the best way to do this is using ellipsometry, where you measure the change in polarization of light because of the reflection from the thin film

https://en.wikipedia.org/wiki/Ellipsometry

In principle you first need to determine the optical properties of your substrate, and then of your substrate + thin film. This is more or less a routine approach and instruments are often available in different departments at universities and in the coating industry.

There are also spectro-ellipsometers that combine ellipsometry with spectroscopy.

Thank you!

I use spectroscopic reflectometry instead of ellipsometric measurements. However, i think that the basic problem is somewhat the same, because the constants and thickness seem to be very dependent on the initial guess of the thickness and Cauchy coefficients. This prevents me from reaching a fine and repeatedly-reliable fit.

The optical properties of the substrate (BK7) is well-determined to me, but those of the film are, of course, not!

Regards,

If you measure only R you should use dispersion models for n and k and nonlinear fitting to obtain the parameters of the dispersion models and thickness by minimization of discrepancies between calculated and measured spectra. To be sure that the solution is the true one you can use transmittance measurements and compared them with the calculated T using already determined n, k and d from R.

The problem with fitting are the initial parameters for unknowns and adequate choice of the dispersion models. Check the literature thoroughly and use the collected data for initial estimates in the fitting.

Also if it is possible, deposit thick film (more than 200 nm), measure T spectra and use Swanepoel's method in the transparent region for n and d determination of your sample. Thus you will have two unkown (n and k) that can be determined simultaneously from T and R spectra by  solving two equations at each wavelength (Tmeas -Tcalc=0 and Rmeas -R calc=0).

Good luck

From your answer I made the conclusion that you use the transparent region for your film. Then it won't be difficult to measure thick film and apply the procedure that I have suggested in my answer.

Similarly we use Wemple-DiDomenico dispersion model for refractive index and exponential function for absorption coefficient. I am sending you a paper in a case if you are interested how we determine the initial guess. Also we used a grid of initial guess and calculated the error function of each solution (see the attached paper). The solution with the smallest error is used for the second run of the fit and so on until the error function value stop to change.

I suggest to measure the T and R  spectra by spectroscopic method for thick films ( >300 nm) and using the swanepoel,s  method to calculate n and d , hence you can obtain the k vales. The calculations must be done in the transparent region of the spectrum

You can take transmission measurements in the wavelength range where you get interference fringes and then use Swanepoule method to calculate optical parameters including band gap.You can also get the thickness of the film. You need not to take refection measurements if you get interference fringes.

Dear Dr. Ahmed Hijazi

Peace be upon you,

I agree with our colleagues in what they mentioned. If you measure either transmission or reflection using the spectrophotometer or UV measurements at wavelengths larger than 200 nm, then by using the swanepoel,s method you can calculate the refractive index (n) and the thickness (d). You can also calculate the absorption coefficient (Alpha) and then the extinction coefficient (k) can be obtained.

For more details, the attached paper shows how the thickness and refractive index can be measured.

Good Luck

Dear Dr. Ahmed Hijazi

Have a nice time and good day

Herein also three attached papers may help you in calculating the optical constants.

https://www.researchgate.net/publication/281495242_Influence_of_composition_on_optical_and_dispersion_parameters_of_thermally_evaporated_non-crystalline_Cd50_S50-xSex_thin_films

https://www.researchgate.net/publication/283338024_Effect_of_Se_addition_on_optical_and_electrical_properties_of_chalcogenide_CdSSe_thin_films

https://www.researchgate.net/publication/294426798_Studies_on_dielectric_properties_opto-electrical_parameters_and_electronic_polarizability_of_thermally_evaporated_amorphous_Cd50S50-xSex_thin_films

Good Luck

Article Influence of composition on optical and dispersion parameter...

Article Effect of Se addition on optical and electrical properties o...

Article Studies on dielectric properties, opto-electrical parameters...

Dear colleague Dr. Ahmed Hijazi

Have a nice time and good day.

Alsalamo Alicom wa rahmatao allah wa barakatoh

Firstly herein fined three methods to calculate the thickness of the film

One of the following you may use according to  condition of the film.

Film thickness can be checked by:

1-    An interferometric method, more details is available at: TOLANSKY S. Multiple beam interferometry of surfaces and films. London: Oxford University Press, 1988: 147.

2-    Using mass of the film, m according to the next details::

 W1 is the weight of cleaned substrate

W2 is the weight of cleaned substrate plus film

W2-W1= the weight of the film

ΔW=mg

m = ΔW/g

m=ρXV   ;   ρ, is the density of film material and V is the volume of the film                   

the Volume  = (Width X Length X Thickness) of the film

3-    SEM or AFM  (film-substrate ) cross section (paper attached)

Secondly herein fined how to calculate the optical constants of the film also the references are mentioned and all you need to overcame the all problems of optical properties [five items]. Due to the format of Equations it is attached in a separate file, just you know the equation number please check it through the attach file.

1- How to calculate the correction of T and R?

