AlGaN material model

1. Introduction

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

Wurtzite Al(x)Ga(1-x)N, x = Al fraction ~ 0.25

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 AlGaN (read before using):
--
-- Al(x)Ga(1-x)N is a wide-gap wurtzite III-nitride, DIRECT gap across the whole
-- range (both GaN and wurtzite AlN are direct), tunable from ~3.4 eV (GaN) to
-- ~6.1 eV (AlN). Al fraction x is the design knob; used in UV LEDs and HEMT
-- barriers. Modelled at a representative x = 0.25.
--
-- Two things this bulk file does NOT capture, both central to real nitride
-- devices:
--   * Spontaneous + piezoelectric POLARISATION. Wurtzite nitrides have large
--     built-in polarisation; the AlGaN/GaN interface sheet charge is what
--     forms the HEMT 2DEG. That is a heterostructure/interface effect, not a
--     bulk material parameter.
--   * Very high dislocation densities in heteroepitaxial GaN/AlGaN, which
--     dominate SRH lifetime - set ss_srh_* from your own material.
--
-- Eg, lattice constant and thermal conductivity are computed from x; the rest
-- are given at x = 0.25 with GaN/AlN end points noted for interpolation.
--
-- REFERENCES
--
-- [1]  I. Vurgaftman, J. R. Meyer, "Band parameters for nitrogen-containing
--      semiconductors", J. Appl. Phys. 94, 3675 (2003); and I. Vurgaftman,
--      J. R. Meyer, L. R. Ram-Mohan, J. Appl. Phys. 89, 5815 (2001).
--      [end-point gaps, Varshni parameters, bowing b ~ 0.7 eV]
-- [2]  S. Adachi, "Properties of Semiconductor Alloys", Wiley (2009).
--      [transport / dielectric / thermal]
-- [3]  O. Madelung, "Semiconductors: Data Handbook", Springer (2004).
-- ---------------------------------------------------------------------------

function material.name()
	local enabled = true

	return "AlGaN", enabled
end

function material.description()
	local enabled = true

	return "Wurtzite Al(x)Ga(1-x)N, x = Al fraction ~ 0.25", enabled
end

function material.formula()
	local enabled = true

	return "AlGaN", enabled
end

function material.Eg(state)
	-- Units: eV
	-- Refs: [1]
	--
	-- Direct gap. End-point Varshni fits [1]:
	--   GaN: 3.510 - 0.909e-3*T^2/(T+830)
	--   AlN: 6.250 - 1.799e-3*T^2/(T+1462)
	-- interpolated with bowing b = 0.7 eV:
	--   Eg(x,T) = x*Eg_AlN + (1-x)*Eg_GaN - 0.7*x*(1-x).
	-- x = 0.25 -> ~3.99 eV at 300 K. Wurtzite AlGaN stays direct for all x.
	-- Change al_fraction here and in the functions noted in the header.

	local enabled = true
	local al_fraction = 0.25
	local x = al_fraction
	local T = state.T
	local eg_gan = 3.510 - 0.909e-3*T*T/(T + 830.0)
	local eg_aln = 6.250 - 1.799e-3*T*T/(T + 1462.0)
	local value = x*eg_aln + (1.0 - x)*eg_gan - 0.7*x*(1.0 - x)

	return value, enabled
end

function material.Xi(state)
	-- Electron affinity
	-- Units: eV
	-- Refs: [2],[3]
	--
	-- ~3.55 eV at x = 0.25 (linear GaN 4.1 -> AlN ~1.9 eV; the AlN value
	-- is itself uncertain). The strong variation with x makes chi a key
	-- band-offset parameter for AlGaN/GaN heterostructures.

	local enabled = true
	local value = 3.55

	return value, enabled
end

function material.me(state)
    local enabled = true
    local x = state.x     -- Al fraction, AlxGa1-xN, c-plane
    -- GaN 0.20 -> AlN 0.32 (Vurgaftman 2003)
    local value = 0.20 + 0.12*x
    return value, enabled
end

function material.mh(state)
    local enabled = true
    local x = state.x
    -- HH along c: GaN 1.9 -> AlN 3.6. WARNING: multiband VB, approximate.
    local value = 1.9 + 1.7*x
    return value, enabled
end

function material.Nc(state)
	-- Effective conduction-band density of states
	-- Units: m^-3
	-- Refs: [2],[3]
	--
	-- ~3.0e24 m^-3 at 300 K for x = 0.25 (m_e* rises from ~0.20 m0 in GaN
	-- to ~0.32 m0 in AlN). Endpoints: GaN ~2.3e24, AlN ~6.3e24 m^-3.

	local enabled = true
	local T = state.T
	local value = 3.0e24*(T/300.0)^1.5

	return value, enabled
end

function material.Nv(state)
	-- Effective valence-band density of states
	-- Units: m^-3
	-- Refs: [2],[3]
	--
	-- ~5e25 m^-3 at 300 K (heavy valence bands). Endpoints: GaN ~4.6e25,
	-- AlN ~4.9e25 m^-3.

	local enabled = true
	local T = state.T
	local value = 5.0e25*(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]
	--
	-- ~0.03 m^2/V/s (300 cm^2/V/s) bulk at x = 0.25, alloy-scattering
	-- reduced from GaN (~1000 cm^2/V/s bulk). NOTE: 2DEG mobilities at an
	-- AlGaN/GaN interface are far higher (>1000-2000 cm^2/V/s) - a
	-- heterostructure effect, not this bulk value. (300/T)^1.5 approximate.

