Curve Fitting in PyARPES

Why curve fit

Curve fitting is an extremely important technique in angle resolved-photoemission because it provides a coherent way of dealing with sometimes noisy data, allows for simple treatment of backgrounds, avoids painful questions of interpretation inherent with some techniques, and grants access to the rich information ARPES provides of the single particle spectral function.

Simple curve fitting

PyARPES uses lmfit in order to provide a user friendly, compositional API for curve fitting. This allows users to define more complicated models using operators like + and *, but also makes the process of curve fitting transparent and simple.

Here we will prepare an EDC with a step edge, and fit it with a linear density of states multipled by the Fermi distribution and convolved with Gaussian instrumental broadening (AffineBroadenedFD). In general in PyARPES, we use extensions of the models available in lmfit, which provides an xarray compatible and unitful fitting function guess_fit. This has more or less the same call signature as fit except that we do not need to pass the X and Y data separately, the X data is provided by the dataset coordinates.

A simple curve fitting example

The results of a fit should also provide a useful summary table if you print them in Jupyter.

Jupyter will print tables summarizing curve fitting results

Using the params= keyword you can provide initial guess with value, enforce a max or min, and request that a parameter be allowed to vary or not. In this case, we will force a fit with the step edge at 10 millivolts, obtaining a substantially worse result.

An example holding parameters constant

A number of models already exist including lineshapes, backgrounds, and step edges. All of these can also be easily composed to handle several lineshapes, or convolution with instrumental resolution:

• arpes.fits.fit_models.GaussianModel

• arpes.fits.fit_models.VoigtModel

• arpes.fits.fit_models.LorentzianModel

• arpes.fits.fit_models.AffineBackgroundModel

• arpes.fits.fit_models.GStepBModel - for a Gaussian convolved low temperature step edge

• arpes.fits.fit_models.ExponentialDecayModel

• arpes.fits.fit_models.ConstantModel

• arpes.fits.fit_models.LinearModel

• arpes.fits.fit_models.FermiDiracModel

• arpes.analysis.gap.AffineBroadenedFD - for a linear density of states with Gaussian convolved Fermi edge

Adding additional models is very easy, especially if they are already part of the large library of models in lmfit. If you are interested, have a look at the definitions in arpes.fits.fit_models.

Broadcasting fits

While curve fitting a single EDC or MDC is useful, often we will want to repeat an analysis across some experimental parameter or variable, such as the binding energy to track a dispersion, or across temperature to understand a phase transition.

PyARPES provides a tool, broadcast_model that allows for automatic and compositional curve fitting across one or more axes of a Dataset or DataArray. As before, you can use the constraints= keyword to enforce constraints or specify initial guesses for the fitting parameters. Broadcasts can be performed over a single dimension (str) or a list of dimensions (pass [str]). Here we demonstrate performing the fitting procedure as a function of the sample temperature, and then plot the step edge location onto the data.

A broadcasted curve fitting example

In the above, we also used the .F extension to xarray in order to get the concrete values of the fd_center fit parameter as an array. This is necessary because the result of a broadcast fit is a Dataset containing the full data, the residual, and the results. The results attribute is itself a DataArray whose values are the full results of the fit, rather than any single of the values.

Because of the rich information provided by a broadcast, PyARPES also has facilities for interacting with the results of an array of fit results more simply, furnished by the .F attribute.

The .F attribute

You can get all the parameter names with .parameter_names.

Getting fit values

Using the .F attribute we can obtain the values for (p) as well as the fit error of (s) any fit parameters we like.

The p and s getters

Quickly plotting a fit

We can also quickly plot a fit result with plot_param. This is sometimes useful for immediately plotting a fit result onto data or another plot sequence.

The plot_param function

Examining fit quality interactively

Fit results can also be explored interactively with .F.show(), similar to .S.show for spectra.

The plot_param function

Fitting complex lineshapes semi-interactively

In addition to looking at the results of fits interactively, you can also lay down lineshapes for one or more dispersive bands with .S.show_band_tool(). Follow roughly the information provided for masking to get started. The “Center Constraint” value dictates how much the lineshape is allowed to vary from the approximate location you lay down.

Using the “Mode” setting, you can choose whether EDCs or MDCs will be fit. Should more than one band (or the same band more than once) cross a given EDC or MDC during the fit, the appropriate number and location of lineshapes will be used. As a result of one band crossing an EDC or MDC more than once, the fit parameters will be postfixed with _{number} to indicate the index of the crossing.

Band tool