/
strain_balancing.py
executable file
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/
strain_balancing.py
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from scipy.optimize import bisect
from numpy import sqrt, log, pi, linspace, array, ones
from solcore import convert, asUnit
from solcore import ParameterSystem
from solcore.science_tracker import science_reference
# Zero stress criterion from:
# Ekins-Daukes et al. Strain-balanced Criteria for Multiple Quantum Well Structures and Its
# Signature in X-ray Rocking Curves. Crystal growth & design (2002) vol. 2 (4) pp. 287-292
# Critical Thickness from:
# Matthews, J., & Blakeslee, A. (1974). Defects in epitaxial multilayers: I. Misfit dislocations.
# Journal of Crystal Growth, 27, 118-125.
def optimise(well_material, well_thickness, well_fraction, barrier_material, barrier_thickness, barrier_fraction,
lattice_material, lattice_material_fraction=0, parameter_to_optimise="well_thickness", T=300):
""" This can't handle sol-core materials yet. Needs layers."""
options = ["barrier_thickness", "well_thickness", "barrier_fraction", "well_fraction"]
if parameter_to_optimise not in options:
print('ERROR: Wring option in strain_balancing.optimise. '
'The only valid options are: {}, {}, {}, {}'.format(*options))
science_reference("strain-balancing",
"Ekins-Daukes et al. Strain-balanced Criteria for Multiple Quantum Well Structures and Its "
"Signature in X-ray Rocking Curves. Crystal growth & design (2002) vol. 2 (4) pp. 287-292")
get_vurgaftman_parameter = ParameterSystem().get_parameter
t1 = well_thickness
t2 = barrier_thickness
x1 = well_fraction
x2 = barrier_fraction
C11_well = get_vurgaftman_parameter(well_material, "c11", x=x1, T=T)
C12_well = get_vurgaftman_parameter(well_material, "c12", x=x1, T=T)
A1 = C11_well + C12_well - 2 * C12_well ** 2 / C11_well
a1 = get_vurgaftman_parameter(well_material, "lattice_constant", x=x1, T=T)
C11_barrier = get_vurgaftman_parameter(barrier_material, "c11", x=x2, T=T)
C12_barrier = get_vurgaftman_parameter(barrier_material, "c12", x=x2, T=T)
A2 = C11_barrier + C12_barrier - 2 * C12_barrier ** 2 / C11_barrier
a2 = get_vurgaftman_parameter(barrier_material, "lattice_constant", x=x2, T=T)
a0 = get_vurgaftman_parameter(lattice_material, "lattice_constant", x=lattice_material_fraction, T=T)
a0 = convert(a0, "angstrom", "m")
a1 = convert(a1, "angstrom", "m")
a2 = convert(a2, "angstrom", "m")
if parameter_to_optimise == "barrier_thickness":
t2 = - (A1 * t1 * a2 ** 2 * (a0 - a1)) / (A2 * a1 ** 2 * (a0 - a2))
if parameter_to_optimise == "well_thickness":
t1 = - (A2 * t2 * a1 ** 2 * (a0 - a2)) / (A1 * a2 ** 2 * (a0 - a1))
if parameter_to_optimise == "well_fraction":
def func(x):
C11_well = get_vurgaftman_parameter(well_material, "c11", x=x, T=T)
C12_well = get_vurgaftman_parameter(well_material, "c12", x=x, T=T)
A1 = C11_well + C12_well - 2 * C12_well ** 2 / C11_well
a1 = get_vurgaftman_parameter(well_material, "lattice_constant", x=x, T=T)
a1 = convert(a1, "angstrom", "m")
misfit = - (A1 * t1 * a2 ** 2 * (a0 - a1)) / (A2 * a1 ** 2 * (a0 - a2)) - t2
return misfit
print(func(linspace(0, 1, 10)))
x1 = bisect(func, 0, 1)
if parameter_to_optimise == "barrier_fraction":
def func(x):
C11_barrier = get_vurgaftman_parameter(barrier_material, "c11", x=x, T=T)
C12_barrier = get_vurgaftman_parameter(barrier_material, "c12", x=x, T=T)
A2 = C11_barrier + C12_barrier - 2 * C12_barrier ** 2 / C11_barrier
a2 = get_vurgaftman_parameter(barrier_material, "lattice_constant", x=x, T=T)
a2 = convert(a2, "angstrom", "m")
if a0 == a2 and x == 0:
a2 = get_vurgaftman_parameter(barrier_material, "lattice_constant", x=0.001, T=T)
a2 = convert(a2, "angstrom", "m")
misfit = - (A1 * t1 * a2 ** 2 * (a0 - a1)) / (A2 * a1 ** 2 * (a0 - a2)) - t2
return misfit
x2 = bisect(func, 0, 1)
return {
"well_material": well_material,
"well_thickness": t1,
"well_fraction": x1,
"barrier_material": barrier_material,
"barrier_thickness": t2,
"barrier_fraction": x2,
"lattice_material": lattice_material,
"lattice_fraction": lattice_material_fraction
}
def critical_thickness(layer_material="GaInAs", lattice_material="GaAs", layer_fraction=0, lattice_fraction=None, T=300,
during_growth=True, final_unit="m", bowed_material="In"):
science_reference("critical thickness",
"Matthews, J., & Blakeslee, A. (1974). Defects in epitaxial multilayers: I. Misfit dislocations. "
"Journal of Crystal Growth, 27, 118-125.")
