Optical Characterization of Oxide-Based Materials Using Ellipsometry

Peter Petrik

INTRODUCTION

Since ellipsometry measures surface and interface properties with sub-nanometer precision, an inherent requirement is that the surfaces and interfaces must be of optical quality, i.e. flat on the scale of nanometers or at most tens of nanometers. This is a challenge in many cases even today, but back to the beginnings of the method only very few surfaces could be investigated - they were typically polished metal surfaces (Rothen, 1945). As the technology for microelectronics and thin film preparation developed, the precision, speed, and non-destructive nature of ellipsometry made sense, and it could be applied very effectively for the testing and the development of materials and structures for microelectronics (Irene, 1993). Since the energies of the electronic transition in silicon and other semiconductors are in the photon energy range that could easily be covered by standard instrumentation, ellipsometry was proved to be a powerful tool for structural characterization of semiconductors (Petrik et ah, 2014) except the first applications being the measurement of thickness of dielectric materials, such as the thickness of thermal oxide layers on silicon wafers (Irene, 2006).

For ellipsometry, as for any measurement method, the development of evaluation and instrumentation goes hand in hand. In case of optical modeling, the direction of development in ellipsometry is from well-defined dielectric layers with simple dispersions toward complex systems both in terms of dispersion models (semiconductors with multiple oscillators; see for example Petrik et al. (2009)) and layer structures (inhomogeneous multi-layer structures (Petrik et ah, 2005)). Because of the versatile structures and properties of oxides, the modeling of all major categories of the dielectric functions will be discussed from dielectric to semiconducting features (Fried et ah, 2011; Petrik et ah, 2001).

In case of instrumentation, the main driving forces are the capabilities to measure in a broader wavelength range (Petrik, et ah, 2019), in different environments (vacuum chamber (Collins, 1990) or flow cells (Kalas. et ah. 2019)), to measure anisotropic materials or large surfaces (Fried et ah, 2011; Shan et ah, 2014).

In this chapter, the aspects of modeling and instrumentation will be discussed for oxide- based materials. Since the field is already huge and covers a broad range of characteristics (dielectrics, semiconductors or even metallic properties - as for transparent conductive oxides (TCOs)), this summary is far from being comprehensive, only discussing some important issues, mainly demonstrated by our own results, for those who are interested in oxides but who are no experts of ellipsometry.

OPTICAL CHARACTERIZATION BY ELLIPSOMETRY

Although the method of ellipsometry has been known since the end of the 19th century (Drude, 1889), the word “ellipsometer” has only been coined in the middle of the 20th century (Rothen, 1945), and the first major applications of ellipsometry are connected to the microelectronics technology (Irene, 1993) later on. The number of publications in ellipsometry increased rapidly in the years from -1980 to =2000, reaching a kind of saturation by the year of 2010, as shown, for example, in Figure 2.2 of Kalas et al. (2019). At the beginning, most publications have dealt with the development of the method, which gradually turned to a situation in which most of the publications are about the application of ellipsometry, rather than its development. Of course, it is not because the activities on development decreased, but because the number of papers on the application increased rapidly.

Compared to the numerous spectroscopic measurement methods, ellipsometry has some important features, (i) First of all, it doesn’t require references, because it compares the reflection of two different polarizations of the same beam. Therefore, it is also very robust in terms of instrumentation, because it is not sensitive to intensity fluctuations from the lamp or vibrations. (ii) In spite all of these factors, it features a sub-nanometer sensitivity that can even be achieved using simple optical components like regular lamps and film polarizers, (iii) But most importantly, ellipsometry provides many data (e.g. as a function of wavelength and/or angles of incidence) which can be fitted by optical models. This fact ensures fully quantitative analysis as soon as proper optical models exist for the measured system. Consequently, although it is an indirect method, it is highly sensitive, quantitative, robust, fast, simple - in terms of instrumentation - and non-destructive. The latter feature is very important, because with proper hardware extensions, such as vacuum chambers (Fried et al., 2011), flow cells (Kalas, et al., 2017) or environmental cells (Petrik et al., 2019), real-time measurements can be performed with high sensitivity and good time resolution.

The literature on the theory and applications of ellipsometry is already huge, covering most areas of the field from the mathematical basics (Azzarn & Bashara, 1977) and applications (Tompkins & Irene, 2005 or Fried, et al., (2001)) to a comprehensive practical guide with many excellent graphs and visualizations (Fujiwara, 2007). Therefore, in this chapter only some “oxides-related” aspects of hardware and evaluation will briefly be discussed, primarily based on first-hand experiences.

Hardware

The general components and working principle of an ellipsometer is shown in the example of a mapping ellipsometer in Figure 2.1 (Fried et al., 2011). The minimum requirement is

FIGURE 2.1 Beam path of a divergent light source mapping ellipsometer using components of a (1) light source, (2) film polarizer, (3) spherical mirror, (4) convergent beam, (5) sample, (6) cylindrical mirror. (7) corrected beam, (8) analyzer, (9) pinhole, (10) divergent beam, (11) spectrograph.

a light source, two polarizers, and a detector. The light source doesn't have to be coherent. One of the polarizers is in the beam path of the light source, serving as a polarization state modulator, sometimes combined with an additional compensator that changes the phase shift (Lee, Rovira, An. & Collins, 1998). The light reflected from the sample changes its polarization state, which can be measured by having another polarizer (called as analyzer in this case) in the detector path, sometimes also combined with a compensator to change the phase shift, increase the sensitivity, and allow measuring all the Muller matrix parameters. There are many configurations from null ellipsometry to rotating component ellipsometry (Aspnes & Studna, 1975; Collins, 1990), the latter of which utilizes one or two rotating polarizers or compensators. The most advanced Muller matrix ellipsometry uses simultaneously rotating compensators in both arms of the ellipsometer to allow full Muller matrix determination (Hauge, 1978; Lee et ah, 2001). This kind of modulation ellipsometry results in a sinusoidal intensity signal in the detector. The analysis of this line shape provides the measured ellipsometric angles or elements of the Muller matrix (Fujiwara, 2007).

