Spectroscopic Ellipsometry (SE)

Spectroscopic ellipsometryis a characterization technique which is used to determine the thickness of the layers, their optical properties and composition. It is based on the polarization difference (incident vs. reflected) measurement over a spectral region from 235 nm to 10000 nm.

A probe beam, whose polarization state is known, is launched on the sample under test and the reflected beam is analysed for its polarization state. The two important parameters measured for samples are the wave function ( ) and the phase ( ). Probe beam is a linearly polarized which gets modified to an elliptically polarized beam (reflected), as depicted in Fig. 2.13 The polarization ellipse can be described using a typical p-s coordinate system, where the s-direction is parallel to the sample surface and perpendicular to the propagation direction. However the p-direction is considered to be contained in the plane of incidence and perpendicular to the propagation direction. The plain containing the output beam, the input beam, and the direction normal to the sample surface, is defined as the plane of incidence. The angle between the directions of input beam and normal to the sample surface is defined as the angle of incidence.

Light is a transverse electromagnetic wave which is made up of fluctuating magnetic and electric fields which are perpendicular to each other. Ordinary white light consist of waves fluctuating at possibly all angles. If the light contains waves fluctuating in only a specific plane, it is considered to be "linearly polarized" (Fig. 2.13).

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Fig. 2.13 Geometry of an ellispometric experiment showing the p- and s- directions.

It can generate light with a specific angle of vibration, by using polarization effect, which allow only light with a specific vibration angle [11].

Linear polarization can be understood as a special case of circular polarization. It can consider a case where we have one XY plane polarized wave and one YZ plane polarized wave. If the waves are in phase that means they reach their maximum and minimum points at the same time, then their vector sum will lead to a 45 linearly polarized wave (Fig. 2.14a). However, if the two waves are considered 180 out of phase, the resultant will be 45 linearly polarized in the opposite manner (Fig. 2.14b). Further, if the two waves are 90 degrees out of phase and of the same magnitude, the resultant will be circularly polarized (Fig. 2.14c). In general, the two field components Ex and Ey do not have to be equal in magnitude, and therecan be any phase relationship (not necessarily 90 or 180 ). An ellipse as a function of time is traced by the tip of the total E-field vector for a general polarization state.As we know that a line segment and a circle are two special types of ellipse, therefore, linearly and circularly polarized light can be achieved by elliptically polarized light. In ellipsometry, and are measured by determining the polarization of ellipse of the probe beam, that’s why this is called ellipsometry. The values and

are expressed using the ratio of Fresnel reflection coefficients R~_s

and R~_p

for polarized light s- and p- , respectively according to the relation:

40 a)

b)

c)

Fig. 2.14 The schematic representation for a)-b) linear and c) circular polarized light.

) exp(

~ tan

~

~ i

R R

s

p (2.7)

In ellipsometry the ratio of two values is estimated which makes it precisely accurate and highly reproducible. Furthermore, the ratio is complex number which contains the information about

“phase” ( ) making the measurement quite sensitive.

A J.A. Woollam Co., Inc. VASE (Variable Angle Spectroscopic Ellipsometry) which is based on the traditional rotating analyzer design canacquire data even at multiple angles of incidence.

The basic reason of using a variable angle of incidence instrument is to gather data at angles of incidence around the pseudo-Brewster angle for each and every sample. When the incidence angle is close to pseudo-Brewster angle, the measured values are found to be around 90 , and this range of values gives the sample’s most sensitive measurement. It is observed that for semiconductor samples, the pseudo-Brewster angle typically comes out to be around 75 . The ellipsometric system WVASE32^TM is used in the Electrical Engineering & Electronics Department; it has the possibility to cover the 245-1000 nm spectral range (470 wavelengths) at a

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fixed incidence angle of 75 and also has fast acquisition time, ~ 390 wavelengths in less than 1 second. Within WVASE32^TM, optical constants can be represented as a complex refractive index,

~

~ n ik

n or as a complex dielectric function,~ ₁ i ₂ WVASE32^TM considers any optical constants with a positive imaginary part to represent an absorbing material [11].

Using SE, we can determine thin film thicknesses as well as optical constants. To achieve best results with ellipsometry the film thickness should not vary too much in comparison to the wavelength of the light used. This is due the working principle of ellipsometry in which measurement is wavelength limited and also periodic, so if the variation in thickness is much the accurate thickness cannot be estimated. Ellipsometry is especially for dielectric films and limited to one layer analysis with single wavelength ellipsometry. Also, for a valid spectroscopy ellipsometry analysis, roughness of the sample surface or film interfaces should be smaller than ~ 10% of the input beam wavelength. Larger features have concerns of causing non-specular scattering of the incident beam and depolarization of the specularly reflected beam. Finally, the films thicknessunder study should not vary by more than ~ 10% over the spot width on the sample surface, else the assumption of parallel filminterfaces will not be valid, and the calculated data cannot be expected to be close to the experimental data.

SE is an optical technique which requires aprecisely formulated model of the measurement process to analyse the measured data. The main measurements/ parameters analysed by ellipsometric models are the thickness of the layers, andthe optical constants of the substrate and oxide layers. To get good model fits to the measured data we need highly accurate optical constants.It is evident that on the visible, near-UV, and near-IR wavelength ranges, the premier technique for measuring optical constants is ellipsometry.

Precise models are made in such a way that the calculated ellipsometric data matches closely with the experimental data ( , ), which helps in determining the structure and composition of the samples.Maximum likelihood estimator defines the goodness of match between the data calculated using the model and the one acquired experimentally. The value of maximum likelihood estimator must be a smallest possible positive number approaching to zero, in the condition of a perfect match in the case of calculated and the experimental data. WVASE32^TM uses the mean-squared error (MSE) value as maximum likelihood estimator which is given as:

2

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where M is the total number of variable parameters used in the model, N is the number of ( , ) pairs, and are the standard deviations calculated on the experimentally calculated data points.

Further to estimate the set of values for the variable model parameters which yields a single unique absolute minimum of the MSE fitting is required. This problem is a minimization problem where we need to estimate the minimum value of the MSE. The minimum value should happen at a small value of the MSE, and it should be fairly sharp as a function of the variables. Only one out of many sets of variable parameters will lead to the lowest MSE from the given model. To obtain calibration parameters, WVASE32^TM implements the Levenberg-Marquardt algorithm for all model minimizations and for analysis of the calibration data. The primary drawback or limitation of used algorithm is that sometimes it tends to freeze on local minima of the MSE surface, which leads to an incorrect result. Mentioned limitation can be addressed by fitting from widely separated initial guesses for the variables ensuring the best suitable-fit minimum located by the fitting algorithm is actually the true MSE surface’s minimum[11].

2.7 X-Ray Diffraction