The reflectance was measured at normal incidence with an aluminum reference mirror. The absolute values of  transmittance, T and reflectance R of the substrates after correcting the  are given by [[**]A. A. Sagade,  R. Sharma, Sensors and Actuators B: Chemical, 133 (2008) 135-143 [**] M. Dongol, M. M. El-Nahass, A. El-Denglawey, A.F. Elhady, A.A. Abuelwafa. Current Applied. Physics 12 (2012) 1178.]

Equation No.……………………………...... (1)

Ift and Iq are the intensities of the light passing through the film-quartz system and the reference quartz, respectively, and Rq is the reflectance of quartz. In addition, if the intensity of light reflected from the sample mirror reaching the detector is Ifr and that reflected from the reflectance reference mirror is Im, then

Equation No.…………………… (2)

Ifr is the intensity of light reflected from the sample reaching the detector and Rm is the mirror reflectance.

T and R spectra were performed on thin film samples with single face. Extinction coefficient, k and refractive index, n were calculated using T, R and thickness d taking into account the experimental error of   the film thickness. T and R.

2- How to calculate  Absorption and extinction coefficient?

T, R and d are used to calculate the absorption coefficient, α

according to [[**]A. El-Denglawey, M. Dongol, M.M  El-Nahass, J. Lumin 130 (2010) 801.]:

Equation No.………………………….…….…… (3)

α is given by:

Equation No. …………………………………… (4)

The calculated α is included within high absorption region (α ≥ 104 (cm)-1).  

Extinction coefficient, k of TiO2 film is calculated by using:

Equation No. ……………………………………….…. (5)

3- How to calculate  Optical gap?

            An absorption edge of semiconductors corresponds to the threshold of charge transition between the highest nearly filled band and the lowest nearly empty band. According to inter-band absorption theory, the film’s optical gap can be calculated according to [[**]J. Tauc (Ed.). Amorphous and Liquid Semiconductors, Plenum, New York, 1976].

Equation No.…………………………. (6)

 A is a parameter of transition probability, it measures the disorder of material 

A = 4πσmin / nc∆E, Where  σmin is the minimum metallic conductivity, n is the refractive index, c is the light-velocity, and  ∆E = ∆Ec - ∆Ev   represents the  band tailing

[[**]M. M. El-Nahass, M. H. Ali, A. El-Denglawey Trans. Nonferrous Met. Soc.      

       China 22(2012) 3003−3011].  

is the optical gap of the material, hυ is the incident photon energy and r is the transition coefficient. The reported values of r are 2 for the measurement of indirect optical gap and 1/2 for direct optical band gap.

4- To determine the value of r plot the relation

ln ahn  = ln B+ r ln(hn- Egopt) and find the value of r from the slope of the line for Ex.

 May you have both direct 

 and indirect band gap

The indirect optical gap, is evaluated by extrapolating the straight line part of (αhν)1/2 curves with energy axes (hυ) i.e (αhυ)1/2 = 0  according to eq:

Equation No.……………………………. (7)

Direct optical gap,  is evaluated by extrapolating the straight line part of (αhν)2 curves with energy axes (hυ) i.e (αhυ)2 = 0  according to eq:

Equation No.……………………………. (8)

Both direct, and indirect, optical gaps of TiO2 films may be found.

5- How to calculate refractive index and dielectric constant?

Both R, and k at different λ were used to calculate refractive index, n according to:

[[**] T. S. Moss. Optical Process in semiconductor. Butter Worths, London, 1959.]

Equation No.     ………………………….… (9)

The dispersion of refractive index of the films is analyzed by the concept of the single oscillator and can be expressed by the Wemple–DiDomenico (WDD) relationship[ [**][S. H. Wemple, M. DiDomenico. Phys. Rev. B 3 (1971) 1338-1351] as: 

Equation No.          ……………….(10)

 Eo is the single-oscillator energy and Ed is the dispersion energy which is a measure of the intensity of the inter-band optical transition, it does not depend significantly on the band gap. The oscillator parameters can be determined  by fitting a straight line to the experimental points according to [[**][A. El-Denglawey. J. Non-Cryst. Solids 357 (2011) 1757-1763].

Plotting  vs (hν)2 allows to determine the oscillator parameters by fitting a straight line to the experimental points. By extrapolating the linear part of WDD optical dispersion relationship towards the infrared spectral region (hν = 0), static refractive index n(0), could be defined by the infinite wavelength dielectric constant ε∞ or n(0)2,

The relation between the lattice dielectric constant εL, and the refractive index, n as

[[**] A. El-Denglawey. J. Non-Cryst. Solids 357 (2011) 1757-1763.]:

Equation No. ………………………………….(11)

where εL is the lattice dielectric constant, N/m* is the ratio of the carrier concentration to the effective mass, c is the speed of light, and e is the electronic charge, εo is the permittivity of free space. The linearity of the plots of n2 versus λ2, verifying of Eq. (11). The value of εL is determined from the extrapolation of these plots to λ2 = 0 and N/m* from the slope of the graph.

Good luck

A. El-Denglawey