	local enabled = true
	local T = state.T
	local value = 0.03*(300.0/T)^1.5

	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]
	--
	-- ~0.002 m^2/V/s (20 cm^2/V/s) at x = 0.25 - low, as in all nitrides
	-- (heavy holes, hard p-doping). (300/T)^1.5 approximate.

	local enabled = true
	local T = state.T
	local value = 0.002*(300.0/T)^1.5

	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
	-- Refs: [2],[3]
	--
	-- ~8.9 at x = 0.25 (GaN ~8.9, AlN ~8.5; weak x dependence).

	local enabled = true
	local value = 8.9

	return value, enabled
end

function material.free_to_free_recombination(state)
	-- Radiative recombination coefficient
	-- Units: m^3 s^-1
	-- Refs: representative (see note)
	--
	-- Direct-gap value ~2e-11 cm^3/s = 2e-17 m^3/s (order of GaN). In
	-- practice GaN/AlGaN efficiency is limited by dislocation-mediated SRH
	-- and, at high current, Auger. Treat as representative.

	local enabled = true
	local value = 2.0e-17

	return value, enabled
end

function material.auger_Cn(state)
	-- Electron Auger recombination coefficient
	-- Units: m^6 s^-1
	-- Refs: representative (see note)
	--
	-- ~1e-31 cm^6/s = 1e-43 m^6/s (order of GaN). Auger is the leading
	-- candidate for LED "efficiency droop" at high injection.

	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
	-- Refs: representative (see note)
	--
	-- As auger_Cn: representative 1e-31 cm^6/s = 1e-43 m^6/s.

	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).
	--
	-- Defect-dependent. In heteroepitaxial AlGaN/GaN, threading
	-- dislocations dominate SRH; set level and density from your own
	-- material. 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; realistic GaN/AlGaN films carry high
	-- dislocation densities (~1e8-1e10 cm^-2). Set from your material.

	local enabled = true
	local value = 1.0e22

	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]
	--
	-- Alloy-disorder model: GaN (~130) and AlN (~285 W/m/K) are both high,
	-- but mass-disorder scattering collapses the alloy in between. Modelled
	--   1/kappa = (1-x)/130 + x/285 + 1.6*x*(1-x)   (W/m/K)^-1
	-- giving ~130 (GaN), ~2.6 (x=0.25), ~2.3 (x=0.5), ~285 (AlN). The deep
	-- minimum is a well-known feature of AlGaN. (300/T)^0.5 is a weak,
	-- alloy-appropriate dependence; near the pure endpoints ~T^-1.4 applies.

	local enabled = true
	local al_fraction = 0.25
	local x = al_fraction
	local T = state.T
	local inv_k = (1.0 - x)/130.0 + x/285.0 + 1.6*x*(1.0 - x)
	local value = (1.0/inv_k)*(300.0/T)^0.5

	return value, enabled
end

function material.heat_capacity(state)
	-- Specific heat capacity
	-- Units: J kg^-1 K^-1
	-- Refs: [3]
	--
	-- ~500 J/kg/K at x = 0.25 (GaN ~490, AlN ~600 J/kg/K).

	local enabled = true
	local value = 500.0

	return value, enabled
end

function material.density(state)
	-- Mass density
	-- Units: kg m^-3
	-- Refs: [3]
	--
	-- ~5420 kg/m^3 at x = 0.25 (GaN 6150, AlN 3230 kg/m^3; interpolated).

	local enabled = true
	local value = 5420.0

	return value, enabled
end

function material.lattice_constant(state)
	-- Cubic lattice constant
	-- Units: m
	-- Refs: [1],[3]
	--
	-- AlGaN is WURTZITE (not cubic); this routine returns the in-plane
	-- a lattice constant. Vegard: a(x) = 3.189 - 0.077*x (angstrom), i.e.
	-- GaN a = 3.189 A -> AlN a = 3.112 A; c ~ 5.185 A (GaN) -> 4.982 A
	-- (AlN). x = 0.25 -> a = 3.170 A. Linear thermal expansion ~4.5e-6 /K
	-- (a-axis).

	local enabled = true
	local al_fraction = 0.25
	local x = al_fraction
	local T = state.T
	local a300 = (3.189 - 0.077*x)*1.0e-10
	local expansion = 4.5e-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: Hydrodynamic literature/family value
	-- Confidence: Medium
	--
	-- Reference:
	-- https://doi.org/10.1063/1.1383260
	--
	-- Comments:
	-- 0.2 ps is a commonly used GaN/AlGaN hydrodynamic energy-relaxation value;
	-- extended to related nitrides.

	local enabled = true
	local value = 2.000000e-13

	return value, enabled
end

function material.thermal_tau_h(state)
	-- Hole energy relaxation time towards the lattice temperature
	-- Units: s
	--
	-- Value basis: Hydrodynamic literature/family value
	-- Confidence: Medium
	--
	-- Reference:
	-- https://doi.org/10.1063/1.1383260
	--
	-- Comments:
	-- 0.2 ps is a commonly used GaN/AlGaN hydrodynamic energy-relaxation value;
	-- extended to related nitrides.

	local enabled = true
	local value = 2.000000e-13

	return value, enabled
end


-- Al_x Ga_(1-x) N .  x = state.x = Al mole fraction (x=0 GaN, x=1 AlN).
-- Linear (Vegard) interpolation of the k.p parameters; no bowing is
-- applied because no published k.p bowing basis exists. S1/S2 are NOT
-- interpolated directly - they are recomputed from the interpolated
-- masses/gap/crystal-field/Ep (see qw_S1), which is more faithful for a
-- quantity this nonlinear and reproduces both binary endpoints exactly.
-- Both endpoints are traced; every function carries both references.

