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Abstract
The optical properties of several commonly used single-crystal oxide substrates were explored by spectroscopic ellipsometry over a wide spectral range from 0.74 eV to 8.8 eV. The crystals examined are (100) SrTiO3, 0.7 % wt Nb-doped (100) SrTiO3,(100) (LaAlO3)0.29(SrAl0.5Ta0.5O3)0.7, (011) DyScO3, (100) MgAl2O4, (100) MgO, and (100) LaAlO3, all of which enable epitaxial growth of numerous perovskite-type and other optical thin films. An analytic form for the complex dielectric function was derived from ellipsometric data through a physically consistent modeling process. The obtained dielectric spectra were further utilized to calculate the complex index of refraction and absorption coefficient for each substrate material. The absorption spectra and optical band gap were analyzed using Tauc plots. The parameters for reconstructing the dielectric functions are given in detail, allowing for extensive applications of the results of this work.
© 2017 Optical Society of America
1. Introduction
Many ABX3 materials possessing perovskite-type structure exhibit functional responses that are intriguing fundamentally and useful for practical applications. In particular, there is a variety of perovskite metal oxides exhibiting such unique phenomena as colossal magnetoresistance [1, 2], charge ordering [2], spin dependent transport [3], and formation of 2D electron gases [4] to name a few. In addition, perovskites are often mentioned as key materials in the advancement of photovoltaics [5, 6], and are also incorporated in the development of applications in photonic devices [7, 8], lasing media and LEDs [5, 9–11], fuel cell technology [12], memory devices [1, 13–15], catalytic chemistry, superconductivity [16–18], and much more.
Among all perovskite oxides, ferroelectrics are of particular interest for their remarkable optical properties which include large index of refraction, high transparency in the visible range, strong linear and nonlinear electro-optic effects, photovoltaic properties, and large permittivity [8]. Some of ferroelectrics are available as high quality crystals. Progress in preparation of thin films, nearly entirely devoid of defects, has made these materials increasingly more accessible in the form of single-crystal-type epitaxial films, whose perfect quality often result in the best performance of devices based thereon [14, 19]. Likewise, advancements in non-destructive characterization techniques such as Raman spectroscopy, reflectometry, and ellipsometry have made it possible to determine the optical functions of these materials, either as bulk crystals or as thin films, accurately in a broad spectral range [20]. This knowledge is often a key component necessary for tailoring response functions of thin films and multilayer heterostructures of such materials. It is worth noting that lattice strain induced by a mismatch between crystal lattices of substrates and thin films is a powerful tool to manipulate many of the thin-film properties [21–25].
Both the development of heterostructures for optical applications and the fundamental investigations of thin films require accurate knowledge of the optical properties of the substrates, whereon the heterostructures or films are grown. Here we investigate the properties of a number of crystals frequently used as substrates in epitaxial growth, providing an analytic form for the optical constants of these crystals with a wide range of lattice parameters. The substrates examined here are single-crystal (100) SrTiO3 (STO), 0.7 % wt Nb-doped (100) SrTiO3, (100) (LaAlO3)0.29(SrAl0.5Ta0.5O3)0.7 (LSAT), (011) DyScO3 (DSO), (100) MgAl2O4 (MAS), (100) MgO, and (100) LaAlO3 (LAO). We note that because of increasing use of these substrates, their synthesis and surface processing have been significantly improved in the last decades. This allows for better accuracy in measuring optical constants of these crystals. Excellent data on STO, MgO and MAS are available [26–28]; however, few if any sources covering a complete set of optical properties of the other substrates listed are available. Additionally, STO substrate is well-analyzed by Zollner et al. [26]; nevertheless, it is necessary to present our spectra here in order to observe the variation of optical parameters between undoped and Nb-doped STO substrates. Optical properties of LSAT [29, 30] and LAO [31–33] are well-defined in the IR and VIS-NUV spectral range up to 6 eV, but this spectral range is not sufficient for complete characterization as these substrates are optically transparent through the visible range and the onset of absorption occurs at photon energies up around 5 eV. In general, optical studies of epitaxial thin films in the vacuum ultra-violet range are becoming increasingly more accessible [34, 35], and therefore accurate studies of these substrates are required up through the VUV range to photon energies of at least 8.8 eV.
