Basic Ellipsometric Theory

A monochromatic collimated linearly polarised light beam is directed at an angle of incidence <f> on to the surface of the sample under study. Upon reflection the beam becomes elliptically polarised and is analysed using a polariser and photodetector. Fig.3.2 shows the geometric arrangement. <f>, is required to be accurately determined. Linearly polarised light can be described by two vector components, one of which lies in the plane of incidence (p polarisation) while the other is perpendicular to the plane of incidence (s polarisation).

Fig.3.2 Geometrical arrangement of ellipsometric measurement.

By applying Maxwell’s equations to the boundary between the sample and ambient the complex reflectances, rp and r$, can be determined.

Incident Beam Reflected Beam n (sam ple) Transm itted Normal Beam to Surface (3.2) Page-94

where = n C0S(t> ~ n o c o s ^ g / ^ _(3.3)

(3.4) n0 cos(|) + n cos( | E ginc

E is the electric field strength for incident (inc) and reflected (ref) waves, n and no are the complex refractive indices as defined in Fig.3.2. <f> and <£ are related by Snell’s law,

Ellipsometric instruments measure the property rp/r s. This ratio can be redefined as

Tan^ and c o s a are the parameters most used in SE to describe the optical properties of a sample because these are the data that are actually collected. However it is necessary to convert these to more common optical functions such as the refractive index. In the case of a bare substrate,

This can be done with ease only in this the simplest of cases. For even the situation of a single layer on a substrate the expressions become very complex. A detailed discussion of the ambient-film-substrate case can be found in chapter 4 of the book by Azzam and Bashara. An alternative to direct solution of the theoretical expressions for the optical functions is to model the experimental data using an assumed sample structure and reference spectra for the constituents of the model. The optical properties of a layer of more than one constituent can be calculated using the Bruggeman effective medium approximation (see appendix C). Thus the density of a film can be simulated by mixing positive or negative "voids'’ for lower or higher density respectively. When the model is sufficiently close to the actual data it is assumed to be representative of the sample under investigation Typically a regression analysis is applied to the model. Various means are used to determine the "quality of fit" (defined in detail below). The main method is the minimisation of the unbiased estimator (a).

nQ sin<|> = n sunj/ (3.5)

£. = p s tanijj exp(iA) _(3.6)

Calculation of optical functions such as n by the fitting route requires an accurate knowledge of tani/> and c o s a (or and A in the case of conventional

SWE instruments) to obtain a high confidence result. However, there arises another difficulty, one which so far has attracted little attention but is high-lighted by Bu-Abbud et al. very well in their study of 1986. There are as with all analyses theoretical bounds on the sensitivity of both instrument and technique. Given that instrumental sensitivity is sufficient to "see" a change caused by an arbitrarily small perturbation in an experimental parameter (e.g. wavelength or <f>) then tan^ and c o s a can be determined within certain limits. Thus the smallest value of these limits can be defined as the precision of the instrument. These values are known for the ellipsometer used in the present study. This leaves the issue of the theoretical limit on the sensitivity of the method (particularly with respect to data extraction) which is a critical difficulty with ellipsometry particularly for situations when very thin films or multi-layer structures are under study. Typically in SWE the accurate calculation of both refractive index and oxide thickness is not possible for layers of less than 50nm. The cause of this is that the changes in V and a with respect to layer thickness and refractive index are very similar i.e.

3tam|r ^ dtanijj . 8 cos A ^ dcosA p g )

dn dx dn dx

In particular the sensitivity of V> for thin oxides is very low and therefore one is attempting to calculate n and x from just one well defined datum. These points are illustrated in the Fig.3.3 to 3.9 below showing the effect on c o s a and tan^> when small variations in the layer are introduced. In multi-angle SE (MASE) calculations an initial sensitivity (or response surface) analysis is particulary important because it allows one to chose the wavelength and angle ranges so as to maximise measurement sensitivity. In the present discussion a brief introduction and the conclusions of the analysis of Bu-Abbud et al. (1986) are noted. The aim of this work being to both verify their methods and obtain information on thin oxide films.