function material.qw_delta1(state)
-- Crystal-field delta1 (Rinke 2008 Delta_CR). Units: eV. Linear.
--
-- GaN endpoint:
-- P. Rinke, M. Winkelnkemper, A. Qteish, D. Bimberg, J. Neugebauer,
-- and M. Scheffler, "Consistent set of band parameters for the
--  group-III nitrides AlN, GaN, and InN," Physical Review B 77,
--  075202 (2008). DOI: 10.1103/PhysRevB.77.075202
--
-- AlN endpoint:
-- P. Rinke, M. Winkelnkemper, A. Qteish, D. Bimberg, J. Neugebauer,
-- and M. Scheffler, "Consistent set of band parameters for the
--  group-III nitrides AlN, GaN, and InN," Physical Review B 77,
--  075202 (2008). DOI: 10.1103/PhysRevB.77.075202
	local x = state.x
	local GaN = 0.034
	local AlN = -0.295
	local enabled = true
	local value = (1.0-x)*GaN + x*AlN
	return value, enabled
end

function material.qw_delta2(state)
-- Spin-orbit delta2 = Dso/3 (quasi-cubic). Units: eV. Dso linear.
--
-- GaN endpoint: Dso=0.017 eV
-- I. Vurgaftman and J. R. Meyer, "Band parameters for nitrogen-
--  containing semiconductors," Journal of Applied Physics 94,
--  3675-3696 (2003). DOI: 10.1063/1.1600519
--
-- AlN endpoint: Dso=0.019 eV
-- I. Vurgaftman and J. R. Meyer, "Band parameters for nitrogen-
--  containing semiconductors," Journal of Applied Physics 94,
--  3675-3696 (2003). DOI: 10.1063/1.1600519
	local x = state.x
	local Dso_GaN = 0.017
	local Dso_AlN = 0.019
	local Dso = (1.0-x)*Dso_GaN + x*Dso_AlN
	local enabled = true
	local value = Dso/3.0
	return value, enabled
end

function material.qw_delta3(state)
-- Spin-orbit delta3 = Dso/3 (quasi-cubic). Units: eV. Dso linear.
--
-- GaN endpoint: Dso=0.017 eV
-- I. Vurgaftman and J. R. Meyer, "Band parameters for nitrogen-
--  containing semiconductors," Journal of Applied Physics 94,
--  3675-3696 (2003). DOI: 10.1063/1.1600519
--
-- AlN endpoint: Dso=0.019 eV
-- I. Vurgaftman and J. R. Meyer, "Band parameters for nitrogen-
--  containing semiconductors," Journal of Applied Physics 94,
--  3675-3696 (2003). DOI: 10.1063/1.1600519
	local x = state.x
	local Dso_GaN = 0.017
	local Dso_AlN = 0.019
	local Dso = (1.0-x)*Dso_GaN + x*Dso_AlN
	local enabled = true
	local value = Dso/3.0
	return value, enabled
end

function material.qw_A1(state)
-- Wurtzite valence-band k.p parameter A1. Dimensionless. Linear.
--
-- GaN endpoint:
-- P. Rinke et al., Physical Review B 77, 075202 (2008).
-- DOI: 10.1103/PhysRevB.77.075202
--
-- AlN endpoint:
-- P. Rinke et al., Physical Review B 77, 075202 (2008).
-- DOI: 10.1103/PhysRevB.77.075202
	local x = state.x
	local GaN = -5.947
	local AlN = -3.991
	local enabled = true
	local value = (1.0-x)*GaN + x*AlN
	return value, enabled
end

function material.qw_A2(state)
-- Wurtzite valence-band k.p parameter A2. Dimensionless. Linear.
--
-- GaN endpoint:
-- P. Rinke et al., Physical Review B 77, 075202 (2008).
-- DOI: 10.1103/PhysRevB.77.075202
--
-- AlN endpoint:
-- P. Rinke et al., Physical Review B 77, 075202 (2008).
-- DOI: 10.1103/PhysRevB.77.075202
	local x = state.x
	local GaN = -0.528
	local AlN = -0.311
	local enabled = true
	local value = (1.0-x)*GaN + x*AlN
	return value, enabled
end

function material.qw_A3(state)
-- Wurtzite valence-band k.p parameter A3. Dimensionless. Linear.
--
-- GaN endpoint:
-- P. Rinke et al., Physical Review B 77, 075202 (2008).
-- DOI: 10.1103/PhysRevB.77.075202
--
-- AlN endpoint:
-- P. Rinke et al., Physical Review B 77, 075202 (2008).
-- DOI: 10.1103/PhysRevB.77.075202
	local x = state.x
	local GaN = 5.414
	local AlN = 3.671
	local enabled = true
	local value = (1.0-x)*GaN + x*AlN
	return value, enabled
end

function material.qw_A4(state)
-- Wurtzite valence-band k.p parameter A4. Dimensionless. Linear.
--
-- GaN endpoint:
-- P. Rinke et al., Physical Review B 77, 075202 (2008).
-- DOI: 10.1103/PhysRevB.77.075202
--
-- AlN endpoint:
-- P. Rinke et al., Physical Review B 77, 075202 (2008).
-- DOI: 10.1103/PhysRevB.77.075202
	local x = state.x
	local GaN = -2.512
	local AlN = -1.147
	local enabled = true
	local value = (1.0-x)*GaN + x*AlN
	return value, enabled
end

function material.qw_A5(state)
-- Wurtzite valence-band k.p parameter A5. Dimensionless. Linear.
--
-- GaN endpoint:
-- P. Rinke et al., Physical Review B 77, 075202 (2008).
-- DOI: 10.1103/PhysRevB.77.075202
--
-- AlN endpoint:
-- P. Rinke et al., Physical Review B 77, 075202 (2008).
-- DOI: 10.1103/PhysRevB.77.075202
	local x = state.x
	local GaN = -2.51
	local AlN = -1.329
	local enabled = true
	local value = (1.0-x)*GaN + x*AlN
	return value, enabled
end

