SiO2 reflectivity
SiO2 ~100 nm
on Al2O3 wafer
sample length 5 mm
measured at a CuKalpha source
beam width ~0.05 mm
[1]:
import nexus as nx
import numpy as np
import matplotlib.pyplot as plt
load data
[2]:
# data are saved as theta and intensity in column 0 and 1
angles_data, intensity_data = nx.data.Load('SiO2_refl.txt',
x_index = 0,
intensity_index = 1,
# x_start=None,
# x_stop=None,
# intensity_threshold=None
)
plt.semilogy(angles_data, intensity_data)
plt.show()
sample
[3]:
# fit parameters
#SiO2
t_SiO2 = nx.Var(100, min = 90, max = 110, fit = True, id = "t SiO2")
d_SiO2 = nx.Var(2.65, min = 2.0, max = 2.65, fit = True, id = "d SiO2")
r_SiO2 = nx.Var(0.2, min = 0.05, max = 2.0, fit = True, id = "r SiO2")
#substrate
r_subs = nx.Var(0.1, min = 0.05, max = 0.4, fit = True, id = "r subs")
SiO2 = nx.Material(id = "SiO2",
composition = [("Si", 1), ("O", 2)],
density = d_SiO2
)
lay_SiO2 = nx.Layer(id = "lay SiO2",
material = SiO2,
thickness = t_SiO2,
roughness = r_SiO2
)
# silicon substrate
substrate = nx.Layer(id = "Si substrate",
material = nx.Material.Template(nx.lib.material.Al2O3),
thickness = nx.inf, # set the substrate to infinite thickness
roughness = r_subs
)
# define sample
# in grazing incidence the beam profile and the sample length get important
sample = nx.Sample(layers = [lay_SiO2,
substrate], # define layers from top to bottom
geometry = "r", # set the sample geometry to reflectivity
roughness = "a", # analytical roughness model
length = 5 # in mm
)
experiment
[4]:
beam_fwhm = nx.Var(0.03, min = 0.02, max = 0.06, fit = True, id = "b fwhm")
beam = nx.Beam(profile = 'g', # Gaussian beam profile
fwhm = beam_fwhm # FWHM of the beam in the scattering plane, in mm
)
exp = nx.Experiment(beam = beam,
objects = [sample]
)
reflectivity
[5]:
res = nx.Var(0.01, min = 1e-7, max = 0.02, fit = True, id = "resolution")
reflectivity = nx.Reflectivity(experiment = exp,
sample = sample, # sample in the experiment for the reflectivity calculation
energy = nx.lib.energy.CuKalpha, # Cu K alpha line
angles = angles_data, # loaded dataset
intensity_data = intensity_data, # loaded dataset
# scaling = 'auto',
# background = 0.0,
# offset = 0.0,
resolution = res,
# residual = nx.lib.residual.Log10() # or change to another nx.lib.residual. method
)
reflectivity()
reflectivity.Plot()
fit
[6]:
fit = nx.Fit(id = "fit reflectivity",
measurements = [reflectivity]
# external_fit_variables = [] # for advanced fitting
# einequalities = None # for advanced fitting
)
fit.options.file_output = False # no putput file is generated for fit
fit.options.method = "PagmoDiffEvol" # or any other method of choice
# for refelctivities it is often good to increase the iterations of global methods
fit.options.iterations = 200
fit.Evaluate()
Run Fit instance with id: fit reflectivity
Starting fit with 1 measurement data set(s) and 7 fit parameter(s):
no. | id | initial value | min | max
0 | Refl scaling | 916932 | 0 | 9.16932e+07
1 | resolution | 0.01 | 1e-07 | 0.02
2 | b fwhm | 0.03 | 0.02 | 0.06
3 | t SiO2 | 100 | 90 | 110
4 | r SiO2 | 0.2 | 0.05 | 2
5 | d SiO2 | 2.65 | 2 | 2.65
6 | r subs | 0.1 | 0.05 | 0.4
Using 0 equality constraint(s) on parameter(s):
Using 0 inequality constraint(s).
Calling Pagmo solver with fit method PagmoDiffEvol
population: 80
iterations: 200
cost = 1.310547e-01
Calling ceres solver with fit method LevMar
Ceres Solver Report: Iterations: 1, Initial cost: 1.310547e-01, Final cost: 1.310547e-01, Termination: CONVERGENCE
Gradient error analysis.
Fit performed with algorithm:
PagmoDiffEvol
Local algorithm:
LevMar
Error analysis:
Gradient
Using 7 fit parameter(s):
no. | id | fit value | +/- std dev | initial value | min | max
0 | Refl scaling | 3.62628e+06 | 2.92717e-08 | 916932 | 0 | 9.16932e+07
1 | resolution | 0.019984 | 0.000298631 | 0.01 | 1e-07 | 0.02
2 | b fwhm | 0.0598533 | 0.000494428 | 0.03 | 0.02 | 0.06
3 | t SiO2 | 96.2317 | 0.0141121 | 100 | 90 | 110
4 | r SiO2 | 0.0557617 | 0.0727758 | 0.2 | 0.05 | 2
5 | d SiO2 | 2.49841 | 0.00894172 | 2.65 | 2 | 2.65
6 | r subs | 0.0884143 | 0.158201 | 0.1 | 0.05 | 0.4
Correlation matrix:
no. | 0 1 2 3 4 5 6
-----|---------------------------------------------------------------
0 | 1.000 0.566 0.309 -0.096 0.549 -0.218 -0.825
1 | 0.566 1.000 0.191 -0.147 0.696 -0.302 -0.728
2 | 0.309 0.191 1.000 0.069 0.090 0.087 -0.200
3 | -0.096 -0.147 0.069 1.000 0.087 0.637 -0.013
4 | 0.549 0.696 0.090 0.087 1.000 0.080 -0.926
5 | -0.218 -0.302 0.087 0.637 0.080 1.000 0.047
6 | -0.825 -0.728 -0.200 -0.013 -0.926 0.047 1.000
Using 0 equality constraint(s) on parameter(s):
and 0 inequality constraint(s).
Total cost = 1.311e-01
cost for each FitMeasurement is:
no. | id | cost | %
0 | | 1.311e-01 | 100.000
Fit instance with id "fit reflectivity" finished.
[7]:
reflectivity.Plot()
