4H-SiC material model

1. Introduction

This page contains the OghmaNano material model for 4H-SiC (SiC).

Bulk crystalline silicon carbide (4H polytype)

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 "4H-SiC", enabled
end

function material.description()
	local enabled = true

	return "Bulk crystalline silicon carbide (4H polytype)", enabled
end

function material.formula()
	local enabled = true

	return "SiC", enabled
end

function material.Eg(state)
	-- Units: eV
	--
	-- Reference:
	-- Y. P. Varshni,
	-- "Temperature dependence of the energy gap in semiconductors",
	-- Physica, 34, 149-154, 1967.
	--
	-- 4H-SiC parameter set (Eg(0) = 3.265 eV, alpha = 6.5e-4 eV/K,
	-- beta = 1300 K) as tabulated in the Ioffe NSM database.
	-- Gives Eg(300 K) = 3.23 eV. Indirect gap.
	--
	-- Note: polytype dependent. 6H-SiC ~3.0 eV, 3C-SiC ~2.36 eV
	-- at 300 K; do NOT use these parameters for other polytypes.

	local enabled = true
	local T = state.T
	local value = 3.265 - 6.5e-4*T*T/(T + 1300.0)

	return value, enabled
end

function material.Xi(state)
	-- Electron affinity
	-- Units: eV
	--
	-- Reference:
	-- Ioffe NSM database (4H-SiC) / device literature.
	--
	-- Note: reported 4H-SiC electron affinity scatters (~3.17-3.6 eV);
	-- 3.3 eV is a common device-modelling value. Verify against the
	-- band-alignment convention used in your simulation.

	local enabled = true
	local value = 3.3

	return value, enabled
end

function material.me(state)
    local enabled = true
    -- Approx c-axis (M-L) electron mass ~0.29-0.33; CB is anisotropic multivalley (M-point).
    -- WARNING: scalar parabolic model is marginal for 4H-SiC. Ref: Persson JAP 82,5496(1997).
    local value = 0.31
    return value, enabled
end

function material.mh(state)
    local enabled = true
    -- Anisotropic VB; c-axis HH ~1.0-1.75 in literature. Approximate scalar.
    local value = 1.0
    return value, enabled
end

function material.Nc(state)
	-- Effective conduction-band density of states
	-- Units: m^-3
	--
	-- Reference:
	-- Ioffe NSM database (4H-SiC).
	-- Nc(300 K) = 1.7e19 cm^-3 = 1.7e25 m^-3.
	--
	-- Note: the (T/300)^1.5 form is the simple parabolic-band model.

	local enabled = true
	local T = state.T
	local value = 1.7e25*(T/300.0)^1.5

	return value, enabled
end

function material.Nv(state)
	-- Effective valence-band density of states
	-- Units: m^-3
	--
	-- Reference:
	-- Ioffe NSM database (4H-SiC).
	-- Nv(300 K) = 2.5e19 cm^-3 = 2.5e25 m^-3.

	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:
	-- Ioffe NSM database (4H-SiC), experimental compilation.
	-- mu_n(300 K) ~ 950 cm^2/V/s = 0.095 m^2/V/s (low doping),
	-- phonon-limited temperature dependence approximately (300/T)^2.6.
	--
	-- Note: 4H-SiC electron mobility is anisotropic about the c-axis
	-- (mu_parallel is ~20% higher than mu_perp). The value here is a
	-- representative average; intrinsic (lattice) mobility only, no
	-- doping dependence.

	local enabled = true
	local T = state.T
	local value = 0.095*(300.0/T)^2.6

	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
	--
	-- Reference:
	-- Ioffe NSM database (4H-SiC), experimental compilation.
	-- mu_p(300 K) ~ 120 cm^2/V/s = 0.012 m^2/V/s, phonon-limited
	-- temperature dependence approximately (300/T)^2.5.

	local enabled = true
	local T = state.T
	local value = 0.012*(300.0/T)^2.5

	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:
	-- Ioffe NSM database (4H-SiC).
	-- Static value ~9.7 perpendicular to c-axis (~10.3 parallel);
	-- 9.7 used here as a representative value.

	local enabled = true
	local value = 9.7

	return value, enabled
end

function material.free_to_free_recombination(state)
	-- Radiative (band-to-band) recombination coefficient
	-- Units: m^3 s^-1
	--
	-- Reference:
	-- Representative value for 4H-SiC, ~1.5e-12 cm^3/s = 1.5e-18 m^3/s.
	--
	-- SiC is an indirect-gap semiconductor, so B is small and
	-- radiative recombination is rarely dominant. This value is
	-- poorly constrained in the literature; verify if radiative
	-- recombination matters for your device.

