from math import pi
import numpy as np
from simudo.physics import Material
from .helpers import thermal_velocity
[docs]
class AluminumArsenideMaterial(Material):
""" AlAs material data based on Palankovski
V. Palankovski and R. Quay, "Analysis and Simulation of Heterostructure Devices",
Springer-Verlag (2004).
Both gamma and X valley properties are included here.
The CB energy level is still taken to be that of lowest valley
"""
name = "AluminumArsenide"
[docs]
def get_dict(self):
d = super().get_dict()
U = self.unit_registry
d.update(
{
# Static dielectric constant, Palankovski Table 3.3
"poisson/permittivity": U("10.1 vacuum_permittivity"),
# Springer's handbook (2017) , Table 30.12
"CB/Eg_300K": U("3.01 eV"),
"CBX/Eg_300K": U("2.15 eV"),
"CBL/Eg_300K": U("2.37 eV"),
# Table 3.15 band edge alignment
"VB/E_off": U("-1.008 eV"),
# Table 3.21 Parameter for energy minima in the DOS model
"CB/MC": U("1"),
"CBX/MC": U("3"),
"CBL/MC": U("4"),
# Springer handbook Table 30.17 Effective electron masses
"CB/mn": U("0.124"),
"CBX/mn": U("0.71"),
"CBL/mn": U("0.78"),
# Springer Table 3.20 Effective hole masses
"VB/mp": U("0.81"),
# conductivity mass, Springer Table 30.17
"CBX/mc": U("0.26"),
"CBL/mc": U("0.21"),
# Include other parameters for SRH etc. here.
# SRH recombination lifetimes, Table 3.38
"SRH/CB/tau": U("0.3e-9 s"),
"SRH/CBX/tau": U("0.3e-9 s"),
"SRH/CBL/tau": U("0.3e-9 s"), #TODO - find appropriate lifetimes for these valleys
"SRH/VB/tau": U("10e-9 s"),
# Springer 2017, Table 30.1
"a": U("5.66139e-10 m"), # Lattice parameter
# Palankovski Table 3.22, lattice scattering mobility, not L-valley
"CB/mu_L": U("250 cm^2/V/s"),
"CBX/mu_L": U("410 cm^2/V/s"),
"CBL/mu_L": U("410 cm^2/V/s"),
"VB/mu_L": U("130 cm^2/V/s"),
# Palankovski Table 3.22, mobility temperature dependence power term
"CB/y_0": U("-2.1"),
"CBX/y_0": U("-2.1"),
"CBL/y_0": U("-2.1"),
"VB/y_0": U("-2.2"),
# Palankvoski Table 3.23, mobility parameters to factor in ionized impurities
"CB/mu_min" : U("10 cm^2/V/s"),
"CBX/mu_min" : U("10 cm^2/V/s"),
"VB/mu_min" : U("5 cm^2/V/s"),
"CB/C_ref" : U("5e17 elementary_charge/cm^3"),
"CBX/C_ref" : U("1e17 elementary_charge/cm^3"),
"VB/C_ref" : U("2.9e17 elementary_charge/cm^3"),
"CB/alpha_mu" : U("1.0 dimensionless"),
"CBX/alpha_mu" : U("0.5 dimensionless"),
"VB/alpha_mu" : U("1.0 dimensionless"),
}
)
T = self.temperature
E_off = d["VB/E_off"]
# Effective electron and hole mass
mn = d["CB/mn"]
mnX = d["CBX/mn"]
mnL = d["CBL/mn"]
mp = d["VB/mp"]
# Effective DOS. Palankovski eq. 3.111 & 3.112
m_e = U.electron_mass
k_B = U.boltzmann_constant
h = U.planck_constant
DOS_term = lambda m : (2 * pi * m * m_e * k_B * T / h ** 2) ** (3 / 2)
NC = 2 * d["CB/MC"] * DOS_term(mn)
NCX = 2 * d["CBX/MC"] * DOS_term(mnX)
NCL = 2 * d["CBL/MC"] * DOS_term(mnL)
