1. Simulation of Thickness and Optical Constants from transmission Spectrum of Thin Film by Envelope Method: Practical Constraints and Their Solution
Sekh Maidul*1, D. D. Shinde1, J.S. Misal1, Nisha Prasad1, P. R. Sagdeo1, N. K. Sahoo2
  1 Optics & Thin Film Laboratory, BARC-VIZAG Facility, Visakhapatnam , India
2 Applied Spectroscopy Division, Bhabha Atomic Research Centre, Mumbai , India *
Introduction
By estimating the order number of second extrema instead of first.
Calculation of ‘n’ at first
extrema involves extrapolation of any one of the envelopes, which leads to inaccurate
estimation of order number
in some cases.
Transmittance spectrum measured at normal incidence is considered as
an important tool for the optical characterization of thin film.
Substrate
(s)
I0(l)
I(l)
Thin Film
(d, n, k)
Transmittance
Single layer thin film deposited on a thick substrate
d=?
n=?
k=? ,
Instead, we suggest to estimate second order extrema, which only relies upon the interpolation of envelopes.
where ,
and A ,B, C, D are
functions of n and s.
Estimated order number considering different set of extrema’s
By considering minimum standard deviation of ‘d’ values.
This problem is, in principle, under-determined since corresponding to one measured quantity T, we have three unknown namely d , n and k.
The problem is often solved by envelope method.
Mismatch of calculated order number with the actual one should be sorted
out by the consideration of minimum standard deviation.
Estimated order number, m2
3.5
4.5
5.5
Standard deviation
14.26
14.57
42.86
Calculated Thickness
612
729
847
Algorithm of envelope method
* Actual thickness as calculated from PUMA is 610 nm.
The envelope of extrema are
drawn by parabolic interpolation of the
tangent points.
Discarding The Solution Corresponding To Unphysical Response Function.
Set of solution, which leads to increasing function of refractive index (not a normal dispersion case), should be discarded.
2. The refractive indices at extrema
points are calculated using the
magnitude of the extrema by using
following relation.
To Cope-up With Negative Values of Attenuation Coefficient
(-ve) value of ‘k’ is obtained when transmission for substrate-thin film assembly
exceeds that of substrate alone at that wavelength.
This is generally the case of antireflective thin film. Here it is caused by inhomogeniety.
It is recommended to skip such points from calculating attenuation coefficient.
where
To Decide The Range of Wavelength For Reliable Output Parameter
Plot of a typical transmission spectrum
of bare substrate and thin film
3. The condition of interference
extrema is given by-
The method is strictly valid in the wavelength range where interference fringes occur.
The choice of highest wavelength could be at wavelength corresponding to first extrema and the lowest wavelength could be at wavelength where T=0.8*TMax.
Slope=2d.
Intersection on
ordinate axis= -m1
4. The condition of interference
extrema can be written as-
Conclusion
Although envelope method suffers from the problem of multiple solution, it’s careful implementation to practical thin film reveals correct set of solution.
where l= 0,1,2,3….and m1 is the order
number of the first extrema.
5. The calculated values of ‘d’ and ‘m1’ is
then used to refine the values of
refractive indices at different extrema.
References
1.J. C. Manifacier, J. Gasiot, and J. P. Fillard, “A simple method for the determination of the optical constants n, k and the thickness of a weakly absorbing thin film”, J. Phys. E 9, 1002–1004(1976).
2. R. Swanepoel, “Determination of the thickness and optical constants of amorphous silicon”, J. Phys. E. 16, 1214–1222(1983).
3. N.K.Sahoo and K.V.S.R.Apparao, “Non-Destructive Characterization Technique for Rapid Optical and Structural Analysis of Dielectric Thin Films, BARC Report, BARC/1993/E/024, 1993.
4. E. G. Birgin, I. Chambouleyron, and J. M. Martínez, “Estimation of optical constants of thin films using unconstrained optimization”, Journal of Computational Physics 151, pp , 1999.
Plot of l/2 vs n/l
CRITICAL ISSUES FOR PRACTICAL THIN FILM AND THEIR SOLUTION:
Critical issues arise due to in-homogeneity and surface roughness that are not included in the model.
Fixing Order Number
Since entire spectrum is not available, the identification of order number of extrema is not obvious.