A User's Guide to Ellipsometry

by Harland G. Tompkins

Overview

This text on optics for graduate students explains how to determine material properties and parameters for inaccessible substrates and unknown films as well as how to measure extremely thin films. Its 14 case studies illustrate concepts and reinforce applications of ellipsometry — particularly in relation to the semiconductor industry and to studies involving corrosion and oxide growth.
A User's Guide to Ellipsometry will enable readers to move beyond limited turn-key applications of ellipsometers. In addition to its comprehensive discussions of the measurement of film thickness and optical constants in film, it also considers the trajectories of the ellipsometric parameters Del and Psi and how changes in materials affect parameters. This volume also addresses the use of polysilicon, a material commonly employed in the microelectronics industry, and the effects of substrate roughness. Three appendices provide helpful references.

Product Details

ISBN-13:	9780486151922
Publisher:	Dover Publications
Publication date:	03/21/2013
Series:	Dover Civil and Mechanical Engineering
Sold by:	Barnes & Noble
Format:	eBook
Pages:	272
File size:	13 MB
Note:	This product may take a few minutes to download.

About the Author

Author Harland G. Tompkins is a Mesa, Arizona-based scientist associated with Motorola, Inc.

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A User's Guide to Ellipsometry

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By Harland G. Tompkins

Dover Publications, Inc.

Copyright © 1993 Harland G. Tompkins
All rights reserved.
ISBN: 978-0-486-15192-2

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CHAPTER 1

Theoretical Aspects

Electromagnetic waves and polarized light are treated in textbooks and reference books on optics. We review here some of the salient features that are directly applicable to ellipsometry.

1.1 Description of an Electromagnetic Wave

The electromagnetic wave is a transverse wave consisting of both an electric field vector and a magnetic field vector, which are mutually perpendicular and are perpendicular to the propagation direction of the wave. It can be specified with either the magnetic field vector or the electric field vector. For simplicity, we shall consider the electric vector only. The light wave can be represented mathematically as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where A is the electric field strength of the wave at any given time or place. Ao is the maximum field strength and is called the "amplitude," x is the distance along the direction of travel, t is time, v is the velocity of the light, λ is the wavelength, and [xi] is an arbitrary phase angle, which will allow us to offset one wave from another when we begin combining waves.

If we consider the wave at a fixed time, the variation of the electric field with position can be represented as shown in Figure 1-1. We can identify locations where the electric field is maximum, minimum, and zero. In this particular case, the electric field variation occurs only in the vertical direction.

Waves transport energy, and the amount of energy per second that flows across a unit area perpendicular to the direction of travel is called the "intensity" of the wave. It can be shown that the energy density is proportional to the square of the amplitude and that the intensity I is given by

I = cA2/8π (2)

where c is the speed of light.

1.2 Interaction of Light with Material

1.2.1 The Complex Index of Refraction

Suppose we have light passing from one medium (e.g., ordinary room air) into another medium that is not totally transparent, as suggested by Figure 1-2. Several phenomena occur when the light passes the interface. One phenomenon that occurs is some of the light is reflected back and does not enter the second medium. We shall deal with this reflected component later. For the moment, let us consider the light that enters the second medium.

The parameter we use to describe the interaction of light with the material is the complex index of refraction Ñ, which is a combination of a real part and an imaginary part adnd is given as

Ñ = b - jk (3)

"n" is also called the "index of refraction" (this sometimes leads to confusion), and "k" is called the "extinction coefficient". (In the scientific literature, the imaginary number, the square root of -1, is denoted either as i or j. We will use j.) For a dielectric material such as glass, none of the light is absorbed and k = 0. In this case, we are only concerned with n.

The velocity of light in free space is usually designated as "c" and its value is approximately c = 3x10 cm/sec. The velocity of light in air is not significantly different from this value. When light enters another medium, however, the velocity will be different. Let us designate its value in the medium as "v." We define the index of refraction "n" to be

n = c/v. (4)

Clearly, the value of n for free space is unity.

