Elevated temperature spectroscopic ellipsometry analysis of the dielectric function, exciton, band-to-band transition, and high-frequency dielectric constant properties for single-crystal ZnGa₂O₄

Ultrawide bandgap metal oxide semiconductors, such as gallium oxide, have attracted remarkable attention for their use in high-power electronic devices. These materials promise a significant increase in the Baliga's figure of merit, which is proportional to the bandgap cubed (B_FOM $∝$  $Eg3$⁠), over current high-power device materials, such as GaN and SiC.^1–3 However, unlike Ga₂O₃, zinc gallate (ZnGa₂O₄) contains the isotropic spinel structure. The bandgap is, thus, isotropic as well. Previous findings for the onset of absorption were reported at 4.6 eV in transmission experiments, while reflection ellipsometry reported a bandgap value of E_g = 5.27(3) eV, making it attractive for device design and implementation.^4,5 Recently, controllable n-type conductivity in ZnGa₂O₄ has been demonstrated, and signatures of p-type conductivity were observed in the high-temperature Hall effect on one ZnGa₂O₄ layer grown on β-Ga₂O₃.^4,6 This further amplifies ZnGa₂O₄ as a promising material for applications, including ultraviolet photodetectors, light emitting diodes, gas sensors, photocatalysts, and photovoltaics.^7–12 Multiple growth methods have demonstrated the ability to deposit ZnGa₂O₄, including chemical vapor deposition,^6,13,14 melt growth,^4,15,16 thermal evaporation,¹² magnetron sputtering,¹⁷ and synthesis.^18–20 Galazka et al. reported high-quality single crystal growth of ZnGa₂O₄ containing the free electron concentrations approaching of 10²⁰ cm⁻³ with the high electron Hall mobility of 100 cm²/V s.^4,15,16 The production of large single crystals with minimal impurity concentration provides an avenue for ZnGa₂O₄ to be used in device applications, which are based on ultrawide bandgap materials.

The operation of the semiconductor devices present in power electronic circuit designs at high voltage and high frequency values lead them to commonly be exposed to elevated temperatures.²¹ For example, SiC metal-oxide semiconductor field effect transistors (MOSFETs) have been developed to operate at $≥$ 400 °C.²² As attractive alternative GaN/AlGaN based bipolar transistors for various satellite, radar, and communication systems are designed for operating at a high frequency range (1–5 GHz) and temperatures larger than 400 °C,²³ the SiC material is also utilized for micro-electromechanical sensors that play a critically important role in the electrical subsystems of emerging vehicle and aircraft propulsion technologies, which can operate at high temperatures (500 °C).^24,25

High power electronic device designs of ultra-wide bandgap materials and their response at the elevated temperatures also require precise knowledge on their optical bandgap information. For metal oxides, it has been shown that increasing the temperature above room temperature frequently results in a linear decrease in the bandgap.^26–28 In most semiconductors, this linear dependence between temperature and bandgap is mainly attributed to both the lattice thermal expansion and the electron-phonon interaction, which shrinks the bandgap with temperature increase.^26,28–31 The temperature rise triggers the escalation of both thermal energy and thermal expansion, which leads to larger inter-atomic space and results in a decrease in potential energy of electrons and, thus, reduction of the bandgap energy.²⁸ 

Albeit its obvious necessity, for ZnGa₂O₄ to be used in contemporary device applications, an understanding of its temperature characteristics has been rarely discussed in the literature. For ZnGa₂O₄ current studies are limited to room temperature optical measurements and density functional theory (DFT) calculations.^32–37 Recently, Hilfiker et al. used a combined density functional theory calculation and spectroscopic ellipsometry analysis approach to determine the optical properties for room temperature ZnGa₂O₄. Spectroscopic ellipsometry analysis relies on a line shape analysis of the dielectric function, where multiple line shape methods may be applied to interpret the electronic transitions. In their letter, density functional theory calculations of the valence band structure showed positive curvature for bands associated with the electronic transitions at Γ, indicative of an M₀-type singularity in the density of states. These calculations, thereby, justified selection of an M₀ CP line shape to describe the lowest band-to-band transition. Additionally, further examination of the dielectric function indicated the presence of a strong excitonic absorption. A nonsymmetrically broadened Lorentz was applied to account for this effect; however, at room temperature, excitonic contributions are sufficiently broadened where ambiguity exists in assigning specific line shapes for individual features.⁵ In this work, we implement a similar critical point dielectric function line shape model for elevated temperatures. Here, we determine not only temperature dependent characteristics of band-to-band transitions and excitonic contributions but also provide a further justification for the line shape features used to describe ZnGa₂O₄.

