Table of ContentsIntroductionPolarization of LightThe Proper Coordinate SystemReflection at SurfacesOptical Components Used For EllipsometryPolarizersOptical RetardersNuling EllipsometryOptical ModelingImaging EllipsometryScannerNanofilm Imaging EllipsometerConclusionAbout Accurion
Ellipsometry is a highly sensitive optical technique that has been used for about a hundred years to obtain information about surfaces. It works on the principle that the polarization state of light may change when the light beam is reflected from a surface. If the surface is covered by a thin film or a stack of films, the entire optical system of film & substrate affects the change in polarization. The elliptical state of polarization, where the electrical field vector moves along an ellipse when seen at a fixed point in space, is the most general state of polarization. The basic components of an ellipsometer are a light source, some optical components to change the polarization and a detector. With the help of imaging technology, it is possible to extend the classical ellipsometer to a new form of visualization tool or a microscope with high sensitivity to thin films.
Figure 1. Historical setup of an ellipsometer [Paul Drude, Lehrbuch der Optik, Leipzig, 1906]
Polarization of Light
In order to describe light, which is an electromagnetic wave, the direction and strength of the electric field E is considered as this has a stronger interaction with matter than the magnetic field. Monochromatic light may at a point in space E, be split into three independent harmonic oscillations along an x,y,z-coordinate system. If the light wave is a plane wave that travels along the z-axis, the E vector is always orthogonal to z, thus it can be described by two harmonic oscillations along x and y. These oscillations have an identical frequency, but a different amplitude and phase. Consequently, the E vector moves along an ellipse at a specified point in space. The manner in which a vector field varies with time at a fixed point in space is known as polarization. Hence the most general polarization of monochromatic light is elliptical. If the x and y oscillations are equal, the resulting ellipse forms into a straight line. If the phase difference is +/-90° the ellipse forms into a circle. Thus, linear and circular polarization are specialized cases of the general elliptical state. For all other phase differences, a "true” ellipse evolves.
Figure 2. Polarization state of light.
The Proper Coordinate System
When a light beam illuminates a surface under oblique incidence, a plane may be specified by the wave vector K pointing in the direction of the light and the surface normal n. This is known as the plane of incidence. The directions of x and y are defined in such a way that x is parallel to the plane of incidence and y is perpendicular. These directions are designated as p for parallel and s for perpendicular that replace the x,y notation. Thus, the electric field E is resolved into its p and s components.
Reflection at Surfaces
The light is reflected by the sample surface. The sample comprises a complex optical system with several layers having different optical properties. Multiple reflection at the layer interfaces superimpose to form a reflected light wave with a modified state of polarization. Specifically, the p and s components will be subject to a range of phase shifts and also exhibit different reflective properties. Thus, the shape and the size of the ellipse of polarization are changed. This change is the quantified values of the properties of the optical system or sample. The incident and reflected E vectors are linked by the reflection matrix R of the sample as shown in Equation 1:
Figure 3. Reflection from the sample (film/substrate system) changes the ellipse of polarization.
Optical Components Used For Ellipsometry
The main optical components used in most types of ellipsometers are described in the following sections and include polarizers and retarders.
A polarizer produces light in a special state of polarization at the output. Linear polarizers work by suppressing one component of the incident light and allow only the other component to pass. The rotation of this polarizer causes a beam of linearly polarized light to be produced from unpolarized incident light with the direction of polarization corresponding to the angle of rotation of the axis of the polarizer. In case the incident beam is already polarized, the transmitted intensity will depend on the amplitude of the component of E along the axis of the polarizer. In such a case the polarizer is called an analyser it allows one to measure the ratio of the p and s components.
Optical retarders are used to shift the phase of one component of the incident light. A typical retarder is a "quarterwave plate” that has a “fast” and a “slow” axis causing a phase shift of 90° in the components of E along these axes. Based on the orientation of the quarter-wave plate it transforms the ellipse of polarization, for instance, linearly polarized light is transformed to circularly polarized light when set to 45° with reference to the linear polarization axis. The retarders are also called compensators. It is essential to note that the combination of a linear polarizer P and a quarter-wave compensator C (PC) in rotatable mounts can act as a variable polarization filter that can generate any desired elliptical state of polarization at the output given the s and p amplitude are equal at the input.
When linearly polarized light having an axis pointing anywhere except the s or p direction is incident on a sample, the reflected light will show an elliptical state of polarization. The same elliptical state of polarization but with a reversed rotation incident on a surface will produce a linearly polarized reflection.
For a linearly polarized beam it is possible to extinguish the beam by setting the analyzer to a 90° position with reference to the axis of the linear polarization. This is known as “finding the Null” or ”nulling”. The recipe a nulling ellipsometer in a PCSA arrangement is given below:
- Light is made to pass through a PC combination, while recording the angular setting of P and C.
- P and C are changed in such a way that the reflection from the sample S is linearly polarized.
