A User's Guide to Ellipsometry

Copyright

Copyright © 1993 by Harland G. Tompkins All rights reserved.

Bibliographical Note

This Dover edition, first published in 2006, is an unabridged republication of the work originally published in 1993 by Academic Press, Inc., San Diego, CA.

9780486151922

Manufactured in the United States of America

Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y 11501

Table of Contents

Title Page

Copyright Page

Preface

Chapter 1 - Theoretical Aspects

1.1 Description of an Electromagnetic Wave

1.2 Interaction of Light with Material

1.3 Polarized Light

1.4 Reflections

1.5 Ellipsometry Definitions

1.6 References

Chapter 2 - Instrumentation

2.1 Fundamentals and History

2.2 Optical Elements

2.3 The Manual Null Instrument

2.4 Rotating Element Instruments

2.5 References

Chapter 3 - Using Optical Parameters to Determine Material Properties

3.1 Del/Psi and n and k for Substrates

3.2 The Calculation of Del/Psi Trajectories for Films on Substrates

3.3 Trajectories for Transparent Films

3.4 Trajectories for Absorbing Films

3.5 Two-Film Structures

3.6 References

Chapter 4 - Determining Optical Parameters for Inaccessible Substrates and Unknown Films

4.1 Inaccessible Substrates and Unknown Films

4.2 Determining Film-Free Values of Del and Psi

4.3 Determining the Complex Index of Refraction of the Film

4.4 Summary

4.5 References

Chapter 5 - Extremely Thin Films

5.1 General Principles

5.2 Some Examples of Extremely Thin Films

5.3 Summary

5.4 References

Chapter 6 - The Special Case of Polysilicon

6.1 General

6.2 Range of the Optical Constants

6.3 Del/Psi Trajectories in General

6.4 Effect of the Coefficient of Extinction

6.5 Effect of the Index of Refraction

6.6 Requirements

6.7 Measuring the Thickness of Oxide on Polysilicon

6.8 Simplifications

6.9 References

Chapter 7 - The Effect of Roughness

7.1 General

7.2 Macroscopic Roughness

7.3 Microscopic Roughness

7.4 Perspective

7.5 Substrate Roughness

7.6 Film Growth with Roughness

7.7 References

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: - Adsorption on Metal Surfaces

Case 12: - Silicon-Germanium Thin Films

Case 13: - Profiling of HgCdTe

Case 14: - Oxides and Nitrides of Silicon

Appendices

INDEX

Astronomy

DOVER BOOKS ON ENGINEERING

Preface

Ellipsometry is a very old technique for studying surfaces and thin films. It is also used extensively in integrated circuit (IC) manufacture and development, in corrosion science, and various other areas. The fundamental principles of the technique are well known. Although the literature is extensive, the few available books deal with the subject on a level that is too mathematical or theoretical for most casual users. Most users are skilled in microelectronics IC fabrication, corrosion science, etc., and are not particularly skilled in physical optics.

Many commercial instruments are available and most present-day instruments utilize an associated microcomputer of some form. Software is normally available so that for certain specified tasks, the system can be used in a turnkey manner without knowing very much about how the ellipsometer does what it does. However, the tasks which can be done in a turnkey manner represent only a small subset of the possible applications of the technique.

This book is intended for the technologist who has an instrument available and who wants to use it in its present form without significant modification. We will assume that the instrument is kept in working condition by the instrument manufacturer’s technical representative. Correspondingly, we will not spend any time on alignment of instruments or on instrument design aspects. The angle of incidence on most turnkey instruments is 70° and a He-Ne laser with a wavelength of 6328 Å is used as the light source. These values will be generally used in this text unless stated otherwise.

This book is specifically designed for the technologist who wishes to stretch the use of the technique beyond the turnkey applications. We shall deal with situations where all of the critical information is not available and those where approximations are required.

