Optical Properties of Metal-Oxide Thin Films: Influence of Microstructure Manuscript Received 2-6-89 Kim F. Ferris, Michael R. Thompson, Gregory J. Exarhos, Wendy S. Frydrych, and Charles B. Nancy J. Hess, University of Washington, Department of Geology, Seattle, WA 98195 The optical response of a dielectric film is perturbed from that of the analogous bulk material by the inherent film microstructure. A mean field approach has been developed for modelling the real part of the dielectric constant in terms of film microstructure. On this basis, eventual determination of film microstructure parameters such as void density and grain size could be determined from the measured optical response of the film. A semiempirical 'fragment' method has also been used to model the imaginary part of the refractive index for titania. Differences in the electronic transition energies for the anatase and rutile phases of titania are ascribed to dielectric field variation rather than changes in the localized chemical bonding around the titanium site. Keywords: Dielectric Function; Electronic Structure; Microstructure; Optical Properties 1.0 Introduction The optical response of dielectric films is influenced by phase homogeneity, interfacial strain, grain morphology, and the presence and distribution of voids. Different film deposition techniques have been found to generate distinct microstructures which perturb physical properties from their bulk single crystal values. For example, recent transmission electron microscopy (TEM) results for sol-gel prepared thin films of TiO2 have identified the presence of spherical grains, with nearly the same size as the film thickness [1]. In contrast, plasma vapor deposited (PVD) materials are characterized by columnar grain growth [2]. Variation in film microstructure also influences phase transformation phenomena as seen in Raman studies of TiO2 films at high pressures [3]. Thus, the optical properties of these materials may not be equivalent to those of a pure phase single crystal. However, knowledge of the microstructure will allow optical properties to be modelled in terms of perturbation of single crystal values. The aim of this paper is to correlate film microstructure and the dielectric properties of single crystal materials to the optical response of dielectric films. The optical properties of materials are classically defined by an optical response function (equation 1), where n' represents the measured refractive index, and k the extinction coefficient. Both the real and imaginary parts of the optical response are wavelength dependent, particularly in the region of anomalous dispersion. As a first approximation in the absence of strong absorption, the dielectric constant is simply the square of the refractive index. Ideally, one would like to describe dielectric films in terms of their optical response. Our approach has been to relate the real part of the refractive index to the microstructure of the dielectric films, and the imaginary part to the electronic structure of model systems. Based upon microstructural results such as those mentioned above, thin films can be treated as a composite material containing grains and voids. Previously, a white light model (equation 2) of the optical response has been used for an approximation of the microstructure in terms of packing fraction. This model provides a first approximation of the microstructure. Serious discrepancies can result since it is assumed that the measured refractive indices can be represented by an average property, and are often compared to values determined at single wavelengths. Quasi-static models offer the incorporation of finer details into the dielectric response. The classical models for dielectric function define limits for the measured dielectric response. Commonly cited examples of these methods are the Weiner [4] and Hashin-Shtrikman [5] limits which define bounds for both the real and imaginary parts of the dielectric response. The upper Weiner limits (equation 3a) are the linear combination of the component systems, where f¡ is defined as the volume fraction of component i and &; as its dielectric constant, providing no screening and the boundaries parallel to the applied field. The lower limits (equation 3b) are defined by the maximum screening condition. The Hashin-Shtrikman limits are simply the volume defined limits in the Weiner formalism. Bruggeman extended these models to include an effective medium (equation 4) in which models (EMA) can effectively describe the optical dielectric response of the material if packing fraction is desired, but offer no new insight into the material science of the film itself. The implicit assumptions to this model are that the particle size in the microstructure remains small compared to the wavelength of light, and also that the properties of a bulk solid can be used to describe the individual grain properties. In keeping with the effective medium approximation, the Ɛh value is determined from self-consistency. Nearing d~0.25 (d-particle size, λ- light wavelength), waveguiding and propagation effects can become significant. For dielectric films prepared by PVD, TEM results indicate that the microstructure spans the film thickness, with grains growing in a columnar fashion. Dielectric films for optical applications have thicknesses less than 1 μ, and thus will require specific corrections for finite wavelength effects. Stroud and Pan [6] have proposed the dynamic effective medium approximation (DEMA - equation 5) which incorporates corrections for magnetic dipole effects. This model for the microstructure uses a composite solid consisting of spherical grains and voids, in which d; is a spherical diameter, and λ is the wavelength of the probe light source. A simple view of the absorption spectra of solids is a system of coupled oscillators, in which the characteristic resonance frequency of a single mode is coupled to the electric field [7]. Electronic structure methods are often used to determine the nature of allowed transitions in molecules, and by extension, the band structure in condensed phases. However, ab initio molecular orbital calculations typically are both storage and computationally intensive, which restrict their utility in problems of this magnitude. Semiempirical methods such as the INDO/S approximation [8] are commonly used to circumvent these problems, and have been shown to give good agreement with experimental results for a number of inorganic systems [8]. Basically, these methods solve the electronic Schrodinger equation, assuming the Born-Oppenheimer approximation, using a linear combination of atom centered orbital functions to approximate the molecular orbitals. The optical excitation spectra are generated from excitations of electrons in high lying occupied molecular orbitals to low lying unoccupied ones, allowing for configuration mixing of orbitals with the same electronic symmetry. 