AlSb material model

1. Introduction

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

Bulk crystalline aluminium antimonide (indirect gap)

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 AlSb (read before using):
--
-- AlSb is an INDIRECT-gap III-V (conduction-band minimum near X), Eg ~ 1.6 eV
-- at 300 K, zinc-blende, lattice-matched close to GaSb / InAs (a ~ 6.13 A). It
-- is used as a barrier/buffer in antimonide (6.1 A family) heterostructures
-- and as a room-temperature gamma/X-ray detector material. Two consequences:
--   * Being indirect, radiative recombination is weak, which favours long
--     carrier lifetimes - good for detectors, poor for light emission. The
--     free_to_free (radiative) coefficient is correspondingly small.
--   * Unusually, holes are MORE mobile than electrons in AlSb.
-- Practical note: bulk AlSb oxidises/reacts readily in air and is usually
-- capped; this is a handling issue, not a simulation parameter.
--
-- 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  [band parameters; X-valley
--      is the fundamental gap]
-- [2]  S. Adachi, "Properties of Group-IV, III-V and II-VI Semiconductors",
--      Wiley (2005). [transport / thermal / dielectric]
-- [3]  M. Levinshtein, S. Rumyantsev, M. Shur (eds.), "Handbook Series on
--      Semiconductor Parameters, Vol. 1", World Scientific (1996).
-- [4]  Y. P. Varshni, "Temperature dependence of the energy gap in
--      semiconductors", Physica 34, 149-154 (1967). [Varshni form]
--
-- Nc and Nv are computed from density-of-states effective masses
-- (N = 2.509e25 * (m*/m0)^1.5 * (T/300)^1.5 m^-3); Nc uses the multi-valley
-- X-minimum DOS mass.
-- ---------------------------------------------------------------------------

function material.name()
	local enabled = true

	return "AlSb", enabled
end

function material.description()
	local enabled = true

	return "Bulk crystalline aluminium antimonide (indirect gap)", enabled
end

function material.formula()
	local enabled = true

	return "AlSb", enabled
end

function material.Eg(state)
	-- Units: eV
	-- Refs: [1] (primary), [4] (Varshni form)
	--
	-- Fundamental (indirect, X-valley) gap, Varshni form with the
	-- recommended III-V parameter set [1]:
	-- Eg(X,0) = 1.696 eV, alpha = 3.9e-4 eV/K, beta = 140 K.
	-- Gives Eg(300 K) = 1.62 eV.
	--
	-- For reference the direct (Gamma) gap is higher: Eg(Gamma,0) =
	-- 2.386 eV (~2.30 eV at 300 K). This routine returns the fundamental
	-- indirect gap, which is what governs thermal carrier generation.

	local enabled = true
	local T = state.T
	local value = 1.696 - 3.9e-4*T*T/(T + 140.0)

	return value, enabled
end

function material.Xi(state)
	-- Electron affinity
	-- Units: eV
	-- Refs: [2]
	--
	-- ~3.65 eV (reported ~3.6-3.7 eV). Sets the heterojunction band
	-- offset with GaSb / InAs in 6.1 A-family structures.

	local enabled = true
	local value = 3.65

	return value, enabled
end

function material.me(state)
    local enabled = true
    local value = 0.14    -- Gamma mass (material is X-indirect) (Vurgaftman 2001)
    -- NOTE: Gamma mass valid as barrier for a Gamma-confined well; if AlSb is the
    -- well, X valleys apply -> use multivalley solver.
    return value, enabled
end

function material.mh(state)
    local enabled = true
    -- HH[001]: g1=5.18 g2=1.19 -> 1/(5.18-2.38) (Vurgaftman 2001)
    local value = 0.36
    return value, enabled
end

function material.Nc(state)
	-- Effective conduction-band density of states
	-- Units: m^-3
	-- Refs: computed from the X-valley DOS mass ~0.92 m0 [1],[2]
	--
	-- Nc = 2.509e25*(0.92)^1.5*(T/300)^1.5 = 2.2e25 m^-3 at 300 K
	-- (2.2e19 cm^-3). Large because the indirect X minimum has 3
	-- equivalent valleys and a heavy density-of-states mass - quite
	-- unlike the direct antimonides GaSb/InSb.