We have applied variable angle spectroscopic ellipsometry (VASE) in the IR-VIS -VUV spectral range with photon energies ranging from 0.78 to 8.8 eV in order to accurately determine a functional form of the complex dielectric function and, consequently, the complex refractive indices and absorption coefficients of all aforementioned crystal substrates. We further use this data to determine the type and energy level of the optical bandgap for each of these substrates, and this data is tabulated and compared to previous work for easy reference. The results of our studies are summarized as a comprehensive and practical resource to precisely reconstruct the complex dielectric function in a broad optical range, providing a useful link to all optical constants for the substrates. This information is presented within Tables 2–8 in the appendix of this article as a list of all parameters for each substrate material.
2. Experiment
Single-crystal substrates of (100) STO, 0.7 % wt. Nb-doped (100) STO, (100) LSAT, (011) DSO, (100) MAS, (100) MgO, and (100) LAO were purchased from MTI Corp. All substrates are a standard size of 10×10×0.5 mm. These samples were received epitaxially polished on a single side by Chemical Mechanical Planarization/Polishing (CMP) [36] with a specified surface roughness of less than 0.5 nm. However, the surfaces of these substrates were analyzed using an Alpha-Step 500 surface profiler (Tencor; USA), and resulting surface roughness appeared to be greater than specified. Though epi-polishing leads to less sub-surface damage and better plane flatness, this process is the main culprit in producing surface steps, which are observed with profilometry and ellispometry.
The optical properties of these substrates were investigated with a VUV ellipsometer from J. A. Woollam operating in rotating analyzer mode. Before measurements we employ ultrasonic cleaning in a high-purity acetone bath. The ellipsometric angles Ψ and Δ were measured at room temperature over a wide spectral range from 0.74 to 8.8 eV with a step size of 0.02 eV. To reach the VUV range, the sample chamber was purged with dry nitrogen. Measurements were performed in reflection mode at several angles of incidence ranging from 60° to 75°. The optical constants for our crystal substrates were determined using WVASE32 software [37, 38]. The modeling procedure within this software package is initiated through the construction of a layered structure, and for substrates such as those examined in this study the substrate layer is simplified to a semi-infinite slab. This approximation is valid here as backside reflections are mitigated in the measurement process via diffuse reflection, which is achieved through roughening of the back surface. Front side surface roughness and contact with ambient gas are taken into account in the modeling process through additional independent layers.
Generally, it is possible to calculate the complex index of refraction at each wavelength using a point-by-point analysis of ellipsometric data. However here we opt instead for the construction of dispersion models using the complex dielectric function. We feel this route is more complete because the functional form for dielectric spectra can be used more readily to describe a material’s response to incoming radiation. Moreover, it allows us to develop physically consistent models as described below. Development of these models is accomplished for each substrate. The experimental ellipsometric data are collected in order to calculate the pseudo dielectric function over the entire spectral range. A multi-oscillator layer is used to match that function in the modeling process. All fitting parameters used in the multi-oscillator model are determined simultaneously through a fitting process that uses the Levenberg-Marquardt algorithm [39]. These multi-oscillator dielectric models can then be used to generate ellipsometric data, which is then compared directly to experimental ellipsometric data to verify overall consistency.
