How can one compute the refractive index of a thin film by knowing only the bulk refractive index and the film thickness?
In view of the way you asked the question, let me address only its theoretical/physical impications. OK, everything depends of course on how thin is "thin". Real trouble starts when you get to from few tens to a hundred of atomic layers; until that you are more or less safe assuming bulk properties (it is not always so, but let us go with that for now). In semiconductor layers you get something of the order of 100 Angst, in metallic layers you can go down to 10-20 Angst, and in graphen you have just a single atomic layer. Bad news -- there is no general, universal approach to the physics of those layers. You are in a twisted maze of particular conditions and material properties + of course new physical properties, especially when you go to very low temperature. About physical properties -- in semiconductors you can get a so called quantum size effect (due to resonances of wave function in a semi-infinite quantum well); the first work has been done by V. B. Sandomirsky I believe in 1960-ties. In metallic layers below about 30 Angts (even if they are still well-structured) you run into shortened mean scattering length of electrons (they are scatted now more often from the walls than from the ions); I think that the first was my own paper, 50 years back, too
http://psi.ece.jhu.edu/~kaplan/PUBL/AEK.pubs/RUSS/3.pdf
The main thing for metallic layers at the thicknesses below that is that they don't want to stay in a flat layer, by getting instead into small clusters like water dropplets (the so called "island layers"). In the most used semiconductor layers, a lot depends on the substrate and the technology by which they have been grown. (For example, one of the problem -- a match between lattice constants of a layer and substrate. If there is a significant mismatch, the layer will be just broken...). And again a big part of all that goes to the way you deposited the films (partticular technology and the the mode of deposition -- fast, slow, etc.)
So, you are in for a big research adventure (but this is the fun of science, isn't it?), be ready and start reading the literature in your specific application. Best of luck.
The best way is to measure it by Ellipsometry analysis. It will give you the optical constants including refractive index. Then you can fit the obtained data to extend in strong absorption area. Cauchy, selmier and so on are the famous fitting relations. Swanepoel method is reliable for the films whose transmission showes interference exterma. Some methods are used when both R&T are measured though I'm not sure about that because those don't regard the substrate!
With transmission and reflection spectra and mathematical calculations
Ellipsometry is almighty. Reflection and transmission spectra in UV-VIS-NIR can be used when a thickness is in the range of about 50-400 nm or around.
On basis of known index of refraction of bulk material cannot be calculated index of refraction of thin film because during deposition the matter is usually completely rearranged and thus there is no general and universal relationship between bulk and thin film.
However, if thin film properties (composition, structure, crystallinity, …) are close to the bulk properties, the index of refraction is usually similar.
To obtain refractive index only for given wavelength, probably the best choice is direct measurement via coupling prism.
To obtain spectral dependence of refractive index, Spectroscopic Ellipsometry (based on measurement of light polarization) is universal technique for that. However, evaluation consist in parameters fitting, thus it may led to misinterpretation of measured data and incorrect results. More direct (and thus less sensitive for interferring effects) and also more simple (but less universal) is measurement of transmittance and/or reflectance spectra within UV-Vis(-NIR) range and utilize interference fringes (if occurs) for index of refraction calculation. There are several techniques, e.g. well known Swanepoel technique may be used:
(R. Swanepoel, J. phys. E: Sci. Instrum., 16 (1983) p.1214-1222. ;
R. Swanepoel, J. phys. E: Sci. Instrum., 17 (1984) p. 897-903.).
Usually thin film possesses different refractive index than those of the bulk materials. Optical constants of thin films are usually determined by optical methods like spectrophotometry, prism-coupling technique and ellipsometry. The prism-coupling technique and ellipsometry possess higher accuracy for determination of refractive index and thickness of thin layers in comparison with the spectrophotometric methods. The spectrophotometric methods are easier. Several spectrophotometric methods have been developed for the determination of refractive index, n and extinction coefficient, k) and film thickness, d. (1. J. A. Dobrowolsky, F. C. Ho, A. Waldorf, Appl. Opt. 22 (1983) 3191.; 2. F. Abeles, M. L. Theye, Surf. Sci. 5 (1966) 325; 3. J.C. Manifacter, P. Filard, J. Phys. E 9 (1976) 1002; 4. R. Swanepoel, J. Phys. E: Sci. Instrum. 16 (1983) 1214.). The comparison of the different spectrophotometric methods and their applicability at different thicknesses for thin films with same composition and can be seen in R Todorov, J Tasseva, Tz Babeva, K. Petkov, J.Phys. D: Appl. Phys. 43, art. num. 505103, 2010
The choice and accuracy of the method depend on many factors - the film’s thickness, whether the thin film absorbs the light in the investigated spectral range, kind of the substrate (transparent or non-transparent).
