TiOx material model

1. Introduction

This page contains the OghmaNano material model for TiOx (TiOx).

Titanium oxide (sub-stoichiometric TiO2-x / amorphous TiOx), electron-transport/hole-blocking layer

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 "TiOx", enabled
end


function material.description()
	local enabled = true

	return "Titanium oxide (sub-stoichiometric TiO2-x / amorphous TiOx), electron-transport/hole-blocking layer", enabled
end


function material.formula()
	local enabled = true

	return "TiOx", enabled
end


function material.Eg(state)
	-- Units: eV
	--
	-- Reference:
	-- J. Robertson, "High dielectric constant oxides",
	-- Eur. Phys. J. Appl. Phys. 28, 265-291, 2004.
	--
	-- TiO2 gap: anatase ~3.2 eV, rutile ~3.0 eV. Device TiOx is
	-- typically anatase-like / amorphous with an optical gap ~3.2 eV.
	--
	-- Note: sub-stoichiometric TiOx carries a significant density of
	-- O-vacancy gap states below the CB; the nominal gap here does not
	-- capture those. Constant value; T-dependence weak/uncharacterised.

	local enabled = true
	local value = 3.2

	return value, enabled
end


function material.Xi(state)
	-- Electron affinity
	-- Units: eV
	--
	-- Reference:
	-- CB position from device-alignment literature
	-- (J. Robertson, Eur. Phys. J. Appl. Phys. 28, 265, 2004).
	--
	-- TiO2/TiOx electron affinity ~4.0-4.2 eV (CB ~ -4.0 to -4.2 eV
	-- vs vacuum). This shallow CB, combined with a deep valence band,
	-- is what makes TiOx a good electron-transport / hole-blocking
	-- layer. Primary device-relevant parameter.

	local enabled = true
	local value = 4.1

	return value, enabled
end


function material.Nc(state)
	-- Effective conduction-band density of states
	-- Units: m^-3
	--
	-- Reference:
	-- From anatase CB effective mass m_e* ~ 1 m0 (TiO2 band-structure
	-- literature): Nc(300 K) ~ 2.5e19 cm^-3 = 2.5e25 m^-3. Approximate.
	--
	-- Note: amorphous TiOx transport is dominated by gap states rather
	-- than a clean parabolic band.

	local enabled = true
	local T = state.T
	local value = 2.5e25*(T/300.0)^1.5

	return value, enabled
end


function material.Nv(state)
	-- Effective valence-band density of states
	-- Units: m^-3
	--
	-- Reference:
	-- Heavy (O 2p) valence band, m_h* of order a few m0:
	-- Nv(300 K) ~ 1e26 m^-3. Poorly constrained placeholder.

	local enabled = true
	local T = state.T
	local value = 1.0e26*(T/300.0)^1.5

	return value, enabled
end


function material.mu_e(state)
	-- Low-field electron mobility
	-- Units: m^2 V^-1 s^-1
	--
	-- Note: strongly morphology-dependent. Amorphous / sol-gel TiOx
	-- device layers have very low electron mobility ~1e-4 cm^2/V/s
	-- (= 1e-8 m^2/V/s), used here as the device-representative value.
	-- Crystalline anatase is far higher (~1-20 cm^2/V/s). Transport is
	-- trap-limited in the disordered film, so no phonon (300/T)^n law
	-- is applied.

	local enabled = true
	local value = 1.0e-8

	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
	--
	-- Note: TiOx is deliberately a HOLE BLOCKER; hole transport is
	-- intentionally negligible and poorly constrained. Low placeholder.

	local enabled = true
	local value = 1.0e-10

	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
	--
	-- Reference:
	-- TiO2 dielectric constants, CRC Handbook of Chemistry and
	-- Physics; J. Robertson, Eur. Phys. J. Appl. Phys. 28, 265, 2004.
	--
	-- Note: TiO2 permittivity is large and strongly polymorph- and
	-- axis-dependent: anatase static ~30-45, rutile ~86 (perp c) to
	-- ~170 (parallel c). Amorphous device TiOx is typically lower
	-- (~20-40) and process-dependent. A representative (approximate)
	-- anatase-like value is used; adjust to your film if the local
	-- field matters.