function material.qw_A6(state)
-- Wurtzite valence-band k.p parameter A6. Dimensionless. Linear.
--
-- GaN endpoint:
-- P. Rinke et al., Physical Review B 77, 075202 (2008).
-- DOI: 10.1103/PhysRevB.77.075202
--
-- AlN endpoint:
-- P. Rinke et al., Physical Review B 77, 075202 (2008).
-- DOI: 10.1103/PhysRevB.77.075202
	local x = state.x
	local GaN = -3.202
	local AlN = -1.952
	local enabled = true
	local value = (1.0-x)*GaN + x*AlN
	return value, enabled
end

function material.qw_Ep1(state)
-- Kane energy || c. Units: eV. Linear.
--
-- GaN endpoint:
-- P. Rinke et al., Physical Review B 77, 075202 (2008).
-- DOI: 10.1103/PhysRevB.77.075202
--
-- AlN endpoint:
-- P. Rinke et al., Physical Review B 77, 075202 (2008).
-- DOI: 10.1103/PhysRevB.77.075202
	local x = state.x
	local GaN = 17.292
	local AlN = 16.972
	local enabled = true
	local value = (1.0-x)*GaN + x*AlN
	return value, enabled
end

function material.qw_Ep2(state)
-- Kane energy perp c. Units: eV. Linear.
--
-- GaN endpoint:
-- P. Rinke et al., Physical Review B 77, 075202 (2008).
-- DOI: 10.1103/PhysRevB.77.075202
--
-- AlN endpoint:
-- P. Rinke et al., Physical Review B 77, 075202 (2008).
-- DOI: 10.1103/PhysRevB.77.075202
	local x = state.x
	local GaN = 16.265
	local AlN = 18.165
	local enabled = true
	local value = (1.0-x)*GaN + x*AlN
	return value, enabled
end

function material.qw_S1(state)
-- Conduction remote+free-electron parameter || c (kz^2). Dimensionless.
-- Recomputed from interpolated m/Eg/delta1/Ep (see AlN.lua qw_S1
-- for the derivation); reproduces the binary S at x=0 and x=1.
-- Eg here is linear (no bowing); for strongly-bowed alloys feed
-- the physical Eg if high accuracy in S is needed.
--
-- GaN endpoint:
-- P. Rinke et al., Physical Review B 77, 075202 (2008).
-- DOI: 10.1103/PhysRevB.77.075202
--
-- AlN endpoint:
-- P. Rinke et al., Physical Review B 77, 075202 (2008).
-- DOI: 10.1103/PhysRevB.77.075202
	local x = state.x
	-- interpolate the underlying Rinke quantities, then compute S:
	local mpar  = (1.0-x)*0.186 + x*0.322
	local mperp = (1.0-x)*0.209 + x*0.329
	local Eg    = (1.0-x)*3.24 + x*6.47
	local d1    = (1.0-x)*0.034 + x*-0.295
	local Ep1   = (1.0-x)*17.292 + x*16.972
	local Ep2   = (1.0-x)*16.265 + x*18.165
	local Egt = Eg
	if d1 < 0.0 then Egt = Eg - d1 end   -- Egtilde = Eg + |d1|
	local enabled = true
	local value = 1.0/mpar - Ep1/(Egt + d1)
	return value, enabled
end

function material.qw_S2(state)
-- Conduction remote+free-electron parameter perp c (kx^2+ky^2). Dimensionless.
-- Recomputed from interpolated m/Eg/delta1/Ep (see AlN.lua qw_S1
-- for the derivation); reproduces the binary S at x=0 and x=1.
-- Eg here is linear (no bowing); for strongly-bowed alloys feed
-- the physical Eg if high accuracy in S is needed.
--
-- GaN endpoint:
-- P. Rinke et al., Physical Review B 77, 075202 (2008).
-- DOI: 10.1103/PhysRevB.77.075202
--
-- AlN endpoint:
-- P. Rinke et al., Physical Review B 77, 075202 (2008).
-- DOI: 10.1103/PhysRevB.77.075202
	local x = state.x
	-- interpolate the underlying Rinke quantities, then compute S:
	local mpar  = (1.0-x)*0.186 + x*0.322
	local mperp = (1.0-x)*0.209 + x*0.329
	local Eg    = (1.0-x)*3.24 + x*6.47
	local d1    = (1.0-x)*0.034 + x*-0.295
	local Ep1   = (1.0-x)*17.292 + x*16.972
	local Ep2   = (1.0-x)*16.265 + x*18.165
	local Egt = Eg
	if d1 < 0.0 then Egt = Eg - d1 end   -- Egtilde = Eg + |d1|
	local enabled = true
	local value = 1.0/mperp - Ep2/Egt
	return value, enabled
end

function material.qw_a1(state)
-- Conduction-band deformation potential || c (= acz-D1). Units: eV.
-- Linear.
--
-- GaN endpoint:
-- I. Vurgaftman and J. R. Meyer, "Band parameters for nitrogen-
--  containing semiconductors," Journal of Applied Physics 94,
--  3675-3696 (2003). DOI: 10.1063/1.1600519
--
-- AlN endpoint:
-- I. Vurgaftman and J. R. Meyer, "Band parameters for nitrogen-
--  containing semiconductors," Journal of Applied Physics 94,
--  3675-3696 (2003). DOI: 10.1063/1.1600519
	local x = state.x
	local GaN = -4.9
	local AlN = -3.4
	local enabled = true
	local value = (1.0-x)*GaN + x*AlN
	return value, enabled
end

