PTB7 material model

1. Introduction

This page contains the OghmaNano material model for PTB7 (C41H53FO4S4).

PTB7 low-bandgap donor polymer (benzodithiophene-alt-fluorothienothiophene)

The model is written in Lua and provides simulation-ready material parameterisations for use within OghmaNano. For documentation, licensing, references, and information about the scope and accuracy of these models, see the material scripting documentation.

2. Lua material model

-- See end of file for copyright, licensing and documentation links.

local material = {}

function material.name()
	local enabled = true

	return "PTB7", enabled
end


function material.description()
	local enabled = true

	return "PTB7 low-bandgap donor polymer (benzodithiophene-alt-fluorothienothiophene)", enabled
end


function material.formula()
	local enabled = true

	-- Nominal repeat-unit formula of the benzodithiophene-alt-
	-- fluorothieno[3,4-b]thiophene copolymer (MW ~725 g/mol).
	return "C41H53FO4S4", enabled
end


function material.Eg(state)
	-- Units: eV
	--
	-- The value returned is the electrochemical HOMO-LUMO gap
	-- (5.15 - 3.31 = 1.84 eV), chosen so that Eg = HOMO - Xi is
	-- self-consistent with the frontier levels below. The OPTICAL gap is
	-- lower, ~1.6-1.65 eV (absorption onset ~750-770 nm), the difference
	-- being roughly the exciton binding energy. Optical absorption itself is
	-- set separately via the n/k database. The Varshni model does not apply
	-- to a disordered polymer; a constant value is returned.
	--
	-- Reference:
	-- Y. Liang, Z. Xu, J. Xia, S.-T. Tsai, Y. Wu, G. Li, C. Ray, L. Yu,
	-- "For the Bright Future - Bulk Heterojunction Polymer Solar Cells with
	-- Power Conversion Efficiency of 7.4%", Adv. Mater., 22, E135-E138, 2010.

	local enabled = true
	local value = 1.84

	return value, enabled
end


function material.Xi(state)
	-- Electron affinity (LUMO level below vacuum)
	-- Units: eV
	--
	-- Reported LUMO of PTB7 is -3.31 eV and HOMO -5.15 eV (cyclic
	-- voltammetry). With Eg = 1.84 eV this reproduces the measured HOMO at
	-- -5.15 eV.
	--
	-- Reference:
	-- Y. Liang et al., Adv. Mater., 22, E135-E138, 2010.

	local enabled = true
	local value = 3.31

	return value, enabled
end


function material.Nc(state)
	-- Effective conduction-band (LUMO) density of states
	-- Units: m^-3
	--
	-- Disordered organic (approximately Gaussian DOS): a constant effective
	-- DOS is used; the crystalline (T/300)^1.5 scaling does not apply. The
	-- polymer site density is ~1e27 m^-3; the effective transport-level DOS
	-- used in drift-diffusion is normally set lower (1e25-1e27 m^-3). A
	-- representative value is used.
	--
	-- Reference:
	-- Add the precise reference used for the effective density of states.

	local enabled = true
	local value = 1.0e26

	return value, enabled
end


function material.Nv(state)
	-- Effective valence-band (HOMO) density of states
	-- Units: m^-3
	--
	-- See material.Nc. Constant effective DOS; representative value.
	--
	-- Reference:
	-- Add the precise reference used for the effective density of states.

	local enabled = true
	local value = 1.0e26

	return value, enabled
end


function material.mu_e(state)
	-- Low-field electron mobility
	-- Units: m^2 V^-1 s^-1
	--
	-- PTB7 is a donor (hole-transporting) polymer; neat-film electron
	-- transport is poorly defined (in blends electrons travel through the
	-- fullerene phase). A low representative value is used; adjust for the
	-- specific system.
	--
	-- Note: organic mobility is thermally activated, field dependent
	-- (Poole-Frenkel) and carrier-density dependent; the crystalline
	-- (300/T)^n form is not used.
	--
	-- Reference:
	-- Add the precise reference used for this value.