	local enabled = true
	local value = 40.0

	return value, enabled
end


function material.free_to_free_recombination(state)
	-- Radiative (band-to-band) recombination coefficient
	-- Units: m^3 s^-1
	--
	-- Note: TiOx is an indirect wide-gap oxide used as a transport
	-- layer, not an absorber; bulk radiative recombination is weak and
	-- not the operative loss. Poorly constrained placeholder.

	local enabled = true
	local value = 1.0e-19

	return value, enabled
end


function material.auger_Cn(state)
	-- Electron Auger recombination coefficient
	-- Units: m^6 s^-1
	--
	-- Note: not characterised. Order-of-magnitude placeholder.

	local enabled = true
	local value = 1.0e-43

	return value, enabled
end


function material.auger_Cp(state)
	-- Hole Auger recombination coefficient
	-- Units: m^6 s^-1
	--
	-- Note: not characterised. Order-of-magnitude placeholder.

	local enabled = true
	local value = 1.0e-43

	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 band).
	-- Negative values are below mid-gap (towards the valence band).

	local enabled = true
	local value = 0.0

	return value, enabled
end


function material.ss_srh_Nt(state)
	-- SRH trap density
	-- Units: m^-3
	--
	-- Note: for a defect-rich TiOx ETL this is often the DOMINANT and
	-- most important parameter (interfacial / bulk recombination). It
	-- is strongly quality/processing dependent; the value below is a
	-- placeholder to be set from measured lifetime or J-V fitting.

	local enabled = true
	local value = 1.0e22

	return value, enabled
end


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

	local enabled = true
	local value = 1.0e-19

	return value, enabled
end


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

	local enabled = true
	local value = 1.0e-19

	return value, enabled
end


function material.thermal_conductivity(state)
	-- Thermal conductivity
	-- Units: W m^-1 K^-1
	--
	-- Reference:
	-- Bulk anatase TiO2 ~ 4-10 W/m/K (CRC Handbook of Chemistry and
	-- Physics; TiO2 thermal-transport literature).
	--
	-- Note: thin amorphous TiOx films are lower (~1-3 W/m/K). A low
	-- film-appropriate value is used. Approximate / poorly constrained.

	local enabled = true
	local value = 3.0

	return value, enabled
end


function material.heat_capacity(state)
	-- Specific heat capacity
	-- Units: J kg^-1 K^-1
	--
	-- Reference:
	-- TiO2 (anatase), c_p(300 K) ~ 700 J/kg/K (CRC Handbook of
	-- Chemistry and Physics).

	local enabled = true
	local value = 700.0

	return value, enabled
end


function material.density(state)
	-- Mass density
	-- Units: kg m^-3
	--
	-- Reference:
	-- TiO2 anatase rho = 3.78 g/cm^3 (CRC Handbook of Chemistry and
	-- Physics). Amorphous TiOx films are often less dense
	-- (~3.2-3.9 g/cm^3).

	local enabled = true
	local value = 3780.0

	return value, enabled
end


function material.lattice_constant(state)
	-- Cubic lattice constant
	-- Units: m
	--
	-- DISABLED: device TiOx is usually AMORPHOUS or nanocrystalline
	-- anatase; anatase itself is tetragonal, NOT cubic. A single cubic
	-- lattice constant is not meaningful.
	--
	-- Reference (crystallography):
	-- R. W. G. Wyckoff, "Crystal Structures".
	-- Anatase TiO2: a = 3.7845 A, c = 9.5143 A.
	-- The a-axis value is returned for reference only.

	local enabled = false
	local value = 3.7845e-10

	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)))
end

return material

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-- This file is part of the OghmaNano Materials Model Library.
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