software to fit optical spectra - Quantum Materials Group

−1
where ( α ) kl is the inverse Hessian matrix also called the ‘covariance matrix’. Note, that all
−1
2 2 ( 0)
( 0)
the diagonal elements ( α ) kk are positive because χ ( 0)
= χ ( p1 , K,
pM
) is a local minimum
point and Equation 2-14 can always be applied.
The reasonable choice of
defined by the condition:
2 ⎛ M δχ ⎞
P ⎜ , ⎟ = p ,
⎝ 2 2 ⎠
2
2
δχ absolute value is quite an issue. Ideally, δχ should be
Equation 2-15
where M is the number of parameters, p is the desired confidence probability limit (typically,
x
∞
p = 0.
95 ) and ⎜
⎛ −t
a−1
≡ ⎟
⎞
⎜
⎛ −t
a−1
P(
a,
x)
⎟
⎞
⎝∫
e t dt /
0 ⎠ ⎝∫
e t dt is the incomplete gamma-function (the
0 ⎠
derivation can be found in Ref. [3]).
However, the Equation 2-15 can be applied, if (i) the data points are statistically
independent, (ii) all weight coefficients w ν are unities, and (iii) the model is absolutely
adequate to the data. However, as was mentioned in the section 2.1.2, the existence of the
systematic error bars invalidates the first two assumptions. One also needs much optimism to
2
heavily rely on the assumption (iii). In this situation the choice of δχ becomes rather
ambiguous and, therefore, human-dependent.
Fortunately, from Equation 2-14 it follows that
2
δχ scales the confidence limits of all
parameters proportionally. One can therefore reliably compare the error bars of different
parameters, even though their absolute values might be ill-defined.
2.2. Modeling of the dielectric functions
The central assumption in the calculations RefFIT does is that all measurable optical
quantities (such as reflectivity, penetration depth etc.) can be expressed in terms of the complex
frequency-dependent dielectric function ε ( ω)
= ε1
( ω)
+ iε
2 ( ω)
of the material under study.
Therefore, the most important issue is the modeling of the dielectric function itself.
2.2.1. Physical properties of the dielectric functions
It is well-known from the textbooks (e.g., [2]) that any realistic dielectric function ought
to satisfy certain physical conditions.
First of all, ε1 ( ω)
= ε1
( −ω)
and ε 2 ( ω)
= −ε
2 ( −ω)
, therefore it is sufficient to model the
ε (ω)
for ω ≥ 0 only.
Secondly, ε 2 ( ω > 0)
≥ 0 , which means that the intensity of light cannot increase in the
direction of propagation.
Guide to RefFIT Page 14