Chicheley Rubrene mobility notebook.

examples/RubreneOrdejonPlutoNotebook.jl

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+### 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