## Basic Theory

For general information on electromagnetic waves and optics, the reader should refer to textbooks (Ref 1, 2, 3, 4, 5) and reference books (Ref 6) on the subjects. Some of the salient features that are directly applicable to reflectometry and ellipsometry (Ref 7, 8) are reviewed here.

The electromagnetic wave is a transverse wave consisting of both an electric field vector and a magnetic field vector that are mutually perpendicular and perpendicular to the propagation direction of the wave. The wave can be specified with either the magnetic field vector or the electric field vector. For simplicity, the electric vector only is considered. The light wave can be represented mathematically as a sine wave with amplitude A. 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 and will be denoted as I. The intensity, or energy density (Ref 7), is proportional to the square of the amplitude.

Reflection. For a single film on a substrate (Fig. 1), reflections rather than transmission, are the primary concern. As shown in Fig. 1, some of the light is reflected and some passes into the material at the air-to-solid interface. At the second interface, again, some is reflected and some is transmitted. The various rays that leave the material from the top surface combine to make the outgoing wave. For reflectometry, the ratio of the intensity of the outgoing wave to the intensity of the incoming wave is measured. Reflectometry measurements are often made at normal (perpendicular) incidence. The various rays give constructive or destructive interference, depending on the wavelength of the light, the thickness of the film, and the optical properties of the various materials. For the reflectometry technique, one measures the reflected intensity versus the wavelength of light to deduce the film thickness.

Fig. 1 Schematic of light reflected and transmitted at film interfaces. The outgoing beam is a combination of all of the rays emerging from film from the top interface. Each material is characterized by the index of refraction N1. The thickness of the film is d. Source: Ref 8

For ellipsometry, the measured parameter is the ratio of the wave amplitude parallel to the plane of incidence versus the wave amplitude perpendicular to the plane of incidence. The reflection process also causes a phase shift between these two waves, and this phase shift is measured during ellipsometry. The amplitude ratios and phase shifts are functions of the wavelength, thickness, optical properties of the various materials, and angle of incidence.

Polarized Light. Most light sources emit unpolarized light, or light with electric-field components oriented in all possible directions perpendicular to the direction of travel. If all the photons in a light beam have the electric field oriented in one direction, the light is referred to as polarized light 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.

Figure 2(a) illustrates 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. In this case, the maxima of the two beams coincide (i.e., the phase is the same). These two beams can be combined to give a resultant light beam that is also linearly polarized. 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 and lies in a plane.

Fig. 2 Linear and elliptical polarization. (a) If two linearly polarized light beams that are in phase are combined, the resultant light beam is linearly polarized. (b) If two linearly polarized light beams that are out of phase are combined, the resultant light beam is elliptically polarized. In this particular example, they are out of phase by 90°. Because the amplitudes are equal, the resultant beam is circularly polarized. Source: Ref 8

Fig. 2 Linear and elliptical polarization. (a) If two linearly polarized light beams that are in phase are combined, the resultant light beam is linearly polarized. (b) If two linearly polarized light beams that are out of phase are combined, the resultant light beam is elliptically polarized. In this particular example, they are out of phase by 90°. Because the amplitudes are equal, the resultant beam is circularly polarized. Source: Ref 8

Figure 2(b) shows two beams where the maxima do not coincide, but are out of phase. 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. This is, in general, elliptically polarized light. The key point is that when two linearly polarized waves with the same wavelength (or frequency) are combined out ofphase, the resultant wave is elliptically polarized or spiraling in three-dimensional space.

In ellipsometry the important fact is that when linearly polarized light makes a reflection on a metal surface, there is a shift in the phases of both the components (parallel and perpendicular to the plane of incidence). For non-normal incidence, the shift is, in general, not the same for both components, and hence the resultant light will be elliptically polarized. The induced amount of elliptical polarization depends on various factors including the optical properties of the substrate as well as the thickness and optical properties of overlying films. From this concept of elliptical polarization, the term ellipsometry takes its name for the measurement of induced ellipticity.

The Complex Index of Refraction. When light passes from one medium (e.g., ordinary room air) into another medium that is not totally transparent (Fig. 1), several phenomena occur at the interface. Some of the light is reflected back and does not enter the second medium, while an unreflected component enters the second medium. The unreflected component will be considered first.

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

where n is also called the index of refraction (sometimes leading to confusion), k is called the extinction coefficient, and j is the imaginary number V-1.

For a dielectric material such as glass, none of the light is absorbed and k= 0. In this case, only n is being considered. Both n and k are functions of the wavelength. It is not uncommon for a material to have k = 0 for a range of wavelengths and k 0 for another wavelength range.

The index of refraction n is defined to be:

where c and v are the velocities of light in free space and in the material, respectively.

The extinction coefficient k is a measure of how rapidly the light is absorbed as a function of depth in the material. A transparent material such as glass has an extinction coefficient of zero. Metals typically have values ranging from k = 2 to about k = 6.

Dispersion. It should be noted that n and k are not simple constants for a given medium, but are in fact functions of the wavelength, x. This is the reason that white light entering a prism emerges with the various colors separated.

The term dispersion is used to describe the way in which the optical constants change with wavelength. Figure 3 shows how n and k vary for a metal such as nickel and for a dielectric such as silicon nitride (Ref 9). The index of refraction n, for both materials, is near 2 for the entire range of wavelength. From an optical point of view, the quantity that differentiates these two materials is not n, but k. In a material with k ^ 0, the intensity of light I decreases as a function of distance into the material. The functional form is:

where z is the distance into the material and a is the absorption coefficient (related to the extinction coefficient, k). Because the intensity continuously decreases, the concept of how far the light goes into the material has no meaning. Instead, to illustrate penetration, the distance z is used where the quantity az is equal to unity and I/I0 = exp (-1) » 0.37.

Fig. 3 Optical constants for (a) nickel and (b) silicon nitride. The value of k for silicon nitride is zero in the wavelength range shown. Source: Ref 9

Nickel has a value of k that is greater than 2 for the entire range. The thickness where the intensity drops to 37% is about 13 nm. For thicknesses three times this value, or about 40 nm, the material is essentially opaque. For silicon nitride, the value of k is zero for the entire range. This material is transparent.

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