Before we define the extinction coefficient "k," let us consider first the absorption coefficient "α." In an absorbing medium, the decrease in intensity I per unit length z is proportional to the value of I. In equation form this is

dI(z)/dz = -α(z) (5)

where α is the absorption coefficient. This integrates out to

I(z) = Io e-αz(6)

where Io is the value of the intensity at the surface of the absorbing medium. The extinction coefficient k is defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

where λ is the vacuum wavelength and λm is the wavelength in the medium. From Equation 6, the value of I(z) is 1/e (approximately 37%) of the value of Io when αz = 1. This occurs when z = 1/α or when

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

1.2.2 Laws of Reflection and Refraction

When the light beam reaches the surface, as suggested in Figure 1-2, some of the light is reflected and some passes into the material. The law of reflection says that the angle of incidence is equal to the angle of reflection, i.e.,

φi = φr(9)

In the figure, both are shown as φ1. The part of the light beam that enters the material at an angle φ1 does not continue in the same direction, but is refracted to a different angle φ2. The law of refraction is called "Snell's law" after its discoverer (in the early 1600s) and is given by

Ñ1 sin φ1 = Ñ2 sin φ2(10)

for materials in general. For dielectrics, k = 0 and Equation 10 becomes

n1 sin φ1 = n2 sin φ2(11)

Equation 11 consists of only real numbers, and hence is reasonably straightforward. In Equation 10, k for the first medium usually is zero, hence Ñ1 = n1. If Ñ2, on the other hand, is complex, i.e., k2 is nonzero, then the angle φ2 is complex and the quantity no longer has the simple significance of the angle of refraction.

1.2.3 Dispersion

Up to this point, we have referred to the real and imaginary parts of the index of refraction as "n" and "k". It should be mentioned that these are not simple constants for a given medium, but are in fact functions of the wavelength λ. This is why when white light enters a prism, it emerges with the various colors separated.

We use the term "dispersion" to describe how the optical constants change with wavelength. The equation for n(λ) is often approximated by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

where n1, n2, and n3 are called the "Cauchy coefficients." The equation for k(λ) is approximated by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

where k1, k2, and k3 are called the "Cauchy extinction coefficients." Figures 1-3 and 1-4 show typical examples of the variation of n and k as a function of wavelength.

1.2.4 The Effect of Temperature

The optical constants are functions of temperature, in addition to being functions of wavelength. Figure 1-5 shows an example of how n varies with temperature. Figure 1-6 shows corresponding data for silicon. Small variations in temperature are insignificant, but for in situ experiments where the material may be at elevated temperatures, one must take into account this variation.

1.3 Polarized Light

Most light sources emit light that has components with electric fields oriented in all of the possible directions perpendicular to the direction of travel. We refer to this as unpolarized light. If all of the photons in a light beam have the electric field oriented in one direction, the light is referred to as polarized or, more completely, linearly polarized light. Some light sources emit polarized light. In addition one can obtain polarized light by passing the light beam through an optical element or by causing the beam to make a reflection under some specific conditions.

1.3.1 Linearly Polarized Light

Let us suppose that we have two light beams with the same frequency moving along the same path, one polarized in the vertical plane and the other polarized perpendicular to the vertical plane, as suggested in Figure 1-7. For simplicity, let us assume that the amplitude of both waves are the same. Suppose that the maxima of the two beams coincide. This is the same as saying that the phase is the same. These two beams can be combined to give a resultant light beam that is also linearly polarized. In our illustration, we have suggested that the intensity of the two beams is the same and hence the resultant beam would be linearly polarized at 45° from each of the original beams. If the intensities were not the same, the beams would still have been linearly polarized, but the angle of polarization would have been different from 45°. The key point here is that when two linearly polarized waves with the same wavelength (or frequency) are combined in phase, the resultant wave is linearly polarized.

1.3.2 Elliptically Polarized Light

Suppose we have two light beams traveling along the same path and that they have the same wavelength. As opposed to the previous example, let us suppose that the maxima do not coincide, but are out of phase by some amount (in Figure 1-8, the maximum of one coincides with the zero of the other, giving a phase difference of 90°). When these two waves are combined, the tips of the arrows do not move back and forth in a plane as in the previous example. In this case, they move around in a manner such that if you looked end on, it would appear to be a circle. We refer to this as circularly polarized light. If the phase difference is other than 90°, we have, in general, elliptically polarized light.