NV = 2 * DOS_term(mp)
# Thermal velocity
vth = lambda m : thermal_velocity(U, T, m)
#Gamma valley
mtc = d["CB/mn"]
vth_c = vth(mtc)
#X valley
mtcX = d["CBX/mc"]
vth_cX = vth(mtcX)
#L valley
mtcL = d["CBL/mc"]
vth_cL = vth(mtcL)
# For valence band, just use the DOS effective mass
mtv = d["VB/mp"]
vth_v = vth(mtv)
Eg = d["CB/Eg_300K"]
EgX = d["CBX/Eg_300K"]
EgL = d["CBL/Eg_300K"]
# Caughey-Thomas impurity scattering model, Palankovski eq. 3.116
def mu_LI(band: str):
'''Band is either `CB`, `CBX`, or `VB`'''
doping = abs(self.pdd.spatial.get('poisson/static_rho'))
mu_min = d[band+'/mu_min']; mu_L = d[band+'/mu_L']; C_ref = d[band+'/C_ref']; alpha = d[band+'/alpha_mu']
return mu_min + (mu_L - mu_min) / (1 + (doping/C_ref)**alpha)
# Palankovski eq. 3.115 incorporates temperature dependence with mobility
mu_T = lambda mu, y_0 : mu * (T/U("300 K"))**y_0
d.update(
{
# Palankovski Table 3.22 Mobility
"CB/mobility": mu_T(d["CB/mu_L"], d["CB/y_0"]), #Gamma valley
"CBX/mobility": mu_T(d["CBX/mu_L"], d["CBX/y_0"]), #X valley
"CBL/mobility": mu_T(d["CBL/mu_L"], d["CBL/y_0"]),
"VB/mobility": mu_T(d["VB/mu_L"], d["VB/y_0"]),
# "CB/mobility" : mu_LI('CB'),
# "CBX/mobility" : mu_LI('CBX'),
# "VB/mobility" : mu_LI('VB'),
"CB/mDOS": mn,
"CBX/mDOS": mnX,
"CBL/mDOS": mnL,
"VB/mDOS": mp,
"CB/energy_level": E_off + Eg,
"CBX/energy_level": E_off + EgX,
"CBL/energy_level": E_off + EgL,
"VB/energy_level": E_off,
# Midgap trap level for SRH. The material sets the SRH
# lifetimes above, so it should set the level they go with;
# without it a project using SRHRecombination has to supply
# <proc>/energy_level by hand. Midgap is the usual default.
# TODO it should be source- and destination-band dependent.
"SRH/energy_level": E_off + Eg / 2,
"VB/effective_density_of_states": NV,
"CB/effective_density_of_states": NC,
"CBX/effective_density_of_states": NCX,
"CBL/effective_density_of_states": NCL,
"CB/vth": vth_c,
"CBX/vth": vth_cX,
"CBL/vth": vth_cL,
"VB/vth": vth_v,
}
)
# alpha(E) for a BeerLambert process named 'opt_cv', from the measured
# table below. A top-hat set on a layer or overlay overrides this:
# material rules have the lowest priority.
# Keep this a literal dict -- the GUI detects it by AST scan.
d["opt_cv/alpha_function"] = {
"type": "wavelength_table",
"data": self.optical_properties_table()[:, 0:2],
"arg_unit": "nm",
"value_unit": "1/cm",
}
return d
# Optical data sourced from:
# A. D. Rakić and M. L. Majewski. Modeling the optical dielectric function of GaAs and AlAs: Extension of Adachi’s model. J. Appl. Phys. 80, 5909-5914 (1996)
# Data was collated into CSV files from:
# https://refractiveindex.info/?shelf=main&book=AlAs&page=Rakic
[docs]
def optical_properties_table(self):
'''Array of arrays, where each element is of the form [wl, alpha, n, k].'''