	local enabled = true
	local T = state.T
	local value = 2.2e25*(T/300.0)^1.5

	return value, enabled
end

function material.Nv(state)
	-- Effective valence-band density of states
	-- Units: m^-3
	-- Refs: computed from m_h(DOS) ~ 0.98 m0 [1],[2]
	--
	-- Nv = 2.509e25*(0.98)^1.5*(T/300)^1.5 = 2.4e25 m^-3 at 300 K
	-- (2.4e19 cm^-3).

	local enabled = true
	local T = state.T
	local value = 2.4e25*(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],[3]
	--
	-- ~200 cm^2/V/s = 0.02 m^2/V/s at 300 K. Modest because the
	-- fundamental minimum is the heavy indirect X valley. The
	-- (300/T)^1.7 factor is an approximate lattice-limited dependence.

	local enabled = true
	local T = state.T
	local value = 0.02*(300.0/T)^1.7

	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],[3]
	--
	-- ~400 cm^2/V/s = 0.04 m^2/V/s at 300 K - notably HIGHER than the
	-- electron mobility, a distinctive feature of AlSb. (300/T)^1.8
	-- approximate lattice-limited dependence.

	local enabled = true
	local T = state.T
	local value = 0.04*(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 12.0 (high-frequency 10.24).

	local enabled = true
	local value = 12.0

	return value, enabled
end

function material.free_to_free_recombination(state)
	-- Radiative recombination coefficient
	-- Units: m^3 s^-1
	-- Refs: representative (see note)
	--
	-- Small representative value ~1e-11 cm^3/s = 1e-17 m^3/s. AlSb is
	-- indirect, so band-to-band radiative recombination is weak and
	-- carrier lifetime is governed by SRH (and Auger at high injection).
	-- This weakness is exactly why AlSb is attractive for detectors.

	local enabled = true
	local value = 1.0e-17

	return value, enabled
end

function material.auger_Cn(state)
	-- Electron Auger recombination coefficient
	-- Units: m^6 s^-1
	-- Refs: representative (see note)
	--
	-- Representative ~1e-30 cm^6/s = 1e-42 m^6/s. Auger is modest in
	-- this wide-ish indirect gap and normally sub-dominant to SRH at
	-- device injection levels.

	local enabled = true
	local value = 1.0e-42

	return value, enabled
end

function material.auger_Cp(state)
	-- Hole Auger recombination coefficient
	-- Units: m^6 s^-1
	-- Refs: representative (see note)
	--
	-- As auger_Cn: representative 1e-30 cm^6/s = 1e-42 m^6/s.

	local enabled = true
	local value = 1.0e-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).
	--
	-- SRH sets the lifetime in indirect AlSb, so this block matters.
	-- Native defects (Al vacancies, antisites) and residual impurities
	-- dominate; set the level and density from your own DLTS / lifetime
	-- data. Mid-gap is 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. Detector-grade AlSb targets low trap densities for long
	-- lifetimes / good charge collection.

	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
	--
	-- 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]
	--
	-- 300 K value ~57 W/m/K (0.57 W/cm/K) - high for a III-V antimonide,
	-- reflecting the light Al atom. The (300/T)^1.2 factor approximates
	-- the phonon-limited decrease above the Debye temperature; treat the
	-- exponent as approximate.

	local enabled = true
	local T = state.T
	local value = 57.0*(300.0/T)^1.2

	return value, enabled
end

function material.heat_capacity(state)
	-- Specific heat capacity
	-- Units: J kg^-1 K^-1
	-- Refs: [2]
	--
	-- ~330 J/kg/K near 300 K, consistent with the Dulong-Petit limit
	-- for AlSb (M = 148.7 g/mol).

	local enabled = true
	local value = 330.0

	return value, enabled
end

function material.density(state)
	-- Mass density
	-- Units: kg m^-3
	-- Refs: [2]
	--
	-- 4.26 g/cm^3 (crystallographic: Z = 4, M = 148.74 g/mol,
	-- a = 6.1355 A).

	local enabled = true
	local value = 4260.0

	return value, enabled
end

function material.lattice_constant(state)
	-- Cubic lattice constant
	-- Units: m
	-- Refs: [1],[2]
	--
	-- a(300 K) = 6.1355 A - close to GaSb (6.096 A) and InAs (6.058 A),
	-- hence its role in 6.1 A-family heterostructures. Linear thermal
	-- expansion ~4.2e-6 /K.