	local enabled = true
	local value = 1.5e-18

	return value, enabled
end

function material.auger_Cn(state)
	-- Electron Auger recombination coefficient
	-- Units: m^6 s^-1
	--
	-- Reference:
	-- Representative value for 4H-SiC, ~5e-31 cm^6/s = 5e-43 m^6/s.
	--
	-- Note: Auger coefficients for SiC are poorly constrained and
	-- scatter widely. Treat as an order-of-magnitude placeholder and
	-- verify against a measurement for your regime.

	local enabled = true
	local value = 5.0e-43

	return value, enabled
end

function material.auger_Cp(state)
	-- Hole Auger recombination coefficient
	-- Units: m^6 s^-1
	--
	-- Reference:
	-- Representative value for 4H-SiC, ~5e-31 cm^6/s = 5e-43 m^6/s.
	--
	-- Note: as for Cn, SiC Auger coefficients are poorly constrained.
	-- Treat as an order-of-magnitude placeholder.

	local enabled = true
	local value = 5.0e-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
	--
	-- This is entirely material-quality dependent and should be set
	-- from the intended bulk lifetime. The value below is a
	-- representative placeholder for device-grade material.

	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
	--
	-- Representative value ~1e-15 cm^2 = 1e-19 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
	--
	-- Representative value ~1e-15 cm^2 = 1e-19 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:
	-- Ioffe NSM database (4H-SiC).
	-- kappa(300 K) ~ 370 W/m/K; near room temperature kappa decreases
	-- with T with an effective exponent of about -1.3.
	--
	-- Note: anisotropic (higher parallel to the c-axis, ~490 W/m/K);
	-- 370 W/m/K is a representative value.

	local enabled = true
	local T = state.T
	local value = 370.0*(300.0/T)^1.3

	return value, enabled
end

function material.heat_capacity(state)
	-- Specific heat capacity
	-- Units: J kg^-1 K^-1
	--
	-- Reference:
	-- Ioffe NSM database (4H-SiC). c_p(300 K) ~ 690 J/kg/K.

	local enabled = true
	local value = 690.0

	return value, enabled
end

function material.density(state)
	-- Mass density
	-- Units: kg m^-3
	--
	-- Reference:
	-- Ioffe NSM database (SiC). rho = 3.21 g/cm^3.

	local enabled = true
	local value = 3210.0

	return value, enabled
end

function material.lattice_constant(state)
	-- Lattice constant (hexagonal a-axis)
	-- Units: m
	--
	-- Reference:
	-- Ioffe NSM database (4H-SiC).
	-- a(300 K) = 3.073 A; linear expansion ~4.0e-6 /K near 300 K.
	--
	-- IMPORTANT: 4H-SiC is hexagonal, with two lattice constants
	-- a = 3.073 A and c = 10.053 A. This single scalar field holds
	-- the a-axis constant only; the c-axis constant is not
	-- representable here.

	local enabled = true
	local T = state.T
	local a300 = 3.073e-10
	local expansion = 4.0e-6
	local value = a300*(1.0 + expansion*(T - 300.0))

	return value, enabled
end

function material.thermal_tau_e(state)
	-- Electron energy relaxation time towards the lattice temperature
	-- Units: s
	--
	-- Value basis: Conventional semiconductor estimate
	-- Confidence: Medium
	--
	-- Reference:
	-- https://doi.org/10.1109/16.381985
	--
	-- Comments:
	-- Representative sub-ps energy relaxation for a conventional semiconductor
	-- near room temperature. Actual value is field and carrier-energy
	-- dependent.

	local enabled = true
	local value = 5.000000e-13

	return value, enabled
end

function material.thermal_tau_h(state)
	-- Hole energy relaxation time towards the lattice temperature
	-- Units: s
	--
	-- Value basis: Conventional semiconductor estimate
	-- Confidence: Medium
	--
	-- Reference:
	-- https://doi.org/10.1109/16.381985
	--
	-- Comments:
	-- Representative sub-ps energy relaxation for a conventional semiconductor
	-- near room temperature. Actual value is field and carrier-energy
	-- dependent.

