Forward simulation of Scatterometry for a nanostructure

Simulation of Scatterometry Measurement

The given data are a simualtion of a scatterometry measurement for a silicone line-grating with a thin oxide-layer. You can see an image of such nanostructure for example in [Lohr et al. (2023)]. The structure was modeled as a silicon bulk with a thin silicon oxid layer on top.

Forward calculation f

For the forward calculations we used the finite element solver from JCMsuite and made the measurement simulations for three different energies. The measurement setup is the following:

energies: 95.4 eV, 105.0 eV, 109.9 eV

s-polarization

incidence angle alpha_i = 30°

azimuthal angle phi  ∈ [0°,6°] in steps of 1°

The shape of the line-grating is defined by geometric parameters, and the behavoir of the material by the optical constants for silicon (Si) and silicon-oxide (SiO2), wich are given as a function of the material densities using information from databases. For SiO2 the information comes from OCDB from the PTB and for Si the values were taken from CXRO-Database . So the forward model depends on 5 parameter describing the geometry and the line-edge roughness and the two densities for Si and SiO2. These x-parameter are uniformly sampled from the following domains:

height : h ∈[120 ,140] nm

width: w ∈[80 ,100] nm

side-wall-angle : swa ∈ [0°,5°]

thickness of oxid layer: thick_sio2 ∈ [1.5 ,2.5] nm

density for SiO2: rho_sio2 ∈ [0,6] g/cm³

density for Si: rho_si ∈ [0,6] g/cm³

roughness parameter: sigma_roughness ∈ [0,2]

The parameter for the pitch and the courner roundings are fixed to the following values:

pitch: 160 nm

courner rounding top: 5 nm 

courner rounding bottom: 10 nm

The forward calculation determines for a set of x-parameter the intensities of the diffraction orders from -5 to 5 and the azimuth angles defined above. Hence for one parameterset x the forward evaluation f(x) has the shape (7,11).

Data

The uploaded data contain:

parameter_samples_uniform : uniform samples from the parameter domain of x

intensitiy_95.4eV, intensity_105.0eV, intensity_109.9eV: forward evualuation of the samples 

parameter_domain: boundaries of parameter domains

parameter_names: sequence of parameter names

Please note, that the data does not contain any modeling error and measurement noise as it is the case for real measurements. To get simulated data for a scatterometry measurement including these errors/noise for this grating line please run the python script: generate_measurement_simulation_atmoclg.py.

Python Script

For a random chosen sample x out of parameter_samples_uniform the python script generate_measurement_simulation_atmoclg.py will generate one set of intensities containing error/noise like it is expected to appear in a real measurement. So the measurement data y is described by a forward model f plus the model error epsilon and the measurement noise eta which are both given as random variables. The model error comes from inaccuracies of the forward model and the measurement noise is caused by deviations of the measuring instrument. We assume here, that both random variables are Gaussian distributed such that the measurement data y is given by

y = f(x) + epsilon + eta   for 

f(x) --  forward evaluation of x

epsilon ~ N(0,sigma_me²) -- model error Gaussian distributed with zero mean and variance  sigma_me²

eta ~ N(0,sigma_noise²) -- measurement noise Gaussian distributed with zero mean and variance  sigma_noise².

Of course you are free to choose different values for the variances of epsilon and eta, but here are some Reasonable values:

Model Error: about 3% of the measurement plus a small approximation error

sigma_me² = (a*f(x))² + b²  ,   for a = 0.03, b = 1e-4

Measurement Nosie : about 1% of the measurement

sigma_noise² = (0.01*f(x))²

Acknowledgement

This simulation of a scatterometry measurement is part of the ATMOC project . This project (21IND04 ATMOC) has received funding from the EMPIR programme co-financed by the Participating States and from the European Union’s Horizon 2020 research and innovation programme.

References

Lohr, Leonhard M., Richard Ciesielski, Sven Glabisch, Sophia Schröder, Sascha Brose, and Victor Soltwisch. 2023. “Nanoscale Grating Characterization Using EUV Scatterometry and Soft x-Ray Scattering with Plasma and Synchrotron Radiation.” Applied Optics 62 (1): 117. https://doi.org/10.1364/AO.475566.