Our recent development of high-speed mapping ellipsometry (Figure 2.1) is only one of the many instrumental possibilities. There are also micro imaging ellipsometry (Wurstbauer et ah, 2010), versatile in situ configurations in vacuum chambers (Cobet et ah, 2018; Lee et ah, 1998; Mikhaylova et ah, 2007; Petrik et ah. 2001), flow cells (Arwin. 2005; Kalas et ah, 2017; Kozma et ah, 2011; Poksinski & Arwin, 2004), or environmental cells (Petrik et ah, 2019; Welch et ah, 2017). The mapping or macro imaging configuration in Figure 2.1 also has numerous versions with or without a mirror after the sample (see also Figure 2.2), a line- scan spectroscopic version and many configurations to integrate in vacuum chambers or in a roll-to-roll environment. The advantage is (as in multichannel ellipsometry - see Collins et ah (1998)) that utilizing a detector matrix, both the spectral and lateral points can be measured simultaneously, by analyzing the line shapes of the signal at each pixel of the detector, replacing a one-by-one measurement with a simultaneous acquisition. In this way, the measurement speed of a typical (30-points by 30-points) matrix can be reduced from minutes to seconds, revealing an orders of magnitude improvement in terms of speed - but still maintaining a sensitivity of thickness in the range of nanometers.

One of the most important advantages of ellipsometry is its speed and non-destructive character, which results in applications such as in situ ellipsometry (Collins et ah, 2000; Tompkins & Irene, 2005). First, the integration of ellipsometry in vacuum chambers was a major in situ application, but later, bioellipsometry (Arwin, 2005; Kalas et ah. 2019), electrochemical in situ ellipsometry, and ellipsometry in environmental cells gained increasing interest. The reason is that the sub-nanometer sensitivity is usually possible even at high acquisition speeds (seconds or even better). Advanced configurations of environmental cells (Figure 2.3) even allow multiple angles of incidence or mapping measurements on heated samples and in controlled ambient conditions (both liquid and gas are possible).

Finally, besides sophisticated configurations, many developments of ellipsometry expand the wavelength range from the shortest ones (synchrotron - BESSY) through vacuum UV, near UV, and extending the visible range towards the near infrared, mid infrared and terahertz regions. A crucial problem in these fields is the availability of reliable references for the refractive indices. In case of the in situ ellipsometry during annealing, the problem is even larger due to the need of high temperature references. There are databases provided by hardware manufacturers; however, global databases are currently lacking.

Evaluation

From the analysis of the sinusoidal function generated on the detector as a function of the angle of rotating element(s) (polarizer, analyzer or compensator), the change of polarization

FIGURE 2.2 Prototype of a roll-to-roll compatible mapping ellipsometer using a configuration schematically plotted in Figure 2.1. The optical components of the ellipsometer are located on the left-hand side of the picture, except for the mirror on the right-hand side at the back, partly covered by an iron block. The mirror serves for imaging the divergent source on the pinhole camera and spectrograph, both located at the right-hand side. The apparatus only uses the central line of the image for mapping - the other index on the CCD detector is used for spectral information. This means that spectroscopic mapping can be done by moving the sample (i.e. the line with spectroscopic information in each point). The advantage of this arrangement is that it can easily be integrated in a vacuum chamber with only the mirror located in the chamber - all optical components are outside.

as a result of the reflection from the sample surface is calculated, described by the complex reflection coefficient

where у and Д are the ellipsometry angles related to the amplitude and the phase, respectively, whereas r_p and r_s denote the reflection coefficients of light polarized parallel and perpendicular to the plane of incidence. The evaluation is only possible if one has an idea about the structure of the sample. It is required in order to construct an optical model that is good enough to supply a set of initial parameters from which spectra of у and Д can be calculated. They should be close enough to the measured one, so that a proper parameter search or gradient fit algorithm can find suitable model parameters by minimizing the difference between the measured and the fitted spectra.

FIGURE 2.3 Environmental cell to study high-temperature processes using real-time ellipsometry in controlled ambient temperatures up to 600 °C.

There are three major evaluation issues of ellipsometry: (1) calibration of the instrument, (2) modeling the structure in terms of layers, lateral and vertical features, components, and phases, and (3) modeling the wavelength dependence of the refractive index of layers and components. The most important task of instrument calibration is the determination of the offsets (optical zero position regarding the plane of incidence) or the rotating components, and the polarization dependence of either the light source or the detector, depending on whether the source or detection part contains the rotating element. The structural modeling is getting more and more relevant as samples of ever-increasing complexity can be studied - due to the increasing computational power and experiences in complex modeling using many model parameters. The modeling of the dispersion (wavelength dependence of the refractive index) is mainly the materials science part of ellipsometry. From these parameters numerous material properties can indirectly be detected, such as the band structure, crystallinity, conductance, carrier density, physical density, porosity, band gap, and many more, limited only by the physical models, the wavelength region, and the sensitivity in the given optical configuration. Most importantly, the dispersion is a fingerprint of the materials in terms of whether it is a conductor, a semiconductor, or an isolator. Since the properties of oxides cover all of these categories, the modeling of these will briefly be discussed below.