Trying to answer your question, I would say that the refraction index of a thin film does not depend on its thickness if it is larger than the material structure (atomic bond distance). The question is how to compare the bulk material with the material deposited in a thin film because, as Jan Gutwirth mentioned, the deposited material has generally a different arrangement than the already known bulk material. For instance: the bulk is crystalline and the film is amorphous: in that case you cannot use the bulk refractive index to determine the film refractive index (example: cSi and aSi). In other cases, if you verify that the film material has already the crystal structure (by XRD or Raman), you can use the bulk refractive index, but you will perhaps have to reduce the film refractive index by a factor taking into account the existence of spaces between the microcrystals forming the film. In conclusion, every situation should be analyzed and, to be sure that your computation is correct, checked it using experimental characterization as it was mentioned by the colleagues.
you can use the parav program
"Computer program PARAV for calculating optical
constants of thin films and bulk materials: Case study of
amorphous semiconductors
A. GANJOO R. GOLOVCHAK
http://www.chalcogenide.org/downloads/software/Ganjoo-JOAM-2008.pdf
http://www.chalcogenide.org/downloads/software/Ganjoo-JOAM-2008.pdf
An approximation could be made using a UV-VIS spectrum, determining the exact thickness of the layer and later using the transfer matrix formalism. The idea is that a thin layer (less than 1 um) would produce an optical interference that would help you to estimate the refractive index.
Firstly you need to deposit the thin layer and exactly determine its thickness, for instance using AFM. Later you measure its UV-VIS transmittance and then you need to write a code, using the matrix formalism (theory background http://sjbyrnes.com/fresnel_manual.pdf), for calculating such transmittance with the given thickness and bulk refractive index (you can find such code here http://www.stanford.edu/group/mcgehee/transfermatrix/ ). Finally, you can manually modify the refractive index to match the experimental and theory estimation. This procedure would give you an approximation of the refractive index of the deposited material. To have a better result you can use a bilayer where the second material has a known refractive index (different to the first material) and a well determined thickness. Doing this you for sure will find interference peaks that will make easier the experimental-theoretical matching.
I think the best technique to determine both thickness and optical properties (complex refractive index) of thin films is spectroscopic ellipsometry. You have to interpret the experimental measurements assuming a physical model of the film you are working on and an appropriate model for the refractive index dependence on the wavelength. The simplest physical model is to consider of an homogeneous and isotropic film. However, depending on the fabrication procedure you can expect some surface roughness, substrate-film interlayers, oxide overlayers, etc. For films with high absorption peaks, you can use a transparency region in order to determine the thickness and use such thickness as fixed parameter in order to determine the imaginary part of the refractive index.
In view of the way you asked the question, let me address only its theoretical/physical impications. OK, everything depends of course on how thin is "thin". Real trouble starts when you get to from few tens to a hundred of atomic layers; until that you are more or less safe assuming bulk properties (it is not always so, but let us go with that for now). In semiconductor layers you get something of the order of 100 Angst, in metallic layers you can go down to 10-20 Angst, and in graphen you have just a single atomic layer. Bad news -- there is no general, universal approach to the physics of those layers. You are in a twisted maze of particular conditions and material properties + of course new physical properties, especially when you go to very low temperature. About physical properties -- in semiconductors you can get a so called quantum size effect (due to resonances of wave function in a semi-infinite quantum well); the first work has been done by V. B. Sandomirsky I believe in 1960-ties. In metallic layers below about 30 Angts (even if they are still well-structured) you run into shortened mean scattering length of electrons (they are scatted now more often from the walls than from the ions); I think that the first was my own paper, 50 years back, too
http://psi.ece.jhu.edu/~kaplan/PUBL/AEK.pubs/RUSS/3.pdf
The main thing for metallic layers at the thicknesses below that is that they don't want to stay in a flat layer, by getting instead into small clusters like water dropplets (the so called "island layers"). In the most used semiconductor layers, a lot depends on the substrate and the technology by which they have been grown. (For example, one of the problem -- a match between lattice constants of a layer and substrate. If there is a significant mismatch, the layer will be just broken...). And again a big part of all that goes to the way you deposited the films (partticular technology and the the mode of deposition -- fast, slow, etc.)