An electrically insulating, double-sided polished bulk substrate with dimensions of 10 × 10 × 0.5 mm² with a principal (100) surface was grown using the vertical gradient freeze (VGF) method at Leibniz-Institut für Kristallzüchtung, Berlin, Germany. The sample was further annealed at 1100 °C for 10 h resulting in an electrically insulating substrate. Further growth details are reported by Galazka et al. and can be found in Refs. 4, 15, and 16.

Spectroscopic ellipsometry provides a contactless method for optically characterizing material properties. This highly sensitive measurement technique can determine material properties, such as optical phonon modes, dielectric functions, band-to-band transitions, and excitonic contributions, for example. For reflection-mode ellipsometry, the ratio of the p- and s-polarized Fresnel coefficients is related to $Ψ$ and Δ³⁸ 

In this Letter, a single-rotating compensator ellipsometer (M-2000, J. A. Woollam Co., Inc.) is used to measure ZnGa₂O₄ over the spectral range of 1–6.42 eV. The ellipsometer is attached to a vacuum chamber such that the angle of incidence on the sample within the chamber is obtained at 75°. The sample is mounted within the chamber onto a sample heater stage. The vacuum chamber is equipped with a turbomolecular pump allowing for all temperature measurements to be performed under 6.5 × 10⁻⁵Torr. The heater stage contains electrical resistance heating elements. The sample temperature as a function of the electric heating power is calibrated using a single crystalline epi-ready silicon wafer. The ellipsometric parameters from the silicon waver are measured over a large range of power, and the changes in the ellipsometric parameter were matched to a temperature-dependent model dielectric function for silicon from Sik et al.³⁹ From this model, we determine the temperature of the sample stage. This temperature is then plotted vs the applied power, and a second-order power law is determined between the applied voltage and the resulting sample stage temperature. Then, a control of the temperature of a different sample placed onto the stage is possible by adjusting the electrical heating power according to the previous calibration. Measurements for the ZnGa₂O₄ crystal are then performed at room temperature (⁠$22$ °C) and from $50$ to $600$ °C in steps of $50$ °C.

The spectroscopic ellipsometry data are analyzed using a model-based approach rendering the sample by a substrate and overlayer model to describe the optical constants of the (substrate) crystal and possibly existing nanoscale surface roughness, respectively.⁴⁰ The overlayer is described using an effective medium approximation, where the optical constants consist are the average of ambient (⁠$εr=1$⁠) and the underlying ZnGa₂O₄ crystal, where volume fractions were considered temperature independent. The thickness of the overlayer (t_ov) is determined for all temperature measurements individually, using a Cauchy dispersion equation to calculate the refractive index, n, of the ZnGa₂O₄ substrate in the below bandgap (transparent) region (⁠$≤$3.5 eV), where then the dielectric function is obtained from $ε=n2$⁠. With the thickness of the nanoscale roughness determined, a Cauchy dispersion equation is augmented

where $A(T)=A0+mA(T−T0), B(T)=B0+mB(T−T0)$ are the zero and second order coefficients with A₀ and B₀ the respective room temperature (T₀) coefficients; m_A and m_B are the linear temperature slopes for the A and B terms, respectively; and T is the temperature.

