MoOx material model

1. Introduction

This page contains the OghmaNano material model for MoOx (MoO3-x).

Sub-stoichiometric molybdenum oxide (MoO3-x), oxygen-deficient high-work-function hole-injection/extraction oxide

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 "MoOx", enabled
end


function material.description()
	local enabled = true

	return "Sub-stoichiometric molybdenum oxide (MoO3-x), oxygen-deficient high-work-function hole-injection/extraction oxide", enabled
end


function material.formula()
	local enabled = true

	return "MoO3-x", enabled
end


function material.Eg(state)
	-- Units: eV
	--
	-- Reference:
	-- J. Meyer et al., "Transition Metal Oxides for Organic
	-- Electronics", Adv. Mater. 24, 5408-5427, 2012.
	--
	-- Host alpha-MoO3 optical gap ~3.0 eV.
	--
	-- Note: this is the OXYGEN-DEFICIENT oxide. Sub-stoichiometry
	-- introduces a dense manifold of O-vacancy gap states below the
	-- conduction band; near the Fermi level the transport-relevant
	-- states are these defect states, not the nominal 3.0 eV gap.
	-- Constant value.

	local enabled = true
	local value = 3.0

	return value, enabled
end


function material.Xi(state)
	-- Electron affinity
	-- Units: eV
	--
	-- Reference:
	-- M. Kroeger et al., Appl. Phys. Lett. 95, 123301, 2009;
	-- J. Meyer et al., Adv. Mater. 24, 5408, 2012;
	-- M. T. Greiner et al., Nat. Mater. 11, 76, 2012.
	--
	-- This is the KEY device parameter. The host oxide has a very deep
	-- conduction band (electron affinity ~6.7 eV) which enables
	-- electron extraction from the HOMO of adjacent organics
	-- (charge-transfer hole-injection mechanism).
	--
	-- IMPORTANT: real reduced MoOx films have a LOWER effective work
	-- function than pristine MoO3, typically ~5.3-5.7 eV, because
	-- filled O-vacancy gap states pin the Fermi level below the deep
	-- CB. Choose the value to match your film's processing/air
	-- exposure.

	local enabled = true
	local value = 6.7

	return value, enabled
end


function material.Nc(state)
	-- Effective conduction-band density of states
	-- Units: m^-3
	--
	-- Note: conduction in MoOx is via an O-vacancy-derived defect band
	-- rather than a clean parabolic CB, so this parabolic Nc is an
	-- order-of-magnitude placeholder only.

	local enabled = true
	local T = state.T
	local value = 1.0e25*(T/300.0)^1.5

	return value, enabled
end


function material.Nv(state)
	-- Effective valence-band density of states
	-- Units: m^-3
	--
	-- Note: order-of-magnitude placeholder; see Nc note.

	local enabled = true
	local T = state.T
	local value = 1.0e25*(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
	--
	-- Reference:
	-- Evaporated/amorphous molybdenum-oxide transport discussed in
	-- J. Meyer et al., Adv. Mater. 24, 5408, 2012.
	--
	-- Note: electron mobility is very low and extremely
	-- process-dependent, reported ~1e-6 to ~1e-1 cm^2/V/s. An
	-- effective 1e-4 cm^2/V/s = 1e-8 m^2/V/s is used (approximate).
	-- Transport is trap/hopping limited, so no phonon (300/T)^n law is
	-- applied. Tune to your layer.

	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: MoOx extracts holes from the adjacent organic by electron
	-- transfer into its deep CB, NOT by hole conduction within MoOx,
	-- so intrinsic hole mobility is not the operative channel and is
	-- poorly constrained. Low placeholder.

	local enabled = true
	local value = 1.0e-9

	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:
	-- Molybdenum-oxide optical constants / anisotropy, J. Meyer et
	-- al., Adv. Mater. 24, 5408, 2012, and ellipsometry studies.
	--
	-- Note: layered, anisotropic and poorly constrained for reduced
	-- films. High-frequency epsilon_inf ~5.5; static value larger and
	-- axis-dependent. Representative (approximate) value used.

	local enabled = true
	local value = 5.5

	return value, enabled
end


function material.free_to_free_recombination(state)
	-- Radiative (band-to-band) recombination coefficient
	-- Units: m^3 s^-1
	--
	-- Note: thin charge-selective buffer, not an absorber; bulk
	-- radiative recombination is not the operative physics. 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: MoOx is deliberately oxygen-deficient, so its O-vacancy
	-- gap-state density is high (higher than stoichiometric MoO3) and
	-- very process-dependent. Placeholder; set from measurement.

	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:
	-- Low, anisotropic values from the molybdenum-oxide
	-- thermal-transport literature (CRC Handbook; layered-oxide
	-- studies).
	--
	-- Note: layered/amorphous evaporated films have low thermal
	-- conductivity. A low representative value is used. Approximate.

	local enabled = true
	local value = 1.0

	return value, enabled
end


function material.heat_capacity(state)
	-- Specific heat capacity
	-- Units: J kg^-1 K^-1
	--
	-- Reference:
	-- From alpha-MoO3 molar heat capacity ~75 J/mol/K and
	-- M = 143.94 g/mol, c_p ~ 520 J/kg/K (CRC Handbook of Chemistry
	-- and Physics). MoOx is similar. Approximate.

	local enabled = true
	local value = 520.0

	return value, enabled
end


function material.density(state)
	-- Mass density
	-- Units: kg m^-3
	--
	-- Reference:
	-- alpha-MoO3 rho = 4.69 g/cm^3 (CRC Handbook of Chemistry and
	-- Physics).
	--
	-- Note: reduced MoO3-x is denser than MoO3 and approaches MoO2
	-- (6.47 g/cm^3) as x increases. A near-MoO3 value is used for
	-- small x. Approximate.

	local enabled = true
	local value = 4700.0

	return value, enabled
end


function material.lattice_constant(state)
	-- Cubic lattice constant
	-- Units: m
	--
	-- DISABLED: evaporated MoOx device films are typically AMORPHOUS,
	-- and the crystalline host alpha-MoO3 is orthorhombic/layered, NOT
	-- cubic, so a single cubic lattice constant is not meaningful.
	--
	-- Reference (crystallography):
	-- R. W. G. Wyckoff, "Crystal Structures".
	-- alpha-MoO3: a = 3.963 A, b = 13.855 A, c = 3.696 A.
	-- The a-axis value is returned for reference only.

	local enabled = false
	local value = 3.963e-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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