-
Notifications
You must be signed in to change notification settings - Fork 8
Commit
This commit does not belong to any branch on this repository, and may belong to a fork outside of the repository.
Chicheley Rubrene mobility notebook.
- Loading branch information
Showing
1 changed file
with
292 additions
and
0 deletions.
There are no files selected for viewing
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
Original file line number | Diff line number | Diff line change |
---|---|---|
@@ -0,0 +1,292 @@ | ||
### A Pluto.jl notebook ### | ||
# v0.18.1 | ||
|
||
using Markdown | ||
using InteractiveUtils | ||
|
||
# ╔═╡ 3df3da8e-2bc9-44f5-bbe0-2cabdf3a2a81 | ||
begin | ||
import Pkg | ||
# careful: this is _not_ a reproducible environment | ||
# activate the global environment | ||
Pkg.activate("../") | ||
|
||
using PolaronMobility | ||
using QuadGK | ||
end | ||
|
||
|
||
# ╔═╡ 69e7d9cd-7900-474d-aeba-06b38e62d756 | ||
using Gnuplot | ||
|
||
# ╔═╡ 4df7f468-544d-11ec-06a3-0b757c39e918 | ||
# Rubrene Ordejon | ||
# Rubrene electron-phonon matrix elements | ||
|
||
# Ordejón, P., Boskovic, D., Panhans, M., Ortmann, F., 2017. Ab initio study of | ||
# electron-phonon coupling in rubrene. Phys. Rev. B 96, 035202. | ||
# https://doi.org/10.1103/PhysRevB.96.035202 | ||
|
||
# ╔═╡ 78d3ef38-fd4a-4957-bc9b-4523cb641df4 | ||
begin | ||
CM1inTHz = 33.356 | ||
hbar=ħ=1.0545718E-34 | ||
q=1.6E-19 | ||
const Boltzmann = const kB = 1.3806504e-23 | ||
const me=MassElectron = 9.10938188e-31 | ||
end | ||
|
||
# ╔═╡ c4d0bfad-d837-4814-aa60-d71bd41ce28b | ||
# Table II | ||
# Units: cm^-1 meV | convert to THz meV | ||
EffectiveH=[1208.9 106.8] # Holstein | ||
|
||
# ╔═╡ 43f24329-1a03-4012-9b31-cf72d273c3d6 | ||
cm1tomeV(cm1)=1000 * 2π* ħ*cm1/CM1inTHz*1E12 / q | ||
|
||
# ╔═╡ 139b3aed-b4f8-4705-8eff-d1e0d6ea9d12 | ||
cm1tomeV(EffectiveH[1]) # I got 149.88 meV by hand | ||
|
||
# ╔═╡ e1573de4-450a-4ed8-a9db-003fedcc9cc3 | ||
g0 = EffectiveH[2] / cm1tomeV(EffectiveH[1]) # I got 0.71 by hand | ||
|
||
# ╔═╡ cc9153d2-86d6-4272-a266-053165197d99 | ||
EffectiveP=[117.9 21.9] | ||
|
||
# ╔═╡ d303b91b-567c-4705-b51e-5f32c91b3d80 | ||
gi = EffectiveP[2] / cm1tomeV(EffectiveP[1]) | ||
|
||
# ╔═╡ f192deb9-5de9-40e1-959b-a518aee7a6a3 | ||
α=gi+g0 # I got 2.21 by hand; this is essentially the Frohlich alpha | ||
|
||
# ╔═╡ b1c36a20-9c5b-4fc2-9907-8061ed78d88f | ||
v,w=PolaronMobility.feynmanvw(α) | ||
|
||
# ╔═╡ 7f6aedfa-700b-47b5-8030-14d1448c4737 | ||
# We've just solved the Feynman polaron theory! | ||
|
||
# ╔═╡ 9829e516-9240-48b2-b4f6-9eff06bf5018 | ||
EffectiveFreq=(g0*EffectiveH[1] + gi*EffectiveP[1])/α | ||
|
||
# ╔═╡ edacbf0c-959f-4a2e-b406-afe7fdfa98bc | ||
ω=1E12 * EffectiveFreq / CM1inTHz | ||