function material.qw_a2(state)
-- Conduction-band deformation potential perp c (= act-D2). Units: eV.
-- Linear. (AlN endpoint -11.8 eV; VM2003 misprints it as meV.)
--
-- GaN endpoint:
-- I. Vurgaftman and J. R. Meyer, "Band parameters for nitrogen-
--  containing semiconductors," Journal of Applied Physics 94,
--  3675-3696 (2003). DOI: 10.1063/1.1600519
--
-- AlN endpoint:
-- I. Vurgaftman and J. R. Meyer, "Band parameters for nitrogen-
--  containing semiconductors," Journal of Applied Physics 94,
--  3675-3696 (2003). DOI: 10.1063/1.1600519
	local x = state.x
	local GaN = -11.3
	local AlN = -11.8
	local enabled = true
	local value = (1.0-x)*GaN + x*AlN
	return value, enabled
end

function material.qw_D1(state)
-- Valence-band deformation potential D1. Units: eV. Linear.
--
-- GaN endpoint:
-- I. Vurgaftman and J. R. Meyer, "Band parameters for nitrogen-
--  containing semiconductors," Journal of Applied Physics 94,
--  3675-3696 (2003). DOI: 10.1063/1.1600519
--
-- AlN endpoint:
-- I. Vurgaftman and J. R. Meyer, "Band parameters for nitrogen-
--  containing semiconductors," Journal of Applied Physics 94,
--  3675-3696 (2003). DOI: 10.1063/1.1600519
--
-- Original parameter source cited therein:
-- K. Shimada, T. Sota, and K. Suzuki, "First-principles study on
--  electronic and elastic properties of BN, AlN, and GaN,"
--  Journal of Applied Physics 84, 4951-4958 (1998).
--  DOI: 10.1063/1.368739
	local x = state.x
	local GaN = -3.7
	local AlN = -17.1
	local enabled = true
	local value = (1.0-x)*GaN + x*AlN
	return value, enabled
end

function material.qw_D2(state)
-- Valence-band deformation potential D2. Units: eV. Linear.
--
-- GaN endpoint:
-- I. Vurgaftman and J. R. Meyer, "Band parameters for nitrogen-
--  containing semiconductors," Journal of Applied Physics 94,
--  3675-3696 (2003). DOI: 10.1063/1.1600519
--
-- AlN endpoint:
-- I. Vurgaftman and J. R. Meyer, "Band parameters for nitrogen-
--  containing semiconductors," Journal of Applied Physics 94,
--  3675-3696 (2003). DOI: 10.1063/1.1600519
--
-- Original parameter source cited therein:
-- K. Shimada, T. Sota, and K. Suzuki, "First-principles study on
--  electronic and elastic properties of BN, AlN, and GaN,"
--  Journal of Applied Physics 84, 4951-4958 (1998).
--  DOI: 10.1063/1.368739
	local x = state.x
	local GaN = 4.5
	local AlN = 7.9
	local enabled = true
	local value = (1.0-x)*GaN + x*AlN
	return value, enabled
end

function material.qw_D3(state)
-- Valence-band deformation potential D3. Units: eV. Linear.
--
-- GaN endpoint:
-- I. Vurgaftman and J. R. Meyer, "Band parameters for nitrogen-
--  containing semiconductors," Journal of Applied Physics 94,
--  3675-3696 (2003). DOI: 10.1063/1.1600519
--
-- AlN endpoint:
-- I. Vurgaftman and J. R. Meyer, "Band parameters for nitrogen-
--  containing semiconductors," Journal of Applied Physics 94,
--  3675-3696 (2003). DOI: 10.1063/1.1600519
--
-- Original parameter source cited therein:
-- K. Shimada, T. Sota, and K. Suzuki, "First-principles study on
--  electronic and elastic properties of BN, AlN, and GaN,"
--  Journal of Applied Physics 84, 4951-4958 (1998).
--  DOI: 10.1063/1.368739
	local x = state.x
	local GaN = 8.2
	local AlN = 8.8
	local enabled = true
	local value = (1.0-x)*GaN + x*AlN
	return value, enabled
end

function material.qw_D4(state)
-- Valence-band deformation potential D4. Units: eV. Linear.
--
-- GaN endpoint:
-- I. Vurgaftman and J. R. Meyer, "Band parameters for nitrogen-
--  containing semiconductors," Journal of Applied Physics 94,
--  3675-3696 (2003). DOI: 10.1063/1.1600519
--
-- AlN endpoint:
-- I. Vurgaftman and J. R. Meyer, "Band parameters for nitrogen-
--  containing semiconductors," Journal of Applied Physics 94,
--  3675-3696 (2003). DOI: 10.1063/1.1600519
--
-- Original parameter source cited therein:
-- K. Shimada, T. Sota, and K. Suzuki, "First-principles study on
--  electronic and elastic properties of BN, AlN, and GaN,"
--  Journal of Applied Physics 84, 4951-4958 (1998).
--  DOI: 10.1063/1.368739
	local x = state.x
	local GaN = -4.1
	local AlN = -3.9
	local enabled = true
	local value = (1.0-x)*GaN + x*AlN
	return value, enabled
end

