UNIFORM STRUCTURES
From the three major evaluation issues of calibration, structural models, and dispersion, the two latter ones will be discussed in the next sections, regarded as the most important aspects of ellipsometry on complex oxides. First, we consider structurally uniform materials, and only deal with the dispersion. Then these dispersions will be included in more complex structural models that take into account many sublayers, superlattice structures, or just unknown vertical non-uniformities, multicomponent materials, and other complexities in the lateral and vertical structure.
Models for Dielectric Oxides
Most oxides are dielectric materials. At the dawn of microelectronics, the primary application of ellipsometry was the accurate measurement of the Si02 layer on silicon wafers. The accuracy, speed, and non-destructive capabilities were unprecedented, and ellipsometry has been used for quality checking and structural analysis for microelectronics until today - now used for many more kinds of oxides and thin films. The refractive index of thermally grown Si02 (Paulson, et al., 2002) is one of the most accurate references besides the single-crystalline silicon even today, the latter being the most studied and cited of all optical references (Aspnes & Studna, 1983; Humlicek & Sik, 2015; Jellison. 1992; Paulson et ah, 2002).
In the case of thermal oxide studies, the refractive index of oxide can usually be taken from a database or literature reference, because it is very reproducible. However, this is unfortunately exceptional, because most thin film properties largely depend on the parameters of the preparation. Even for thermal oxide, the properties deviate from the bulk value when its thickness is approaching the low-dimensional case, as discussed below (in section 2.4.2).
Being wide band gap materials, the oxides can be described by simple dispersion models in the visible wavelength range used by the majority of commercial ellipsometers. One of the most widely used approaches is the Cauchy dispersion, which is a polynomial model in which the absorption is either modeled by another polynomial or using an exponential dependence on the photon energy for the absorption. The real part of the refractive index (/?) is described by the equation
where Ac, Bc and Cc are the Cauchy parameters, and X denotes the wavelength. The extinction coefficient (к) is usually described by an exponential function:
where A* and BA. are the fitted parameters. In order to obtain physically relevant values, the real and imaginary parts of the optical functions must be Kramers-Kronig consistent. Consequently, the real part of the dielectric function (e.) = n2 - k2) must satisfy the following equation:
where e2 denotes the imaginary part of the dielectric function. The Cauchy equations are not Kramers-Kronig (KK) consistent, and therefore, in many cases rather the KK-consistent Sell- rneier equation is used:
where As, Bs, and £«, are the three fit parameters of the Sellmeier model.
Figure 2.4 shows an example of using the Cauchy dispersion for the determination of the refractive index of a BaxSri.xTiOj (BST) layer prepared by metal organic chemical vapor deposition (MOCVD). к was here zero for most deposited films, and it revealed a non-zero value only for lower quality layers.
The fact that the dispersion of the refractive index can be described by simple polynomials in certain wavelength ranges can be used for the determination of the layer thickness and to use this value as a fixed parameter later for more complex optical models in the wavelength range that shows absorption and a more complex dispersion. In this case, the accurate knowledge of the layer thickness is very helpful for a successful determination of the optical properties using В-Spline models or model-independent point-by-point inversions (Agocs et al.. 2014; Collins & Ferlauto, 2005).
Zr02 was also modeled by the Cauchy dispersion in both for sputtered layers and for oxidized Zr surfaces for nuclear applications. The latter was also an environmental ellipsometry study (Figure 2.5). The comparison shown here demonstrates very well the difference between the optical properties of the same material but using different preparation methods. The anodic oxide and the cubic zirconium oxide had the highest refractive index, the thermal oxidation being the next, and the sputtered layers the lowest. This behavior also depends on the
FIGURE 2.4 Fitted tan 4* and cos Д spectra using the Cauchy model shown in the inset for the dispersion of a BST layer.
FIGURE 2.5 Zr02 references from different kinds of layers. The thicker solid line is from Petrik et al. (2017). the anodic oxide reference from Patrito and Macagno (1993), the cubic reference from Synowicki and Tiwald (2004) whereas the sputtered reference is from Lehninger et al. (2015).
layer thickness, since the structure also changes with the layer thickness. Besides other factors, this is the main reason that by calculating refractive index reference data, modeling of the surface (nano-roughness, contamination) is of crucial importance, because it substantially modifies the determined refractive index for the bulk layers. Taking into account all the above experiences, reference refractive index data should always be accepted with healthy skepticism. The best practice in ellipsometry is to try to collect a lot of measured data (broad wavelength range and many angles of incidence). This allows more fit parameters to be used and to parameterize the dispersion of the refractive index together with modeling the vertical non-uniformity within the layer that usually involves the application of preferably advanced global parameter search methods.