Each multi-oscillator model of the complex dielectric function ε = ε1 + iε2 is composed of a linear combination of several oscillators combined with a purely real offset to ε1 matching the limit of material permittivity at high frequency, often denoted ε∞. This offset is not used as a fitting parameter and kept at ε∞ = 1 for all models presented here. For most materials in this study, it is sufficient to use a combination of n Gaussian oscillators to construct a functional form for the imaginary component of the dielectric function:
where An is the scaled amplitude of the nth oscillator, Brn corresponds to oscillator broadening (FWHM), and En is the central energy of the nth oscillator. The amplitude, broadening and central energy for each Gaussian oscillator are all used as fitting parameters in the fitting process. The real component of the dielectric function for each model is determined through a physically consistent causality relationship known as the Kramers-Kronig relations [40]. and For Gaussian oscillators with the form given above, we get the following contribution to the real component for the nth oscillator: where P denotes the Cauchy principle value. The sum of these terms is often insufficient in fully describing the real component of the dielectric function. Therefore, in order to take into account the dispersion from absorption peaks in the UV range outside of our measured spectral range, we include a single UV pole into the sum: where again Apole is the pole amplitude and Epole is the pole central energy, and both of these constants are used as fitting parameters in the fitting process. This type of pole is often considered as a Lorentz oscillator with no broadening, and thus has no impact on the sum determining ε2. Finally, putting all of this together we arrive at the functional form for the complex dielectric function ε = ε1 + iε2 used here:Optical properties for each substrate were obtained through a physically consistent causality relationship (Kramers-Kronig) between the real and imaginary parts of the complex dielectric function and complex index of refraction. The optical properties of STO and Nb-doped STO were constructed using nine Gaussian oscillators and a UV pole. Moreover, a weak contribution from a Drude oscillator was observed in Nb-doped STO. We used five Gaussian oscillators and a UV pole for the LAO and DSO substrates. A UV pole was combined with three and four Gaussian oscillators for the models for MAS and LSAT, respectively. The optical properties of MgO were obtained using a single Lorentz and two Tauc-Lorentz oscillators with a UV pole. While these types of oscillators differ in functional form from that of a Gaussian oscillator, the general idea of constructing a physically consistent model for the dielectric function presented above is easily repeated [41].
Depolarization of reflected light has been determined to be less than 0.01 % for all angles of incidence, and is therefore neglected for all models presented. The thickness of the surface roughness layer for each sample was calculated in a spectral range below the bandgap energies where absorption was low or completely absent and was then fixed for the entire range. Surface roughness was represented as a mixture of 50 % solid and 50 % voids, as prescribed by the Bruggeman effective medium approximation [42]. As stated above, surface roughness is included in the models constructed as an independent layer, and the depth or thickness of this layer is determined as an additional fitting parameter. In order to increase the accuracy of the absorption coefficients for each model we fixed the thickness of the surface roughness layer and all oscillator parameters, and then we used numerical inversion (point-by-point) to extract the optical constants over the full measured spectral range. Direct and indirect band gap energies were determined through the absorption coefficients for each material using the widely employed Tauc relations [43]. Indirect bandgap energies E are extracted by plotting the zero intercept of the fitting function (αE)1/2 in the range of α from 2×103 cm−1 to 45×103 cm−1, whereas direct band gap energies are determined by using (αE)2 as the fitting function in the range of α from 1.2 × 105 cm−1 to 3×105 cm−1 [43].
3. Results and discussion
3.1. Strontium titanate
Single crystal strontium titanate, SrTiO3 (STO), is frequently used as a standard substrate as it meets the lattice-matching condition necessary for epitaxial growth of many thin films quite well. The room temperature crystal structure of both undoped and Nb-doped STO is cubic perovskite with a lattice parameter of 3.905 Å. Undoped STO is an insulating material with a high dielectric constant and optical transparency through a wide wavelength range [43]. In contrast, even slight doping of STO with niobium greatly modifies the electronic properties [15, 44, 45].
The dielectric functions were constructed using a multi-oscillator model which required nine Gaussian oscillators and a single UV pole to obtain excellent matches to the expected functions for each material, as illustrated in Fig. 1(a) for Nb-doped STO. For Nb-doped STO a weak increase in absorption at energies lower than 1 eV is observed, indicating a Drude-type contribution from charge carriers. The positions (central wavelengths), widths, and amplitudes for these oscillators were all used as fitting parameters and are presented in the appendix of this article in Tables 2 and 3. The thickness of the surface roughness layers for both undoped and Nb-doped STO were also allowed to vary in the fitting process and found to have a depth in the range of 1.5 – 1.8 nm. We used a profiler to directly measure surface roughness, and the depth profile of Nb-doped STO can be seen in Fig. 1(b). Considering this data obtained from a direct measurement of the sample, it is clear that the surface roughness depth obtained in the fitting process is reasonable.
Fig. 1 (a) The expected imaginary component of the dielectric function, ε2 plotted with the Gaussian oscillators used to construct the model substrate layer and (b) the surface roughness profile of Nb-doped STO measured with a profiler. These graphs represent many of the fitting parameters used, and when combined with the fitting parameters of the UV pole, all necessary information is present for the construction of the full material model developed for Nb-doped STO.