return np.array([
[223.45, 1.84252406e+06, 1.47110, 3.27630e+00],
[224.48, 1.85679770e+06, 1.49780, 3.31690e+00],
[225.53, 1.87149938e+06, 1.52670, 3.35880e+00],
[226.59, 1.88670254e+06, 1.55820, 3.40200e+00],
[227.65, 1.90242652e+06, 1.59250, 3.44640e+00],
[228.73, 1.91849631e+06, 1.62970, 3.49200e+00],
[229.81, 1.93507125e+06, 1.67040, 3.53880e+00],
[230.91, 1.95181188e+06, 1.71470, 3.58650e+00],
[232.02, 1.96874223e+06, 1.76310, 3.63500e+00],
[233.14, 1.98580349e+06, 1.81580, 3.68420e+00],
[234.27, 2.00277705e+06, 1.87350, 3.73370e+00],
[235.41, 2.01950186e+06, 1.93630, 3.78320e+00],
[236.56, 2.03571379e+06, 2.00490, 3.83220e+00],
[237.72, 2.05115393e+06, 2.07960, 3.88020e+00],
[238.89, 2.06541076e+06, 2.16070, 3.92640e+00],
[240.08, 2.07804682e+06, 2.24870, 3.97010e+00],
[241.28, 2.08859663e+06, 2.34370, 4.01020e+00],
[242.49, 2.09657164e+06, 2.44580, 4.04570e+00],
[243.71, 2.10139046e+06, 2.55490, 4.07540e+00],
[244.94, 2.10238141e+06, 2.67050, 4.09790e+00],
[246.19, 2.09880185e+06, 2.79200, 4.11180e+00],
[247.45, 2.09024779e+06, 2.91840, 4.11600e+00],
[248.72, 2.07624009e+06, 3.04820, 4.10940e+00],
[250.00, 2.05641115e+06, 3.17970, 4.09110e+00],
[251.30, 2.03072149e+06, 3.31100, 4.06100e+00],
[252.61, 1.99939657e+06, 3.44020, 4.01920e+00],
[253.94, 1.96284591e+06, 3.56550, 3.96650e+00],
[255.28, 1.92172729e+06, 3.68540, 3.90390e+00],
[256.63, 1.87694951e+06, 3.79920, 3.83310e+00],
[258.00, 1.82904011e+06, 3.90650, 3.75520e+00],
[259.39, 1.77864116e+06, 4.00770, 3.67140e+00],
[260.78, 1.72603278e+06, 4.10350, 3.58190e+00],
[262.20, 1.67086720e+06, 4.19420, 3.48630e+00],
[263.63, 1.61261193e+06, 4.27980, 3.38310e+00],
[265.07, 1.55061248e+06, 4.35830, 3.27080e+00],
[266.53, 1.48450360e+06, 4.42660, 3.14860e+00],
[268.01, 1.41492991e+06, 4.48020, 3.01770e+00],
[269.50, 1.34378566e+06, 4.51550, 2.88190e+00],
[271.01, 1.27356162e+06, 4.53070, 2.74660e+00],
[272.54, 1.20684004e+06, 4.52670, 2.61740e+00],
[274.08, 1.14568180e+06, 4.50650, 2.49880e+00],
[275.64, 1.09110052e+06, 4.47380, 2.39330e+00],
[277.22, 1.04358626e+06, 4.43240, 2.30220e+00],
[278.82, 1.00302912e+06, 4.38540, 2.22550e+00],
[280.44, 9.69184246e+05, 4.33540, 2.16290e+00],
[282.07, 9.41709299e+05, 4.28410, 2.11380e+00],
[283.73, 9.20211054e+05, 4.23300, 2.07770e+00],
[285.40, 9.04699429e+05, 4.18340, 2.05470e+00],
[287.09, 8.95171707e+05, 4.13640, 2.04510e+00],