|
||
# ╔═╡ 6d6d286a-2f36-4762-803b-59299d7373a1 | ||
EffectiveMass(;a=1E-9, J=0.030q)=ħ^2/(2J*a^2) | ||
# Tight-binding model, then cos(2θ) expansion around extremum | ||
|
||
# ╔═╡ 88f68d4c-4782-47da-ab8f-6f7ea728f23c | ||
|
||
|
||
# ╔═╡ 79c95049-89a3-42cc-962a-e5266d8902a1 | ||
# Rubrene crystals have an orthorhombic unit cell with lattice parameters a = 1.44 nm, b = 0.718 nm, and c = 2.69 nm. | ||
# https://www.lehigh.edu/~inlo/rubrene.html#:~:text=Rubrene%20crystals%20have%20an%20orthorhombic,the%20a%20and%20c%20directions. | ||
EffectiveMass(a=0.718E-9, J=0.134*q)/MassElectron | ||
|
||
# ╔═╡ 58828fe1-12c4-499d-837d-ae45049758e7 | ||
mb = 0.88 * MassElectron | ||
|
||
# ╔═╡ ac1f76d8-1cdd-4de3-b1e1-049ba33374d2 | ||
βred = (q/1000 * cm1tomeV(EffectiveFreq) ) / (kB*300) | ||
|
||
# ╔═╡ 691ace17-eb19-444a-9807-887156a813aa | ||
|
||
|
||
# ╔═╡ 5f62b499-3363-4d0f-8b70-6dabe5e579ed | ||
cm1tomeV(EffectiveH[1]) | ||
|
||
# ╔═╡ 755da976-d29f-4c82-9645-7acef0a140c5 | ||
EffectiveH[1] | ||
|
||
# ╔═╡ 78695321-c088-4e99-987c-fca646aa7af3 | ||
# Hellwarth1999 - directly do contour integration in Feynman1962, for | ||
# finite temperature DC mobility | ||
# Hellwarth1999 Eqn (2) and (1) - These are going back to the general | ||
# (pre low-T limit) formulas in Feynman1962. to evaluate these, you | ||
# need to do the explicit contour integration to get the polaron | ||
# self-energy | ||
R=(v^2-w^2)/(w^2*v) # inline, page 300 just after Eqn (2) | ||
|
||
# ╔═╡ 76794859-df83-4c35-bb51-3f34774798e0 | ||
#b=R*βred/sinh(b*βred*v/2) # This self-references b! What on Earth? | ||
# OK! I now understand that there is a typo in Hellwarth1999 and | ||
# Biaggio1997. They've introduced a spurious b on the R.H.S. compared to | ||
# the original, Feynman1962: | ||
b=R*βred/sinh(βred*v/2) # Feynman1962 version; page 1010, Eqn (47b) | ||
|
||
# ╔═╡ 10adfb5e-d9ed-4f5b-bc1e-98b8a452e886 | ||
a=sqrt( (βred/2)^2 + R*βred*coth(βred*v/2)) | ||
|
||
# ╔═╡ 1e2a33b0-624c-41a5-8c94-5f1f51fbe835 | ||
k(u,a,b,v) = (u^2+a^2-b*cos(v*u))^(-3/2)*cos(u) # integrand in (2) | ||
|
||
# ╔═╡ 4a36c74a-d44d-4a92-9d24-35ec22d8ae57 | ||
K=quadgk(u->k(u,a,b,v),0,Inf)[1] # numerical quadrature integration of (2) | ||
|
||
# ╔═╡ 6f49c66c-82b8-428c-8420-4646a6394a9e | ||
#Right-hand-side of Eqn 1 in Hellwarth 1999 // Eqn (4) in Baggio1997 | ||
RHS= α/(3*sqrt(π)) * βred^(5/2) / sinh(βred/2) * (v^3/w^3) * K | ||
|
||
# ╔═╡ 7255cc2c-4680-4cde-93da-1ba2154e2d22 | ||
μ=RHS^-1 * (q)/(ω*mb) #, "m^2/Vs" | ||
|
||
# ╔═╡ e6fe559e-46f1-4336-9513-7306c32f2515 | ||
μ*100^2 , "cm^2/Vs" | ||
|
||
# ╔═╡ ed7cb9bb-4b83-4ed6-a8fd-9936eb02643b | ||
function FHIPmob(T; v=v, w=w, ω=ω) | ||
#βred = (q/1000 * cm1tomeV(EffectiveP[1]) ) / (kB*T) | ||