So, you are in for a big research adventure (but this is the fun of science, isn't it?), be ready and start reading the literature in your specific application. Best of luck.
If you know the thickness of the film you can use both Transmittance and reflectance measurements for simultaneous determination of refractive index (n) and extinction coefficient (k) using the bulk value as an initial values and the Newton-Raphson minimization algorithm. In the case your film is deposited onto opaque substrate you can determine n, k and d simultaneously using non-linear curve fitting and using adequate dispersion models for n and k (it will be very easy to choose the right models because you already know how the dispersion curves of the bulk material look like). The bulk values of n and k can be used as initial estimates for the minimization procedure and the known thickness will help you to pick the physically correct solution (when the thickness is not known the solution with the smallest error could be used)
Please; I want to know the exact equations to calculate the n and k if i have transmission and reflection values of the thin film? and what is the exact equations that the ellipsometry used to calculate T and the thickness of the thin film?
Program PARAV if you interesting http://www.chalcogenide.org/computer-program-parav-for-calculating-optical-constants-of-thin-films-and-bulk-materials-case-study-of-amorphous-semiconductors/
The refractive index (η) at different wavelengths can be evaluated according to the formula
Dear Md. Ferdous,
Thin solid films have proved a very useful vehicle to understand the properties and structure of a solid material especially in its noncrystalline form. Indeed, the determination of the value of refractive index (nf) and its dispersion, (i.e. variation of nf vs. the wavelength of the incident light (lambda λ) of non-metallic thin solid films have wide applications in designing different optical components and modeling optical coatings. When a monochromatic electromagnatic radiation of angular frequency (omega) interacts with a material the reflected, transmitted and absorbed radiation components formula,
A(ω)+ R(ω)+T (ω)=1
The equation T + R + A = 1 describes the theory, where T=transmittance, R=reflectance, and A=absorptance. When designing a thin film, though the wavelength of light and angle of incidence are usually specified, the index of refractionand thickness of layers can be varied to optimize performance.
The refractive index of thin films have shown variation with different parameters such as deposition techniques, substrate temperature, annealing temperature, degree of oxidation, mixing ratio, density, doping, hydrogen content, films thickness, non-stoichiometry, inhomogeneity, and anisotropy of the deposited films. Moreover, the variation of the refractive index on other external effects such as temperature, pressure, electric field, light intensity, exciton modulation, radiation, fatigue, etching and aging effects.
The dispersion equations can give a good fit with the transmission spectrum over a wide range of wavelengths for many thin film materials. The dispersion curve of nf(λ), however, varies at different regions of the electromagnetic spectrum.
Methods to determine the refractive index
(1) Abelés Method (2) Reflectance and transmittance Method (3) Interference fringes method(4) The prism spectrometer and other methods
For thin film (coating)
Use this site to understand (https://www.iasj.net/iasj?func=fulltext&aId=61809)
The simplest approach to achieve this layer of refractive index nAR and thickness d is obtained from:
AR= √nhnI , d=Lambda/4nAR . phi/△
Lastly, for antireflection thin films, we can use Cauchy equation , shelmier equation for calculating refractive index. The real part of the wavelength-dependent refractive index obtained from transmission spectrum(for wide band semiconductor)…
Hope content is helpful for you.
Ashish
Hello sir,
If you have transmittance data, first you need to calculate the absorption
coefficient (alpha-a)
alpha = 2.303 log (1/T)/t where t-thickness of the sample
then, reflectance(R) in terms of absorption
coefficient (alpha-a)
R = (exp(−at)+(or)-((exp(−at)T -exp(-3at)T+exp(-2at)T2)1/2)/exp(−at)+exp(-2at)T
then
n = (-(R + 1)+(or)- 2R1/2)/R-1
where n is refractive index
I hope this information may help you.
Regards,
Aravind Rudrarapu
Please, is there any easiest way of calculating the thickness of a thin films from transmittance data
You need to measure the permittivity and permeability of your material and then calculate n.
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