[288.81, 8.91929058e+05, 4.09370, 2.04990e+00],
[290.54, 8.95527995e+05, 4.05780, 2.07050e+00],
[292.30, 9.06601517e+05, 4.03270, 2.10880e+00],
[294.07, 9.25502278e+05, 4.02550, 2.16580e+00],
[295.87, 9.51471371e+05, 4.04700, 2.24020e+00],
[297.69, 9.80564423e+05, 4.11110, 2.32290e+00],
[299.54, 1.00341341e+06, 4.22930, 2.39180e+00],
[301.40, 1.00580997e+06, 4.39530, 2.41240e+00],
[303.29, 9.77043706e+05, 4.57300, 2.35810e+00],
[305.21, 9.22107520e+05, 4.71290, 2.23960e+00],
[307.14, 8.59115225e+05, 4.79260, 2.09980e+00],
[309.11, 8.03798906e+05, 4.82630, 1.97720e+00],
[311.09, 7.62045008e+05, 4.84360, 1.88650e+00],
[313.11, 7.31121885e+05, 4.87140, 1.82170e+00],
[315.15, 7.02624200e+05, 4.92700, 1.76210e+00],
[317.21, 6.64942879e+05, 5.00940, 1.67850e+00],
[319.31, 6.09487275e+05, 5.09370, 1.54870e+00],
[321.43, 5.38770972e+05, 5.14450, 1.37810e+00],
[323.58, 4.64355317e+05, 5.14420, 1.19570e+00],
[325.76, 3.96865241e+05, 5.10010, 1.02880e+00],
[327.97, 3.40767541e+05, 5.02980, 8.89370e-01],
[330.20, 2.95933842e+05, 4.94820, 7.77610e-01],
[332.47, 2.60436491e+05, 4.86440, 6.89040e-01],
[334.77, 2.32175869e+05, 4.78300, 6.18520e-01],
[337.11, 2.09376140e+05, 4.70630, 5.61680e-01],
[339.47, 1.90703660e+05, 4.63470, 5.15170e-01],
[341.87, 1.75143308e+05, 4.56850, 4.76480e-01],
[344.30, 1.61975882e+05, 4.50730, 4.43790e-01],
[346.77, 1.50657292e+05, 4.45080, 4.15740e-01],
[349.28, 1.40788827e+05, 4.39850, 3.91320e-01],
[351.82, 1.32067976e+05, 4.35000, 3.69750e-01],
[354.39, 1.24259524e+05, 4.30510, 3.50430e-01],
[357.01, 1.17177244e+05, 4.26330, 3.32900e-01],
[359.66, 1.10681613e+05, 4.22430, 3.16780e-01],
[362.36, 1.04644617e+05, 4.18800, 3.01750e-01],
[365.09, 9.89710539e+04, 4.15410, 2.87540e-01],
[367.87, 9.35671453e+04, 4.12250, 2.73910e-01],
[370.69, 8.83534267e+04, 4.09280, 2.60630e-01],
[373.55, 8.32465285e+04, 4.06510, 2.47460e-01],
[376.45, 7.81621910e+04, 4.03910, 2.34150e-01],
[379.41, 7.29751044e+04, 4.01480, 2.20330e-01],
[382.41, 6.75063287e+04, 3.99190, 2.05430e-01],
[385.45, 6.12425145e+04, 3.97040, 1.87850e-01],
[388.55, 5.33961598e+04, 3.93900, 1.65100e-01],
[391.69, 4.99491496e+04, 3.91020, 1.55690e-01],
[394.89, 4.63748687e+04, 3.88580, 1.45730e-01],
[398.14, 4.26285732e+04, 3.86360, 1.35060e-01],
[401.45, 3.86147087e+04, 3.84300, 1.23360e-01],
[404.80, 3.40887638e+04, 3.82250, 1.09810e-01],
[408.22, 2.87026839e+04, 3.79680, 9.32410e-02],