	local enabled = true
	local value = 5.000000e-13

	return value, enabled
end

function material.lattice_a(state)
	-- Lattice constant, a-axis
	-- Units: m
	--
	-- Reference:
	-- M. E. Levinshtein, S. L. Rumyantsev, M. S. Shur (eds.),
	-- "Properties of Advanced Semiconductor Materials:
	-- GaN, AlN, InN, BN, SiC, SiGe," Wiley, New York (2001).
	-- See also O. Madelung, "Semiconductors: Data Handbook,"
	-- 3rd ed., Springer (2004), DOI: 10.1007/978-3-642-18865-7.
	--
	-- Notes:
	-- Hexagonal 4H-SiC polytype (space group P6_3mc). This is the
	-- in-plane a-axis lattice constant a = 3.0730 Angstrom at ~300 K.
	-- The c-axis (c = 10.053 Angstrom) is deliberately NOT returned here.
	-- No useful thermal dependence applied; room-temperature value.

	local enabled = true
	local value = 3.0730e-10

	return value, enabled
end

function material.C11(state)
    -- Elastic stiffness constant C11
    -- Units: Pa
    --
    -- Crystal phase: 4H-SiC is hexagonal (P6_3mc), NOT cubic. The value is
    -- the hexagonal-tensor component C11 (in-plane longitudinal).
    --
    -- Reference:
    -- Kamitani et al. (1997), Brillouin-scattering study of 4H and 6H SiC
    -- single crystals, as attributed in the Ioffe NSM archive (SiC
    -- mechanical properties). Full bibliographic details were not
    -- independently verified.
    --
    -- Notes:
    -- Source reports 501 GPa at 300 K; converted to 501e9 Pa.
    -- 4H and 6H constants are equal within experimental uncertainty.
    -- Arlt & Schodder (1965) give 500(20) GPa, consistent.

    local enabled = true
    local value = 501e9

    return value, enabled
end

function material.C12(state)
    -- Elastic stiffness constant C12
    -- Units: Pa
    --
    -- Crystal phase: hexagonal 4H-SiC tensor component C12.
    --
    -- Reference:
    -- Kamitani et al. (1997), Brillouin-scattering study of 4H and 6H SiC
    -- single crystals, as attributed in the Ioffe NSM archive (SiC
    -- mechanical properties). Full bibliographic details were not
    -- independently verified.
    --
    -- Notes:
    -- Source reports 111 GPa at 300 K; converted to 111e9 Pa.
    -- Arlt & Schodder (1965) give 92(28) GPa; Kamitani preferred (smaller
    -- uncertainty).

    local enabled = true
    local value = 111e9

    return value, enabled
end

function material.C44(state)
    -- Elastic stiffness constant C44
    -- Units: Pa
    --
    -- Crystal phase: hexagonal 4H-SiC tensor component C44 (shear in a plane
    -- containing the c axis).
    --
    -- Reference:
    -- Kamitani et al. (1997), Brillouin-scattering study of 4H and 6H SiC
    -- single crystals, as attributed in the Ioffe NSM archive (SiC
    -- mechanical properties). Full bibliographic details were not
    -- independently verified.
    --
    -- Notes:
    -- Source reports 163 GPa at 300 K; converted to 163e9 Pa.
    -- Arlt & Schodder (1965) give 168(2) GPa, consistent within ~3%.

    local enabled = true
    local value = 163e9

    return value, enabled
end

function material.D_ac_e(state)
    -- Effective scalar acoustic deformation potential for electrons
    -- Units: eV
    --
    -- Reference:
    -- S. G. Muller, D. Hofmann, A. Winnacker,
    -- "Experimental and theoretical analysis of the Hall-mobility in n-type
    -- bulk 6H- and 4H-SiC,"
    -- MRS Online Proc. Libr. 572, 275 (1999).
    -- DOI: 10.1557/PROC-572-275
    --
    -- Notes:
    -- Effective scalar acoustic deformation potential for electrons,
    -- Xi_ac = 14.8 +/- 0.5 eV, obtained by fitting a full relaxation-time
    -- Hall-mobility model (acoustic deformation-potential + polar optical +
    -- intervalley + impurity scattering) to temperature-dependent Hall data
    -- on n-type bulk 4H-SiC. This is exactly the scalar equipartition
    -- acoustic parameter (not an intervalley or optical potential).
    -- It is a transport-fit value: it absorbs the anisotropy of the M-point
    -- conduction valleys into a single number, and it is correlated with the
    -- other fitted scattering strengths. Iwata & Itoh, J. Appl. Phys. 89,
    -- 6228 (2001) also fitted 4H-SiC Hall mobilities with an acoustic
    -- deformation potential; the Muller et al. value is used here because
    -- its numerical value was verified.
    -- Sign: reported as a magnitude; enters the scattering rate squared.