Ellipsometry is an optical technique. In order for the user to understand how the ellipsometer does what it does, we shall begin by providing some of the fundamentals of optics that are particularly applicable. We take these from physics textbooks on optics, taking only those ideas applicable to our purposes here. These will lead us to the definitions of Del and Psi, which are the parameters an ellipsometer measures. We shall refer to the change in values of Del and Psi as a function of film thickness, index of refraction, etc., as the Del/Psi trajectories. Understanding these trajectories is key to understanding the technique and stretching it beyond the turnkey applications. We have tried to keep the theoretical aspects short and to the point, including only those ideas that are directly related to the subject at hand.

As indicated above, this book is intended for users who already have an operating ellipsometer. It consists of seven chapters followed by 14 case studies. Chapter 2, on Instrumentation briefly describes how the commercially available instruments work. A chapter entitled Using Optical Parameters to Determine Material Properties deals with how one goes from the measured optical parameters to deducing information about the material being studied. The chapter on Determining Optical Parameters for Inaccessible Substrates and Unknown Films, shows what to do when all of the necessary information is not available.

A question often asked is How thin a film can be measured with ellipsometry? Chapter 5 considers this question and uses some examples of the adsorption of single layers of atoms. In the microelectronics industry, a commonly used material is polysilicon. Chapter 6 deals with the difficulties specific to this material when using ellipsometry. Some suggestions are made for dealing with some of the difficulties. Finally, the last chapter discusses the effect of substrate roughness.

Examples are the best way to show how ellipsometry can be stretched beyond the turnkey applications. Fourteen case studies are included to illustrate various uses of ellipsometry. These are works of others taken from the reviewed literature.

Finally, we include an appendix on calculating the Del/Psi trajectories discussed in the text, an appendix on Effective Medium Considerations in which we discuss how one calculates an effective index of refraction with mixtures of materials or rough surfaces, and an appendix where we list some of the literature values of the optical constants of various materials.

No one writes a book such as this alone, without significant input from others. Correspondingly, I would like to acknowledge the management at Bell Laboratories and at Motorola for allowing me the necessary time for the exploration needed to understand and use this method and to acknowledge my co-workers at these institutions for stimulating me to do so. In addition, I would like to acknowledge my wife, Rose Ann Tompkins, for encouraging me to write the book and for spending many hours alone without complaint while I sat in front of a computer, buried in the intricacies of composing the book.

Chapter 1

Theoretical Aspects

Electromagnetic waves and polarized light are treated in textbooks ¹, ², ³, ⁴, ⁵ and reference books⁶ 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

(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 ξ 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.

Figure 1-1. An electromagnetic wave at a fixed time, represented schematically.

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 shown⁷ that the energy density is proportional to the square of the amplitude and that the intensity I is given by

(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

(3)

Figures 1-2. Light reflecting from and passing through an interface between air and a material characterized by the complex index of refraction Ñ2.

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 = 3x10¹⁰ 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

(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

(5)

where α is the absorption coefficient. This integrates out to

(6)

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

(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

(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.,

(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

(10)

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

(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.⁸

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

(12)

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

(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.

Figure 1-3. Index of refraction n as a function of wavelength for silicon nitride.⁹

Figure 1-4. Extinction coefficient k as a function of wavelength for polycrystalline silicon.¹⁰

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.

Figure 1-5. The value of the refractive index of silicon dioxide as a function of temperature in °C. (After Yu¹¹)

Figure 1-6. Temperature dependence of the refractive index of silicon. Circles are determined ellipsometrically in present experiments, the dashed curve is that given by Ibrahim and Bashara.¹² (After Hopper¹³)

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.

Figure 1-7. If two linearly polarized light beams which are in phase are combined, the resultant light beam is linearly polarized.

Figure 1-8. If two linearly polarized light beams which are out of phase are combined, the resultant light beam is elliptically polarized. In this particular example, they are out of phase by 90°, hence the resultant beam is circularly 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