software to fit optical spectra - Quantum Materials Group

The second possibility is to extract the dielectric function ε (ω)
itself from the measured
spectra as careful as possible without any model assumptions (the model-dependent analysis of
ε (ω)
can be done afterwards). For example, we might want to obtain ε 1( ω)
and ε 2 ( ω)
from
the measured ellipsometric quantities ψ (ω)
and ∆ (ω)
, which is rather trivial (in the isotropic
case), or from reflectivity R (ω)
alone, which is not trivial, as it heavily relies on the Kramers-
Kronig relations. It is remarkable, that in this case the problem can be again formulated in
terms of fitting! Then, one has to find the best dielectric function that matches the available
optical data. In other words, we have to minimize the chi-square (Equation 2-1 or Equation
2-12) in the whole Hilbert space of physically sensible dielectric functions, which formally
means that we have an infinite number of the fitting parameters. As it is technically impossible
to deal with an infinite number of parameters, one has to restrict somehow the set of considered
dielectric functions to a finite-dimensional subspace 4 . One way to do it is to represent ε (ω)
again by a formula (section 2.2.3). The particular meaning of the formula is not important,
provided that it gives a physically allowed result (see section 2.2.1) and is ‘flexible’ enough to
fit all important features of the experimental spectra. However, the formula-given dielectric
functions are usually too ‘stiff’ for this purpose. In this case a better solution might be to use
the so-called variational dielectric functions (section 2.2.4).
2.2.3. Formula-defined dielectric functions
There is a variety of formulas corresponding to different models of the dielectric
functions. Perhaps, the most famous example is the Drude-Lorentz (DL) model, which we will
refer to many times later on:
2
ω pi
+ ∑ 2
i ωoi
− ω − iγ iω
,
ε ( ω)
= ε ∞
2
Equation 2-18
It describes the optical response of a set of harmonic (damped) oscillators. Here ε ∞ is the socalled
‘high-frequency dielectric constant’, which represents the contribution of all oscillators
at very high frequencies (compared to the frequency range under consideration). The
parameters ω pi , ω 0i
and γ i are the ‘plasma’ frequency, the transverse frequency
(eigenfrequency) and the linewidth (scattering rate) respectively of the i -th Lorentz oscillator.
For the Drude term, which describes to response of the unbound (free) charge carriers, the ω 0i
is zero.
We shall distinguish between the formulas that satisfy the KK relation and the ones that
do not. Of course, the safest way is to always fit the data with the KK-compliant functions.
However, non-KK functions are also often used. For instance, the formula for an asymmetric
oscillator:
ε ( ω)
2 iπφ
ω pe
2
ω − ω − iγω
= 2
o
4 Moreover, it makes no sense to have more parameters than the total number of experimental points.
Guide to RefFIT Page 16