	local enabled = true
	local value = 1.0e-10

	return value, enabled
end

function material.mue_x(state)
	return material.mu_e(state)
end

function material.mue_y(state)
	return material.mu_e(state)
end

function material.mue_z(state)
	return material.mu_e(state)
end


function material.mu_h(state)
	-- Low-field hole mobility
	-- Units: m^2 V^-1 s^-1
	--
	-- Neat PTB7 hole mobility is ~1e-3 cm^2/V/s (= 1e-7 m^2/V/s) by
	-- time-of-flight; space-charge-limited-current values are somewhat lower
	-- (~4e-4 cm^2/V/s neat, ~2e-4 cm^2/V/s in PTB7:PC71BM blends). The
	-- neat-film value is returned; reduce it for blends.
	--
	-- Note: organic mobility is thermally activated, field dependent and
	-- carrier-density dependent.
	--
	-- References:
	-- "Charge carrier mobility of the organic photovoltaic materials PTB7 and
	-- PC71BM and its influence on device performance", Organic Electronics,
	-- 2015 (time-of-flight, neat ~1e-3 cm^2/V/s).
	-- Appl. Phys. Lett., 108, 073505, 2016 (SCLC, PTB7 ~3.9e-4 cm^2/V/s).

	local enabled = true
	local value = 1.0e-7

	return value, enabled
end

function material.muh_x(state)
	return material.mu_h(state)
end

function material.muh_y(state)
	return material.mu_h(state)
end

function material.muh_z(state)
	return material.mu_h(state)
end


function material.epsilonr(state)
	-- Relative static permittivity
	-- Dimensionless
	--
	-- A value of ~3 is the standard assumption for conjugated polymers such
	-- as PTB7 and is commonly used when extracting SCLC mobilities.
	--
	-- Reference:
	-- Add the precise reference used for this value (eps_r ~= 3 is the typical
	-- assumed value for conjugated polymers).

	local enabled = true
	local value = 3.0

	return value, enabled
end


function material.free_to_free_recombination(state)
	-- Bimolecular (band-to-band) recombination coefficient
	-- Units: m^3 s^-1
	--
	-- Free-carrier recombination is Langevin-type (non-radiative). The
	-- Langevin coefficient q*(mu_e + mu_h)/(eps0*eps_r) is ~6e-16 m^3/s for
	-- the values above; PTB7:PC71BM is well known to show strongly REDUCED
	-- (sub-)Langevin recombination (prefactor << 1), so a smaller effective
	-- value is used.
	--
	-- Reference:
	-- Add the precise reference used for this value.

	local enabled = true
	local value = 1.0e-17

	return value, enabled
end


function material.auger_Cn(state)
	-- Electron Auger recombination coefficient
	-- Units: m^6 s^-1
	--
	-- Negligible in organic semiconductors; a very small representative value
	-- is used so the channel is effectively inactive.
	--
	-- Reference:
	-- Not applicable / negligible for organic semiconductors.

	local enabled = true
	local value = 1.0e-45

	return value, enabled
end


function material.auger_Cp(state)
	-- Hole Auger recombination coefficient
	-- Units: m^6 s^-1
	--
	-- See material.auger_Cn. Very small representative value.
	--
	-- Reference:
	-- Not applicable / negligible for organic semiconductors.

	local enabled = true
	local value = 1.0e-45

	return value, enabled
end


function material.ss_srh_trap_energy(state)
	-- SRH trap energy relative to the middle of the band gap.
	-- Units: eV
	--
	-- Positive values are above mid-gap (towards the conduction/LUMO band).
	-- Negative values are below mid-gap (towards the valence/HOMO band).
	--
	-- Placed at mid-gap as a representative single-level approximation.