	local enabled = true
	local T = state.T
	local a300 = 6.1355e-10
	local expansion = 4.2e-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 AlSb.

	local enabled = true
	local value = 0.676

	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 AlSb.

	local enabled = true
	local value = 18.7

	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 AlSb.

	local enabled = true
	local value = 5.18

	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 AlSb.

	local enabled = true
	local value = 1.19

	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 AlSb.

	local enabled = true
	local value = 1.97

	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 AlSb.
	--
	-- 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 = -4.5

	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 AlSb.
	--
	-- 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 = 1.4

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

	local enabled = true
	local value = -1.35

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

	local enabled = true
	local value = -4.3

	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 AlSb.
	-- a(300K)=6.1355 Angstrom, da/dT=2.60e-05 Angstrom/K.
	-- Linear thermal expansion: a(T)=a300+da/dT*(T-300).

	local enabled = true
	local T = state.T
	local value = (6.1355 + 2.60e-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 AlSb.
	-- C11=87.69 GPa, converted to Pa.

	local enabled = true
	local value = 87.69e9

	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 AlSb.
	-- C12=43.41 GPa, converted to Pa.

	local enabled = true
	local value = 43.41e9

	return value, enabled
end

function material.C44(state)
    -- Elastic stiffness constant C44
    -- Units: Pa
    --
    -- Reference:
    -- D. I. Bolef and M. Menes (1960), ultrasonic single-crystal measurement
    -- on AlSb at room temperature (journal details not independently
    -- verified).
    -- Value as listed in the experimental column (with this attribution) of:
    -- Experimental elastic-constant table (Table 2.67, values with original
    -- references) in the AFLOW review arXiv:1811.08464.
    --
    -- Notes:
    -- C44 = 40.76 GPa -> 40.76e9 Pa (300 K).
    -- The original 1960 paper was not independently accessed; the value and
    -- attribution are taken from the experimental table cited above.

    local enabled = true
    local value = 40.76e9

    return value, enabled
end

function material.phonon_lo_energy(state)
    -- Representative LO phonon energy for polar optical (Frohlich) scattering
    -- Units: eV
    --
    -- Reference:
    -- M. Hass and B. W. Henvis,
    -- "Infrared lattice reflection spectra of III-V compound semiconductors,"
    -- J. Phys. Chem. Solids 23, 1099 (1962).
    --
    -- Notes:
    -- AlSb, room-temperature data reanalysed: TO = 318 +/- 8 cm^-1,
    -- LO = 345 +/- 5 cm^-1. 345 x 1.239842e-4 = 0.04277 eV.

    local enabled = true
    local value = 0.04277

    return value, enabled
end

function material.epsilon_static(state)
    -- Static relative dielectric constant (lattice + electronic)
    -- Dimensionless
    --
    -- No sufficiently reliable value/reference identified.
    -- Disabled rather than estimated.
    --
    -- Notes:
    -- Sources conflict: the microwave value 10.9 (K. Seeger and E. Schonherr,
    -- Semicond. Sci. Technol. 6, 301 (1991), DOI: 10.1088/0268-1242/6/4/013)
    -- is inconsistent via LST with eps_inf ~10.2 and the Hass & Henvis LO/TO
    -- ratio (which would need ~12). No verified self-consistent pair.

    local enabled = false
    local value = 0.0

    return value, enabled
end

function material.epsilon_inf(state)
    -- High-frequency (electronic) relative dielectric constant
    -- Dimensionless
    --
    -- No sufficiently reliable value/reference identified.
    -- Disabled rather than estimated.
    --
    -- Notes:
    -- See epsilon_static; no verified primary value.

    local enabled = false
    local value = 0.0

    return value, enabled
end

function material.e14(state)
    -- Zincblende piezoelectric stress coefficient e14
    -- Units: C m^-2
    --
    -- No sufficiently reliable value/reference identified.
    -- Disabled rather than estimated.
    --
    -- Notes:
    -- A magnitude ~0.07 C/m^2 appears in secondary tables but neither value
    -- nor sign could be verified.

    local enabled = false
    local value = 0.0

    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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