LOW-DIMENSIONAL AND COMPOSITE STRUCTURES
On the scale of ellipsometric sensitivity, most thin films are vertically non-uniform, having at least a surface layer (typically nanoroughness) and a bottom interface layer (e.g. nucleation layer in case of deposition techniques), but vertically graded optical properties within the films aren’t rare as well. In case of very thin films, the layer itself can be considered as an interface, the optical properties of which are very much different from the bulk layers. This effect and the formation of interface layers have been studied by in situ ellipsometry already in the late 1980s for the deposition of amorphous and nanocrystalline silicon films (Nguyen et al., 1994).
Also, in case of one of key materials of microelectronics, the above mentioned Si02, the direction of technological development points towards ever-decreasing oxide thicknesses, for which the proper modeling of the interface between an amorphous oxide and the singlecrystalline substrate is a crucial issue (Irene, 1993, 2006). Si02 is a technologically critical material not only on the surface of silicon, but also on SiC. The proper modeling of ultra-thin oxide is of primary importance for SiC as well (Petrik, Szilagyi, et al., 2009). Ellipsometry is also actively used for the characterization of two-dimensional oxide layers or layer stacks (Zheng et al., 2017). However, the sample preparation, the interface, and interpretation issues largely influence the reliability of the data, as shown in the example of graphene (Papa et al., 2017).
The second subtopic of this section will be the case of a 3D matrix with low-dimensional inclusions, in which the oxides are either the host materials (e.g. embedded low-dimensional semiconductor nanoparticles) or the inclusions (e.g. for changing the refractive index of the layers). The applications of semiconductor nanocrystals in dielectric matrices cover a broad range starting from silicon in silicon oxide (Agocs et al., 2011) or silicon nitride (Basa et al., 2007) for memory application to germanium in oxides such as Hf02 and Zr02 (Agocs et al., 2017).
2.4.1 Low-Dimensional Oxide Structures
Oxide nanodots can be embedded in thin films in order to modulate the refractive index. In case the size of inclusions is much smaller than the wavelength of measuring light, the EMA can be used. If both components are dielectrics, such as in the case of Ti02 nanoparticles in polyimide (Petrik et al., 2012), simple polynomial models (e.g. Cauchy dispersion) can be applied, only fitting the imperfections at the interfaces, or the graded refractive index within the layer.
In the Ti02/polyimide system (Petrik et al., 2012), the refractive index of the composite (at the wavelength of 633 nm) can be varied from 1.65 to 1.80 when increasing the volume fraction of Ti02 nanoparticles from 0 to 60%. The intrinsic refractive indices of the components are 1.65 and 2.5. Even at the wavelength of 1300 nm (important for waveguide applications), the variation with the composition is close to 0.1. However, as expected, although the composite layer created by doctor blading was very homogeneous on electron microscopy images with uniformly distributed nanoparticles (average size of approximately 10 nm), the fit result of ellipsometry was much better when assuming a thin layer at the interface between the composite layer and the silicon substrate. Nevertheless, the layer quality was very good, taking into account that doctor blading is a roll-to-roll compatible, no-vacuum, low-cost thin film preparation technique being widely used in the industry. In this approach, the solution of the film material is placed in front of the blade, and the blade is moved over the sample to produce a layer behind the blade. The parameters of the film can be controlled among others by the concentration, the speed of the blade, the gap between the blade and the substrate as well as the temperature.
FIGURE 2.8 Refractive indices and extinction coefficients measured by ellipsometry on approximately 300 nm thick polyimide layers with embedded 10-nm size ТЮ2 nanoparticles up to 60 vvt% ТЮ2, at the wavelength of 633 nm.
Besides polyimide, industrial prototype epoxy polymer can also be used as host material for Ti02 nanoparticles of sizes below 10 nm (Landwehr et al., 2014) targeting applications such as the high throughput production of micro-optical elements (Figure 2.9). Due to the small size of the components, the effective refractive index of the layer can also be described by a simple Cauchy dispersion with the imaginary part of the refractive index set to k=0, meaning a low absorption of the composite. Since the refractive index of the starting material is lower here, and the volume fraction of ТЮ2 was only increased to the value of 23 wt%, the refractive index could be varied in the range from 1.54 to 1.63. In case of this material, an important feature was the stability and low shrinkage at high temperatures (shown up to 220°C).