βred = ħ*ω / (kB*T) | ||
R=(v^2-w^2)/(w^2*v) # inline, page 300 just after Eqn (2) | ||
b=R*βred/sinh(βred*v/2) # Feynman1962 version; page 1010, Eqn (47b) | ||
a=sqrt( (βred/2)^2 + R*βred*coth(βred*v/2)) | ||
k(u,a,b,v) = (u^2+a^2-b*cos(v*u))^(-3/2)*cos(u) # integrand in (2) | ||
K=quadgk(u->k(u,a,b,v),0,Inf)[1] # numerical quadrature integration of (2) | ||
#Right-hand-side of Eqn 1 in Hellwarth 1999 // Eqn (4) in Baggio1997 | ||
RHS= α/(3*sqrt(π)) * βred^(5/2) / sinh(βred/2) * (v^3/w^3) * K | ||
μ=RHS^-1 * (q)/(ω*mb) #, "m^2/Vs" | ||
μ=μ*100^2 # convert to cm^2/Vs, from SI | ||
end | ||
|
||
# ╔═╡ 084ac351-17e6-4916-b0b7-12aa593984b2 | ||
Trange=90:10:400 | ||
|
||
# ╔═╡ 322962f3-c1f5-47d1-882b-f4efed0c2fa0 | ||
mobs=[FHIPmob(T) for T=Trange] | ||
|
||
# ╔═╡ 5e996c83-88dc-4569-a435-4d00684de2af | ||
FHIPmob(300) | ||
|
||
# ╔═╡ 19b69d34-367e-4794-abb1-3f76e1356561 | ||
begin | ||
@gp Trange mobs " w lp title 'Rubrene - b axis'" | ||
@gp :- "set yrange [0:]" | ||
@gp :- "set xlabel 'Temperature (K)'" | ||
@gp :- "set ylabel 'Mobility (cm^2/Vs)'" | ||
end | ||
|
||
# ╔═╡ bbe9a49b-6495-4c13-91d0-3a4dbd0a2800 | ||
begin | ||
@gp Trange mobs " w lp title 'Rubrene - b axis'" | ||
#@gp :- "set yrange [0:]" | ||
@gp :- "set xlabel 'Temperature (K)'" | ||
@gp :- "set ylabel 'Mobility (cm^2/Vs)'" | ||
@gp :- "set logscale y" | ||
end | ||
|
||
# ╔═╡ 444d7454-48a9-44dd-87a3-f4cf1a495072 | ||
@printf("\n\tμ(Hellwarth1999)= %f m^2/Vs \t= %.2f cm^2/Vs",μ,μ*100^2) | ||
|
||
# ╔═╡ b0f72692-d76c-44f1-91aa-2e9b7b882d1b | ||
# Jan's COF | ||
Jperylenecoff=0.204 | ||
|
||
# ╔═╡ 0c043a99-18bf-4b75-95fa-ce8aea8f8bef | ||
relevantmode=200 #cm^-1 | ||
|
||
# ╔═╡ 06ccb204-694d-4c64-9ecd-742c009fb972 | ||
perylenecofmass=EffectiveMass(a=1.0E-9, J=Jperylenecoff*q) | ||
|
||
# ╔═╡ 2ba19614-f2ef-4dca-9f61-b95908b7c81c | ||
perylenecofmass/MassElectron | ||
|
||
# ╔═╡ 27ee4adf-b7d9-41e0-b0a0-f414422b7b73 | ||
perylenecofg0 = 128 / cm1tomeV(relevantmode) | ||
|
||
# ╔═╡ ed6d3a1a-3fcd-4ef1-a57f-a394895d4b29 | ||
pv,pw= feynmanvw(perylenecofg0) | ||
|
||
# ╔═╡ fbee75ec-96f0-41b6-8c14-a724872476fb | ||
FHIPmob(300; v=pv, w=pw, ω=relevantmode/CM1inTHz*1E12*2π) | ||
|
||
# ╔═╡ 7d7d3ff6-30d4-477a-8f03-e4c08ac8a608 | ||
md""" | ||
# Stuff below here for later | ||
""" | ||
|
||
# ╔═╡ ba64c0a2-8cd4-421a-b494-a70f1f92ca0f | ||
EffectiveHGirlando=[1277 99] | ||
|
||
# ╔═╡ d30b10f4-2c62-4596-aa73-e4f2796b0541 | ||
EffectivePGirlando=[77 20] | ||
|
||
# ╔═╡ 04b0a526-b808-4c61-a565-6c60e12be2b7 | ||
# Table IV | ||
# ωp ωpg0 ωpgi | ||
# cm^-1 meV meV | ||
PerModeOrdejon= | ||
[57.8 -1.7 0.85 | ||
59.6 1.4 -0.83 | ||
89.0 1.6 -4.8 | ||
107.3 -0.14 2.8 | ||
139.1 -2.3 -3.7 | ||
