A Quasi-Exact Invertible Equation for Absorption Coefficient from Reflectance and Transmittance Measurements
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A QUASI-EXACT INVERTIBLE EQUATION FOR ABSORPTION COEFFICIENT FROM REFLECTANCE AND TRANSMITTANCE MEASUREMENTS
C.SUMMONTE CNR-LAMEL, via Castagnoli 1, 1-40126 Bologna, Italy
ABSTRACT Exact formulas describing reflectance and transmittance R and T of the system absorbing film/substrate/back ambient are examined. Basing on the consideration that, unlike R and T, the quantity T/(1-R) does not oscillate with wavelength, the entity of terms entering the full equation are evaluated, and some approximations are made. As a result, a non-oscillating analytical equation is obtained, that can be inverted and therefore utilized to compute the film 3 absorption coefficient with an accuracy that is better than 3% for ax>10 cm-1. The effect of further approximations, like fixing the refractive index of the film and the substrate, is also evaluated. A comparison with a normally used, simplified formula, as well as an example of application, are also reported. INTRODUCTION The determination of thin films spectral absorption coefficient cc through photometric measurement, namely transmittance and reflectance Tmp, R•,, is a normal practice in the characterization of amorphous silicon. Typical applications are the determination of the Tauc gap and the normalization of photothermal deflection spectroscopy measurements, in turn utilized to determine the defect density. Now, the formulas linking Tma,, RExp, to oxfor the case of supported film suffer for the characteristic of being non-invertible; to solve them for ca requires the use of numerical methods, involving algorithms based on successive approximationsl, 2. Alternatively, one can use the approximated formula Tx = (1 -RE,,v)e-da(1 where d is the film thickness, that is rigorously correct only for the case of non-supported single films. Advantages, as computational simplicity or improved accuracy of results, are therefore expected from the use of an invertible formula for a. APPROACH The quantities R~u and TFxp oscillate with wavelength, the oscillating period strongly of the film. On the contrary, the quantity depending on thickness and refractive index Ti=TEJ(1-R,) does not show this effect 3,4. (See for example Fig. 1, where R• andTF measured on a 0.4gtm a-Si:H film on glass, along with the computed Ti, are reported.) This suggested us to investigate the behavior of oscillating components in TE/(1-RE,), in order to identify an admissible set of approximations leading to an analytically invertible form for Tf, the criterium for admissibility being the evaluation of errors deriving from its use. The wavelength region of interest for this procedure is around the absorption edge of the material constituting the film; at the high energy side of the spectrum it is limited at the wavelengths at which, due to film thickness, transmittance vanishes, while its limit toward the low absorption region, at which the sum TExp+Ra,: approaches unity, does not come from theoretical limitations, but rather from the accuracy of photometric measurements. In the following we fully report the relation
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