- A photo-detector is arranged behind an analyzer A to detect this as a minimum in the signal.
It is now possible to determine iterative routines to actually find the right angle settings for P, C and A to satisfy the Null condition. The most commonly used technique is the “fixed compensator nulling scheme” in which the compensator is fixed at a specific angle and that P and A are rotated. It can be shown that a rotation of P followed by a rotation of A while keeping P at its minimum signal position causes a Null. This technique has to be repeated iteratively to obtain the required precision. One benefit of nulling ellipsometry is the fact that one measures angles instead of light flux, thus partially avoiding problems of the stability of the light source or non-linearity of the detectors.
Figure 4. Setup of a nulling imaging ellipsometer.
Figure 5. State of polarisation during nulling ellipsometry.
For isotropic materials, where R is diagonal (Rsp, Rps = 0), two so-called ellipsometric angles Y and D can be defined, defining the ratio of the complex reflection coefficients Rpp and Rss, which are actually measured by the ellipsometer:
Y is an angle and the tangent gives the ratio of amplitude change for the p and s components, while D denotes the relative phase shift of the p and s component upon reflection. The result of nulling is a set of angles of P, C and A. There are formulas that relate these numbers to the ellipsometric angles Y and D and thus to the reflection matrix R as shown in Equation 5.
It is important to be able to determine the physical quantities of the sample being examined for instance, the film thickness on a substrate. Based on R, these parameters cannot be measured directly so it becomes necessary to develop an optical model and fit the output of the model until it is equal to the measured values of Y and D. The optical modeling is considered the most critical point in ellipsometry.
Figure 6. Angle of incident (AOI) spectra of Y and D: Air | substrate and Air | Surface coating | substrate.
A single nulling results in two measurable real quantities. Thus, principally speaking, a complex index of refraction or the real index of refraction along with a film thickness or another combination of two real numbers is possible. But normally for a double layered system, two thicknesses plus two refractive indices for a double layer system need to be measured. This is possible by either doing multiple-angle-of-incidence measurements or measure at different wavelengths where every wavelength introduces a new unknown refractive index due to dispersion but provides two new values for Y and D. This leads to spectroscopic ellipsometry. The calculations and mathematics involved are too complicated especially if the sample is anisotropic. In that case even defining Y and D is not enough.
Figure 7. Wavelength spectra of Y and D. at different AOI – air | graphene | SiO2 | Si.
In order to add imaging to an ellipsometer
there is a need for an objective and a spatially resolving detector, such as a sensitive CCD camera. The objective images the illuminated area of the sample onto the camera. As a result, regions that have different optical properties cause a different signal in the camera image. The regions that satisfy the condition of the ellipsometric "Null” is extinguished for that particular setting of P, C and A, and will appear dark in the image.
Figure 8. Additional optical components of an imaging ellipsometer
Where this condition is not met higher light intensities are incident at the detector, producing brighter image regions. By altering the settings of P,C, and A it is now possible to determine the Null for these regions, which will result in the former dark areas to appear bright. The main benefit of such an imaging ellipsometer is that the signal is spatially resolved to show the details of the sample and not the average over an entire laser beam spot on the sample. One receives not only immediate qualitative information, but ellipsometric analysis is restricted to a specific region of interest within the field-of-view. The application of proprietary algorithms allows one to map the Nulls for the entire image. This yields a two-dimensional map of the ellipsometric data that can be modified into a thickness map of the sample or another quantity.
Figure 9. Mapping: Thickness map – air | SiO2 | Si
One normally faces the problem of an inclined observation angle in imaging ellipsometry. Only a limited area of the image appears to be well-focused when using traditional optics. The Imaging Ellipsometer overcomes this limitation by using a motorized focusing mechanism to collect a series of images with different focal lengths within the field-of-view. A digital image processing system then superimposes only the focused parts of an image series, resulting in a digitized image that is sharp from edge to edge. Since the movement of the objects under observation is crucial, a variable scanner speed is provided to adapt the system to a wide range of experimental situations.
Figure 10. Overall focused images by using a focus scanner.
Nanofilm Imaging Ellipsometers
The Nanofilm Imaging Ellipsometers enable study of the surface in three steps that include the following:
- generating high contrast images from the surface
- ellipsometry with highest lateral resolution (1µm)
- generating of 3D thickness maps
Typical applications exist in the fields of biophysics, surface chemistry and Nanotechnology.
Figure 11. Spectroscopic imaging ellipsometer.
Ellipsometry is a well-known non-destructive optical method for measuring film thickness and optical properties. Imaging Ellipsometry combines the power of ellipsometry with microscopy and overcomes the limits of conventional ellipsometers.
Accurion is a high-tech company providing advanced instrumentation in the field of surface analysis and active vibration isolation.
This information has been sourced, reviewed and adapted from materials provided by Accurion.
For more information on this source, please visit Accurion.