The photon energy (⁠$ℏω$⁠) dependence of $ε(ℏω,T)$ across the full spectral region (1–6.42 eV) is then analyzed using a point-by-point (PBP) regression analysis resulting in a wavelength-by-wavelength (photon energy-by-photon energy) dielectric function. In this step, the thickness of the surface roughness overlayer is kept constant for each independent temperature as determined from the Cauchy analysis. To gain information on the band-to-band transitions and excitonic contributions, a critical point model dielectric function (MDF) approach is applied using the PBP dielectric function as reference. As determined in our previous work, DFT calculations indicate that the direct bandgap of ZnGa₂O₄ can be modeled using a M₀-type critical point line shape function^5,41,42

Here, A_CP is the amplitude, E is the center energy, and Γ is the broadening parameter. The dielectric function of ZnGa₂O₄ reveals strong excitonic contributions as shown in previous room temperature measurements.⁵ Multiple model line shape functions have been proposed to describe exciton contributions in crystalline materials.^43,44 Recently, Mock et al. applied a mathematically equivalent form of the four-parameter semi-quantum model developed by Gervais and Piriou using an anharmonic broadened Lorentz to describe exitonic contributions in the visible to ultraviolet spectral range.^45,46 This model has been readily applied to ultrawide bandgap materials,^45,47,48 including room-temperature ZnGa₂O₄.⁵ Here, we apply the anharmonic broadened Lorentz oscillator to describe the excitonic contribution to the dielectric function as follows:

where b and $R⋆$ denote the anharmonic broadening parameter and effective Rydberg energy, respectively. An estimation of $R⋆$ = 14.8 meV was provided by our previous work and not further varied during our analysis in this Letter.⁵ We note that while also $R⋆$ could reveal a temperature dependence, such modifications should be small. For example, a recent temperature study on exciton properties in CuO₂ revealed very small changes for temperatures from 4 to 100 K.⁴⁹ Furthermore, because of $R⋆≪Γ,b$⁠, the exciton broadening parameters, our data do not permit to determine its actual value.⁵ Hence, we have assumed $R⋆$ independent of temperature here. Higher energy electronic transitions are described using a Gaussian oscillator for the imaginary (⁠$ℑ$⁠) component of the dielectric function^38,42

from which Kramers–Kronig integration allows us to acquire the real (⁠$ℜ$⁠) part of the dielectric function

The evolution of Cauchy terms A and B as a function of temperature is displayed in Figs. 1(a) and 1(b), respectively. Note that the zero order Cauchy term is related to the high-frequency dielectric constant, i.e., $ε∞=A2(T)$⁠. The high-frequency dielectric constant is obtained from the Lyddane–Sachs–Teller relationship and measurement of static dielectric constant and polar optical phonon modes, see, e.g., Stokey et al.³⁷ Hence, we have also determined the index of refraction for spectral energies below the bandgap as well as the high-frequency dielectric constant as a function of temperature here. Linear slopes for the A and B terms are found to be m_A = 3.8852 × 10⁻⁵ K⁻¹ and m_B = 1.1501 × 10$−7μ$m² K⁻¹, respectively. Accordingly, we find A₀ = 1.899(5) and B₀ = 0.0150(4) μm².

The temperature dependent dielectric function of ZnGa₂O₄ determined from both the PBP and critical point model approaches is shown in Fig. 2. We note that the results of both model approaches are in excellent agreement with each other. The bandgap energy is identified by the onset of the imaginary part and the associated dispersion in the real part. As the temperature increases, two main effects are seen: a red shift of the bandgap features and an increase in broadening. This is indicative of phonon interaction with the electronic transitions and an increase in the lattice expansion. The bandgap shift to lower energy is typical for many semiconductor materials.^50,51