There are several ways to obtain elliptically polarized light in practice. Of primary interest to us is the fact that when linearly polarized light makes a reflection on a metal surface, there is a shift of the phases of both the components (parallel and perpendicular to the plane of incidence). The shift is, in general, not the same for both components, hence the resultant light will be elliptically polarized. The amount of ellipticity induced depends on various things including the optical properties of the substrate as well as thickness and optical properties of overlying films.

Another way to change the ellipticity of polarized light is to pass the beam through an optical element called a quarter-wave plate. We shall discuss this optical element as well as the polarizer later in Chapter 2.

1.4 Reflections

1.4.1 Coordinate System for Reflections

Ellipsometry invariably involves the reflection of light from a surface. To describe this reflection, let us suppose that we have an incident plane wave moving in the direction suggested in Figure 1-9 and suppose that this wave makes a reflection from the surface as suggested in the figure. Let us define the plane of incidence as being that plane that contains the light beam prior to and after the reflection. The plane of incidence also contains the normal to the surface.

We shall refer to plane polarized waves that are in the plane of incidence as "p waves" and plane waves polarized perpendicular to the plane of incidence as "s waves" ("s" comes from the German word "senkrecht").

1.4.2 Fresnel Reflection Coefficients and the Brewster Angle

Suppose that we have a light beam making a reflection at an interface between medium 1 and medium 2 as shown in Figure 1-10. Some of the light is reflected and some is transmitted. The Fresnel reflection coefficient r is the ratio of the amplitude of the reflected wave to the amplitude of the incident wave for a single interface. The Fresnel reflection coefficients are given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

where the superscript refers to waves parallel or perpendicular to the plane of incidence and the subscript refers to either medium 1 or medium 2. Corresponding equations exist for transmission, but these are not used in ellipsometry.

Recall that the intensity of the light is proportional to the square of the amplitude. We define the reflectance R as the ratio of the reflected intensity to the incident intensity. For a single interface, this can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

If both media are dielectric in nature (i.e., k = 0) then the complex index of refraction Ñi becomes simply ni. It is useful to illustrate some of the features of these equations with an example. Let us suppose that medium 1 is air, with n1 = 1.0, and medium 2 is a dielectric such as silicon nitride (n2 = 2.0). Figure 1-11A shows a plot of the Fresnel coefficients (Equation 14) as a function of the angle of incidence. Figure 1-11B shows the resulting reflectances Rp and Rs from Equation 15.

At normal incidence, φ1 = 0 and from Snell's law, φ2 = 0 and all of the cosines in equation 14 are equal to +1. We have, then,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

For the example mentioned, the reflectance for both the parallel and perpendicular component is about 11%. Note that at normal incidence the distinction between parallel and perpendicular disappears.

For other than normal incidence, we see from Figure 1-11A that rs is always negative and nonzero. This follows from the fact that for 0 < φ < 90°, cos φ is always positive and hence the denominator of both equations is always positive and nonzero. In our example, we have chosen n2 > n1. One of the results of this from Snell's law, Equation 10, is that φ2< φ1 and hence cos φ2 > cos φ1. From this we can see that n2 cos φ2 > n1 cos φ1, which implies that the numerator of the equation for the s wave is always negative and nonzero. Hence Rs is always positive and nonzero.

Figure 1-11A shows that rp changes from positive to negative, i.e., goes through zero, as the angle of incidence increases. At this point, the reflectance Rp is also zero. This too can be shown mathematically. Let us consider the numerator for the p wave. From Snell's law,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

We can use this equation to eliminate φ2 from the numerator and express it in terms of φ1, n2, and n1. For the specific angle φ1 such that

tan φ1 = n2/n1(18)

it can be readily shown by algebraic manipulation that the numerator is zero.