FIGURE 2.9 Refractive index at the wavelength of 635 nm of KATIOBOND OM VE 1107/07 composite containing with trimethoxy(7-octen-l-yl)silane (TMOS). trioctylphosphine oxide (TOPO) or 4-tert- butylcatechol (TBC) stabilized Ti02 nanoparticles in weight fractions ranging from 0% (pure organic resist) to 23%. Layer thickness ranges from 100 to 300 nm.
Due to its sub-nanometer sensitivity, ellipsometry is a powerful tool for the measurement of low-dimensional, ultrathin layers. Since the optical constants can be measured with a precision down to 10 4, the deviation of the optical properties from that of the bulk value can sensitively be characterized. The first high-quality and high-importance oxide investigated by ellipsometry was Si02 on Si surfaces for electronic applications. The models in this case (and later on SiC) can be based on tabulated references, since the refractive index of thermal and native oxide is dense and reproducible (this is an important feature that makes silicon- based microelectronics technology so powerful).
The thickness of gate oxides in microelectronics has been continuously decreasing, requiring more and more the sub-nanometer precision of ellipsometry, which holds even in case of embedded layers. Figure 2.10 shows the optical model and simulation with and without the oxide layer. The high sensitivity is clearly seen from the difference between the simulated curves, especially in the region of larger wavelength, in which the polysilicon layer on the top is transparent. In this application, the oxide layer was used as a diffusion barrier that doesn’t completely hinder the diffusion; it only allows a shallow dopant profile after a high temperature diffusion process for bipolar transistors. Consequently, the thickness of this barrier oxide was only a few nanometers. However, in spite of this challenge, its thickness could be measured with high accuracy, depending on the crystallinity of the silicon layer on top, as it influences the absorption, and the range of wavelengths that can reach the embedded oxide.
The tabulated oxide reference for low-dimensional films, however, requires density correction in many cases. A prominent example is the ultrathin oxide on the surface of native and oxidized SiC, which also depends on the termination of the hexagonal 6H-SiC surfaces. Model 1 of Figure 2.11 from Petrik, et al. (2009) applies a Bruggeman EMA model (Aspnes, 1982; Bruggeman, 1935) using “voids” as a second component and also a transition layer (Model 2) that significantly improves the fit quality when dealing with ultra-thin layers. For most oxides, the Cauchy or Sellmeier dispersion has to be combined with a transition layer model, in which the transition layer is usually an effective medium with components using the optical functions of the layers above and below the transition layer.
Density correction is a major issue for porous films (Fodor et al., 2016; Fried et al., 1996; Petrik, Fried, et al., 2009). In these cases, the effective dielectric function is in almost all cases calculated using EMA. The reason is that EMA provides a quantitative result in terms of volume fractions of the constituents - in case of porous materials it is the volume fraction of void. EMA is a sophisticated method, in which the isotropy of the boundaries of the phases and the kind of host material should be taken into account. However, in the majority of the studies, the Bruggeman EMA is used in which the materials are spatially isotropic, and the constituents are equivalent - neither of the components is considered as a host material. In this case, it holds that
where e is the effective dielectric function of the composite layer; f and Cj denote volume fraction and the dielectric function of the ;'th component. If there is a preferential direction of the phases of the components, the so-called screening effect must be considered and the evaluations get more complicated (Aspnes, 1982; Fodor et al., 2016).
Using ellipsometry, the porosity can be determined with high accuracy that can be used for sophisticated studies like in porosimetry (Baklanov, et al.. 2000), where not only the volume fraction but also the pore size can be calculated. The method can also be used for the study of the saturation of pores with water or other gases and, for example, for the degradation of glasses (Ngo et al., 2018).
FIGURE 2.10 Simulation of ellipsometry spectra using the optical models shown in the inset. The dielectric function of pc-Si was taken from Jellison, et al., (1993).
FIGURE 2.11 (a) Optical models to describe ultra-thin SiCK layers on SiC. (b) Volume fraction of void
when using Model 1. MSE denotes the mean squared error, i.e. the quality of the fit.
In case of many fit parameters (as shown, for example, in Petrik, Fried, et al. (2009) for Figure 2.11), the multi sample approach is very useful, in which the measurements on different samples are fitted together, using all the results as one collective data set. The idea is that there are many model parameters (for example the dispersion of the layers or the optical properties of the substrate) that are the same in all measurements. Therefore, the number of fitted parameters doesn't increase proportionally to the number of measured data - the parameters/data ratio decreases with each newly introduced sample. A typical example of the multi sample method (Herzinger, et al., 1998) is a series of samples for which only the layer thicknesses are different. This method can effectively be used for ultra-thin layers as well (Petrik, Fried, et al., 2009. Petrik, Szilagyi, et al., 2009).