get_vurgaftman_parameter = ParameterSystem().get_parameter
other_params = {
"T": T,
bowed_material: layer_fraction,
}
c11 = get_vurgaftman_parameter(layer_material, "c11", **other_params)
c12 = get_vurgaftman_parameter(layer_material, "c12", **other_params)
a = get_vurgaftman_parameter(layer_material, "lattice_constant", **other_params)
a_lattice = get_vurgaftman_parameter(lattice_material, "lattice_constant", **other_params)
b = array(a / sqrt(2))
v = array(abs(c12 / (c11 + c12)))
f = array(abs(a / a_lattice - 1))
longest = max([each.size for each in [v, b, f]])
if longest != 1:
if b.size != longest:
assert b.size == 1, "array arguments do not match length"
b = ones(longest) * b
if v.size != longest:
assert v.size == 1, "array arguments do not match length"
v = ones(longest) * v
if f.size != longest:
assert f.size == 1, "array arguments do not match length"
f = ones(longest) * f
else:
v, b, f = [v], [b], [f]
result = []
for vi, fi, bi in zip(v, f, b):
roots_at_soln = lambda hc_over_b: (hc_over_b) / (log(hc_over_b) + 1) - (1 - vi / 4) / (
fi * pi * (1 + vi)) # matthews & blakeslee
try:
hc = bisect(roots_at_soln, 1e-10, 1e10) * bi
except ValueError:
raise ValueError("Critical thickness infinite?")
result.append(convert(hc / 4, "Ang", final_unit)) if during_growth else result.append(
convert(hc, "Ang", final_unit))
return array(result)
if __name__ == "__main__":
T = linspace(300, 1000, 10)
from solcore.graphing import *
result = critical_thickness(layer_material="GaInAs", lattice_material="GaAs", layer_fraction=0.24, T=T,
final_unit="nm")
g = Graph(
GraphData(T, asUnit(result, 'nm'), label="Matthews-Blakeslee $\\frac{h_c}{4}$"), # , "-", 1,"red"),
# yscale="log",
# xlim=(0.03,0.53),
# ylim=(1,1000),
xlabel="Temperature (K)",
ylabel="Critical Thickness (nm)",
# grid=True,
title="InGaAs critical thickness"
# labels = ([(x_at_024,y_at_024,"x=0.24, $\\frac{h_c}{4}$=%.2fnm"%y_at_024)])
)
g.draw()
T = 300
x = linspace(0, 1, 100)[1:]
result = critical_thickness(layer_material="GaInAs", lattice_material="GaAs", layer_fraction=x, T=T,
final_unit="nm")
print(result)
g = Graph(
GraphData(x, asUnit(result, 'nm'), label="Matthews-Blakeslee $\\frac{h_c}{4}$"),
# (f[0], f[1], "Previous calculation", "-g")
# yscale="log",
# size="4 square",
# figsize=(4, 8),
xlim=(0.03, 0.53),
ylim=(0, 10),
xlabel="Indium Fraction",
ylabel="Critical Thickness (nm)",
# grid=True,
title="InGaAs critical thickness"
# labels = ([(x_at_024,y_at_024,"x=0.24, $\\frac{h_c}{4}$=%.2fnm"%y_at_024)])
)
g.draw()