Figure 3 shows the evolution of critical point MDF contributions to the imaginary parts for select elevated temperatures. Each MDF consists of an excitonic contribution (CP^0x), a M₀-type critical point (CP⁰), and a Gaussian transition (CP¹). As the temperature increases, excitons become thermally ionized and a reduction of their contribution to the dielectric function is expected. This correlation between exciton contributions and temperature has also been shown for other semiconductors, such as GaAs.^41,52 The change in excitonic contribution for ZnGa₂O₄ is further examined in Fig. 4. Both a reduction in amplitude and an increase in broadening of the MDF CP^0x term occur. This observation is consistent with the theoretical understanding of the exciton nature. Thermal exciton evaporation and lifetime reduction are due to increased scattering at elevated temperature with longitudinal optical phonons (LO) via Frölich interaction.⁴¹ This phenomenon is characteristic for polar semiconductors. There, the LO phonons create an electric field, which is macroscopic with respect to the interatomic distances. Known as Fröhlich interaction, both holes and electrons forming the exciton interact with the macroscopic field, with coupling over long ranges. The coupling leads to the formation of quasi-particles also known as polarons. Thermal excitations leads to an increase in phonon population, and enhancement of disorder and scattering. Hence, the lifetime of the excitons and the associated polarons reduce, and so does the population of excited electron–hole pairs. We clearly observe here that the excitonic contribution on the physical line shape of the ZnGa₂O₄ dielectric function reduces while smaller and more broadened contributions can be still discerned within the data at elevated temperatures [see Figs. 4(a) and 4(b)]. It is of interest here to note the contribution of the exciton MDF to the imaginary part. Often we have observed this contribution to be negative in limited spectral regions.^{5,37,45,48,53,54} The negative contribution is the numerical result of the anharmonic broadening introduced in our MDF for the exciton. The observation must be appropriately discussed in the context of coupled oscillations within an harmonic potential. In a classical mechanical picture, such coupling, e.g., between two different mass particles, leads to an energy exchange that induces a continuously varying phase shift between the individual particles' oscillations. Consequently, when observing the motion of one oscillator, while ignoring the other, there appears to be gain at times while loss at others. However, if observed as entity, there is only loss and no gain. When the displacement contributions are transformed into the frequency domain, the sum of all contributions to the dielectric displacement then adds up to non-negative imaginary parts throughout the spectral range. One can only require that the total response of a system conforms with energy conservation, while individual contributions in coupled systems can possess negative imaginary parts. Therefore, passivity principles are regained. We have observed this phenomenon specifically at the bandgap of many metal oxides recently, for example, in rhombohedral α-Ga₂O₃,⁴⁸ and α-(Al,Ga)₂O₃,⁵⁴ monoclinic β-Ga₂O₃,⁴⁵ and β-(Al,Ga)₂O₃,⁵³ as well as for the room temperature response of ZnGa₂O₄.⁵ A similar observation is frequently made when comparing anharmonically broadened oscillators of the same mathematical form than used here for excitons to phonon modes in anharmonic crystals.⁵⁵ Indeed, individual contributions for the phonon modes in ZnGa₂O₄ to the dielectric function reveal similar negative imprints in the imaginary part, while the sum of all contributions renders the infrared dielectric function positive in its imaginary part throughout the spectral range.³⁷ We note further that additional numerical constraints on the parameters of the CP^0x and CP⁰ model functions may be obtainable requiring that their imaginary parts cancel throughout the wavelength range below the bandgap energy, for example.

The temperature evolution of the direct bandgap energy of ZnGa₂O₄ is shown in Fig. 5. A linear slope of $−0.72(4)$ meVK⁻¹ is found. The temperature dependence of the bandgap energy is often described by the Bose–Einstein model⁵⁶ 