The significance of this is that when unpolarized light makes a reflection with this specific angle of incidence, none of the light polarized parallel to the plane of incidence is reflected, i.e., it is all transmitted. The resulting reflected light is polarized perpendicular to the plane of incidence. This specific angle given by Equation 18 is called various names, including the "polarizing angle," the "Brewster angle," and the "principal angle." For the example shown in Figure 1-11, with n2 = 2.0, the Brewster angle is about 63.43°. This is indicated by the dashed line.

This is the principle by which polarized sunglasses remove the glare from the highway or water surface in front of the user. Most of the light polarized parallel to the plane of incidence has been removed by the reflection. Glare is caused by the remaining reflected light polarized perpendicular to the plane of incidence. The sunglasses are simply polarizers, which remove most of this component. Light reflected from a water surface will be more intense than light transmitted from below the surface. This glare often inhibits a person above the surface from viewing features (fish or alligators) below the water's surface. A photographer may use a polarizer to remove this component thus allowing the light transmitted from below the surface to be used for a good photo.

It might be noted that although all of the reflected light is polarized in one direction, the transmitted light still has components from both polarizations, with the amount of perpendicularly polarized light reduced. Since about 36% of the perpendicularly polarized light was reflected, the transmitted beam contains only 64% of the incident perpendicularly polarized light.

Multiple interfaces can be used to remove successively more and more of the perpendicularly polarized light until the transmitted light is virtually pure polarized light in the plane of incidence. Historically, this is one method of obtaining polarized light.

When the reflecting surface is not a dielectric, i.e., k is nonzero, the situation becomes more complicated. The Fresnel reflection coefficients are, in general, complex numbers, so the concept of being greater than zero, passing through zero, and being less than zero is no longer appropriate. Because of this, there is no plot analogous to Figure 1-11A for metals. The reflectance, however, is a real number. Figure 1-12 shows the reflectance for both the p wave and the s wave for Ni. This is similar to the corresponding plot for dielectrics, except that the reflectance does not go to zero. The angle for which Rp is minimum is called the principal angle of incidence and is denoted by [??]. At the principal angle of incidence the phase difference between the two components is 90°.

It might be noted that high reflectance is obtained when the index of the substrate is significantly different from that of the ambient. The numerator of the Fresnel coefficients will be larger under these conditions. This will occur when either ns is significantly different from unity or when ks is large.

1.4.3 Total Reflection Coefficients for Multiple Interfaces

The above discussion concerned a single interface. We simply disregarded the light transmitted through the interface. Many real world situations involve multiple interfaces as suggested by Figure 1-13.

(Continues...)

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Excerpted from A User's Guide to Ellipsometry by Harland G. Tompkins. Copyright © 1993 Harland G. Tompkins. Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

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Table of Contents

1. Theoretical Aspects
2. Instrumentation
3. Using Optical Parameters to Determine Material Properties
4. Determining Optical Parameters for Inaccessible Substrates and Unknown Films
5. Extremely Thin Films
6. The Special Case of Polysilicon
7. The Effect of Roughness
Case Studies
Case 1. Dissolution and Swelling of Thin Polymer Films
Case 2. Ion Beam Interaction with Silicon
Case 3. Dry Oxidation of Metals
Case 4. Optical Properties of Sputtered Chromium Suboxide Thin Films
Case 5. Ion-assisted Film Growth of Zirconium Dioxide
Case 6. Electrochemical/Ellipsometric Studies of Oxides on Metals
Case 7. Amorphous Hydrogenated Carbon Films
Case 8. Fluoropolymer Films on Silicon from Reactive Ion Etching
Case 9. Various Films on InP
Case 10. Benzotriazole and Benzimidazole on Copper
Case 11. Gas Adsorption on Metal Surfaces
Case 12. Silicon-Germanium Thin Films
Case 13. Profiling of HgCdTe
Case 14. Oxides and Nitrides of Silicon
Appendices

Editorial Reviews

For technologists without much background in physical optics, explains some of the less elementary applications and methods of ellipsometry, a technique for studying surfaces and thin films that is used extensively in integrated circuit manufacture and development, corrosion science, and other areas. Assumes the reader has an instrument in working order, and uses the standard settings and values. Annotation c. Book News, Inc., Portland, OR (booknews.com)

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