	local enabled = true
	local value = 0.0

	return value, enabled
end


function material.ss_srh_Nt(state)
	-- SRH trap density
	-- Units: m^-3
	--
	-- SCLC-extracted trap density for neat PTB7 is ~1.1e17 cm^-3
	-- (= 1.1e23 m^-3).
	--
	-- Reference:
	-- Appl. Phys. Lett., 108, 073505, 2016.

	local enabled = true
	local value = 1.1e23

	return value, enabled
end


function material.ss_srh_sigma_n(state)
	-- Electron capture cross section
	-- Units: m^2
	--
	-- Representative value.

	local enabled = true
	local value = 1.0e-20

	return value, enabled
end


function material.ss_srh_sigma_p(state)
	-- Hole capture cross section
	-- Units: m^2
	--
	-- Representative value.

	local enabled = true
	local value = 1.0e-20

	return value, enabled
end


function material.thermal_conductivity(state)
	-- Thermal conductivity
	-- Units: W m^-1 K^-1
	--
	-- Not specifically characterised for PTB7; organic semiconducting films
	-- are typically ~0.1-0.3 W/m/K. A representative value is used and the
	-- crystalline phonon (300/T)^n scaling is not applied.
	--
	-- Reference:
	-- Add the precise reference used for this value.

	local enabled = true
	local value = 0.2

	return value, enabled
end


function material.heat_capacity(state)
	-- Specific heat capacity
	-- Units: J kg^-1 K^-1
	--
	-- Not specifically characterised for PTB7; organic polymers are typically
	-- ~1000-2000 J/kg/K near room temperature. A representative value is used.
	--
	-- Reference:
	-- Add the precise reference used for this value.

	local enabled = true
	local value = 1200.0

	return value, enabled
end


function material.density(state)
	-- Mass density
	-- Units: kg m^-3
	--
	-- Representative organic-polymer film density (~1.1 g/cm^3); not
	-- precisely characterised for PTB7.
	--
	-- Reference:
	-- Add the precise reference used for this value.

	local enabled = true
	local value = 1100.0

	return value, enabled
end


function material.lattice_constant(state)
	-- Characteristic structural spacing
	-- Units: m
	--
	-- PTB7 is weakly ordered / largely amorphous, so a single cubic lattice
	-- constant is not meaningful. GIWAXS typically shows a lamellar (100)
	-- spacing of ~1.9-2.2e-9 m (set by the branched 2-ethylhexyl side chains)
	-- and a pi-pi (010) stacking distance of ~0.37-0.39e-9 m. The lamellar
	-- spacing is returned as a representative value.
	--
	-- Reference:
	-- Add the precise reference used for this value.

	local enabled = true
	local value = 2.0e-9

	return value, enabled
end

function material.Ntrape(state)
	-- Electron tail (exponential band-tail) trap density
	-- Units: m^-3
	--
	-- Representative magnitude for a disordered organic; adjust for the
	-- specific film. (Distinct from the deep SCLC trap density in ss_srh_Nt.)

	local enabled = true
	local value = 1.0e26

	return value, enabled
end


function material.Ntraph(state)
	-- Hole tail (exponential band-tail) trap density
	-- Units: m^-3
	--
	-- See material.Ntrape. Representative value.

	local enabled = true
	local value = 1.0e26

	return value, enabled
end


function material.Etrape(state)
	-- Electron tail characteristic (Urbach) energy
	-- Units: eV
	--
	-- Representative disordered-organic value (Urbach energy ~60-70 meV).

	local enabled = true
	local value = 0.06

	return value, enabled
end


function material.Etraph(state)
	-- Hole tail characteristic (Urbach) energy
	-- Units: eV
	--
	-- See material.Etrape. Representative value.

	local enabled = true
	local value = 0.06

	return value, enabled
end


function material.srhsigman_e(state)
	-- Electron-to-electron capture cross section
	-- Units: m^2
	--
	-- Representative value for the tail-state SRH model.