where $Eg(0)$ is the gap energy at zero temperature, a_B represents the strength of the exciton-phonon interaction, and θ_B is the average phonon temperature energy equivalent. In this approximation, the exciton, which is the quasi particle formed in association with a band-to-band transition, and its interaction with the lattice is considered by an effective phonon mode. This effective mode, thereby, represents the much larger ensemble of phonons, which form the bath of constant lattice interaction at a given temperature. For temperatures large against the effective phonon temperature, one can approximate the shift of bandgap energy with a linear slope, $γ=δEg/δT≈−2aB/θB$ (⁠$T≫θB$⁠). For low temperatures, parameter $Eg(0)$ can be obtained from monitoring the bandgap energy at low temperatures. The curvature of $Eg(T)$ vs T permits determination of a_B from θ_B. Often, θ_B is found consistent with an average over all IR-active phonon modes.³ Our data set only permits to determine parameter γ. Using the average of IR-active phonon modes determined previously by Stokey et al. (⁠$≈$451 cm⁻¹),³⁷ we can estimate the phonon coupling parameter $aB≈$ 0.162 cm⁻¹. We note that the slope parameter γ found here is similar to monoclinic Ga₂O₃ for the three lowest band-to-band transitions determined by Sturm et al. from low-temperature measurements (−0.9, −0.9, and −0.47 meV) and determined by Mock et al. from elevated-temperature measurements (−0.83, −1.03, and −0.60 meV).^47,57 We also note that the high-frequency (below-bandgap) refractive index shifts linearly across our measured temperature range. The precise origin of this shift can only be speculated about at this point. There are two processes, which can be brought into the discussion. The first process involves the Kramers–Kronig relationship and the sum rules, which descend from the Kramers–Kronig relationship.^58,59 Briefly, by inspecting $ℜε{(ω→∞)}$⁠, and by making use of the Kramers–Kronig relationship, one can show that the integral over the imaginary part can be associated with an effective plasma frequency, or electron density. It is thought that modifications to a material system, e.g., under strain or stress, or photon excitation, or phase transition, e.g., from superconductive to normal conductive etc.,⁶⁰ the total of this integral remains constant for as long as the total charge density in the material is not changing. Yet, specific spectral features of the dielectric function are, thereby, permitted to change. Because temperature induced changes in band-to-band transitions shift the entire dielectric function, one can show by numerical integration over the imaginary part of the dielectric function using the Kramers–Kronig relationship that a red shift of the bandgap energy will result in an increase in the static dielectric constant and, hence, in the below-bandgap index of refraction. In this step, one performs the Kramers–Kronig integration over the imaginary part at frequency of the real part approaching zero. Also in this picture, one ignores phonon mode contributions. However, one also needs to assume that the transition strengths (amplitudes) of the band-to-band transitions (and all higher energy transitions) remain unchanged. Thus, this viewpoint is not conclusive because this information is lacking since we do not have access to information at higher photon energies outside our investigated spectral range. Another process that can be brought into argumentation is that the change in any physical property of a given system should be following the Bose Einstein relationship with an effective phonon bath temperature, because the physical property is affected by the interaction with the phonon bath (the increase in lattice vibration of the entire system). Hence, by argument of similarity, the index of refraction, being a fundamental physical property, should also show a high temperature shift according to the Bose Einstein model. The fact that we see this linear behavior here is actually a nice demonstration that indeed, this property also follows the same statistical trend as band-to-band transitions, or phonon modes, for example. However, neither explanation is satisfying since for the sum rule consideration too many unknowns remain, which could tilt the index-vs-temperature trend in any direction, and because the Bose Einstein model remains heuristic as it does not point to the actual physical origin since it uses an effective phonon temperature.

Finally, we point out that ZnGa₂O₄ is an indirect semiconductor. Theoretical calculations suggest this transition at approximately 100 meV below the direct band-to-band transition.⁵ Contributions to optical absorption are rather small. A previous transmission experiment by Galazka et al. revealed an onset of absorption at approximately 4.2 eV with an absorption coefficient $α≈$ 100 cm⁻¹.⁴ The onset of absorption detected in ellipsometry at E_g is on the order of about 10 000 cm⁻¹. While an increase in phonon population is expected at elevated temperatures, and while this process will increase the probability of the indirect transitions to take place, their contributions to the dielectric function are still insignificant to be detected in our experiment. Hence, indirect transitions are not considered in our analysis here.

In summary, the critical point and Cauchy dispersion models were developed to analyze the dielectric function properties of ZnGa₂O₄ at elevated temperatures. The Cauchy dispersion relation indicated that the high-frequency dielectric function behaves linearly. Furthermore, the direct bandgap also exhibited a linear behavior with the increase in temperature and a corresponding best-fit slope was determined to be $−0.72(4)$ meV⁻¹. A reduction in the excitonic contribution for elevated temperatures was further verified and provided evidence toward the application of the anharmonic broadened Lorentz to accurately describe these effects.

See the supplementary material for spectroscopic ellipsometry Psi and Delta experimental and best-match model data.

This work was supported in part by the National Science Foundation (NSF) under Award No. DMR 1808715 and NSF/EPSCoR RII Track-1: Emergent Quantum Materials and Technologies (EQUATE) No. OIA-2044049, in the framework of GraFOx, a Leibniz-Science Campus partially funded by the Leibniz Association-Germany, by Air Force Office of Scientific Research under Award Nos. FA9550-18-1-0360, FA9550-19-S-0003, and FA9550-21-1-0259, by the Swedish Knut and Alice Wallenbergs Foundation supported grant “Wide-bandgap semi-conductors for next generation quantum components,” and by the American Chemical Society/Petrol Research Fund. Mathias Schubert acknowledges the University of Nebraska Foundation and the J. A. Woollam Foundation for financial support.