The ellipsometer

In this section:
1. Introduction
2. Principle of operation
3. Derivation of ellipsometer equations and curves
4. Table of ellipsometer data for different materials
5. Problems

The Ellipsometer

Introduction

An ellipsometer enables to measure the refractive index and the thickness of semi-transparent thin films. The instrument relies on the fact that the reflection at a dielectric interface depends on the polarization of the light while the transmission of light through a transparent layer changes the phase of the incoming wave depending on the refractive index of the material. An ellipsometer can be used to measure layers as thin as 1 nm up to layers which are several microns thick. Applications include the accurate thickness measurement of thin films, the identification of materials and thin layers and the characterization of surfaces.

Principle of operation

The principle of operation of an ellipsometer is illustrated by the schematic drawing of the ellipsometer shown in the figure below:

ellips1.gif

Fig.1: Schematic drawing of an ellipsometer

It consists of a laser (commonly a 632.8 nm helium/neon laser), a polarizer and a quarter wave plate which provide a state of polarization which can be varied from linearly polarized light to elliptically polarized light to circularly polarized light by varying the angle of the polarizer. The beam is reflected off the layer of interest and then analyzed with the analyzer. The operator changes the angle of the polarizer and analyzer until a minimal signal is detected. This minimum signal is detected if the light reflected by the sample is linearly polarized, while the analyzer is set so that only light with a polarization which is perpendicular to the incoming polarization is allowed to pass. The angle of the analyzer is therefore related to the direction of polarization of the reflected light if the null condition is satisfied. In order to obtain linearly polarized light after reflection, the polarizer must provide an optical retardation between the two incoming polarizations which exactly compensates for the optical retardation caused by the polarization dependent reflections at each dielectric interface. Since the amplitude of both polarizations was set to be equal, the ratio of the amplitudes after reflection equals the tangent of the angle of the analyzer with respect to the normal.

Derivation of the ellipsometer equations and curves

The calculation of the expected angles of the polarizer and analyzer corresponding to the null condition starts with the reflection coefficients at each of the dielectric interfaces for each polarization or:

(ell1)

(opt4)

(ell5)

(ell6)

with

(ell7)
where the subscripts, 0, 1 and 2 refer to air, the thin layer and the substrate and f0 is the angle of the transmitted wave with respect to the normal to the interface as shown in the figure below:

ellips2.gif

Fig.2: Incident, reflected and transmitted waves at a dielectric interface.

RTE = rTE rTE* is the reflectivity if the electric field is transverse to the propagation direction and parallel to the interface while RTM = rTM rTM* is the reflectivity if the magnetic field is transverse to the propagation direction and parallel to the interface. These reflectivities are angle dependent as well as being different for each polarization. An example of the reflection as a function of the incident angle is shown in the figure below:

refl.xls - ellips3.gif

Fig.3: Reflectivity at an air-GaAs interface (ignoring the imaginary part of the refractive index) as a function of the incident angle for both polarizations (TE and TM) of the incident wave. The reflectivity, RTM, goes to zero if the incident angle equals tan-1(n1/n0)

The two dielectric interfaces yield a combined reflection coefficient which can be obtained using the Fabry-Perot equations:

(ell8)
and

(ell9)
with

(ell10)
Combining the above equations yields the expression for rTM and rTE, or the reflection coefficient of the dielectric stack for each polarization. Based on the above discussion the ratio of the two reflection coefficients can be split into an amplitude and phase factor, thereby defining the ellipsometer parameters Y and D:

(ell11)
The angles Y and D are related to the measured angles, P1, A1, P2 and A2 in the following way:

D = 3p/2 - 2 P1 = 5p/2 - 2 P2 = 2p - P1 - P2

Y = A1 = p - A2 = (A1 - A2 + p)/2

The minimal signal is obtained when both polarizations incident on the analyzer are in phase. This can be obtained for two different positions of the polarizer, hence the two values P1 and P2. In principle one could measure either one. In practice both values are measured to eliminate any possible misalignment of the instrument thereby yielding a more accurate result. Y - D curves are typically used to visualize the ellipsometer parameters for different layer thickness and refractive index. An example of such curves as obtained for silicon dioxide layers (n1 = 1.455) on silicon (n2 = 3.875 - 0.018 i ) using a helium-neon laser (l = 632.8 nm) is shown below.

ellips.xls - ellips4.gif
Fig.4: Y - D curves for silicon dioxide on silicon. The layer thickness increases counter clock wise from 0 (square marker on the left) in steps of 10 nm (black diamonds) and in steps of 100 nm (squares). The incident angle of the He/Ne laser beam (632.8 nm) is 70 degrees.
Since the silicon dioxide was assumed to be transparent, one finds the values for both Y and D to be identical for layers which differ in thickness by l/(2 n1 cos f1) = 284 nm. The corresponding curves for the measured values P1, A1, P2 and A2 are also shown in the figure below:

ellips5.gif

Fig.5: A1 - P1 and A2 - P2 curves for silicon dioxide on silicon. The thickness increases counter clock wise from 0 (at the square marker on the left) for A1 versus P1 and counter clock wise from 0 (square marker on the right) for A2 versus P2 , both in steps of 10 nm (black diamonds) and in steps of 100 nm (squares). The incident angle of the He/Ne laser beam (632.8 nm) is 70 degrees from the normal to the surface.

The table below lists the refractive index n* = n - i k for different materials as well as the minimum or maximum angle of the analyzer and the half wavelength thickness when using a He/Ne laser and an incident angle of 70 degrees.

The refractive indices from this table can be used to generate the Y - D curves for any material combination. In addition the minimum/maximum value of A1 can be used to help identify an unknown material, since it is directly related to the refractive index which is unique for each material. However when using this technique one should realize that the surface of the material should be polished and clean. A thin oxide layer (1 - 3 nm) which naturally grows on most materials must therefore be removed before measuring A1. Since the cleaning procedures vary from material to material, this technique is rather limited when trying to identify unknown materials. However it is a very sensitive and therefore valuable technique to verify that one has a clean surface.

Table: containing refractive indices of various materials.

Problems:

1. Derive equations for r01,TE and r01,TM
2. Derive expressions for Y and D
3. Prove that the reflectivity, RTM, goes to zero if the incident angle, f0, equals tan-1(n1/n0) and only if the imaginary part of the refractive index is zero.