FTO material model

1. Introduction

This page contains the OghmaNano material model for FTO (SnO2:F).

Fluorine-doped tin oxide (SnO2:F), transparent conducting 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

Supporting definitions

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

local material = {}

function material.name()
	local enabled = true

	return "FTO", enabled
end

function material.description()
	local enabled = true

	return "Fluorine-doped tin oxide (SnO2:F), transparent conducting oxide", enabled
end

function material.formula()
	local enabled = true

	return "SnO2:F", enabled
end

function material.Eg(state)
	-- Units: eV
	--
	-- Reference:
	-- M. Batzill, U. Diebold, "The surface and materials science of
	-- tin oxide", Prog. Surf. Sci. 79, 47-154, 2005.
	--
	-- Fundamental (direct) gap of SnO2 ~3.6 eV at 300 K.
	--
	-- Note: FTO is a DEGENERATE n-type semiconductor. Heavy doping
	-- fills the conduction band, so the *optical* (apparent) gap is
	-- widened by the Burstein-Moss shift to ~4.0-4.6 eV. This function
	-- returns the fundamental gap, not the optical gap.
	--
	-- Temperature dependence is weak and not well characterised for
	-- device films, so a constant value is used rather than a fitted
	-- Varshni expression.

	local enabled = true
	local value = 3.6

	return value, enabled
end

function material.Xi(state)
	-- Electron affinity
	-- Units: eV
	--
	-- Reference:
	-- M. Batzill, U. Diebold, Prog. Surf. Sci. 79, 47, 2005;
	-- and device-alignment literature.
	--
	-- SnO2 electron affinity ~4.5 eV. Because FTO is degenerate, the
	-- Fermi level sits in/near the conduction band and the effective
	-- work function (~4.4 eV, treatment-dependent, ~4.4-4.8 eV) nearly
	-- coincides with the affinity. The work function is the number
	-- that actually sets contact alignment when FTO is used as an
	-- electrode.

	local enabled = true
	local value = 4.5

	return value, enabled
end

function material.Nc(state)
	-- Effective conduction-band density of states
	-- Units: m^-3
	--
	-- Reference:
	-- Estimated from SnO2 conduction-band effective mass
	-- m_e* ~ 0.3 m0 (M. Batzill, U. Diebold, Prog. Surf. Sci. 79,
	-- 47-154, 2005): Nc(300 K) ~ 4e18 cm^-3 = 4e24 m^-3. Approximate.
	--
	-- Note: FTO is degenerate (free-electron density ~1e20-1e21
	-- cm^-3), so this parabolic-band Nc is a formal quantity only; the
	-- carrier density is set by doping, not by Nc*exp(...). The
	-- (T/300)^1.5 form is the simple non-degenerate model.

	local enabled = true
	local T = state.T
	local value = 4.0e24*(T/300.0)^1.5

	return value, enabled
end

function material.Nv(state)
	-- Effective valence-band density of states
	-- Units: m^-3
	--
	-- Reference:
	-- Estimated from a heavy valence-band mass (m_h* ~ 1 m0):
	-- Nv(300 K) ~ 2.5e19 cm^-3 = 2.5e25 m^-3.
	--
	-- Note: the SnO2 hole mass is poorly constrained (see M. Batzill,
	-- U. Diebold, Prog. Surf. Sci. 79, 47, 2005); treat as approximate.
	-- Rarely relevant for an n-type TCO.

	local enabled = true
	local T = state.T
	local value = 2.5e25*(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:
	-- Representative FTO thin-film Hall mobility from the
	-- transparent-conducting-oxide literature (~10-30 cm^2/V/s); a
	-- mid value of 15 cm^2/V/s = 0.0015 m^2/V/s is used. Approximate.
	--
	-- Note: transport is ionised-impurity limited in this degenerate
	-- material, NOT phonon limited, so no (300/T)^n phonon power law
	-- is applied. Temperature dependence is weak; a constant is used.

	local enabled = true
	local value = 0.0015

	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: FTO is an n-type electrode; hole transport is essentially
	-- irrelevant to device operation and is not experimentally
	-- constrained. A small placeholder is used. Enabled left true so
	-- the solver has a finite value, but do not rely on it.