function material.qw_D5(state)
-- Valence-band deformation potential D5. Units: eV. Linear.
--
-- GaN endpoint:
-- I. Vurgaftman and J. R. Meyer, "Band parameters for nitrogen-
--  containing semiconductors," Journal of Applied Physics 94,
--  3675-3696 (2003). DOI: 10.1063/1.1600519
--
-- AlN endpoint:
-- I. Vurgaftman and J. R. Meyer, "Band parameters for nitrogen-
--  containing semiconductors," Journal of Applied Physics 94,
--  3675-3696 (2003). DOI: 10.1063/1.1600519
--
-- Original parameter source cited therein:
-- K. Shimada, T. Sota, and K. Suzuki, "First-principles study on
--  electronic and elastic properties of BN, AlN, and GaN,"
--  Journal of Applied Physics 84, 4951-4958 (1998).
--  DOI: 10.1063/1.368739
	local x = state.x
	local GaN = -4.0
	local AlN = -3.4
	local enabled = true
	local value = (1.0-x)*GaN + x*AlN
	return value, enabled
end

function material.qw_D6(state)
-- Valence-band deformation potential D6. Units: eV. Linear.
--
-- AlN D6 from VM2003 quasi-cubic (D5=D6); GaN D6 tabulated.
--
-- GaN endpoint:
-- I. Vurgaftman and J. R. Meyer, "Band parameters for nitrogen-
--  containing semiconductors," Journal of Applied Physics 94,
--  3675-3696 (2003). DOI: 10.1063/1.1600519
--
-- AlN endpoint:
-- I. Vurgaftman and J. R. Meyer, "Band parameters for nitrogen-
--  containing semiconductors," Journal of Applied Physics 94,
--  3675-3696 (2003). DOI: 10.1063/1.1600519
--
-- Original parameter source cited therein:
-- K. Shimada, T. Sota, and K. Suzuki, "First-principles study on
--  electronic and elastic properties of BN, AlN, and GaN,"
--  Journal of Applied Physics 84, 4951-4958 (1998).
--  DOI: 10.1063/1.368739
	local x = state.x
	local GaN = -5.5
	local AlN = -3.4
	local enabled = true
	local value = (1.0-x)*GaN + x*AlN
	return value, enabled
end

function material.qw_C13(state)
-- Elastic stiffness constant C13. Units: Pa. Linear.
--
-- GaN endpoint:
-- I. Vurgaftman and J. R. Meyer, Journal of Applied Physics 94,
--  3675-3696 (2003). DOI: 10.1063/1.1600519
--  (original: A. Polian, M. Grimsditch, and I. Grzegory, Journal
--  of Applied Physics 79, 3343-3344 (1996)).
--
-- AlN endpoint:
-- A. F. Wright, Journal of Applied Physics 82, 2833-2839 (1997).
--  DOI: 10.1063/1.366114 (adopted in Vurgaftman and Meyer 2003,
--  DOI: 10.1063/1.1600519).
	local x = state.x
	local GaN = 106.0e9
	local AlN = 108.0e9
	local enabled = true
	local value = (1.0-x)*GaN + x*AlN
	return value, enabled
end

function material.qw_C33(state)
-- Elastic stiffness constant C33. Units: Pa. Linear.
--
-- GaN endpoint:
-- I. Vurgaftman and J. R. Meyer, Journal of Applied Physics 94,
--  3675-3696 (2003). DOI: 10.1063/1.1600519
--  (original: A. Polian, M. Grimsditch, and I. Grzegory, Journal
--  of Applied Physics 79, 3343-3344 (1996)).
--
-- AlN endpoint:
-- A. F. Wright, Journal of Applied Physics 82, 2833-2839 (1997).
--  DOI: 10.1063/1.366114 (adopted in Vurgaftman and Meyer 2003,
--  DOI: 10.1063/1.1600519).
	local x = state.x
	local GaN = 398.0e9
	local AlN = 373.0e9
	local enabled = true
	local value = (1.0-x)*GaN + x*AlN
	return value, enabled
end

function material.qw_Psp(state)
-- Spontaneous polarization Psp. Units: C/m^2. Linear.
--
-- NOTE: a well-established nonlinear (bowing) composition
--  dependence exists for this quantity but is NOT applied here;
--  see -- SKIPPED MATERIALS -- for the reasoning and reference.
--
-- GaN endpoint:
-- F. Bernardini, V. Fiorentini, and D. Vanderbilt, "Spontaneous
--  polarization and piezoelectric constants of III-V nitrides,"
--  Physical Review B 56, R10024-R10027 (1997).
--  DOI: 10.1103/PhysRevB.56.R10024
--
-- AlN endpoint:
-- F. Bernardini, V. Fiorentini, and D. Vanderbilt, "Spontaneous
--  polarization and piezoelectric constants of III-V nitrides,"
--  Physical Review B 56, R10024-R10027 (1997).
--  DOI: 10.1103/PhysRevB.56.R10024
	local x = state.x
	local GaN = -0.029
	local AlN = -0.081
	local enabled = true
	local value = (1.0-x)*GaN + x*AlN
	return value, enabled
end

function material.e31(state)
-- Piezoelectric coefficient e31. Units: C/m^2. Linear.
--
-- GaN endpoint:
-- F. Bernardini, V. Fiorentini, and D. Vanderbilt, "Spontaneous
--  polarization and piezoelectric constants of III-V nitrides,"
--  Physical Review B 56, R10024-R10027 (1997).
--  DOI: 10.1103/PhysRevB.56.R10024
--
-- AlN endpoint:
-- F. Bernardini, V. Fiorentini, and D. Vanderbilt, "Spontaneous
--  polarization and piezoelectric constants of III-V nitrides,"
--  Physical Review B 56, R10024-R10027 (1997).
--  DOI: 10.1103/PhysRevB.56.R10024
	local x = state.x
	local GaN = -0.49
	local AlN = -0.60
	local enabled = true
	local value = (1.0-x)*GaN + x*AlN
	return value, enabled
end