To illustrate the versatility of our well-constructed model for undoped STO, we convert the dielectric function obtained through the modeling process back to the expected ellipsometric angles Ψ and Δ for the angles of incidence used in our experiments, 60°, 65° and 70°. The experimental data for these substrates is shown together with model-generated data in Figs. 2(a) and 2(b) for Ψ and Δ, respectively. The generated and experimental curves are nearly indistinguishable. A difference graph is utilized in to better visualize the variation between experimental and model-generated data for an angle of incidence of 60° (Figs. 2(c) and 2(d)) and the resulting mean-squared error (MSE) of 0.22 was calculated for all (Ψ, Δ) pairs over the entire optical range. Our results demonstrate an excellent match. The model is further utilized to generate data for both the real and imaginary parts of complex refractive index as well as the absorption coefficients of both undoped and Nb-doped STO (Fig. 3). The positions of the main spectral peaks and absorption edge obtained for the STO substrate agree with previous observations [26, 46, 47].
Fig. 2 The (a, b) experimentally obtained (red dashed lines) and model-generated (solid green lines) ellipsometric data (a) Ψ and (b) Δ in a (100)-oriented STO substrate for 60°, 65° and 70° angles of incidence., (c, d). Difference between the model-generated and experimental ellipsometric data for an angle of incidence of 60°.
Fig. 3 (a) Real and (b) imaginary parts of complex refractive index and (c) absorption coefficient as a function of photon energy E in the STO and Nb-doped STO substrates.
Upon closer examination of the model-generated optical absorption spectra at the onset of absorption (Fig. 4), it is possible to calculate the bandgap energies of our substrates. This onset of absorption is gradual enough such that plotting the square root of absorption versus photon energy results in stepwise linear behavior for both undoped and Nb-doped STO. This behavior is attributed to the presence of an indirect bandgap, which is indeed expected as the crystal structure of STO contains several phonon modes that contribute to indirect absorption of radiation [26]. Bandgap energies of E = 3.2 eV for STO and E = 3.02 eV for Nb-doped STO are determined through the zero intercepts of the linear fits to the Tauc plots in the range α from 2×103 cm−1 to 45×103 cm−1 and are shown as dashed lines in Fig. 4. These values are listed for reference in Table 1 along with previously published data where available.
Fig. 4 Tauc plots for indirect optical transition in STO and Nb-doped STO substrates. Data obtained by point-by-point fitting are shown as symbols. Dashed lines show the Tauc fits.
Table 1. Summary of substrate properties: lattice parameter a, optical surface roughness, photon energy at which absorption coefficient α = 104 cm−1, Tauc bandgap energy and type, previously published bandgap energy, real part n of refraction index at photon energy of 2 eV. Legend: a (Å) is a lattice parameter; SR is a surface roughness; E0 (eV) is energy at α = 104 cm−1; BG is band gap energies from our research; TF is Tauc fit of band gap (D for direct and I for indirect); BG Ref is band gap energies published.
The observed reduction in bandgap energy with Nb doping of STO has been theoretically predicted in [48]. This computational study indicates that the shift occurs due to a disturbance of the top valence band by the Nb-derived impurity charge potential. Further study is needed to more precisely determine the relationship between the Nb dopant level and the magnitude of the observed bandgap shift in this material. The emergence of an Urbach tail with doping is also not surprising as it is often observed in bandgap crystals where long-range order is perturbed. The exact cause for this feature is debated [49], with a general consensus pointing toward the band-tail states in the energy gap. We note that theoretical and experimental studies of optical properties of STO are still ongoing [47]. Heterovalent doping of STO has been reported to change the badgap of this material [50].
3.2. Lanthanum strontium aluminum tantalate
Lanthanum strontium aluminum tantalate, (LaAlO3)0.29(SrAl0.5Ta0.5O3)0.7 (LSAT), was originally developed as a suitable substrate for growth of cuprate superconducting thin films [18]. Due to its high chemical and thermal stability and low electrical conductivity, it has quickly been adapted for growth of a wide range of thin film materials including strain-enabled ferroelectrics [25], and GaN-based LEDs and laser diodes [11, 25]. The crystal structure of LSAT is cubic and belongs to space group with a lattice parameter a=3.868 Å at room temperature. It is transparent over the visible and IR spectral range and has been used as a substrate in, for example, IR spectroscopy experiments examining infrared phonon modes [51].