	local enabled = true
	local value = 1.0e-4

	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:
	-- SnO2 (rutile) static permittivity, M. Batzill, U. Diebold,
	-- Prog. Surf. Sci. 79, 47, 2005.
	--
	-- Note: rutile SnO2 is anisotropic (~9 perpendicular to c, ~14
	-- parallel to c). A representative perpendicular value is used.

	local enabled = true
	local value = 9.0

	return value, enabled
end

function material.free_to_free_recombination(state)
	-- Radiative (band-to-band) recombination coefficient
	-- Units: m^3 s^-1
	--
	-- Note: SnO2 is an indirect/wide-gap oxide and, as a degenerate
	-- electrode, bulk radiative recombination is not the operative
	-- physics (contact recombination dominates). Poorly constrained
	-- placeholder.

	local enabled = true
	local value = 1.0e-21

	return value, enabled
end

function material.auger_Cn(state)
	-- Electron Auger recombination coefficient
	-- Units: m^6 s^-1
	--
	-- Note: not characterised for FTO. 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 for FTO. 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
	--
	-- Material-quality dependent placeholder.

	local enabled = true
	local value = 1.0e21

	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_kl(state)
	-- Thermal conductivity
	-- Units: W m^-1 K^-1
	--
	-- Reference:
	-- Bulk rutile SnO2 ~55-98 W/m/K, anisotropic (CRC Handbook of
	-- Chemistry and Physics; thermal-transport literature).
	--
	-- Note: polycrystalline FTO device FILMS are much lower
	-- (~1-15 W/m/K) owing to grain-boundary and thickness-dependent
	-- scattering. A low film-appropriate value is used. Approximate /
	-- poorly constrained.

	local enabled = true
	local value = 5.0

	return value, enabled
end

function material.heat_capacity(state)
	-- Specific heat capacity
	-- Units: J kg^-1 K^-1
	--
	-- Reference:
	-- SnO2, c_p(300 K) ~ 350 J/kg/K (CRC Handbook of Chemistry and
	-- Physics; thermochemical data). Approximate.

	local enabled = true
	local value = 350.0

	return value, enabled
end

function material.density(state)
	-- Mass density
	-- Units: kg m^-3
	--
	-- Reference:
	-- SnO2 (rutile/cassiterite) rho = 6.95 g/cm^3 (CRC Handbook of
	-- Chemistry and Physics).

	local enabled = true
	local value = 6950.0

	return value, enabled
end

function material.lattice_constant(state)
	-- Cubic lattice constant
	-- Units: m
	--
	-- DISABLED: SnO2 is tetragonal (rutile), NOT cubic, so a single
	-- cubic lattice constant is not meaningful.
	--
	-- Reference (crystallography):
	-- R. W. G. Wyckoff, "Crystal Structures"; CRC Handbook.
	-- Rutile SnO2: a = b = 4.737 A, c = 3.186 A at 300 K.
	-- The a-axis value is returned for reference only.

	local enabled = false
	local value = 4.737e-10

	return value, enabled
end

function material.thermal_tau_e(state)
	-- Electron energy relaxation time towards the lattice temperature
	-- Units: s
	--
	-- Value basis: Numerical fallback
	-- Confidence: Low
	--
	-- Reference:
	-- No specific reference supplied.
	--
	-- Comments:
	-- Normally used as a degenerate conductor/electrode, not as a bipolar
	-- non-degenerate DD layer. Fast fallback only.

	local enabled = false
	local value = 1.000000e-14

	return value, enabled
end

function material.thermal_tau_h(state)
	-- Hole energy relaxation time towards the lattice temperature
	-- Units: s
	--
	-- Value basis: Numerical fallback
	-- Confidence: Low
	--
	-- Reference:
	-- No specific reference supplied.
	--
	-- Comments:
	-- Normally used as a degenerate conductor/electrode, not as a bipolar
	-- non-degenerate DD layer. Fast fallback only.

	local enabled = false
	local value = 1.000000e-14

	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("Electron energy relax.:  %.6e s", material.thermal_tau_e(state)))
	print(string.format("Hole energy relax.:      %.6e s", material.thermal_tau_h(state)))

	print(string.format("Thermal conductivity:   %.6e W/m/K", material.thermal_kl(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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