Frequently Asked Questions

A list of common questions.

What’s the best way to ask for help or submit a bug report?

If you have any questions about using refellips or calculations performed by refellips please contact us or use the GitHub Issues tracker. If you find a bug in the code or documentation, please use GitHub Issues.

What are the ‘fronting’ and ‘backing’ media?

The ‘fronting’ and ‘backing’ media are infinite. The ‘fronting’ medium carries the incident beam of radiation, whilst the ‘backing’ medium will carry the transmitted beam away from the interface. In short, the fronting media is the medium that the radiation interacts with first, and the backing media is the medium which the radiation interacts with last.

For example, consider a system with an oxidised silicon wafer where the ambient material is air; the fronting media would be air and the backing media would be silicon.

What formats/types of ellipsometry data does refellips handle?

refellips has the capability of loading data directly from both Accurion EP3 and EP4 ellipsometers, as well Horiba ellipsometers using the open_EP4file() and open_HORIBAfile() functions, respectively.

Alternatively, users also have the option to load-in other datasets using DataSE. Files loaded using DataSE must contain four columns (with header): wavelength, angle of incidence, psi and delta.

Where do I find dispersion curves for a material?

refellips contains preloaded dispersion curves for select materials, which are accessible by the load_material() function. These materials are sourced from refractiveindex.info, and include air, a void, water, dimethyl sulfoxide, silicon, silica, gold, aluminium oxide, polystyrene, poly(N-isopropylacrylamide) (PNIPAM) and a material that represents a diffuse polymer.

If required, users can download their own dispersion curves from refractiveindex.info and load them into refellips using:

my_material = RI("my_dispersion.csv")

The loaded file must contain at least two columns, assumed to be wavelength (in microns) and refractive index. If three columns are provided, the third is loaded as the extinction coefficient. The refellips maintainers are happy to include additional dispersion curves with the package; please ask if you’d like this to happen.

Alternatively, users have the option to choose from any of the in-built oscillator functions to model the optical properties of their material: Cauchy, Sellmeier, Lorentz, TaucLorentz and Gauss. Both the Cauchy and Sellmeier oscillators monotonically decrease in refractive index with increasing wavelength and are therefore not Kramers-Kronig consistent. These optical models are frequently used to model the optical properties of transparent materials, however, the Sellmeier is more accurate at higher wavelengths, i.e., the infra-red region. Users can specify Cauchy and Sellmeier parameters for their material:

my_cauchy_material = Cauchy(A=a, B=b, C=c)
my_sellmeier_material = Sellmeier(Am, En, P, Einf)

The Lorentz Tauc-Lorentz and Gaussian functions are Kramers-Kronig consistent, and allow users to implement multiple oscillators. Lorentz oscillators are typically employed when working with materials above the fundamental band gap, describing well the optical properties of transparent and weakly absorbing materials. Tauc-Lorentz are often normally used for amorphous materials. Gaussian oscillators are typically used for absorbing materials, where the complex component models the Gaussian absorption and the real component is its Kramers-Kronig relation (a Hilbert transform). Users can implement a one Lorentz or Tauc-Lorentz, or a two Gaussian oscillator model for their material by:

my_lorentz_material = Lorentz([Am], [Br], [En], Einf)
my_TaucLorentz_material = TaucLorentz([Am], [C], [En], Eg, Einf)
my_gaussian_material = Gauss([Am_1, Am_2], [Br_1, Br_2], [En_1, En_2], Einf)

A demonstration on how to implement a user defined oscillator/dispersion curve is presented in the User defined oscillator notebook. Parameter values for Cauchy, Sellmeier, Lorentz and Tauc-Lorentz are provided by Horiba. Cauchy parameters can also be found on refractiveindex.info.

Alternatively, users can simply supply a refractive index (n) and extinction coefficient (k) for a single wavelength measurement:

my_material = RI([n, k])

How do I make my own dielectric function/dispersion curve?

A demonstration on how to implement a user defined oscillator/dispersion curve is presented in the User defined oscillator notebook.

What EMA methods does refellips provide?

refellips offers the three main methods of effective medium approximations (EMA): linear, Maxwell Garnett and Bruggeman. All EMA calculations performed in refellips are based on two-component mixing and done so using the complex dielectric function, not refractive indices and extinction coefficients.

For the examples below, \(\varepsilon_1\) and \(f_1\) relate to the complex dielectric function and volume fraction of the lower material (most commonly the host material) and \(\varepsilon_2\) and \(f_2\) relate to the complex dielectric function and volume fraction of the upper material (most commonly the inclusion material; e.g., solvent). It is important to note that \(f_1 + f_2 = 1\).

For a linear EMA, the dielectric constant of the mixture is simply the sum of the products of the substituent dielectric function and volume fraction (Equation (1)). We hypothesise that the linear EMA will be sufficient for most use cases.

(1)\[\varepsilon_{\text{linear}} = f_1 \varepsilon_1 + f_2 \varepsilon_2\]

For the Maxwell Garnett and Bruggeman EMA methods, a depolarisation factor (\(v\)) is included to account for potential electric field screening by anisotropic inclusions. When (\(v = 1/3\)), Equation (2) and (3) reduce down to the isotropic case, assuming all inclusions are spherical in nature. We anticipate that only expert users will use these EMA methods or alter the depolarisation factor.

The complex dielectric function for a mixed layer using the Maxwell-Garnett EMA is determined using Equation (2),

(2)\[\varepsilon_{\text{MG}} = \varepsilon_1 \frac{\varepsilon_1 + (v f_1 + f_2) (\varepsilon_2 - \varepsilon_1)} {\varepsilon_1 + v f_1 (\varepsilon_2 - \varepsilon_1)}\]

The Bruggeman EMA method is employed using Equation (3),

(3)\[\varepsilon_{\text{BG}} = \frac{b + \sqrt{b^2 - 4 (v - 1) (e_1 e_2 v)}} {2(1 - v)}\]

where \(b = e_1 (f_1 - v) + e_2 (f_2 - v)\).

Further details surrounding these EMA methods and their derivations as well as the depolarisation factor and anisotropy are explored by both Markel and Humlicek.

Can I save models/objectives to a file?

Assuming that you have a ReflectModelSE or ObjectiveSE that you’d like to save to file, the easiest way to do this is via serialisation to a Python pickle:

import pickle
# save
with open('my_objective.pkl', 'wb+') as f:
    pickle.dump(objective, f)

# load
with open('my_objective.pkl', 'rb') as f:
    restored_objective = pickle.load(f)

The saved pickle files are in a binary format and are not human readable. It may also be useful to save the representation, repr(objective).

Alternatively, modelled results can be exported into a .csv file. An example of this is provided in Getting started.