[411.69, 2.38995770e+04, 3.76150, 7.82980e-02],
[415.23, 1.98765525e+04, 3.73130, 6.56780e-02],
[418.82, 1.41196895e+04, 3.70680, 4.70590e-02],
[422.48, 9.88432735e+03, 3.67820, 3.32310e-02],
[426.20, 8.28077238e+03, 3.65050, 2.80850e-02],
[429.99, 7.49382254e+03, 3.62500, 2.56420e-02],
[433.85, 6.85943964e+03, 3.60110, 2.36820e-02],
[437.77, 6.28045829e+03, 3.57860, 2.18790e-02],
[441.77, 5.74656086e+03, 3.55710, 2.02020e-02],
[445.84, 5.25468436e+03, 3.53660, 1.86430e-02],
[449.99, 4.80157729e+03, 3.51690, 1.71940e-02],
[454.21, 4.38485299e+03, 3.49800, 1.58490e-02],
[458.52, 4.00160467e+03, 3.47980, 1.46010e-02],
[462.91, 3.64902797e+03, 3.46220, 1.34420e-02],
[467.38, 3.32536420e+03, 3.44520, 1.23680e-02],
[471.94, 3.02802828e+03, 3.42870, 1.13720e-02],
[476.59, 2.75537827e+03, 3.41280, 1.04500e-02],
[481.33, 2.50533741e+03, 3.39730, 9.59620e-03],
[486.16, 2.27622010e+03, 3.38230, 8.80610e-03],
[491.10, 2.06637601e+03, 3.36770, 8.07550e-03],
[496.14, 1.87436837e+03, 3.35350, 7.40030e-03],
[501.28, 1.69879642e+03, 3.33980, 6.77660e-03],
[506.53, 1.53836510e+03, 3.32640, 6.20090e-03],
[511.88, 1.39195361e+03, 3.31330, 5.67000e-03],
[517.36, 1.25831303e+03, 3.30060, 5.18050e-03],
[522.95, 1.13651222e+03, 3.28830, 4.72960e-03],
[528.67, 1.02557101e+03, 3.27620, 4.31460e-03],
[534.51, 9.24627771e+02, 3.26450, 3.93290e-03],
[540.48, 8.32828958e+02, 3.25300, 3.58200e-03],
[546.59, 7.49443915e+02, 3.24180, 3.25980e-03],
[552.83, 6.73769136e+02, 3.23090, 2.96410e-03],
[559.22, 6.05150675e+02, 3.22030, 2.69300e-03],
[565.76, 5.42982000e+02, 3.20990, 2.44460e-03],
[572.46, 4.86709236e+02, 3.19970, 2.21720e-03],
[579.31, 4.35834904e+02, 3.18980, 2.00920e-03],
[586.33, 3.89874043e+02, 3.18020, 1.81910e-03],
[593.53, 3.48389514e+02, 3.17070, 1.64550e-03],
[600.90, 3.11011922e+02, 3.16150, 1.48720e-03],
[608.46, 2.77345743e+02, 3.15250, 1.34290e-03],
[616.21, 2.47040804e+02, 3.14370, 1.21140e-03],
[624.16, 2.19814846e+02, 3.13500, 1.09180e-03],
[632.31, 1.95388769e+02, 3.12660, 9.83150e-04],
[640.69, 1.73472363e+02, 3.11840, 8.84440e-04],
[649.29, 1.53841222e+02, 3.11030, 7.94880e-04],
[658.12, 1.36278253e+02, 3.10250, 7.13710e-04],
[667.19, 1.20580202e+02, 3.09480, 6.40200e-04],
[676.52, 1.06564873e+02, 3.08720, 5.73700e-04],
[686.12, 9.40682914e+01, 3.07990, 5.13610e-04],
[695.99, 8.29374322e+01, 3.07270, 4.59350e-04],
[706.15, 7.30367461e+01, 3.06560, 4.10420e-04],
