software to fit optical spectra - Quantum Materials Group

The problem can be circumvented by doing the fitting in two steps. Initially, the spectra
( 0)
are fitted in a conventional way by some formula-defined dielectric function ε ( ω)
= ε mod ( ω)
.
If the match is reasonably good, then we can assume that ε mod ( ω)
has correct frequency
dependence outside the considered spectral range, even though some fine details of the
experimental curve are not fitted very well. Then we can fix all parameters of ε mod ( ω)
and add
a variational function to it:
( 1)
ε ω)
= ε ( ω)
+ ε ( ω)
.
( mod
var
Now the ε var ( ω)
acts as a small correction to the initial model ε mod ( ω)
. When we do a
( 1)
variational fitting of the reflectivity spectrum with ε ( ω)
, the ‘KK influence’ of the low- and
high-frequency spectral weights on ε 1( ω)
inside [ ω min , ωmax
] is already accounted for by
ε ( ) .
mod ω
Because ε var ( ω)
is now added to ε mod ( ω)
, which is the dominant contribution to the total
dielectric function, the parameters A i are not necessarily positive (which was initially required
by the condition ε ( ω)
≥ 0 ), but can be negative as well.
2
From the Equation 2-22 we can get a simple analytical formula for the first derivatives of
the dielectric function ε var ( ω)
with respect to the parameters A i , which are required by the
Levenberg-Marquardt procedure (section 2.1.1):
∂ε i ∧
var
( ω)
,
= ε
∂A
i
( ω)
.
At the end we minimize the chi-square with respect to all N − 2 parameters, thus
obtaining ‘the best’ KK-related dielectric function, which fits the reflectivity spectrum.
In the above example we discussed the fitting of a single reflectivity spectrum. As was
mentioned before, in the particular case of a normal-incidence reflectivity spectrum of an
isotropic sample, (almost) the same result could be obtained by the ‘conventional’ KK
technique. However, an important advantage of the new method is that it can be applied to
virtually any type of optical spectra, or a combination of them!
The typical number of parameters that are adjusted in the fitting by variational functions
is very large – up to the number of experimental points, which might amount to few thousands.
Although such an enormous number of parameters seem to make the fitting procedure
prohibitively slow, we found that it nevertheless converges within acceptable time limits. This
is yet another reason to admire, how good the LM method is!
In conclusion, it is worth mentioned that the described method is being successfully
utilized in our group. However, there is a set of not yet well-addressed issues, for instance, a
sensitivity of the output dielectric function to the noise and systematic error bars in the input
data, or the optimal number and distribution of the frequencies ω i forming the mesh. Until we
know more about the possible caveats, one should be especially careful when doing this kind of
analysis and interpreting the output data.
Guide to RefFIT Page 21