2.4.2 Oxides with Embedded Semiconductor Nanodots
Non-stoichiometric oxides with embedded semiconductor nanodots have been the subject of intensive research for decades, due to the applications in microelectronics, most importantly in non-volatile memory devices, but also in a range of many different applications from solar absorbers (Conibeer et al., 2006) to light emitters (Pavesi, et al., 2000). The measurement of such material system is challenging for most of the techniques. For example, the size and amount of nanocrystals in a thin film can be below the detection limit of usual X-ray diffraction devices. In many cases, transmission electron microscopy can only detect such nanoparticles when applying sophisticated correlation techniques (Hristova-Vasileva et al., 2018). Ellipsometry on the other hand can detect crystalline phases in nanometer thin layers and in volume fractions down to a few percent and sizes of a few nanometers (Agocs et al., 2017).
The most frequently used method for the description of composite structures is also the EM A (Aspnes, 1982). The greatest challenge in this case is that it assumes the dielectric function of the components to be known. In case of embedded nanocrystals, however, the optical properties are not known due to the deviations from the bulk properties because of the small size. To solve this problem, a useful method is to prepare homogeneous layers with optical properties possibly identical to those found in the composite material, to determine the reference dielectric function with a higher accuracy. This has been demonstrated for Ge nanocrystals in a ZrCF matrix (Agocs et al., 2017). Figure 2.12 shows the reference dispersion curves for amorphous and crystalline Ge measured in homogeneous thin films and from the literature.
The dielectric function of the embedded semiconductor phase can be modeled using many versatile parameterization options. Each of them has its advantages and disadvantages (Petrik. 2014) including the model dielectric function (MDF) reported in Adachi (1987), Cody-Lorentz model (Ferlauto et al., 2002), Johs-Herzinger generalized critical point model (Johs, et al., 1998), generalized oscillator model (Lautenschlager et al., 1987) or the Forouhi-Bloomer model (Forouhi & Bloomer, 1986). Most of these approaches were also applied for modelling the parameters of semiconductor inclusions in dielectric matrices. In literature (Basa et al., 2007; Petrik, Fried, et al., 2009), the MDF was used to describe Si nanocrystals in porous silicon or in Si3N4 host materials. There were many fit parameters in both cases, for which global optimization was used. Recently, the Cody-Lorentz model has been demonstrated on electron irradiated non-stoichiometric silicon oxide (Hristova-Vasileva et al., 2018). The generalized critical point model was used for silicon nanocrystals in silicon-rich oxides (Agocs et al., 2011).
The MDF parameterization for Si02 (Agocs et al.,2013) and SijN4 (Basa et al., 2007) matrices mentioned above provide a versatile way of describing composite materials with embedded semiconductor inclusions. Among the many advantages, its Kramers-Kronig consistency, the physical relevancy and the analytical description has to be emphasized at first. The MDF model can be programmed easily and the Kramers-Kronig consistent computation is quick and flexible. Due to the large number of fit parameters, limitations of parameters must carefully be considered, for example, by coupling or fixing some of them (Petrik, Fried, et al., 2009).
FIGURE 2.12 Optical properties of amorphous (a-Ge) and single-crystalline (c-Ge) Ge used for the modelling of Ge clusters embedded in ZrCK The reference shown by black lines was determined from a single-layer sample, and compared to other references from the literature (a-Ge from Ref. (Adachi. 1999)) and single-crystalline (c-Ge from Aspnes and Studna (1983)) references.
The structure of MDF oscillators is demonstrated for silicon in SijN4 in Figure 2.13. The (2D)M0 critical point and the photon energy of 3.4 eV is described by
with
where Bx and Г, denote the strength and broadening parameters, respectively. For the (2D)M2 contribution, we use the equation
with
where F and Г are the strength and broadening parameters. Ec is a low energy cut-off. The damped harmonic oscillators (DHO) are described by
FIGURE 2.13 Oscillators of the MDF parameterization used for Si nanocrystals embedded in Si3N4. Two 2D critical points are denoted by (2D)M0 and (2D)M2, described by Eqs. (11-15). DHO denotes damped harmonic oscillators from Eqs. (16) and (17). The excitonic lineshape is closely related to the DHOs.
with
where C and у are the non-dimensional strength and broadening parameters.