Naturally, the optical model depends on the used wavelength range. Even in the case of Si02 which requires only a simple Cauchy model without absorption in the visible-near-UV range, it needs to be described by an absorption towards photon energies as high as 9 eV, usually accessible at synchrotron facilities, such as the BESSY 11 in Berlin (Petrik et al., 2011).
Crystalline oxides are very often optically anisotropic. Rotating compensator (Collins, 1990) or dual rotating compensator (Hauge, 1978) ellipsometers are capable of measuring multiple components of the Muller matrix, and therefore enable generalized ellipsometry (Jellison, 2004; Schubert, 1998) to determine the optical anisotropy. Prominent examples of generalized ellipsometry on oxides include: ZnO (Jellison & Boatner, 1998); thin film and bulk anatase ТЮ2 (using Sell- meier parameterization and sample rotation) (Jellison et al., 2003); «-plane and c-plane one-side polished sapphire phase A1203 (Schubert & Tiwald, 2000); and Ga203 on substrate with different orientations in a broad IR wavelength range (from 50 to 1500 cm-1) (Schubert et al., 2016).
Models for Semiconducting Oxides
Once the band gap becomes lower, reaching the UV-Vis photon energy region accessible by regular ellipsometry, the above model for the description of dielectric oxides gets more complicated. If only the onset region of absorption close to the gap energy is in the focus of the study, applications of the Tauc or Cody approaches dominate to describe the photon energy dependence of the absorption (typically applied for the imaginary part of the dielectric function). In higher energy regions, where interband transitions take place, the descriptions become even more complicated (see section 2.4.2). In many cases, the Tauc gap parameterization is only extended by a couple of Lorentz oscillators in the high photon energy region.
One of the most widely used approaches to parameterize the dielectric function of the gap region is a combination of the Tauc expression for the imaginary part of the dielectric function for photon energies above the band edge
with AT and Eg being the amplitude and the optical band gap, and a Lorentz oscillator of
which, after combining Eqs (5) and (6), results in the expression of
if £g>0, otherwise c2=0, with E0 and C being the peak transition energy and the broadening, respectively, each parameter having the dimension of eV. This approach is called the Tauc-Lorentz (TL) oscillator model (Jellison & Modine, 1996), which among others has two great features. The first one is that to be Kramers-Kronig consistent, there is an analytical formula for cb instead of using the numerical integration through Eq. (3). The use of this formula (although it is almost an entire page in the article) makes the calculation very fast, much faster than the numerical integration. The other important feature is that it provides a numerical value of £j, right from the fit, so the Tauc plot doesn't have to be applied. The TL model is mainly used for amorphous semiconductors, and it can be best used at photon energies around the optical gap.
The TL model has later been improved by Ferlauto and coworkers (2002) to add an Urbach tail to the Tauc gap expression. Here, the photon energy region near the optical gap is already divided in three parts, each having its own photon energy-dependent function for the imaginary part of the dielectric function. Very importantly, the analytical formula for the real part of the dielectric function is provided also in this case (similar to that for the TL model), enabling again the fast calculation.
One of the most investigated semiconducting oxides is ZnO. the properties of which also depend very much on the deposition conditions. The interband transitions can be described by oscillators, including excitonic effects (Major, Juhasz, et al., 2009). This way, the method is sensitive to the electrical properties (conducting, semiconducting) and therefore suitable for a quick and reliable qualification of the materials. The Drude tail analysis towards the infrared wavelength region is a method to detect free carriers in the layer, and therefore correlates very well with the conductance (Nemeth et al., 2008).