The process of obtaining the optical properties of LSAT is identical to the procedure described above for doped and undoped STO. The central wavelengths, widths, and amplitudes for all oscillators used as fitting parameters in this process are presented in the appendix of this article in Table 4. The depth of the surface roughness layer of the model was determined to be approximately 4 nm through the fitting process, and when measured directly with a profiler we obtained a roughness depth of approximately 3 nm.
Fig. 5 shows the real and imaginary components of the refractive index as well as the absorption coefficient as a function of photon energy obtained with the multi-oscillator model developed for the LSAT substrate. Our results are in good agreement with previous results [30]; however, by characterizing the substrate further in the VUV range, we have more complete picture of the optical properties of this material. Upon closer examination of the absorption coefficient in the range α from 2×103 cm−1 to 45×103 cm−1, we find that plotting the square root of absorption versus photon energy results again in stepwise linear behavior, which as stated above for STO is indicative of an indirect bandgap for LSAT. A bandgap energy of E = 5.16 eV, was determined through the zero intercept of the linear fit to the Tauc plot for this material. We include this value for the indirect bandgap energy along with previously published results and other useful information for LSAT in Table 1. As a guide to the reader, we also include graphs in the appendix presenting linear fits to the Tauc plots for both the direct and indirect cases for LSAT (Fig. 10). Using the direct case, we estimate the lowest direct gap energy of 5.5 eV in the range in the range of α from 1.2 × 105 cm−1 to 3×105 cm−1, which is close to the results obtained by Barnes et. al. [29] (5.6 eV), but deviates a bit more from other work [30] (5.8 eV).
Fig. 5 (a) Real and (b) imaginary parts of complex refractive index, and (c) absorption coefficient as a function of photon energy E in the LSAT substrate.
3.3. Dysprosium scandate
The development of thermally-stable single-crystal rare-earth scandates such as dysprosium scandate, DyScO3 (DSO), was initially motivated by studies focused on the strain-induced enhancements of superconducting thin films which required an array of substrate materials with a range of available lattice parameters that fit neatly within the window set by STO and barium titanate (3.905 – 4.036 Å) [22, 23, 52]. At room temperature, DSO has an orthorhombic crystal structure with lattice parameters of a=5.440 Å, b=5.713 Å, and c=7.887 Å; however, (011)-oriented DSO has nearly square surface lattice with the lattice parameter of a=3.944 Å, making it a suitable substrate for growth of perovskite ferroelectrics and other similarly structured materials [53]. This material has since found its way in many other applications such as the formation of an insulating buffer layer within heterostructures designed for nonvolatile memory applications [13] and thus knowledge of the optical constants of DSO is useful for future studies.
We again applied the method described above to obtain the optical properties of DSO. In general, DSO is weakly anisotropic; however, upon inspection of the first few batches of ellipsometric data collected from (011) DSO we found that the off-diagonal elements of the Jones matrix were nearly zero over the entire spectral range from 0.74 eV to 8.8 eV. For this reason, we chose to assume the material was isotropic for subsequent data collection and calculations. All of the fitting parameters used in this process are presented in the appendix of this article in Table 5. Surface roughness calculated through ellipsometric modeling has a depth of 2.2 nm, which is consistent with results obtained from profiler data.
The model-generated complex index of refraction and absorption coefficient spectra are shown in Fig. 6. The bandgap for this material was determined to be direct through the use of Tauc plots, and following the methods described above for STO and LSAT, a bandgap energy of 6.3 eV was determined through the zero intercept of a linear fit to the Tauc plot in the appropriate range. This is higher than what was found in [53], which was determined through the comparison of XAS and XES data. We include this value for the bandgap energy along with other useful information for DSO in Table 1. As a guide to the reader, we include a graph in the appendix presenting the linear fit to the Tauc plot used to derive the bandgap energy for this material, where it is clear that the response in the absorption spectrum is indicative of an direct band gap (Fig. 10).
