GaP material model

1. Introduction

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

Bulk crystalline gallium phosphide

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 "GaP", enabled
end

function material.description()
	local enabled = true

	return "Bulk crystalline gallium phosphide", enabled
end

function material.formula()
	local enabled = true

	return "GaP", 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.
	--
	-- Fundamental (indirect, X-valley) gap parameters
	-- (Eg(0) = 2.35 eV, alpha = 5.771e-4 eV/K, beta = 372 K) from
	-- I. Vurgaftman, J. R. Meyer, L. R. Ram-Mohan,
	-- "Band parameters for III-V compound semiconductors and their
	-- alloys", J. Appl. Phys. 89, 5815-5875, 2001.
	-- Gives Eg(300 K) = 2.27 eV. Indirect gap.
	--
	-- Note: the direct (Gamma) gap is much higher (~2.78 eV at 300 K);
	-- this function returns the fundamental indirect gap.

	local enabled = true
	local T = state.T
	local value = 2.35 - 5.771e-4*T*T/(T + 372.0)

	return value, enabled
end

function material.Xi(state)
	-- Electron affinity
	-- Units: eV
	--
	-- Reference:
	-- Ioffe NSM database (GaP) / device literature.
	--
	-- Note: reported GaP electron affinity scatters (~3.8-4.0 eV);
	-- 3.8 eV used here. Verify against the band-alignment convention
	-- used in your simulation.

	local enabled = true
	local value = 3.8

	return value, enabled
end

function material.me(state)
    local enabled = true
    local value = 0.13    -- Gamma mass (material is X-indirect) (Vurgaftman 2001)
    -- NOTE: Gamma mass valid when GaP is a barrier for a Gamma-confined well.
    -- If GaP is the well, X valleys (m_l=2.0, m_t=0.25) apply -> use multivalley solver.
    return value, enabled
end

function material.mh(state)
    local enabled = true
    -- HH[001]: g1=4.05 g2=0.49 -> 1/(4.05-0.98) (Vurgaftman 2001)
    local value = 0.33
    return value, enabled
end

function material.Nc(state)
	-- Effective conduction-band density of states
	-- Units: m^-3
	--
	-- Reference:
	-- Ioffe NSM database (GaP).
	-- Nc(300 K) = 1.8e19 cm^-3 = 1.8e25 m^-3.
	-- (Enlarged by the multiple equivalent X-valleys.)
	--
	-- Note: the (T/300)^1.5 form is the simple parabolic-band model.

	local enabled = true
	local T = state.T
	local value = 1.8e25*(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 (GaP).
	-- Nv(300 K) = 1.9e19 cm^-3 = 1.9e25 m^-3.

	local enabled = true
	local T = state.T
	local value = 1.9e25*(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 (GaP), experimental compilation.
	-- mu_n(300 K) ~ 250 cm^2/V/s = 0.025 m^2/V/s (low doping),
	-- phonon-limited temperature dependence approximately (300/T)^1.8.
	--
	-- Note: low compared with direct-gap III-Vs because the
	-- conduction minimum is at X. Intrinsic (lattice) mobility only,
	-- no doping / ionised-impurity dependence.

	local enabled = true
	local T = state.T
	local value = 0.025*(300.0/T)^1.8

	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
	--
	-- Reference:
	-- Ioffe NSM database (GaP), experimental compilation.
	-- mu_p(300 K) ~ 150 cm^2/V/s = 0.015 m^2/V/s, phonon-limited
	-- temperature dependence approximately (300/T)^1.8.

	local enabled = true
	local T = state.T
	local value = 0.015*(300.0/T)^1.8

	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 (GaP).
	-- Static value 11.1 (high-frequency value is 9.11).

	local enabled = true
	local value = 11.1

	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 GaP, ~5e-14 cm^3/s = 5e-20 m^3/s.
	--
	-- GaP is an indirect-gap semiconductor, so intrinsic band-to-band
	-- radiative recombination is weak (visible GaP LEDs rely on
	-- isoelectronic N traps, not pure band-to-band emission). This
	-- value is poorly constrained; verify if it matters for your
	-- device.

	local enabled = true
	local value = 5.0e-20

	return value, enabled
end

function material.auger_Cn(state)
	-- Electron Auger recombination coefficient
	-- Units: m^6 s^-1
	--
	-- Reference:
	-- Representative value for GaP, ~1e-31 cm^6/s = 1e-43 m^6/s.
	--
	-- Note: Auger is weak in wide indirect-gap GaP and the
	-- coefficients are poorly constrained. Treat as an
	-- 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
	--
	-- Reference:
	-- Representative value for GaP, ~1e-31 cm^6/s = 1e-43 m^6/s.
	--
	-- Note: as for Cn, poorly constrained. Treat as an
	-- 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
	--
	-- 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 (GaP).
	-- kappa(300 K) ~ 110 W/m/K; near room temperature kappa decreases
	-- with T with an effective exponent of about -1.4.

	local enabled = true
	local T = state.T
	local value = 110.0*(300.0/T)^1.4

	return value, enabled
end

function material.heat_capacity(state)
	-- Specific heat capacity
	-- Units: J kg^-1 K^-1
	--
	-- Reference:
	-- Ioffe NSM database (GaP). c_p(300 K) ~ 430 J/kg/K.

	local enabled = true
	local value = 430.0

	return value, enabled
end

function material.density(state)
	-- Mass density
	-- Units: kg m^-3
	--
	-- Reference:
	-- Ioffe NSM database (GaP). rho = 4.138 g/cm^3.

	local enabled = true
	local value = 4138.0

	return value, enabled
end

function material.lattice_constant(state)
	-- Cubic lattice constant
	-- Units: m
	--
	-- Reference:
	-- I. Vurgaftman, J. R. Meyer, L. R. Ram-Mohan,
	-- "Band parameters for III-V compound semiconductors and their
	-- alloys", J. Appl. Phys. 89, 5815-5875, 2001.
	-- a(300 K) = 5.4505 A; linear expansion ~5.3e-6 /K near 300 K.

	local enabled = true
	local T = state.T
	local a300 = 5.4505e-10
	local expansion = 5.3e-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: Inorganic family estimate
	-- Confidence: Low
	--
	-- Reference:
	-- https://doi.org/10.1109/16.381985
	--
	-- Comments:
	-- Generic inorganic-semiconductor estimate where a direct value was not
	-- identified.

	local enabled = true
	local value = 7.500000e-13

	return value, enabled
end

function material.thermal_tau_h(state)
	-- Hole energy relaxation time towards the lattice temperature
	-- Units: s
	--
	-- Value basis: Inorganic family estimate
	-- Confidence: Low
	--
	-- Reference:
	-- https://doi.org/10.1109/16.381985
	--
	-- Comments:
	-- Generic inorganic-semiconductor estimate where a direct value was not
	-- identified.

	local enabled = true
	local value = 7.500000e-13

	return value, enabled
end

function material.delta_so(state)
	-- Spin-orbit splitting energy (Delta_SO)
	-- Units: eV
	--
	-- Reference:
	-- Vurgaftman, Meyer, Ram-Mohan, J. Appl. Phys. 89, 5815 (2001).
	-- Recommended binary value for GaP.

	local enabled = true
	local value = 0.08

	return value, enabled
end

function material.Ep(state)
	-- Kane energy E_P (optical matrix element parameter)
	-- Units: eV
	--
	-- Reference:
	-- Vurgaftman, Meyer, Ram-Mohan, J. Appl. Phys. 89, 5815 (2001).
	-- Recommended binary value for GaP.

	local enabled = true
	local value = 31.4

	return value, enabled
end

function material.gamma1(state)
	-- Luttinger parameter gamma1
	-- Units: dimensionless
	--
	-- Reference:
	-- Vurgaftman, Meyer, Ram-Mohan, J. Appl. Phys. 89, 5815 (2001).
	-- Recommended binary value for GaP.

	local enabled = true
	local value = 4.05

	return value, enabled
end

function material.gamma2(state)
	-- Luttinger parameter gamma2
	-- Units: dimensionless
	--
	-- Reference:
	-- Vurgaftman, Meyer, Ram-Mohan, J. Appl. Phys. 89, 5815 (2001).
	-- Recommended binary value for GaP.

	local enabled = true
	local value = 0.49

	return value, enabled
end

function material.gamma3(state)
	-- Luttinger parameter gamma3
	-- Units: dimensionless
	--
	-- Reference:
	-- Vurgaftman, Meyer, Ram-Mohan, J. Appl. Phys. 89, 5815 (2001).
	-- Recommended binary value for GaP.

	local enabled = true
	local value = 1.25

	return value, enabled
end

function material.ac(state)
	-- Conduction-band hydrostatic deformation potential a_c
	-- Units: eV
	--
	-- Reference:
	-- Vurgaftman, Meyer, Ram-Mohan, J. Appl. Phys. 89, 5815 (2001).
	-- Recommended binary value for GaP.
	--
	-- Note:
	-- VMR sign convention: interband hydrostatic deformation potential
	-- a_gap = a_c - a_v, with a_c negative and a_v tabulated positive.
	-- Sign preserved from source; no sign flip applied.

	local enabled = true
	local value = -8.2

	return value, enabled
end

function material.av(state)
	-- Valence-band hydrostatic deformation potential a_v
	-- Units: eV
	--
	-- Reference:
	-- Vurgaftman, Meyer, Ram-Mohan, J. Appl. Phys. 89, 5815 (2001).
	-- Recommended binary value for GaP.
	--
	-- Note:
	-- VMR sign convention: a_v tabulated as a POSITIVE number; the
	-- interband hydrostatic deformation potential is a_gap = a_c - a_v.
	-- Sign preserved from source; no sign flip applied.

	local enabled = true
	local value = 1.7

	return value, enabled
end

function material.b(state)
	-- Valence-band shear (tetragonal) deformation potential b
	-- Units: eV
	--
	-- Reference:
	-- Vurgaftman, Meyer, Ram-Mohan, J. Appl. Phys. 89, 5815 (2001).
	-- Recommended binary value for GaP.
	--
	-- Note:
	-- Sign convention as in VMR (b negative). Sign preserved; not flipped.

	local enabled = true
	local value = -1.6

	return value, enabled
end

function material.d(state)
	-- Valence-band shear (rhombohedral) deformation potential d
	-- Units: eV
	--
	-- Reference:
	-- Vurgaftman, Meyer, Ram-Mohan, J. Appl. Phys. 89, 5815 (2001).
	-- Recommended binary value for GaP.
	--
	-- Note:
	-- Sign convention as in VMR (d negative). Sign preserved; not flipped.

	local enabled = true
	local value = -4.6

	return value, enabled
end

function material.lattice_a(state)
	-- Cubic (zincblende) lattice constant a
	-- Units: m
	--
	-- Reference:
	-- Vurgaftman, Meyer, Ram-Mohan, J. Appl. Phys. 89, 5815 (2001).
	-- Recommended binary value for GaP.
	-- a(300K)=5.4505 Angstrom, da/dT=2.92e-05 Angstrom/K.
	-- Linear thermal expansion: a(T)=a300+da/dT*(T-300).

	local enabled = true
	local T = state.T
	local value = (5.4505 + 2.92e-05*(T-300.0))*1e-10

	return value, enabled
end

function material.C11(state)
	-- Elastic stiffness constant C11
	-- Units: Pa
	--
	-- Reference:
	-- Vurgaftman, Meyer, Ram-Mohan, J. Appl. Phys. 89, 5815 (2001).
	-- Recommended binary value for GaP.
	-- C11=140.5 GPa, converted to Pa.

	local enabled = true
	local value = 140.5e9

	return value, enabled
end

function material.C12(state)
	-- Elastic stiffness constant C12
	-- Units: Pa
	--
	-- Reference:
	-- Vurgaftman, Meyer, Ram-Mohan, J. Appl. Phys. 89, 5815 (2001).
	-- Recommended binary value for GaP.
	-- C12=62.03 GPa, converted to Pa.

	local enabled = true
	local value = 62.03e9

	return value, enabled
end

function material.C44(state)
    -- Elastic stiffness constant C44
    -- Units: Pa
    --
    -- Reference:
    -- I. Vurgaftman, J. R. Meyer, L. R. Ram-Mohan,
    -- "Band parameters for III-V compound semiconductors and their alloys,"
    -- J. Appl. Phys. 89, 5815 (2001), recommended binary parameter tables
    -- (Tables I-VI checked for GaAs, AlAs, InAs, GaP, AlP, InP).
    -- DOI: 10.1063/1.1368156
    --
    -- Notes:
    -- Vurgaftman et al. (2001) recommended 703.3 kbar (Yogurtcu et al. 1981 ultrasonic: 70.3 GPa).
    -- Converted: 70.33 GPa -> 70.33e9 Pa (1 kbar = 1e8 Pa).

    local enabled = true
    local value = 70.33e9

    return value, enabled
end

function material.phonon_lo_energy(state)
    -- Representative LO phonon energy for polar optical (Frohlich) scattering
    -- Units: eV
    --
    -- Reference:
    -- M. Levinshtein, S. Rumyantsev, M. Shur (eds.),
    -- Handbook Series on Semiconductor Parameters, Vols. 1 and 2
    -- (World Scientific, 1996 and 1999), as reproduced in the Ioffe
    -- Institute NSM archive (www.ioffe.ru/SVA/NSM/Semicond/).
    --
    -- Notes:
    -- Optical phonon energy 0.051 eV (300 K) as given in the compilation,
    -- corresponding to the Gamma-point LO (~403 cm^-1).

    local enabled = true
    local value = 0.051

    return value, enabled
end

function material.epsilon_static(state)
    -- Static relative dielectric constant (lattice + electronic)
    -- Dimensionless
    --
    -- Reference:
    -- M. Levinshtein, S. Rumyantsev, M. Shur (eds.),
    -- Handbook Series on Semiconductor Parameters, Vols. 1 and 2
    -- (World Scientific, 1996 and 1999), as reproduced in the Ioffe
    -- Institute NSM archive (www.ioffe.ru/SVA/NSM/Semicond/).
    --
    -- Notes:
    -- GaP: eps_s = 11.1 (300 K).

    local enabled = true
    local value = 11.1

    return value, enabled
end

function material.epsilon_inf(state)
    -- High-frequency (electronic) relative dielectric constant
    -- Dimensionless
    --
    -- Reference:
    -- M. Levinshtein, S. Rumyantsev, M. Shur (eds.),
    -- Handbook Series on Semiconductor Parameters, Vols. 1 and 2
    -- (World Scientific, 1996 and 1999), as reproduced in the Ioffe
    -- Institute NSM archive (www.ioffe.ru/SVA/NSM/Semicond/).
    --
    -- Notes:
    -- GaP: eps_inf = 9.11 (300 K).

    local enabled = true
    local value = 9.11

    return value, enabled
end

function material.e14(state)
    -- Zincblende piezoelectric stress coefficient e14
    -- Units: C m^-2
    --
    -- Reference:
    -- M. Levinshtein, S. Rumyantsev, M. Shur (eds.),
    -- Handbook Series on Semiconductor Parameters, Vols. 1 and 2
    -- (World Scientific, 1996 and 1999), as reproduced in the Ioffe
    -- Institute NSM archive (www.ioffe.ru/SVA/NSM/Semicond/).
    -- (magnitudes trace to G. Arlt and P. Quadflieg, phys. stat. sol. 25,
    -- 323 (1968)).
    --
    -- Notes:
    -- GaP: e14 = -0.1 C/m^2.
    -- Sign as tabulated in the compilation (negative for III-V in that
    -- convention). e14 sign conventions differ between sources (orientation of
    -- [111] relative to the cation->anion bond); piezoelectric scattering
    -- depends only on e14^2.

    local enabled = true
    local value = -0.1

    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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