In general, since the imaginary part of the dielectric function (which can be measured with an accuracy of ~ 10-4) is proportional to the joint density of states that is closely related to the crystal structure, ellipsometry provides sensitive indirect information on the crystallinity and related characteristics, such as the conductance (Nemeth et al., 2008). To build optical models for semiconductors, in general, it is a useful approach to fit the spectra first in the photon energy region below the band gap using a few-parameter dispersion model (see, for example, the Cauchy model of Eq. (1)) and determine the layer thickness and the below-gap properties. Because now the thickness is known, the В-Spline, point-by-point, or other parametric models can be applied with a higher reliability (Figure 2.6), especially when checking the “semiconductor criterion,” i.e. choosing a thickness for which the imaginary part of the dielectric function smoothly approaches zero towards the low photon energies around and below the band gap (Agocs et al., 2014; Arwin & Aspnes, 1984; Aspnes et ah, 1984).
It is important to point out that the result very often depends on the photon energy range used for the fit (see Figure 2.7). The obvious reason for the increasing deviations is that lowering the upper limit of the photon energy range, the part of the exciton peak involved in the fit decreases, and the sensitivity of the Tauc-Lorentz parameters responsible for this oscillator decreases. However, usually there is the second, less obvious reason. At the scale of ellipsometric sensitivity, most thin films are vertically non-uniform. Most of the grain growth mechanisms result in evolving and vertically changing structures with nucleation layers, V-shaped grains, surface nanoroughness, etc. On the other hand, in the above-gap region of the spectrum, the light penetration depth in semiconductors can be very small (even in the range of 10 nrn or below) due to the large absorption. Therefore, by decreasing the upper limit of the photon energy, the penetration depth increases and results in a kind of “depth scan,” i.e. variation of the photon energy range changes the depth region of the layer used in the fit. In perfectly uniform films, this wouldn't cause a deviation; however, most thin films reveal this effect due to the vertical gradient in the optical properties. This “penetration depth effect” is not only a disadvantage. In case of proper modeling, it can be used to gain more information from the films, and to achieve a model-independent depth scan by scanning the wavelength range (Petrik et al., 2014).
Models for Conducting Oxides
Transparent conducting oxides have been key materials in microelectronics with indium-tin- oxide as the most important one, and with further efforts to replace it, for example, by ZnO
FIGURE 2.6 Dielectric function of a 50 nm thick ZnO layer prepared by sputtering and annealing. The range below 1.5 eV was described by the Cauchy model, whereas that above 1.5 eV by the B-Spline parameterization.
Reproduced with permission from Agocs et al. (2014). Copyright 2014 Elsevier Inc.
FIGURE 2.7 Imaginary part of the dielectric function determined by fitting the Tauc-Lorentz parametric model to the measured data using different wavelength ranges. The legend shows the upper limits of the range in eV. The lower limit was 0.7 eV in each case.
with optimized deposition and doping. The conductance can indirectly be measured by the analysis of the Drude tail in combination with Lorentz oscillators:
where A(> and v0 are the amplitude and broadening parameter of the Drude term and Aj, v0, and Ej are the amplitude, broadening, and center energy of the Lorentz oscillators, respectively (Roeder et al., 2008). The combination of a Cauchy model and a Lorentz oscillator model is sufficient to properly describe both the visible and the infrared range (Synowicki, 1998). It has been shown that the near gap absorption features correlate well with the electrical properties using either the Cauchy model or Lorentzian line shapes (Major, Nemeth, et al., 2009; Nemeth et al., 2008). More specifically, there was a good correlation between the specific resistance and the gap energy, as well as between the specific resistance and the exciton strength (related to the absorption features in the UV photon energy range) using the parameterization published in Yoshikawa and Adachi (1997).
In case of indium tin oxide (1TO) not only the transparency can be utilized in the visible wavelength range, but also the infrared range for plasmonic effects (Rhodes et al., 2008). Utilizing the Drude dispersion and measurements in the Kretschmann geometry, it was shown that the optimum thickness for surface plasmon polaritons was 160 nm in 1TO, and the thickness of the 1TO layer can be changed in a much broader range than in case of gold plasmonics (Kalas et al., 2017), modifying the resonant photon energy position with the layer thickness significantly.
Plasmonics has also been demonstrated for low-dimensional metal-metal oxide structures utilizing a transparent conductive oxide (TCO) layer (Khosroabadi et al., 2014). Here, the Drude-Lorentz model was combined with the generalized Maxwell-Garnett effective medium approximation (EMA) (Aspnes, 1982; Gilliot et al., 2007), which was found to be most suitable for the description of TCO/Ag core shell structures for plasmonic applications.