function material.e33(state)
-- Piezoelectric coefficient e33. Units: C/m^2. Linear.
--
-- GaN endpoint:
-- F. Bernardini, V. Fiorentini, and D. Vanderbilt, "Spontaneous
--  polarization and piezoelectric constants of III-V nitrides,"
--  Physical Review B 56, R10024-R10027 (1997).
--  DOI: 10.1103/PhysRevB.56.R10024
--
-- AlN endpoint:
-- F. Bernardini, V. Fiorentini, and D. Vanderbilt, "Spontaneous
--  polarization and piezoelectric constants of III-V nitrides,"
--  Physical Review B 56, R10024-R10027 (1997).
--  DOI: 10.1103/PhysRevB.56.R10024
	local x = state.x
	local GaN = 0.73
	local AlN = 1.46
	local enabled = true
	local value = (1.0-x)*GaN + x*AlN
	return value, enabled
end

function material.lattice_a(state)
	-- Lattice constant, a-axis
	-- Units: m
	--
	-- Reference:
	-- I. Vurgaftman and J. R. Meyer,
	-- "Band parameters for nitrogen-containing semiconductors,"
	-- J. Appl. Phys. 94, 3675 (2003). DOI: 10.1063/1.1600519
	-- (binary end-point a-axis constants; wurtzite).
	--
	-- Notes:
	-- Wurtzite Al(x)Ga(1-x)N a-axis by Vegard's law between the binary
	-- end points:
	--   a_GaN = 3.189 Angstrom, a_AlN = 3.112 Angstrom (300 K).
	--   a(x) = (1-x)*a_GaN + x*a_AlN
	-- Bowing of the a lattice parameter is negligible and is neglected
	-- (linear Vegard). The c-axis is NOT returned here.
	--
	-- Composition variable:
	-- x is the Al mole fraction in Al(x)Ga(1-x)N.
	-- IMPORTANT: this assumes the model carries composition in state.x.
	-- Confirm this matches the composition field your existing AlGaN
	-- model already uses. If state.x holds spatial position in your
	-- build, change the marked line to the correct composition field.

	local enabled = true
	local x = state.x            -- <-- Al mole fraction; confirm/rename if needed

	local a_GaN = 3.189e-10      -- m, wurtzite a-axis
	local a_AlN = 3.112e-10      -- m, wurtzite a-axis
	local value = (1.0 - x)*a_GaN + x*a_AlN

	return value, enabled
end

function material.qw_C11(state)
    -- Elastic stiffness constant C11
    -- Units: Pa
    --
    -- Composition: Al_x Ga_(1-x) N, x = state.x
    --
    -- Endpoint GaN:
    -- A. Polian, M. Grimsditch, I. Grzegory,
    -- "Elastic constants of gallium nitride,"
    -- J. Appl. Phys. 79, 3343 (1996). (Brillouin scattering, 300 K)
    -- C11 = 390 GPa.
    --
    -- Endpoint AlN:
    -- A. F. Wright,
    -- "Elastic properties of zinc-blende and wurtzite AlN, GaN, and InN,"
    -- J. Appl. Phys. 82, 2833 (1997). (first-principles)
    -- C11 = 396 GPa (AlN set adopted by I. Vurgaftman and J. R. Meyer,
    -- J. Appl. Phys. 94, 3675 (2003), DOI: 10.1063/1.1600519).
    -- Experimental alternative: McNeil et al. (1993) 410 GPa.
    --
    -- Interpolation:
    -- Linear in state.x: C11(x) = (1-x)*C11_GaN + x*C11_AlN. No bowing applied.

    local enabled = true
    local x = state.x

    local GaN = 390e9
    local AlN = 396e9

    local value = (1.0-x)*GaN + x*AlN

    return value, enabled
end

function material.qw_C12(state)
    -- Elastic stiffness constant C12
    -- Units: Pa
    --
    -- Composition: Al_x Ga_(1-x) N, x = state.x
    --
    -- Endpoint GaN: Polian et al., J. Appl. Phys. 79, 3343 (1996): 145 GPa.
    -- Endpoint AlN: Wright, J. Appl. Phys. 82, 2833 (1997): 137 GPa
    -- (adopted by Vurgaftman & Meyer 2003, DOI: 10.1063/1.1600519).
    --
    -- Interpolation:
    -- Linear in state.x: C12(x) = (1-x)*C12_GaN + x*C12_AlN. No bowing applied.

    local enabled = true
    local x = state.x

    local GaN = 145e9
    local AlN = 137e9

    local value = (1.0-x)*GaN + x*AlN

    return value, enabled
end

function material.qw_C44(state)
    -- Elastic stiffness constant C44
    -- Units: Pa
    --
    -- Composition: Al_x Ga_(1-x) N, x = state.x
    --
    -- Endpoint GaN: Polian et al., J. Appl. Phys. 79, 3343 (1996): 105 GPa.
    -- Endpoint AlN: Wright, J. Appl. Phys. 82, 2833 (1997): 116 GPa
    -- (adopted by Vurgaftman & Meyer 2003, DOI: 10.1063/1.1600519).
    --
    -- Interpolation:
    -- Linear in state.x: C44(x) = (1-x)*C44_GaN + x*C44_AlN. No bowing applied.