    local enabled = true
    local value = 14.8

    return value, enabled
end

function material.D_ac_h(state)
    -- Effective scalar acoustic deformation potential for holes
    -- Units: eV
    --
    -- No sufficiently reliable value/reference identified.
    -- Disabled rather than estimated.
    --
    -- Notes:
    -- Targeted search of hole-mobility fits for p-type 4H-SiC did not yield a
    -- scalar acoustic deformation potential whose value and original source
    -- could both be verified. The electron value (14.8 eV) is not
    -- transferable to the valence band.

    local enabled = false
    local value = 0.0

    return value, enabled
end

function material.phonon_lo_energy(state)
    -- Representative LO phonon energy for polar optical (Frohlich) scattering
    -- Units: eV
    --
    -- Reference:
    -- T. E. Tiwald, J. A. Woollam, S. Zollner, J. Christiansen,
    -- R. B. Gregory, T. Wetteroth, S. R. Wilson, A. R. Powell,
    -- "Carrier concentration and lattice absorption in bulk and epitaxial
    -- silicon carbide determined using infrared ellipsometry,"
    -- Phys. Rev. B 60, 11464 (1999).
    -- DOI: 10.1103/PhysRevB.60.11464
    --
    -- Notes:
    -- Table I, 4H-SiC (lightly doped epilayer fits, samples 8-9):
    -- omega_LO(perp, E1) = 970 cm^-1, omega_LO(par, A1) = 964 cm^-1.
    -- The strong E1/A1 LO modes differ by < 1%; the ordinary (E1) mode is
    -- used for consistency with epsilon_static/epsilon_inf below.
    -- 970 cm^-1 x 1.239842e-4 eV/cm^-1 = 0.12026 eV.
    -- The frequently quoted ~104 meV "optical phonon energy" is a folded
    -- zone-boundary phonon seen in exciton replicas, not the Gamma-point LO
    -- relevant to Frohlich coupling, and is not used.

    local enabled = true
    local value = 0.12026

    return value, enabled
end

function material.epsilon_static(state)
    -- Static relative dielectric constant (lattice + electronic)
    -- Dimensionless
    --
    -- Derived from:
    -- T. E. Tiwald, J. A. Woollam, S. Zollner, J. Christiansen,
    -- R. B. Gregory, T. Wetteroth, S. R. Wilson, A. R. Powell,
    -- "Carrier concentration and lattice absorption in bulk and epitaxial
    -- silicon carbide determined using infrared ellipsometry,"
    -- Phys. Rev. B 60, 11464 (1999).
    -- DOI: 10.1103/PhysRevB.60.11464
    --
    -- Notes:
    -- Lyddane-Sachs-Teller relation with same-source ordinary-ray (E perp c)
    -- parameters, Table I: eps_inf = 6.6, omega_TO = 797 cm^-1,
    -- omega_LO = 970 cm^-1:  eps_s = 6.6*(970/797)^2 = 9.78.
    -- Keeps eps_s, eps_inf and phonon_lo_energy mutually consistent.
    -- Extraordinary-ray equivalent: 6.9*(964/782)^2 = 10.5.
    -- 6H-SiC values (Patrick & Choyke 1970, 9.66 perp) are not used, to
    -- avoid mixing polytypes.

    local enabled = true
    local value = 9.78

    return value, enabled
end

function material.epsilon_inf(state)
    -- High-frequency (electronic) relative dielectric constant
    -- Dimensionless
    --
    -- Reference:
    -- T. E. Tiwald, J. A. Woollam, S. Zollner, J. Christiansen,
    -- R. B. Gregory, T. Wetteroth, S. R. Wilson, A. R. Powell,
    -- "Carrier concentration and lattice absorption in bulk and epitaxial
    -- silicon carbide determined using infrared ellipsometry,"
    -- Phys. Rev. B 60, 11464 (1999).
    -- DOI: 10.1103/PhysRevB.60.11464
    --
    -- Notes:
    -- Ordinary-ray (E perp c) value 6.6 from fits to the lightly doped 4H-SiC
    -- epilayer samples (Table I); extraordinary value 6.9. Heavily doped
    -- substrates give 6.2-6.6 because eps_inf correlates with free-carrier
    -- parameters in the fit. The same paper quotes a literature range of
    -- 6.56-7.54 (perp) for 4H-SiC.

    local enabled = true
    local value = 6.6

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