639.1 -7.5 1.0 | ||
1011.2 -3.6 -0.04 | ||
1344.7 19.8 0.04 | ||
1593.3 -42.0 -0.12 | ||
] .* [1/CM1inTHz 1 1] | ||
|
||
# ╔═╡ b38b707f-6b97-424a-a22a-cf2fdebcfb31 | ||
PerModeGirlando = | ||
[37.4 -0.9 3.4 | ||
66.6 1.6 -6.6 | ||
86.7 -0.6 -9.3 | ||
106.3 0 -4.4 | ||
125.1 1.4 -4.7 | ||
631.2 -10.8 1.3 | ||
1002.3 24.6 0 | ||
1348.6 49.9 0 | ||
1593.8 -45.6 1.6 | ||
] .* [1/CM1inTHz 1 1] | ||
|
||
# ╔═╡ Cell order: | ||
# ╠═4df7f468-544d-11ec-06a3-0b757c39e918 | ||
# ╠═3df3da8e-2bc9-44f5-bbe0-2cabdf3a2a81 | ||
# ╠═78d3ef38-fd4a-4957-bc9b-4523cb641df4 | ||
# ╠═c4d0bfad-d837-4814-aa60-d71bd41ce28b | ||
# ╠═43f24329-1a03-4012-9b31-cf72d273c3d6 | ||
# ╠═139b3aed-b4f8-4705-8eff-d1e0d6ea9d12 | ||
# ╠═e1573de4-450a-4ed8-a9db-003fedcc9cc3 | ||
# ╠═cc9153d2-86d6-4272-a266-053165197d99 | ||
# ╠═d303b91b-567c-4705-b51e-5f32c91b3d80 | ||
# ╠═f192deb9-5de9-40e1-959b-a518aee7a6a3 | ||
# ╠═b1c36a20-9c5b-4fc2-9907-8061ed78d88f | ||
# ╠═7f6aedfa-700b-47b5-8030-14d1448c4737 | ||
# ╠═9829e516-9240-48b2-b4f6-9eff06bf5018 | ||
# ╠═edacbf0c-959f-4a2e-b406-afe7fdfa98bc | ||
# ╠═6d6d286a-2f36-4762-803b-59299d7373a1 | ||
# ╠═88f68d4c-4782-47da-ab8f-6f7ea728f23c | ||
# ╠═79c95049-89a3-42cc-962a-e5266d8902a1 | ||
# ╠═58828fe1-12c4-499d-837d-ae45049758e7 | ||
# ╠═ac1f76d8-1cdd-4de3-b1e1-049ba33374d2 | ||
# ╠═691ace17-eb19-444a-9807-887156a813aa | ||
# ╠═5f62b499-3363-4d0f-8b70-6dabe5e579ed | ||
# ╠═755da976-d29f-4c82-9645-7acef0a140c5 | ||
# ╠═78695321-c088-4e99-987c-fca646aa7af3 | ||
# ╠═76794859-df83-4c35-bb51-3f34774798e0 | ||
# ╠═10adfb5e-d9ed-4f5b-bc1e-98b8a452e886 | ||
# ╠═1e2a33b0-624c-41a5-8c94-5f1f51fbe835 | ||
# ╠═4a36c74a-d44d-4a92-9d24-35ec22d8ae57 | ||
# ╠═6f49c66c-82b8-428c-8420-4646a6394a9e | ||
# ╠═7255cc2c-4680-4cde-93da-1ba2154e2d22 | ||
# ╠═e6fe559e-46f1-4336-9513-7306c32f2515 | ||
# ╠═ed7cb9bb-4b83-4ed6-a8fd-9936eb02643b | ||
# ╠═084ac351-17e6-4916-b0b7-12aa593984b2 | ||
# ╠═322962f3-c1f5-47d1-882b-f4efed0c2fa0 | ||
# ╠═5e996c83-88dc-4569-a435-4d00684de2af | ||
# ╠═69e7d9cd-7900-474d-aeba-06b38e62d756 | ||
# ╠═19b69d34-367e-4794-abb1-3f76e1356561 | ||
# ╠═bbe9a49b-6495-4c13-91d0-3a4dbd0a2800 | ||
# ╠═444d7454-48a9-44dd-87a3-f4cf1a495072 | ||
# ╠═b0f72692-d76c-44f1-91aa-2e9b7b882d1b | ||
# ╠═0c043a99-18bf-4b75-95fa-ce8aea8f8bef | ||
# ╠═06ccb204-694d-4c64-9ecd-742c009fb972 | ||
# ╠═2ba19614-f2ef-4dca-9f61-b95908b7c81c | ||
# ╠═27ee4adf-b7d9-41e0-b0a0-f414422b7b73 | ||
# ╠═ed6d3a1a-3fcd-4ef1-a57f-a394895d4b29 | ||
# ╠═fbee75ec-96f0-41b6-8c14-a724872476fb | ||
# ╠═7d7d3ff6-30d4-477a-8f03-e4c08ac8a608 | ||
# ╠═ba64c0a2-8cd4-421a-b494-a70f1f92ca0f | ||
# ╠═d30b10f4-2c62-4596-aa73-e4f2796b0541 | ||
# ╠═04b0a526-b808-4c61-a565-6c60e12be2b7 | ||
# ╠═b38b707f-6b97-424a-a22a-cf2fdebcfb31 |