GaSb material model

1. Introduction

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

Bulk crystalline gallium antimonide

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 = {}

-- ---------------------------------------------------------------------------
-- REFERENCES
--
-- [1]  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). doi:10.1063/1.1368156
-- [2]  Ioffe Institute, "New Semiconductor Materials: GaSb", NSM Archive,
--      www.ioffe.ru/SVA/NSM/Semicond/GaSb/ (compilation; primary sources
--      for the temperature dependences are named individually below).
-- [3]  M. Levinshtein, S. Rumyantsev, M. Shur (eds.), "Handbook Series on
--      Semiconductor Parameters, Vol. 1", World Scientific (1996) -
--      underlying print compilation for [2].
-- [4]  Y. P. Varshni, "Temperature dependence of the energy gap in
--      semiconductors", Physica 34, 149-154 (1967). [Varshni functional form]
-- [5]  Wu and Chen (1992) - GaSb Eg(T) fit, compiled in [2].
-- [6]  S. Adachi, "Properties of Group-IV, III-V and II-VI Semiconductors",
--      Wiley (2005).
-- [7]  Mathur and Jain (1979) - GaSb electron Hall mobility vs T, in [2].
-- [8]  Poujade and Albany (1969); Okhotin et al. (1972) - GaSb thermal
--      conductivity, compiled in [2].
-- [9]  Piesbergen (1963) - GaSb specific heat, compiled in [2].
-- [10] Novikova and Abrikosov (1963) - GaSb linear thermal expansion,
--      compiled in [2].
-- ---------------------------------------------------------------------------

function material.name()
	local enabled = true

	return "GaSb", enabled
end

function material.description()
	local enabled = true

	return "Bulk crystalline gallium antimonide", enabled
end

function material.formula()
	local enabled = true

	return "GaSb", enabled
end

function material.Eg(state)
	-- Units: eV
	-- Refs: [1] (primary), [4] (Varshni form); alt fit [2],[5]
	--
	-- Varshni form with the recommended III-V parameter set [1]:
	-- Eg(0) = 0.812 eV, alpha = 4.17e-4 eV/K, beta = 140 K.
	-- Gives Eg(300 K) = 0.727 eV.
	--
	-- Alternative fit (Wu & Chen [5], compiled in [2], valid
	-- 0 < T < 300 K): 0.813 - 3.78e-4*T*T/(T + 94).

	local enabled = true
	local T = state.T
	local value = 0.812 - 4.17e-4*T*T/(T + 140.0)

	return value, enabled
end

function material.Xi(state)
	-- Electron affinity
	-- Units: eV
	-- Refs: [2] (Ioffe basic parameters, 4.06 eV), [6]

	local enabled = true
	local value = 4.06

	return value, enabled
end

function material.me(state)
    local enabled = true
    local value = 0.039   -- Gamma electron mass (Vurgaftman 2001)
    return value, enabled
end

function material.mh(state)
    local enabled = true
    -- HH[001]: g1=13.4 g2=4.7 -> 1/(13.4-9.4) (Vurgaftman 2001)
    local value = 0.25
    return value, enabled
end

function material.Nc(state)
	-- Effective conduction-band density of states
	-- Units: m^-3
	-- Refs: [2]
	--
	-- Ioffe band-structure page: Nc = 4.0e13*T^1.5 cm^-3, i.e.
	-- 2.1e17 cm^-3 at 300 K. This effective value lumps the low-mass
	-- Gamma valley (m* = 0.041 m0) with the nearby L valleys
	-- (E_GammaL ~ 0.084 eV), so it exceeds the value implied by the
	-- Gamma mass alone. Relevant above ~200 K.

	local enabled = true
	local T = state.T
	local value = 2.1e23*(T/300.0)^1.5

	return value, enabled
end

function material.Nv(state)
	-- Effective valence-band density of states
	-- Units: m^-3
	-- Refs: [2]
	--
	-- Ioffe band-structure page: Nv = 3.5e15*T^1.5 cm^-3, i.e.
	-- 1.8e19 cm^-3 at 300 K.

	local enabled = true
	local T = state.T
	local value = 1.8e25*(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
	-- Refs: [2] (300 K value), [7] (T dependence), [6]
	--
	-- 300 K value 3000 cm^2/V/s (undoped, upper limit) from [2].
	-- The (300/T)^2.0 factor is a representative phonon-limited
	-- (lattice) temperature dependence for lightly doped GaSb,
	-- consistent with the Hall-mobility data of Mathur & Jain [7];
	-- refine the exponent and magnitude for the target doping.

	local enabled = true
	local T = state.T
	local value = 0.30*(300.0/T)^2.0

	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
	-- Refs: [2] (300 K value), [6]
	--
	-- 300 K value 1000 cm^2/V/s (undoped, upper limit) from [2].
	-- The (300/T)^1.9 factor is a representative lattice-limited
	-- temperature dependence; refine for the target doping.

	local enabled = true
	local T = state.T
	local value = 0.10*(300.0/T)^1.9

	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
	-- Refs: [2] (static 15.7, high-frequency 14.4), [6]

	local enabled = true
	local value = 15.7

	return value, enabled
end

function material.free_to_free_recombination(state)
	-- Radiative recombination coefficient
	-- Units: m^3 s^-1
	-- Refs: [2]
	--
	-- Ioffe recombination page: B ~ 1e-10 cm^3/s = 1e-16 m^3/s
	-- (representative, 300 K). Check against the intended GaSb
	-- material quality and recombination model.

	local enabled = true
	local value = 1.0e-16

	return value, enabled
end

function material.auger_Cn(state)
	-- Electron Auger recombination coefficient
	-- Units: m^6 s^-1
	-- Refs: [2]
	--
	-- Ioffe recombination page quotes a single total Auger
	-- coefficient for GaSb: 5e-30 cm^6/s at 300 K (2e-29 cm^6/s at
	-- 77 K), i.e. 5e-42 m^6/s. Applied here to Cn.
	-- Note: reported GaSb Auger coefficients span a wide range,
	-- with device studies often citing ~1e-28 to 1e-27 cm^6/s.
	-- Auger dominates at high injection in this narrow-gap material,
	-- so tune against the operating regime.

	local enabled = true
	local value = 5.0e-42

	return value, enabled
end

function material.auger_Cp(state)
	-- Hole Auger recombination coefficient
	-- Units: m^6 s^-1
	-- Refs: [2]
	--
	-- As auger_Cn: Ioffe total Auger coefficient 5e-30 cm^6/s at
	-- 300 K = 5e-42 m^6/s. See the note there on the wide spread.

	local enabled = true
	local value = 5.0e-42

	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).
	--
	-- Not an intrinsic constant: the dominant recombination centre
	-- depends on growth and defect chemistry. The residual acceptor
	-- of undoped GaSb is a native defect [2]; set this from your own
	-- DLTS / lifetime data. Mid-gap is used as a neutral default.

	local enabled = true
	local value = 0.0

	return value, enabled
end

function material.ss_srh_Nt(state)
	-- SRH trap density
	-- Units: m^-3
	--
	-- Defect-dependent placeholder; set from the intended material
	-- quality (bulk, epilayer, superlattice constituent, etc.).

	local enabled = true
	local value = 5.0e22

	return value, enabled
end

function material.ss_srh_sigma_n(state)
	-- Electron capture cross section
	-- Units: m^2
	--
	-- Defect-dependent placeholder; set from measurement.

	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
	--
	-- Defect-dependent placeholder; set from measurement.

	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
	-- Refs: [2],[8]
	--
	-- 300 K value 32 W/m/K (0.32 W/cm/K) from [2] (Poujade & Albany,
	-- Okhotin et al. [8]). The (300/T)^1.35 factor approximates the
	-- phonon-limited decrease above ~150 K; treat the exponent as
	-- approximate.

	local enabled = true
	local T = state.T
	local value = 32.0*(300.0/T)^1.35

	return value, enabled
end

function material.heat_capacity(state)
	-- Specific heat capacity
	-- Units: J kg^-1 K^-1
	-- Refs: [2],[9]
	--
	-- 0.25 J/g/K = 250 J/kg/K near 300 K (Piesbergen [9], in [2]).

	local enabled = true
	local value = 250.0

	return value, enabled
end

function material.density(state)
	-- Mass density
	-- Units: kg m^-3
	-- Refs: [2] (5.61 g/cm^3)

	local enabled = true
	local value = 5610.0

	return value, enabled
end

function material.lattice_constant(state)
	-- Cubic lattice constant
	-- Units: m
	-- Refs: [2] (a300), [10] (thermal expansion)
	--
	-- a(300 K) = 6.09593 A [2]. Linear thermal expansion 7.75e-6 /K
	-- (Novikova & Abrikosov [10], in [2]).

	local enabled = true
	local T = state.T
	local a300 = 6.09593e-10
	local expansion = 7.75e-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: III-V family estimate
	-- Confidence: Medium
	--
	-- Reference:
	-- https://www.mdpi.com/2673-3978/3/2/16
	--
	-- Comments:
	-- Representative III-V carrier-to-lattice relaxation time. GaAs-like values
	-- are typically sub-ps to ps and field 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: III-V family estimate
	-- Confidence: Medium
	--
	-- Reference:
	-- https://www.mdpi.com/2673-3978/3/2/16
	--
	-- Comments:
	-- Representative III-V carrier-to-lattice relaxation time. GaAs-like values
	-- are typically sub-ps to ps and field dependent.

	local enabled = true
	local value = 5.000000e-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 GaSb.

	local enabled = true
	local value = 0.76

	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 GaSb.

	local enabled = true
	local value = 27

	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 GaSb.

	local enabled = true
	local value = 13.4

	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 GaSb.

	local enabled = true
	local value = 4.7

	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 GaSb.

	local enabled = true
	local value = 6

	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 GaSb.
	--
	-- 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 = -7.5

	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 GaSb.
	--
	-- 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 = 0.8

	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 GaSb.
	--
	-- Note:
	-- Sign convention as in VMR (b negative). Sign preserved; not flipped.

	local enabled = true
	local value = -2

	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 GaSb.
	--
	-- Note:
	-- Sign convention as in VMR (d negative). Sign preserved; not flipped.

	local enabled = true
	local value = -4.7

	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 GaSb.
	-- a(300K)=6.0959 Angstrom, da/dT=4.72e-05 Angstrom/K.
	-- Linear thermal expansion: a(T)=a300+da/dT*(T-300).

	local enabled = true
	local T = state.T
	local value = (6.0959 + 4.72e-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 GaSb.
	-- C11=88.42 GPa, converted to Pa.

	local enabled = true
	local value = 88.42e9

	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 GaSb.
	-- C12=40.26 GPa, converted to Pa.

	local enabled = true
	local value = 40.26e9

	return value, enabled
end

function material.C44(state)
    -- Elastic stiffness constant C44
    -- Units: Pa
    --
    -- Reference:
    -- 300 K value 4.32e11 dyn/cm^2 (temperature dependence: Boyle & Sladek 1975), as attributed in the Ioffe NSM archive
    -- (GaSb mechanical properties); original not independently accessed.
    -- The Vurgaftman et al. (2001) table value for GaSb could not be
    -- verified in this pass, so the compilation value is retained.
    --
    -- Notes:
    -- Converted: 43.2 GPa -> 43.2e9 Pa (1 dyn/cm^2 = 0.1 Pa).

    local enabled = true
    local value = 43.2e9

    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/).
    -- (phonon frequency table, GaSb mechanical properties).
    --
    -- Notes:
    -- nu_LO(Gamma) = 6.87 THz -> 6.87 x 4.135667e-3 = 0.02841 eV.
    -- Hass & Henvis (1962) give 240.3 cm^-1 (29.8 meV) at 4.2 K.

    local enabled = true
    local value = 0.02841

    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:
    -- GaSb: eps_s = 15.7 (300 K).

    local enabled = true
    local value = 15.7

    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:
    -- GaSb: eps_inf = 14.4 (300 K).

    local enabled = true
    local value = 14.4

    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:
    -- GaSb: e14 = -0.13 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 = -0.13

    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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