2.0 Methods Modelling of the microstructure was performed using the above outlined mathematical relationships on an Apple Macintosh computer. Packing and particle size parameters determined for the effective medium and dynamic effective medium models were found by minimizing the least squares differences from experimental values. Experimental data for pure phase Al2O3 were taken from reference 9, for Al2O3 composite media reference 10. The wavelength dependence to the optical response of TiO2 materials was determined on a Hewlett-Packard HP8451A spectrophotometer; optical parameters were obtained using ellipsometry. LCAO-MO-SCF calculations using the INDO/S approximations were performed on a VAX 11/780 minicomputer at Pacific Northwest Laboratory. All calculations reported here were made using the default values for empirical two-electron repulsion integrals [11] and spectroscopic parameterization for B. Beta resonance parameters for Ti and O were -7.5 (s,p) and-21.0 (d), and -27.0 (s,p), respectively. The electronic spectrum of the [TiO6]8- molecular fragments was used to simulate the response of single crystal material and were assigned from the spectroscopic transition energies composed of single excitations from the 5 highest occupied molecular orbitals, to the 5 lowest energy virtual orbitals. A 10x10 configuration interaction calculation was performed for both crystalline phases (anatase and rutile) of titania to verify that sufficient excitations were included in the interaction scheme. These results indicated no significant lowering in the transition energies with increased number of configurations. 3.0 Results and Discussion Modelling of the optical properties of thin films occurred in two separate efforts, involving microstructure and electronic absorption considerations. The microstructural information provided by dielectric function measurements, along with known properties of ideal crystals were used to model the dielectric function of Al2O3 composite media and rutile phase titania films. In a later section, the electronic structure of molecular fragments as model systems for TiO2, and their optical excitation spectra will be discussed. 3.1 Effects of Microstructure on the Real Refractive Index we In order to better understand the optical properties of dielectric films, we evaluated the various dielectric function models discussed previously. Our ultimate goal is to parameterize these models, and use them as predictive tools for the microstructure of dielectric films. In Table 1, list the measured dielectric values for Al2O3 composite media as reported by Egan and Aspnes [10] along with the values estimated from the various effective medium models at the experimental packing density. The performance of the various models improved only with the development of a model for microstructure in the composite media. The dielectric values for the effective medium model (EMA) proposed by Bruggeman markedly (15-31%) underestimate the experimental values. Using the dynamic effective medium models proposed by Stroud and Pan, we can introduce microstructure into our model of the composite media in terms of spherical particles. Assuming that the diameter of Al2O3 particles was maintained in the compressed pellet, the predicted values (DEMA-1) noticeably overestimated (26-45%) the experimental values. On the other hand, using spherical voids (DEMA-2) severely underestimates the experimental values(45-61%). Table 1 - Dielectric Values for Composite Al2O3 Media using Dynamic Effective Medium Model Wavelength Experimental EMA DEMA-1 DEMA-2 DEMA-3 DEMA-3' The DEMA-3 model has incorporated both spherical Al2O3 particles and voids (both 1.0 μ in diameter) and does markedly better in predicting the dielectric value (6-22%) even though it is physically unrealistic. When we refine this model by estimating a void size (0.732 x di) based upon available volume in a unit box containing a spherical particle, the DEMA-3' model gives even better agreement (±8%). Thus, as a result of incorporating microstructural information into our dielectric models, we have been able to significantly improve on the performance of the model using only particle size and the dielectric values of the pure phase material. In Table 2, we show that the incorporation of this simple refinement could lead to predictions of packing fraction. One reason for the poor performance of the EMA model is the sizeable wavelength effects found when the wavelength of probe light is within an order of magnitude of the particle size. Using this same approach, we measured the dielectric values for TiO2 films with film thickness as indicated in Table 3. We had characterized these films (prepared via ion assisted plasma vapor deposition) as polycrystalline, randomly oriented rutile phase material by Raman spectroscopy. The quasi-static limits as noted by the Hashin-Shtrikman model would estimate the packing fractions for these films between 0.80 and 1.0, the upper bound is physically unlikely considering TEM micrographs and also provides little structural detail. Using the DEMA-3' model, satisfactory fits of the experimental data could be obtained as shown in Figure 3 with trends similar to those for the earlier Al2O3 dielectric models. Packing fractions in the dielectric films decreased as the models incorporate microstructural features, and also with increasing film thickness. In addition, by assuming microstructural details, we are able to reduce the limits of the packing fraction, now being bound by the small particle limit (EMA) and a DEMA model which assumes spherical grains and voids. Table 3 - Calculated Packing Fractions for PVD-prepared TiO2 Thin Films |