Fig. 6 (a) Real and (b) imaginary parts of complex refractive index, and (c) absorption coefficient as a function of photon energy E in the DSO substrate.
3.4. Magnesium aluminate
Ceramic oxides such as MgAl2O4, magnesium aluminate spinel (MAS), have been used for many years in applications such as q-switching of pulsed laser cavities and humidity detection in active thin film sensors [7, 54]. Although MAS possesses many attractive physical properties such as high melting point, excellent chemical inertness, high electrical resistivity, low thermal expansion, and good mechanical strength at high temperature, interest in this material in science and industry was relatively low until recent advancements in synthesis in the last decade have resulted in extremely high quality single crystal spinel that is both inexpensive and readily available [55]. The spinel class is indeed named after the structure of magnesium aluminate, and thus MAS has a face-centered spinel crystal structure belonging to the Fd3m symmetry group with a lattice parameter of 8.083 Å at room temperature. It is highly transparent through the visible and well into the IR spectral range, and is being heralded as transparent armor as MAS finds its way into military and industrial applications simultaneously requiring high strength and optical transparency [56].
The obtained absorption coefficient for MAS is shown in Fig. 7(c) along with complex refractive index shown in Figs. 7(a) and 7(b), and these spectra are in a good agreement with previous results [28]. All of the fitting parameters used in the modelling process to obtain these curves are presented in the appendix of this article in Table 6. Additionally, a depth of 1.5 nm for the surface roughness layer was calculated through analysis of the ellipsometric data, which is in excellent agreement with the value obtained through profiler measurements. The onset of optical absorption lies in ultraviolet range above 7 eV, and through further analysis of the absorption coefficient it was determined that MAS has a direct bandgap at 7.8 eV, which is in good agreement with previously obtained results [57]. We also include this value for the bandgap energy along with other useful information for MAS in table 1 with all of the other materials analyzed in this study. We include a graph in the appendix presenting the linear fit to the Tauc plot used to derive the bandgap energy for this material (Fig. 10)
Fig. 7 (a) Real and (b) imaginary parts of complex refractive index, and (c) absorption coefficient as a function of photon energy E in the MgAl2O4 substrate.
3.5. Magnesium oxide
Magnesium oxide, MgO, shares many similar physical properties with MAS such as a high melting point, excellent chemical inertness, low electrical conductivity and low thermal expansion. In addition, MgO has a relatively high thermal conductivity, making it ideal for its most common use as a refractory material [58]. It has also been suggested for applications ranging from broadband laser emission [10] to the composition of support structures in catalytic chemistry. MgO has a cubic crystal structure with a lattice parameter of 4.216 Å at room temperature and is suitable as a substrate for deposition of rocksalt cubic materials such as transition metal nitrides. MgO is transparent in the near IR and through the visible and UV ranges up to a photon energy of 7.5 eV. In comparison with other substrates, MgO has a low index of refraction throughout the entire visible range and as a result has been proven useful in many optoelectronic applications [59, 60].
The optical properties of MgO shown in Fig. 8 are in good agreement with previously obtained results [27]. The parameters obtained for the oscillators used in the modelling process to obtain these properties are presented in the appendix of this article in Table 7. The depth of the surface roughness layer was determined to be about 2.5 nm, which was confirmed using a profiler. Upon closer examination of the absorption coefficient in Fig. 8(c), we find that the onset of optical absorption lies in ultraviolet range above 7 eV and a direct bandgap at 7.8 eV was calculated through the use of a Tauc plot. We include a graph in the appendix presenting the linear fit to the Tauc plot used to derive the bandgap energy for this material (Fig. 10). This calculation is in good agreement with previously obtained results [61], and we include this value for the bandgap energy along with other useful information for MgO in Table 1.
Fig. 8 (a) Real and (b) imaginary parts of complex refractive index, and (c) absorption coefficient as a function of photon energy E in the MgO substrate.
3.6. Lanthanum aluminate
Lanthanum aluminate, LaAlO3 (LAO), has been regarded as an ideal substrate material for epitaxial growth of 1-2-3 type superconducting thin films due to a close lattice match within 1 % as well as its low dielectric constant and loss tangent compared to other common substrate materials [16]. LAO has a pseudo-cubic crystal structure with a lattice parameter of 3.79 Å at room temperature [62] LAO is transparent from the near IR through the visible spectrum to the UV range up to a photon energy of about 5.5 eV.
