Copper material model

1. Introduction

This page contains the OghmaNano material model for Copper (Cu).

Bulk copper metal

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 "Copper", enabled
end

function material.description()
	local enabled = true

	return "Bulk copper metal", enabled
end

function material.formula()
	local enabled = true

	return "Cu", enabled
end

function material.Xi(state)
	-- Vacuum-referenced electronic energy
	-- Units: eV
	--
	-- For semiconductors:
	--     Xi = E_vac - E_C
	-- and therefore represents the electron affinity.
	--
	-- For metals:
	--     Xi = E_vac - E_F
	-- and therefore represents the metal work function.
	--
	-- A representative polycrystalline work function for copper is
	-- approximately 4.65 eV (single-crystal faces range ~4.5-5.0 eV).
	-- The value depends on crystal orientation, surface oxidation and
	-- contamination.
	--
	-- Reference: Michaelson, H. B., J. Appl. Phys. 48, 4729 (1977) (polycrystalline).

	local enabled = true
	local value = 4.65

	return value, enabled
end

function material.electrical_conductivity(state)
	-- Electrical conductivity
	-- Units: S m^-1
	--
	-- Representative conductivity near 300 K:
	--     sigma(300 K) = 5.96e7 S/m
	--     (equivalently rho(300 K) ~ 1.68 micro-ohm cm)
	--
	-- The temperature dependence is calculated from the approximately
	-- linear temperature dependence of the electrical resistivity:
	--
	--     rho(T) = rho(300 K) *
	--              [1 + alpha_rho*(T - 300 K)]
	--
	-- where alpha_rho is approximately 3.9e-3 K^-1.
	--
	-- Since sigma = 1/rho:
	--
	--     sigma(T) = sigma(300 K) /
	--                [1 + alpha_rho*(T - 300 K)]
	--
	-- This approximation is intended for temperatures reasonably
	-- close to room temperature.
	--
	-- Note: this is the value for high-purity annealed copper. Cold
	-- work, alloying and impurities raise the resistivity appreciably.
	--
	-- Reference: Matula, R. A., J. Phys. Chem. Ref. Data 8, 1147 (1979); CRC Handbook of Chemistry and Physics.

	local enabled = true
	local T = state.T

	local sigma_300 = 5.96e7
	local alpha_rho = 3.9e-3

	local denominator = 1.0 + alpha_rho*(T - 300.0)

	-- Prevent an unphysical negative or zero resistivity if the
	-- linear approximation is evaluated far outside its valid range.
	if denominator <= 0.0 then
		return 0.0, false
	end

	local value = sigma_300/denominator

	return value, enabled
end

function material.epsilonr(state)
	-- Relative permittivity
	-- Dimensionless
	--
	-- A static scalar relative permittivity does not provide a useful
	-- physical description of the electromagnetic response of a metal.
	--
	-- Electrostatic calculations should normally treat the metal as an
	-- equipotential conductor. Optical calculations should use complex,
	-- wavelength-dependent optical constants instead.
	--
	-- This parameter is therefore disabled.

	local enabled = false
	local value = 1.0

	return value, enabled
end

function material.thermal_kl(state)
	-- Thermal conductivity
	-- Units: W m^-1 K^-1
	--
	-- Representative value for bulk copper near 300 K:
	--     kappa = 401 W m^-1 K^-1
	--
	-- The value depends on purity, microstructure and temperature.

	--
	-- Reference: Ho, Powell & Liley, J. Phys. Chem. Ref. Data 1, 279 (1972); CRC Handbook of Chemistry and Physics.

	local enabled = true
	local value = 401

	return value, enabled
end

function material.heat_capacity(state)
	-- Specific heat capacity
	-- Units: J kg^-1 K^-1
	--
	-- Representative constant-pressure value near 300 K:
	--     cp = 385 J kg^-1 K^-1
	--
	-- Reference: CRC Handbook of Chemistry and Physics, 100th ed. (2019).

	local enabled = true
	local value = 385

	return value, enabled
end

function material.density(state)
	-- Mass density
	-- Units: kg m^-3
	--
	-- Representative room-temperature density:
	--     rho = 8960 kg m^-3

	--
	-- Reference: CRC Handbook of Chemistry and Physics, 100th ed. (2019).

	local enabled = true
	local value = 8960

	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:
	-- Metal/electrode. Electron and hole energy equations in a non-degenerate
	-- semiconductor DD model are not appropriate. 1e-14 s is only a fast
	-- harmless fallback.

	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:
	-- Metal/electrode. Electron and hole energy equations in a non-degenerate
	-- semiconductor DD model are not appropriate. 1e-14 s is only a fast
	-- harmless fallback.

	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("Work function (Xi):     %.6f eV", material.Xi(state)))
	print(string.format("Electrical conductivity: %.6e S/m", material.electrical_conductivity(state)))
	print(string.format("Relative permittivity:  %.6f", material.epsilonr(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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-- This file is part of the OghmaNano Materials Model Library.
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