    local enabled = true
    local x = state.x

    local GaN = 105e9
    local AlN = 116e9

    local value = (1.0-x)*GaN + x*AlN

    return value, enabled
end

function material.phonon_lo_energy(state)
    -- Representative LO phonon energy for Al(x)Ga(1-x)N.
    -- Units: eV
    --
    -- Model:
    -- Uses the A1(LO) phonon branch as a single effective polar
    -- optical phonon mode for Frohlich scattering.
    --
    -- Validity:
    -- The A1(LO) branch exhibits approximately one-mode behaviour
    -- across the AlGaN composition range. This makes it a reasonable
    -- choice for a simplified, single-mode LO scattering model.
    --
    -- However, wurtzite AlGaN has anisotropic optical phonons.
    -- The E1 modes are not explicitly represented, and E1(TO)
    -- exhibits two-mode behaviour at higher Al concentrations.
    --
    -- This approximation does not include:
    --   - Separate A1 and E1 polar optical scattering channels.
    --   - Phonon anisotropy or direction-dependent coupling.
    --   - Confined or interface optical phonons in quantum wells.
    --   - Strain-induced shifts in phonon frequencies.
    --   - Plasmon-phonon coupling.
    --
    -- The resulting energy is suitable as a representative input
    -- for an effective bulk-LO Frohlich model, but should not be
    -- interpreted as a complete optical phonon description.
    --
    -- References:
    -- Davydov et al., Physical Review B 65, 125203 (2002).
    -- Schoche et al., Journal of Applied Physics 121, 205701 (2017).
    --
    -- Note: The numerical coefficients below are provisional.
    -- The exact polynomial has not yet been independently verified
    -- against the original Davydov publication.

    local x = state.x

    if x == nil or x < 0.0 or x > 1.0 then
        return 0.0, false
    end

    local omega_cm1 = 734.0 + 153.0*x + 75.0*x*(1.0-x)

    local value = omega_cm1 * 1.239841984e-4
    local enabled = true

    return value, enabled
end

function material.epsilon_static(state)
    -- Static relative dielectric constant (lattice + electronic)
    -- Dimensionless
    --
    -- Composition: Al_x Ga_(1-x) N, x = state.x
    --
    -- Endpoint GaN:
    -- A. S. Barker, Jr. and M. Ilegems,
    -- "Infrared lattice vibrations and free-electron dispersion in GaN,"
    -- Phys. Rev. B 7, 743 (1973).
    -- DOI: 10.1103/PhysRevB.7.743
    -- eps_s(E perp c) = 9.5.
    --
    -- Endpoint AlN:
    -- A. T. Collins, E. C. Lightowlers, P. J. Dean,
    -- "Lattice vibration spectra of aluminum nitride,"
    -- Phys. Rev. 158, 833 (1967).
    -- eps_s = 9.14 (IR reflectivity).
    --
    -- Interpolation:
    -- Linear in state.x. Endpoints differ by ~4%; no bowing applied.

    local enabled = true
    local x = state.x

    local GaN = 9.5
    local AlN = 9.14

    local value = (1.0-x)*GaN + x*AlN

    return value, enabled
end

function material.epsilon_inf(state)
    -- High-frequency (electronic) relative dielectric constant
    -- Dimensionless
    --
    -- Composition: Al_x Ga_(1-x) N, x = state.x
    --
    -- Endpoint GaN: Barker & Ilegems, Phys. Rev. B 7, 743 (1973),
    -- DOI: 10.1103/PhysRevB.7.743: eps_inf(E perp c) = 5.35.
    -- Endpoint AlN: Collins et al., Phys. Rev. 158, 833 (1967): eps_inf = 4.84.
    --
    -- Interpolation:
    -- Linear in state.x; no bowing applied.

    local enabled = true
    local x = state.x

    local GaN = 5.35
    local AlN = 4.84

    local value = (1.0-x)*GaN + x*AlN

    return value, enabled
end

function material.qw_e15(state)
    -- Wurtzite piezoelectric stress coefficient e15
    -- Units: C m^-2
    --
    -- Composition: Al_x Ga_(1-x) N, x = state.x
    --
    -- Endpoint GaN:
    -- K. Adachi, H. Ogi, A. Nagakubo, N. Nakamura, M. Hirao, M. Imade,
    -- M. Yoshimura, Y. Mori,
    -- "Piezoelectric coefficients of GaN determined by hopping conduction
    -- of carriers,"
    -- Appl. Phys. Lett. 109, 182108 (2016).
    -- DOI: 10.1063/1.4966995
    -- e15 = -0.22 +/- 0.02 C/m^2 (resonant ultrasound; e33 = +1.15).
    --
    -- Endpoint AlN:
    -- K. Tsubouchi and N. Mikoshiba,
    -- IEEE Trans. Sonics Ultrason. SU-32, 634 (1985) (AlN acoustic-wave
    -- measurements), as quoted in the nextnano material database; the
    -- original paper was not independently accessed.
    -- e15 = -0.48 C/m^2.
    --
    -- Interpolation:
    -- Linear in state.x.
    --
    -- Notes:
    -- Convention: e33 > 0 (metal-polar +c). Both endpoints negative in this
    -- convention. GaN e15 literature is scattered (about -0.40 to +0.33,
    -- partly from sign conventions); the 2016 RUS value is the most direct.

    local enabled = true
    local x = state.x

    local GaN = -0.22
    local AlN = -0.48

    local value = (1.0-x)*GaN + x*AlN

    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

-- ============================================================================
-- Copyright (C) 2026 The OghmaNano Project
-- All rights reserved.
--
-- This file is part of the OghmaNano Materials Model Library.
--
-- Website:
-- https://www.oghma-nano.com
--
-- Documentation and accuracy statement:
-- https://www.oghma-nano.com/manual/material-scripts.html
--
-- These material models are provided to support scientific research and
-- semiconductor device simulation. If you find them useful, please cite
-- OghmaNano where appropriate. Please do not redistribute these files or
-- incorporate them into other software or databases without permission.
-- ============================================================================