The complex index of refraction and absorption coefficient for LAO are shown in Fig. 9, and the parameters obtained for the oscillators used in the modelling process are presented in the appendix of this article in Table 8. The depth calculated for the surface roughness layer of the constructed model was relatively large at about 4.1 nm; however, this calculation was confirmed using a profiler on our sample. Analysis of the absorption coefficient using a Tauc plot revealed an indirect bandgap at 5.6 eV, which is in a good agreement with previously obtained results [31]. We include a graph in the appendix presenting the linear fit to the Tauc plot used to derive the bandgap energy for this material (Fig. 10). In addition, he index of refraction in the visible range at 2 eV and the bandgap energy calculated here are presented in Table 1.
Fig. 9 (a) Real and (b) imaginary parts of complex refractive index, and (c) absorption coefficient as a function of photon energy E in the LaAlO3 substrate.
3.7. Summary
The key optical parameters of the state-of-art single-crystal substrates are summarized in Table 1. The substrates clearly fall into two groups considering the bandgap energies: the wide-bandgap STO ones and those with the very wide bandgaps of over 5 eV. Being transparent in the visible spectral range, the substrates (except MgO and MAS) exhibit rather large refractive index n > 2 in this range. This index may set restrictions to design of optical heterostructures using such promising materials as ferroelectrics, whose index is n > 2, too. Concurrently, the more attractive relatively small n ≈ 1.7 in MgO and MAS is combined with the lattice parameters, which are larger than in ferroelectrics and, thus, preventing pseudomorphic epitaxy of these and related materials. We also note that the LSAT and LAO substrates, which are used in most studies of epitaxial perovskite oxide films, possess significant surface roughness as revealed by profilometry and ellipsometry. This roughness should be taken into account, especially in studies of ultrathin films (with thicknesses of a few nanometers only). Finally, we stress that beyond the data in Table 1, our work offers the details of analytical dielectric functions of the substrates. We believe that these details are of high and extensive practical importance, which will enable wide applications of the results of this work.
4. Conclusion
In summary, an analytic form for the complex dielectric function in the spectral range of 0.74 eV to 8.8 eV was experimentally derived in the state-of-art single-crystal substrates enabling epitaxy. The STO, Nb-doped STO, LSAT, DSO, MAS, MgO, and LAO substrates were examined using spectroscopic ellipsometry, and the dielectric functions for each material were derived through a physically consistent modeling. The parameters necessary for constructing the dielectric spectra are provided within this work. We encourage application of these parameters as well as expansion of this method for optical studies of thin films and heterostructures on these substrates.
Appendix
Table 2. Parameters of Gaussian oscillators used for bulk (100) SrTiO3. Last line in the table shows parameters of UV oscillator.
Table 3. Parameters of Gaussian oscillators used for bulk Nb-doped (100) SrTiO3. Last two lines in the table show parameters of UV and Drude oscillator.
Table 4. Parameters of Gaussian oscillators used for bulk (100) (LaAlO3)0.29(SrAl0.5Ta0.5O3)0.7. Last line in the table shows parameters of UV oscillator.
Table 5. Parameters of Gaussian oscillators used for bulk (011) DyScO3. Last line in the table shows parameters of UV oscillator.
Table 6. Parameters of Gaussian oscillators used for bulk (100) MgAl2O4. Last line in the table shows parameters of UV oscillator.
Table 7. Parameters of two Tauc-Lorentz and one Lorentz oscillators used for bulk (100) MgO. Last line in the table shows parameters of UV oscillator.
Table 8. Parameters of Gaussian oscillators used for bulk (100) LaAlO3. Last line in the table shows parameters of UV oscillator.
Funding
The Grant Agency of the Czech Republic (Grant Nos. 15-15123S and 15-13778S); ELI - Extreme Light Infrastructure - phase 2 (CZ.02.1.01/0.0/0.0/15_008/0000162) from European Regional Development Fund; and the Grant Agency of the Czech Technical University in Prague (SGS16/244/OHK4/3T/14).
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