10 Ellipsometery

Introduction

Very thin layers of material that are deposited onto the surface of another material are extremely important for many technological industries. Thin films are widely used to provide passivation, insulating layers between conducting electrodes, diffusion barrier and hardness coating for wear and tear resistance, The fabrication of Integrated Circuits consists of deposition and selective removal of a series of thin films. These thin films could be deposited using various deposition techniques including spin coating, vacuum evaporation, sputtering, vapour deposition and dip coating. To perform the functions for which they are designed, thin films must have accurate thickness, composition, roughness and other characteristics important to the particular application. These characteristics must often be measured both during and after thin film deposition. Thin film measurement techniques are classified into two categories, Optical and Mechanical. Mechanical techniques have drawbacks of speed and accuracy and always require a “step” in the film to measure thickness. They are often used to measure the thickness of metallic thin films. Optical techniques determine thin film characteristics by measuring how the films interact with light. These techniques can measure the thickness, roughness and optical parameters of the film. Optical parameters describe the behaviour of light propagating through and reflecting from the material and also provide the information regarding the band gap and composition of the material.

Optical techniques are usually the preferred techniques for the measurement of film thickness as they provide high accuracy, are non destructive and require little or no sample preparation. Ellipsometry is one the optical technique which is widely used for thickness measurement. It measures the reflectance at non-normal incidence and at two different polarizations.

The name ellipsometry comes from the fact that most often light becomes elliptically after passing through the medium. In ellipsometry change in polarization state is studied to infer properties of medium. There are already several other methods to measure thickness of the thin films. But ellipsometric measurements have advantage of their own.

Following are the advantages of the Ellipsometry to measure the properties of thin films:;

1.  It provides high precision in frequency

2.  It provides non destructive measurement

3.  It is faster in measurement

4.  Various optical properties like refractive index and thickness could be measured with high accuracy

5.  Lower limit of thickness of film can be studied.

However it has disadvantages of high cost, low spatial resolution, and complicated data analysis.

Principle of ellipsometry

The electric field component of light coming from laser can be divided in two components i.e. s-polarized and p- polarized component. In ellipsometry s-polarized and p-polarized light is made to incident at Brewster angle on the sample as shown in Fig.1. The final polarized light coming from the sample is observed. Let E_is and E_ip are incident electric filed vectors and E_rs and E_rp are incident electric filed vectors in s-polarized and p-polarized plane respectively. The electric field vectors of incident and reflected light overlap when = 90^o shown in Fig1. Let the incident is polarized at 45^o relative to E_is, thus E_is = E_ip since the amplitudes of p- and s- polarizations are same and phase difference between the polarization is zero. The only way that there can be a change in the reflection coefficients for two different coefficients of the light can be due to contribution of electric dipoles (dielectric) present in the sample.

Thus, when the light is incident on the sample, p- and s-polarizations show different changes in amplitude and phase. Ellipsometry measures the two values (ψ, ∆) that express the amplitude ratio and phase difference between p- and s-polarizations, respectively. If we take sample structure to be simple (homogeneous), the amplitude ratio is characterized by the refractive index ‘n’ and absorption is characterized by extinction coefficient. These two values can be determined by applying the Fresnel equations.

Let the amplitude of reflection for s and p polarization as

Consider an interface of two media having refractive indices of no and n1. Using well known Maxwell’s equations, the normal and parallel electric field components of the materials could be derived. Boundary conditions give the reflection coefficients (ρ) for the electric field components that lie in the plane of incidence (i) and are normal to its (r) as

Here, n1 sinθ1 = n0 sinθ0.

If there is absorption then the refractive index can be a complex quantity, hence the angles in general are complex. In the present case the system has three layers having refractive index n0, n1 and n2, first and third layers are separated by a second layer having thickness ‘d’ as shown in Fig. 2. For multiple reflection, the component of electric field (E_r) will be:

Where, E is the incident component of electric field, ρij and ij are reflection and transmission coefficients respectively for a light travelling from i^th medium to j^th medium.

Where λ is the wavelength of the incident light. If the second medium is absorptive, then can be complex quantity.

Amplitude of reflection ‘P’ is defined as

P= E_r/E = Type equation here.

Solving for this gives two values of X.We choose the value with lower imaginary part.

Methods to determine parameters ψ and ∆: There are two methods to determine these parameters.

1.  Photometric ellipsometry

2.  Null ellipsometry

1. Photometric Ellipsometry

Figure 3 shows the schematic of the set up used for the photometric ellipsometry.

The set up consists of two polarizers, one of which is placed before the sample and another one after the sample (used as Analyzer). The analyzer angle is kept fixed and polarizer angle is changed to obtain the variation in intensity of light at different angle. After linearly polarized light is made to incident on a sample because to different refractive index in two different directions we get an elliptically polarized light. Two methods of finding Ψ and ∆ will be discussed. The electric field at the detector in terms of Jones matrix formalism can be presented as follows:

2. Null Ellipsometry

 

In this technique along with the polarizer and analyzer, a quarter wave plate is also used as shown in Fig. 4. From the Fig. 4 it can be seen that the incident light passes through the polarizer making the light linearly polarized. After passing through the compensator the light becomes elliptically polarised and then made to pass through the sample. For some polarizer angle the elliptical polarization produced by sample can be compensated by the quarter wave plate giving a linearly polarized light after passing through the sample. This light can be made to obtain zereo intensity at some angle of analyzer. If the values of a sample are assumed to be ψ= 450 and ∆=900, then the intensity of detected light will be zero. Form Jones formulation as shown in Fig.5:

For ψ = A (A>0)∆= -2P-90⁰

Similarly two solutions for C=45⁰ are obtained. Therefore the four sets of P and A give zero intensity which gives the thickness of thin film.

Thickness of the film In null ellipsometry can be determined by keeping the compensator at 45⁰ and changing the Analyzer and polarizer angle to obtain the minimum intensity.

List of References and suggestive readings

1. HANDBOOK OF ELLIPSOMETRY, HG Tompkins – 2005
2. Spectroscopic Ellipsometry: Principles and Applications , Fujiwara- 1998
3. I. Ohlidal and D. Franta, Ellipsometry of Thin Film Systems, in Progress in Optics, vol. 41, ed. E. Wolf, Elsevier, Amsterdam, 2000, pp. 181–282
4. M. Schubert, Infrared Ellipsometry on semiconductor layer structures: Phonons, Plasmons, and Polaritons, Series: Springer Tracts in Modern Physics, Vol. 209, Springer (2004)