Determination of h by Spectroscopic Methods

In this section we focus on spectroscopic methods used to determine the real and imaginary parts of h(w). It is not our aim to discuss all the methods used in semiconductor physics, but only those commonly applied to CP, based on reflectance/transmittance and ellipsometric measurements. At the end, we report briefly on other methods that are seldom used or work only in reduced spectral ranges. 

It is clear that (2.5) contains two unknowns, namely n and k. We need an additional independent measurement in order to determine them. 

This problem is solved using the Kramers Kronig (KK) integral equations, which connect the dispersive (real part) and dissipative (imaginary part) reaction processes, by using the fundamental principle of causality and 

The natural extension of this model is to consider a free-standing film, i.e., a thin transmitting sample not deposited on a substrate. In this case we have two interfaces (assumed to be flat) and transmission and reflection Fresnel coefficients at both interfaces (air/material and material/air). Even though it is not easy to produce such films, some examples are reported in the CP literature [13,14,26,27,32], Assuming that the medium is in vacuum (no = 2 = 1) with thickness d, it is easy to calculate the total reflectance R, and transmittance T of the sample as [21-23] 

At this point, both in the case of coherent [(2.8) and (2.9)] and incoherent [(2.11) and (2.12)] 1Z and T spectra, numerical inversion of the corresponding equations delivers n if the thickness is known. The model is easily extended to several flat and parallel interfaces, as in the case of a multilayer deposited on a substrate [21,22], In this case, as the thickness of the layers and h for the substrate are known, numerical inversion of the corresponding equations yields n for the unknown layer [38-41]. The critical feature of this procedure is the accurate determination of the thickness (tens to hundreds of nanometres) of the different layers. This is what limits uncertainty in the determination of n. 

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