Converged? : true
 Polaraon Parameters:  v= 29.1747  w= 25.6749  ||   M=0.291206  k=191.963225	
 Polaron frequency (SI) v= 6.6e+13 Hz  w= 5.8e+13 Hz
 Polaron size (rf), following Schultz1959. (s.d. of Gaussian polaron ψ )
	 Schultz1959(2.4): rf= 0.477463 (int units) = 2.42315e-09 m [SI]
 Polaron Free Energy: A= -6.233788 B= 8.932251 C= 3.534048 F= -6.232511	 = -57.995086 meV
 Polaron Mobility theories:
	μ(FHIP)= 0.118449 m^2/Vs 	= 1184.49 cm^2/Vs
	μ(Kadanoff,via Devreese2016)= 0.018949 m^2/Vs 	= 189.49 cm^2/Vs
		Eqm. Phonon. pop. Nbar: 3.687306 
		Gamma0 = 5.99051e+13 rad/s = 9.53419e+12 /s  
		Tau=1/Gamma0 = 1.04886e-13 = 0.104886 ps
		Energy Loss = 1.42142e-08 J/s = 88.7181 meV/ps
	μ(Hellwarth1999)= 0.011480 m^2/Vs 	= 114.80 cm^2/Vs
T: 475.000000 β: 1.52e+20 βred: 0.227 ħω  = 9.31 meV		Converged? : true
 Polaraon Parameters:  v= 30.7876  w= 27.1991  ||   M=0.281276  k=208.086160	
 Polaron frequency (SI) v= 6.9e+13 Hz  w= 6.1e+13 Hz
 Polaron size (rf), following Schultz1959. (s.d. of Gaussian polaron ψ )
	 Schultz1959(2.4): rf= 0.471099 (int units) = 2.39085e-09 m [SI]
 Polaron Free Energy: A= -6.115064 B= 9.168308 C= 3.629823 F= -6.683067	 = -62.187617 meV
 Polaron Mobility theories:
	μ(FHIP)= 0.124792 m^2/Vs 	= 1247.92 cm^2/Vs
	μ(Kadanoff,via Devreese2016)= 0.018913 m^2/Vs 	= 189.13 cm^2/Vs
		Eqm. Phonon. pop. Nbar: 3.917775 
		Gamma0 = 6.04839e+13 rad/s = 9.62631e+12 /s  
		Tau=1/Gamma0 = 1.03882e-13 = 0.103882 ps
		Energy Loss = 1.43515e-08 J/s = 89.5752 meV/ps
	μ(Hellwarth1999)= 0.011224 m^2/Vs 	= 112.24 cm^2/Vs
T: 500.000000 β: 1.45e+20 βred: 0.216 ħω  = 9.31 meV		Converged? : true
 Polaraon Parameters:  v= 32.2706  w= 28.5944  ||   M=0.273655  k=223.750679	
 Polaron frequency (SI) v= 7.3e+13 Hz  w= 6.4e+13 Hz
 Polaron size (rf), following Schultz1959. (s.d. of Gaussian polaron ψ )
	 Schultz1959(2.4): rf= 0.465122 (int units) = 2.36052e-09 m [SI]
 Polaron Free Energy: A= -5.970137 B= 9.398316 C= 3.717672 F= -7.145851	 = -66.493939 meV
 Polaron Mobility theories:
	μ(FHIP)= 0.130957 m^2/Vs 	= 1309.57 cm^2/Vs
	μ(Kadanoff,via Devreese2016)= 0.018855 m^2/Vs 	= 188.55 cm^2/Vs
		Eqm. Phonon. pop. Nbar: 4.148348 
		Gamma0 = 6.10331e+13 rad/s = 9.71372e+12 /s  
		Tau=1/Gamma0 = 1.02947e-13 = 0.102947 ps
		Energy Loss = 1.44818e-08 J/s = 90.3886 meV/ps
	μ(Hellwarth1999)= 0.010982 m^2/Vs 	= 109.82 cm^2/Vs






    
Out[6]:





PolaronMobility.Polaron(Any[50, 75, 100, 125, 150, 175, 200, 225, 250, 275, 300, 325, 350, 375, 400, 425, 450, 475, 500], Any[834.145, 420.555, 310.309, 263.6, 239.861, 224.13, 214.484, 207.843, 203.085, 199.557, 196.889, 194.83, 193.224, 191.957, 190.947, 190.137, 189.488, 189.129, 188.548], Any[523.577, 314.563, 245.624, 211.106, 190.035, 175.023, 163.89, 155.072, 147.826, 141.702, 136.421, 131.79, 127.678, 123.988, 120.645, 117.596, 114.796, 112.239, 109.816], Any[579.359, 438.147, 431.053, 457.711, 499.791, 544.848, 595.884, 649.612, 705.268, 762.318, 820.502, 879.58, 939.431, 999.937, 1060.98, 1122.51, 1184.49, 1247.92, 1309.57], Any[9.74667, 15.8758, 23.078, 31.1735, 40.2988, 49.6707, 59.9352, 70.8185, 82.283, 94.2973, 106.834, 119.873, 133.394, 147.378, 161.81, 176.677, 191.963, 208.086, 223.751], Any[0.995232, 0.846249, 0.725005, 0.635764, 0.564063, 0.517324, 0.476367, 0.442967, 0.415137, 0.391621, 0.371398, 0.353819, 0.338343, 0.324597, 0.312311, 0.301245, 0.291206, 0.281276, 0.273655], Any[-3.26949, -3.99023, -4.5599, -5.01953, -5.39773, -5.69616, -5.93924, -6.13052, -6.27607, -6.38089, -6.44877, -6.48309, -6.48652, -6.46137, -6.40976, -6.3334, -6.23379, -6.11506, -5.97014], Any[3.42549, 3.9642, 4.45868, 4.91074, 5.32747, 5.71513, 6.07864, 6.42179, 6.74752, 7.05818, 7.35562, 7.64137, 7.91668, 8.18259, 8.44, 8.68968, 8.93225, 9.16831, 9.39832], Any[1.30748, 1.54649, 1.75353, 1.93802, 2.11058, 2.26119, 2.40597, 2.54227, 2.67136, 2.79435, 2.91193, 3.02484, 3.13354, 3.23842, 3.33996, 3.43843, 3.53405, 3.62982, 3.71767], Any[-1.46349, -1.52045, -1.65231, -1.82923, -2.04032, -2.28016, -2.54537, -2.83354, -3.1428, -3.47165, -3.81879, -4.18312, -4.56369, -4.95964, -5.3702, -5.7947, -6.23251, -6.68307, -7.14585], Any[0.713468, 0.332853, 0.229469, 0.184844, 0.160825, 0.145787, 0.135747, 0.128567, 0.123202, 0.11905, 0.115751, 0.113072, 0.110858, 0.109, 0.107421, 0.106063, 0.104886, 0.103882, 0.102947], Any[4.42041, 5.88524, 7.41009, 8.95582, 10.5708, 12.07, 13.6291, 15.1885, 16.7478, 18.3053, 19.8617, 21.4166, 22.9707, 24.5237, 26.0752, 27.6255, 29.1747, 30.7876, 32.2706], Any[3.12944, 4.33131, 5.64194, 7.00237, 8.45245, 9.7987, 11.2168, 12.6441, 14.0786, 15.5173, 16.9604, 18.4065, 19.8559, 21.308, 22.7619, 24.2175, 25.6749, 27.1991, 28.5944], Any[2.15966, 1.43977, 1.07983, 0.863863, 0.719886, 0.617045, 0.539914, 0.479924, 0.431931, 0.392665, 0.359943, 0.332255, 0.308522, 0.287954, 0.269957, 0.254077, 0.239962, 0.227332, 0.215966], Any[4.18591e-9, 3.78443e-9, 3.52208e-9, 3.33155e-9, 3.18343e-9, 3.06401e-9, 2.964e-9, 2.87853e-9, 2.80421e-9, 2.73858e-9, 2.68003e-9, 2.62724e-9, 2.57932e-9, 2.5355e-9, 2.49515e-9, 2.45783e-9, 2.42315e-9, 2.39085e-9, 2.36052e-9], Any[], Any[2.39399], Any[1.09313e-31], Any[1.41372e13])

In [7]:

    

using Plots # Julia meta-plotting package

plot(p.T,p.M,label="Phonon effective-mass",
         markersize=3,marker=:rect,
         xlab="Temperature (K)",ylab="Phonon effective-mass",
         ylim=(0,1.2))


    

    
Out[7]:

In [8]:

    

## Relaxationtime vs. Temperature plot
    plot(p.T,p.Tau,label="Kadanoff relaxation time (ps)",markersize=3,marker=:rect,xlab="Temperature (K)",ylab="Relaxation time (ps)",ylim=(0,1.2))


    

    
Out[8]:

In [9]:

    

plot(p.T,p.M,label="Phonon effective-mass (m\$_b\$)",markersize=3,marker=:rect,
        xlab="Temperature (K)",ylab="Effective-mass / relaxation time",ylim=(0,1.2))
    plot!(p.T,p.Tau,label="Kadanoff relaxation time (ps)",markersize=3,marker=:diamond,
        xlab="Temperature (K)",ylab="Relaxation time (ps)",ylim=(0,1.2))


    

    
Out[9]:

In [10]:

    

## Variational parameters, v and w vs. Temperature plot
    plot(p.T,p.v,label="v",markersize=3, marker=:rect, xlab="Temperature (K)",ylab="\hbar\omega")
    plot!(p.T,p.w,label="w",markersize=3, marker=:diamond)


    

    
Out[10]:

In [11]:

    

## Spring Constants vs. Temperature plot
    plot(p.T,p.k,label="Polaron spring-constant",markersize=3, marker=:uptriangle, xlab="Temperature (K)",ylab="Spring-constant",)


    

    
Out[11]:

In [12]:

    

## Variation Energy vs. Temperature plots
    plot( p.T,p.A,label="A",markersize=3,marker=:downtriangle, xlab="Temperature (K)",ylab="Polaron free-energy")
    plot!(p.T,p.B,label="B",markersize=3,marker=:diamond)
    plot!(p.T,p.C,label="C",markersize=3,marker=:uptriangle)
    plot!(p.T,p.F,label="F",markersize=3,marker=:rect)
    #plot!(Ts,Fs,label="F=-(A+B+C)",markersize=3,marker=:rect)


    

    
Out[12]:

In [13]:

    

## Polaron radius vs. Temperature
    plot(p.T,p.rfsi.*10^10, markersize=3,marker=:rect,
        label="Polaron radius",xlab="Temperature (K)",ylab="Polaron Radius (Angstrom)",ylims=(0,Inf))


    

    
Out[13]:

In [14]:

    

## Calculated mobility comparison plot
    plot(p.T,p.Kμ,label="Kadanoff",markersize=3,marker=:rect,xlab="Temperature (K)",ylab="Mobility (cm\$^2\$/Vs)",ylims=(0,1000))
    plot!(p.T,p.FHIPμ,label="FHIP",markersize=3,marker=:diamond)
    plot!(p.T,p.Hμ,label="Hellwarth1999",markersize=3,marker=:uptriangle)


    

    
Out[14]:

In [15]:

    

println("NaCl Frohlich paper α=",frohlichalpha(2.3, 5.6, (4.9E13/(2*pi)), 1.0))
println(" \t should be ~about 5 (Feynman1955)")
println("CdTe  α=",frohlichalpha(7.1,   10.4,  5.08E12, 0.095))
println("\t Stone 0.39 / Devreese 0.29 ")
println("GaAs  α=",frohlichalpha(10.89, 12.9,  8.46E12, 0.063))
println("\t Devreese 0.068 ")


    

    




NaCl Frohlich paper α=5.262434397679507
 	 should be ~about 5 (Feynman1955)
CdTe  α=0.35054915310975776
	 Stone 0.39 / Devreese 0.29 
GaAs  α=0.07082071131530161
	 Devreese 0.068 

In [16]:

    

# Hellwarth et al. PRB 1999 Table II - BiSiO frequencies and activities
HellwarthII = [
    106.23 8.86
    160.51 9.50
    180.33 20.85
    206.69 10.05
    252.76 27.00
    369.64 61.78
    501.71 52.87
    553.60 86.18
    585.36 75.41
    607.29 98.15
    834.53 89.36
]

println("Attempting to reproduce Hellwarth et al.'s data.")
println("\nB scheme: (athermal)")
HellwarthBScheme(HellwarthII)
println("    ... should agree with values given in Hellwarth(60) W_e=196.9 cm^-1 and Hellwarth(61) Ω_e=500 cm^-1")


    

    




Attempting to reproduce Hellwarth et al.'s data.

B scheme: (athermal)
Hellwarth B Scheme... (athermal)
Hellwarth (58) summation: 0.15505835776181887
Hellwarth (59) summation (total ir activity ^2): 38777.7725
Hellwarth (59) W_e (total ir activity ): 196.92072643579192
Hellwarth (61) Omega (freq): 500.08501275972833
    ... should agree with values given in Hellwarth(60) W_e=196.9 cm^-1 and Hellwarth(61) Ω_e=500 cm^-1

In [ ]: