InSb material model

1. Introduction

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

Bulk crystalline indium 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 = {}

-- ---------------------------------------------------------------------------
-- NOTES ON InSb (read before using):
--
-- InSb is the narrowest-gap common III-V (Eg ~ 0.17 eV at 300 K) and has the
-- highest electron mobility of any bulk semiconductor, with a very small
-- electron effective mass (0.014 m0). Two consequences matter for modelling:
--   * The gap is only ~6.6 kT at 300 K, so ni is huge (~2e16 cm^-3) and the
--     material is nearly intrinsic at room temperature. InSb IR detectors
--     are therefore normally operated cooled (77 K).
--   * The conduction band is strongly non-parabolic (alpha ~ 4.1 eV^-1), so
--     the parabolic Nc ~ T^1.5 below is only an approximation once the Fermi
--     level enters the band (high doping / high T).
-- Near room temperature the minority-carrier lifetime is set by (strong)
-- Auger recombination, not SRH - see auger_Cn / free_to_free_recombination.
--
-- 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: InSb", NSM Archive,
--      www.ioffe.ru/SVA/NSM/Semicond/InSb/ (compilation; primary sources
--      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]  C. L. Littler, D. G. Seiler, Appl. Phys. Lett. 46, 986-988 (1985).
--      [InSb Eg(T) fit, alternative to [1]]
-- [6]  S. Adachi, "Properties of Group-IV, III-V and II-VI Semiconductors",
--      Wiley (2005).
-- [7]  M. Oszwaldowski, M. Zimpel, J. Phys. Chem. Solids 49, 1179-1185
--      (1988). [InSb intrinsic carrier concentration]
-- [8]  D. L. Rode (1971); N. G. Yaremenko et al. (1973) - InSb carrier
--      mobility vs T, compiled in [2].
-- [9]  Kosarev et al. (1971); Busch & Steigmeier (1961) - InSb thermal
--      conductivity, compiled in [2].
-- [10] U. Piesbergen, Z. Naturforschung 18a, 141-147 (1963) - InSb specific
--      heat, compiled in [2].
-- [11] Sparks & Swenson (1967); Gibbons (1958) - InSb linear thermal
--      expansion, compiled in [2].
-- ---------------------------------------------------------------------------

function material.name()
	local enabled = true

	return "InSb", enabled
end

function material.description()
	local enabled = true

	return "Bulk crystalline indium antimonide", enabled
end

function material.formula()
	local enabled = true

	return "InSb", 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) = 0.235 eV, alpha = 3.2e-4 eV/K, beta = 170 K.
	-- Gives Eg(300 K) = 0.174 eV.
	--
	-- Alternative fit (Littler & Seiler [5], compiled in [2], valid
	-- 0 < T < 300 K): 0.24 - 6.0e-4*T*T/(T + 500).

	local enabled = true
	local T = state.T
	local value = 0.235 - 3.2e-4*T*T/(T + 170.0)

	return value, enabled
end

function material.Xi(state)
	-- Electron affinity
	-- Units: eV
	-- Refs: [2] (Ioffe basic parameters, 4.59 eV), [6]

	local enabled = true
	local value = 4.59

	return value, enabled
end

function material.me(state)
    local enabled = true
    local value = 0.014   -- Gamma electron mass, 0.0135 (Vurgaftman 2001)
    return value, enabled
end

function material.mh(state)
    local enabled = true
    -- HH[001]: g1=34.8 g2=15.5 -> 1/(34.8-31.0) (Vurgaftman 2001)
    local value = 0.26
    return value, enabled
end

function material.Nc(state)
	-- Effective conduction-band density of states
	-- Units: m^-3
	-- Refs: [2]
	--
	-- Ioffe band-structure page: Nc = 8.0e12*T^1.5 cm^-3, i.e.
	-- 4.2e16 cm^-3 at 300 K - very small, reflecting the tiny electron
	-- mass (0.014 m0). Parabolic-band approximation; see the header
	-- note on conduction-band non-parabolicity.

	local enabled = true
	local T = state.T
	local value = 4.2e22*(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 = 1.4e15*T^1.5 cm^-3, i.e.
	-- 7.3e18 cm^-3 at 300 K.

	local enabled = true
	local T = state.T
	local value = 7.3e24*(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],[8]
	--
	-- Pure n-InSb, Ioffe: mu ~ 7.7e4*(T/300)^-1.66 cm^2/V/s for
	-- T >= 200 K, i.e. 77000 cm^2/V/s = 7.7 m^2/V/s at 300 K - the
	-- highest electron mobility of any bulk semiconductor. Falls
	-- rapidly with doping; this is the lightly-doped limit.

	local enabled = true
	local T = state.T
	local value = 7.7*(300.0/T)^1.66

	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],[8]
	--
	-- Pure p-InSb, Ioffe: mu ~ 850*(T/300)^-1.8 cm^2/V/s for T > 60 K,
	-- i.e. 850 cm^2/V/s = 0.085 m^2/V/s at 300 K.

	local enabled = true
	local T = state.T
	local value = 0.085*(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
	-- Refs: [2] (static 16.8, high-frequency 15.7), [6]

	local enabled = true
	local value = 16.8

	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 ~ 5e-11 cm^3/s = 5e-17 m^3/s (300 K).
	-- Radiative and Auger both matter in this narrow-gap material; near
	-- and above ~250 K Auger dominates the lifetime (see auger_Cn).

	local enabled = true
	local value = 5.0e-17

	return value, enabled
end

function material.auger_Cn(state)
	-- Electron Auger recombination coefficient
	-- Units: m^6 s^-1
	-- Refs: [2]
	--
	-- Ioffe recombination page gives a single (ambipolar) Auger
	-- coefficient C ~ 5e-26 cm^6/s = 5e-38 m^6/s, defined via
	-- tau_n = tau_p ~ 1/(C*ni^2). Applied here to Cn.
	-- This is large (narrow gap) and sets the intrinsic lifetime at
	-- 300 K to tau ~ 5e-8 s; Auger is the dominant recombination
	-- channel for InSb near room temperature.

	local enabled = true
	local value = 5.0e-38

	return value, enabled
end

function material.auger_Cp(state)
	-- Hole Auger recombination coefficient
	-- Units: m^6 s^-1
	-- Refs: [2]
	--
	-- As auger_Cn: Ioffe ambipolar Auger coefficient 5e-26 cm^6/s
	-- = 5e-38 m^6/s.

	local enabled = true
	local value = 5.0e-38

	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).
	--
	-- Defect-dependent, not an intrinsic constant. Note that near room
	-- temperature InSb lifetime is Auger-limited [2], so SRH matters
	-- mainly for cooled (e.g. 77 K) detector operation, where it is set
	-- by growth quality. Mid-gap is a neutral default; set from your
	-- own DLTS / lifetime data.

	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 (detector-
	-- grade) material quality.

	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],[9]
	--
	-- 300 K value 18 W/m/K (0.18 W/cm/K) from [2] (Kosarev et al.,
	-- Busch & Steigmeier [9]). The (300/T)^1.4 factor approximates the
	-- phonon-limited decrease above the Debye temperature (~160 K);
	-- treat the exponent as approximate.

	local enabled = true
	local T = state.T
	local value = 18.0*(300.0/T)^1.4

	return value, enabled
end

function material.heat_capacity(state)
	-- Specific heat capacity
	-- Units: J kg^-1 K^-1
	-- Refs: [2],[10]
	--
	-- 0.20 J/g/K = 200 J/kg/K near 300 K (Piesbergen [10], in [2]).

	local enabled = true
	local value = 200.0

	return value, enabled
end

function material.density(state)
	-- Mass density
	-- Units: kg m^-3
	-- Refs: [2] (5.77 g/cm^3)

	local enabled = true
	local value = 5770.0

	return value, enabled
end

function material.lattice_constant(state)
	-- Cubic lattice constant
	-- Units: m
	-- Refs: [2] (a300), [11] (thermal expansion)
	--
	-- a(300 K) = 6.479 A [2]. Linear thermal expansion 5.37e-6 /K
	-- (Sparks & Swenson, Gibbons [11], in [2]).

