Spectroscopic ellipsometry

Thin film thicknesses can also be measured with ellipsometry, which is another benign optical technique. In this case, the measurement is not limited to surface information and a variety of samples can be measured. By measuring changes in the polarisation state of reflected light, the optical constants and thicknesses of thin films can be measured with little difficulty. Sample roughness and porosity can also be measured, so ellipsometry applications range from characterising implantation damage to monitoring the growth of superlattice structures in situ [23].

A transverse electromagnetic plane wave propagating in free space consists of orthogonal electric and magnetic fields. The electric field, _E~_{(r, t) at position r} and time t can be expressed as

~

E(r, t) =E~s+E~p =Escos(ωt−k·r+φ)as+Epcos(ωt−k·r)ap. (2.3) In (2.3), ω is the angular frequency and k is the propagation vector of the wave.

The amplitude of the electric field at any (r, t) is separated into two orthogonal components Es and Ep, which are parallel to the respective unit vectors as and

ap, and these are mutually perpendicular to the direction of propagation. The

polarisation state of the wave can be described by these amplitude components and the phase difference, φ. The light is linearly polarised whenφ = 0, circularly polarised if φ = ±π/2 and Es = Ep, and elliptically polarised otherwise. When the wave described in (2.3) is incident on a planar surface, the amplitudes and phases of the reflected components will be governed by the optical properties of the two media, as described by the Fresnel equations. The reflection coefficients

Rp and Rs for the parallel and perpendicular components respectively, can be defined as the ratio of the reflected (r) and incident (i) amplitudes [30]. That is,

Figure 2.8: The J.A. Woollam M-44® system in the Plasma Research Laboratory that was used for ellipsometry over visible wavelengths. The s and p polarisation directions of the incident beam are shown, andS~ is the Poynting vector indicating the propagation direction. θi is the angle of incidence and Θ is the orientation of the analyser.

Rp = E~p,r/ ~Ep,i and Rs = E~s,r/ ~Es,i. The ratio of Rp and Rs yields the complex ellipsometric parameter, or the complex reflection coefficient,ρ. That is,

ρ= Rp Rs

= tan(Ψ) ej∆ _{where Ψ,∆∈}

R. (2.4)

In (2.4), tan(Ψ) is the magnitude of the ratio and ∆ is the phase difference between the p and s reflection coefficients [30, 36]. These real numbers are the parameters determined by ellipsometry measurements. The fact that they are extracted from a ratio means that they are insensitive to the absolute phase and intensity of the incident light. These can be substituted into the Fresnel equations to determine the optical parameters of the sample, which is not a trivial process. For example, when the angle of incidence is θi, the ambient has a refractive index n₀ and the sample is a bare Si substrate with complex refractive index ˜n_Si =n_Si−jκ_Si, then these optical parameters are related to Ψ and ∆ by [37]

n2_Si−κ2_Si = n2₀sin2θi "

1 + tan2θi(cos22Ψ−sin22Ψ sin2∆) (1 + sin2Ψ cos∆)2

#

, (2.5a) 2n_Siκ_Si = n

2

0sin2θitan2θisin4Ψ sin∆

(1 + sin2Ψ cos∆)2 . (2.5b)

Such calculations are further complicated when measuring a thin film deposited on another material, which is generally the case. Instead of solving such equations analytically, numerical methods can be used to compute the materials properties. Most of the ellipsometry results presented in this thesis were measured with a J.A. WoollamM-44® system [36]. This is a rotating-analyser ellipsometer, as shown in

figure2.8, and these allow faster and more accurate measurements than some other configurations [23]. The input polariser ensures that the incident light has a known linear polarisation. The reflected light is elliptically polarised and passes through a rotating analyser onto the detectors. The analyser is effectively another linear polariser that rotates around the beam axis. If the reflected light were linearly polarised then the signal onto the detector would obey Malus’ law. Circularly polarised light would not be affected by the analyser, and in general the intensity I reaching the detector is a superposition of both scenarios. This detected signal has the form I(Θ) = I₀(1 +αcos2Θ +βsin2Θ), where I₀ is the average intensity and Θ is the angle between the polarising plane of the analyser and the plane of incidence at any timet [36]. Iff is the rotational frequency of the analyser and Θ₀ is its initial orientation, then Θ = 2πf t+ Θ₀. The normalised Fourier coefficients of the detected signal are α and β. It can be shown that [36]

tanΨ = s 1 +α 1−α|tanP|, (2.6a) cos∆ = √ β 1−α2 tanP |tanP|, (2.6b)

whereP is the constant orientation of the input polariser with respect to the plane of incidence. So by rotating the analyser and taking the Fourier transform of the detected signal, experimental values for the ellipsometry parameters given in (2.4) can be extracted.

Single-wavelength ellipsometry is limited by the correlation between film thickness T and refractive indexn[30]. TheM-44 performs spectroscopic ellipsometry, which is a very powerful technique. The light that passes through the analyser is actually dispersed onto an array of forty-four Si detectors, which simultaneously perform these measurements at different wavelengths. The data is then compared to a model of the sample, in order to extract film thicknesses and wavelength-dependent optical parameters [36].

Most of the samples measured for this thesis consisted of a transparent thin film deposited on Si. The measured data was fitted to either the Cauchy or the Sellmeier dispersion models, which are given in (2.7) and (2.8), respectively [36, 38]. The former can be less accurate for infrared and ultraviolet wavelengths, however these differences are not significant for the work presented here. In each model, λ is the wavelength, n∈_R is the refractive index of the film and the other parameters are

n = A+ B λ2 + C λ4 . (2.7) n2 = A+ Bλ 2 λ2−C2 + Dλ2 λ2−E2 . (2.8)

From the selected model, the WVASE32 software determines theoretical values for Ψ and ∆. The mean-squared error (MSE) is then calculated, which indicates how different the modelled data is from the experimental data. The Levenberg- Marquardt numerical algorithm is used to iteratively minimise the MSE by tweaking the model parameters. As a rule-of-thumb, MSE < 10 indicates a satisfactory fit between the modelled and experimental data. In this way, accurate dispersion curves and thickness values can be obtained for a wide range of samples. TheM-44 performs ellipsometry over 418–763 nm, however the Ge and CaF₂ films discussed in chapter 4 are used in infrared dielectric filters. To characterise these films, infrared ellipsometry was performed by Drs. Mariusz Martyniuk and Thuyen Nguyen at The University of Western Australia using a Sopra IRSE5E spectroscopic ellipsometer. This system also uses a rotating analyser, but the measurement wavelengths cover 2–17 µm. The IRSE5E uses an HgCdTe detector and a globar light source. Hence, the optical constants of these films were directly measured at mid/long infrared wavelengths. Ellipsometry is clearly a useful technique. In some cases, the data may be difficult to fit if the algorithm cannot be provided with suitable starting values. Generally though, the thickness and refractive index can be estimated beforehand, allowing accurate data to be extracted.