[716.61, 6.42373520e+01, 3.05870, 3.66320e-04],
[727.38, 5.64292891e+01, 3.05200, 3.26630e-04],
[738.49, 4.95072359e+01, 3.04540, 2.90940e-04],
[749.93, 4.33814848e+01, 3.03900, 2.58890e-04],
[761.74, 3.79627328e+01, 3.03260, 2.30120e-04],
[773.93, 3.31788685e+01, 3.02650, 2.04340e-04],
[786.51, 2.89590046e+01, 3.02040, 1.81250e-04],
[799.51, 2.52424500e+01, 3.01460, 1.60600e-04],
[812.94, 2.19734492e+01, 3.00880, 1.42150e-04],
[826.84, 1.91009320e+01, 3.00310, 1.25680e-04],
[841.22, 1.65814785e+01, 2.99760, 1.11000e-04],
[856.10, 1.43743368e+01, 2.99220, 9.79270e-05],
[871.52, 1.24430889e+01, 2.98700, 8.62970e-05],
[887.51, 1.07555593e+01, 2.98180, 7.59620e-05],
[904.10, 9.28335243e+00, 2.97680, 6.67900e-05],
[921.32, 8.00067477e+00, 2.97190, 5.86580e-05],
[939.20, 6.88487790e+00, 2.96700, 5.14570e-05],
[957.80, 5.91556189e+00, 2.96230, 4.50880e-05],
[977.14, 5.07482603e+00, 2.95780, 3.94610e-05],
[997.29, 4.34667470e+00, 2.95330, 3.44960e-05],
[1018.30, 3.71684687e+00, 2.94890, 3.01190e-05],
[1040.20, 3.17312335e+00, 2.94460, 2.62660e-05],
[1063.00, 2.70454776e+00, 2.94050, 2.28780e-05],
[1086.90, 2.30111762e+00, 2.93640, 1.99030e-05],
[1111.90, 1.95440460e+00, 2.93240, 1.72930e-05],
[1138.00, 1.65703829e+00, 2.92850, 1.50060e-05],
[1165.50, 1.40230130e+00, 2.92480, 1.30060e-05],
[1194.20, 1.18455564e+00, 2.92110, 1.12570e-05],
[1224.50, 9.98700631e-01, 2.91750, 9.73160e-06],
[1256.30, 8.40365407e-01, 2.91400, 8.40140e-06],
[1289.70, 7.05780457e-01, 2.91060, 7.24350e-06],
[1325.10, 5.91456798e-01, 2.90730, 6.23680e-06],
[1362.40, 4.94630204e-01, 2.90410, 5.36260e-06],
[1401.90, 4.12747772e-01, 2.90090, 4.60460e-06],
[1443.70, 3.43671130e-01, 2.89790, 3.94830e-06],
[1488.10, 2.85477270e-01, 2.89490, 3.38060e-06],
[1535.30, 2.36578113e-01, 2.89200, 2.89040e-06],
[1585.70, 1.95552602e-01, 2.88930, 2.46760e-06],
[1639.40, 1.61238017e-01, 2.88650, 2.10350e-06],
[1696.90, 1.32587836e-01, 2.88390, 1.79040e-06],
[1758.60, 1.08728475e-01, 2.88140, 1.52160e-06],
[1824.90, 8.88990326e-02, 2.87890, 1.29100e-06],
[1896.50, 7.24628679e-02, 2.87650, 1.09360e-06],
[1973.80, 5.88852054e-02, 2.87420, 9.24910e-07],
[2057.80, 4.76860117e-02, 2.87200, 7.80880e-07],
[2149.20, 3.84808556e-02, 2.86990, 6.58130e-07],
[2249.10, 3.09356991e-02, 2.86780, 5.53680e-07],
[2358.80, 2.47704752e-02, 2.86580, 4.64960e-07],
[2479.70, 1.97493257e-02, 2.86390, 3.89710e-07]]
)