	local enabled = true
	local value = 1.0e-20

	return value, enabled
end


function material.srhsigmap_e(state)
	-- Hole-to-electron capture cross section
	-- Units: m^2
	--
	-- Representative value for the tail-state SRH model.

	local enabled = true
	local value = 1.0e-20

	return value, enabled
end


function material.srhsigman_h(state)
	-- Electron-to-hole capture cross section
	-- Units: m^2
	--
	-- Representative value for the tail-state SRH model.

	local enabled = true
	local value = 1.0e-20

	return value, enabled
end


function material.srhsigmap_h(state)
	-- Hole-to-hole capture cross section
	-- Units: m^2
	--
	-- Representative value for the tail-state SRH model.

	local enabled = true
	local value = 1.0e-20

	return value, enabled
end

function material.print()
	local state = {
		T = 300.0,
		x = 0.0,
		y = 0.0,
		z = 0.0,
		photon_density = 0.0,
	}

	print(string.format("Material:               %s", material.name()))
	print(string.format("Description:            %s", material.description()))
	print(string.format("Formula:                %s", material.formula()))
	print(string.format("Temperature:            %.2f K", state.T))
	print(string.format("Position:               %.6e, %.6e, %.6e m", state.x, state.y, state.z))
	print(string.format("Photon density:         %.6e m^-3", state.photon_density))

	print(string.format("Band gap:               %.6f eV", material.Eg(state)))
	print(string.format("Electron affinity:      %.6f eV", material.Xi(state)))
	print(string.format("Electron mobility:      %.6e m^2/V/s", material.mu_e(state)))
	print(string.format("Hole mobility:          %.6e m^2/V/s", material.mu_h(state)))
	print(string.format("Nc:                     %.6e m^-3", material.Nc(state)))
	print(string.format("Nv:                     %.6e m^-3", material.Nv(state)))
	print(string.format("Relative permittivity:  %.6f", material.epsilonr(state)))

	print(string.format("Radiative coeff.:       %.6e m^3/s", material.free_to_free_recombination(state)))
	print(string.format("Electron Auger coeff.:  %.6e m^6/s", material.auger_Cn(state)))
	print(string.format("Hole Auger coeff.:      %.6e m^6/s", material.auger_Cp(state)))

	print(string.format("SRH trap energy:        %.6f eV", material.ss_srh_trap_energy(state)))
	print(string.format("SRH trap density:       %.6e m^-3", material.ss_srh_Nt(state)))
	print(string.format("SRH sigma n:            %.6e m^2", material.ss_srh_sigma_n(state)))
	print(string.format("SRH sigma p:            %.6e m^2", material.ss_srh_sigma_p(state)))

	print(string.format("Thermal conductivity:   %.6e W/m/K", material.thermal_conductivity(state)))
	print(string.format("Heat capacity:          %.6e J/kg/K", material.heat_capacity(state)))
	print(string.format("Mass density:           %.6e kg/m^3", material.density(state)))

	print(string.format("Electron trap density:  %.6e m^-3", material.Ntrape(state)))
	print(string.format("Hole trap density:      %.6e m^-3", material.Ntraph(state)))
	print(string.format("Electron trap energy:   %.6f eV", material.Etrape(state)))
	print(string.format("Hole trap energy:       %.6f eV", material.Etraph(state)))

	print(string.format("SRH sigma n->e:         %.6e m^2", material.srhsigman_e(state)))
	print(string.format("SRH sigma p->e:         %.6e m^2", material.srhsigmap_e(state)))
	print(string.format("SRH sigma n->h:         %.6e m^2", material.srhsigman_h(state)))
	print(string.format("SRH sigma p->h:         %.6e m^2", material.srhsigmap_h(state)))
end

return material

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-- All rights reserved.
--
-- This file is part of the OghmaNano Materials Model Library.
--
-- Website:
-- https://www.oghma-nano.com
--
-- Documentation and accuracy statement:
-- https://www.oghma-nano.com/manual/material-scripts.html
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-- These material models are provided to support scientific research and
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