	local enabled = true
	local T = state.T
	local a300 = 6.479e-10
	local expansion = 5.37e-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: Narrow-gap family estimate
	-- Confidence: Low
	--
	-- Reference:
	-- https://doi.org/10.1103/PhysRevB.103.245205
	--
	-- Comments:
	-- Narrow-gap semiconductor with strong non-parabolicity. Use 2 ps as a
	-- conservative effective value and test field dependence.

	local enabled = true
	local value = 2.000000e-12

	return value, enabled
end

function material.thermal_tau_h(state)
	-- Hole energy relaxation time towards the lattice temperature
	-- Units: s
	--
	-- Value basis: Narrow-gap family estimate
	-- Confidence: Low
	--
	-- Reference:
	-- https://doi.org/10.1103/PhysRevB.103.245205
	--
	-- Comments:
	-- Narrow-gap semiconductor with strong non-parabolicity. Use 2 ps as a
	-- conservative effective value and test field dependence.

	local enabled = true
	local value = 2.000000e-12

	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 InSb.

	local enabled = true
	local value = 0.81

	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 InSb.

	local enabled = true
	local value = 23.3

	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 InSb.

	local enabled = true
	local value = 34.8

	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 InSb.

	local enabled = true
	local value = 15.5

	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 InSb.

	local enabled = true
	local value = 16.5

	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 InSb.
	--
	-- 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 = -6.94

	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 InSb.
	--
	-- 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 = 0.36

	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 InSb.
	--
	-- 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 InSb.
	--
	-- 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 InSb.
	-- a(300K)=6.4794 Angstrom, da/dT=3.48e-05 Angstrom/K.
	-- Linear thermal expansion: a(T)=a300+da/dT*(T-300).

	local enabled = true
	local T = state.T
	local value = (6.4794 + 3.48e-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 InSb.
	-- C11=68.47 GPa, converted to Pa.

	local enabled = true
	local value = 68.47e9

	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 InSb.
	-- C12=37.35 GPa, converted to Pa.

	local enabled = true
	local value = 37.35e9

	return value, enabled
end

function material.C44(state)
    -- Elastic stiffness constant C44
    -- Units: Pa
    --
    -- Reference:
    -- Slutsky and Garland (1959), ultrasonic, 300 K, as attributed in the
    -- Ioffe NSM archive (InSb mechanical properties); original not
    -- independently accessed.
    --
    -- Notes:
    -- 3.02e11 dyn/cm^2 = 30.2 GPa -> 30.2e9 Pa.

    local enabled = true
    local value = 30.2e9

    return value, enabled
end

function material.phonon_lo_energy(state)
    -- Representative LO phonon energy for polar optical (Frohlich) scattering
    -- Units: eV
    --
    -- Crystal phase: zincblende InSb.
    --
    -- Reference:
    -- D. L. Price, J. M. Rowe, R. M. Nicklow,
    -- Phys. Rev. B 3, 1268 (1971). (InSb lattice dynamics, inelastic neutron
    -- scattering; frequencies as tabulated in the Ioffe NSM archive)
    --
    -- Notes:
    -- nu_LO(Gamma) = 5.90 THz -> 5.90 x 4.135667e-3 = 0.02440 eV.
    -- Hass & Henvis (1962): 197.2 cm^-1 (24.4 meV) at 4.2 K, consistent.

    local enabled = true
    local value = 0.02440

    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:
    -- InSb: 16.8 (300 K).

    local enabled = true
    local value = 16.8

    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:
    -- InSb: 15.7 (300 K).

    local enabled = true
    local value = 15.7

    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 Arlt and Quadflieg (1968)).
    --
    -- Notes:
    -- InSb: e14 = -0.07 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 = -0.07

    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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-- All rights reserved.
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-- This file is part of the OghmaNano Materials Model Library.
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-- https://www.oghma-nano.com
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