Quantification of the strong, phonon-induced Urbach tails in β-Ga₂O₃ and their implications on electrical breakdown

Gallium oxide (Ga₂O₃) is a promising material for next-generation high-power electronics due to its ultrawide wide bandgap (UWBG), expected high breakdown field, and ability to be grown from the melt as well as epitaxially using multiple processes. Ga₂O₃ has many polytypes; among them, β-Ga₂O₃ is thermodynamically stable at ambient conditions and has a bandgap (E_g) estimated to be near 4.8–4.9 eV. Its high melting point is near 1800 °C, and strong bonding suggests β-Ga₂O₃ for power electronic devices in extreme conditions, including, e.g., high voltages, high temperatures, and or radiation.¹ High breakdown field (E_cr) enables higher-efficiency power devices through Baliga's Figure of Merit, which scales as $E cr 2 or E cr 3$ depending on vertical or lateral device geometry. E_cr scales empirically as $∼ E g 2$ for conventional semiconductors, which led to the suggestion that E_cr for Ga₂O₃ may be near 8 MV/cm or ∼2 times higher than conventional WBG materials, such as GaN, SiC, or ZnO with E_g = 3.2–3.4 eV.^2–4 Detailed calculations taking into account phonon and electronic band structures of β-Ga₂O₃ predict anisotropic E_cr 5.5–7 MV/cm or 5.8 MV/cm.^5–7 In terms of electron–phonon coupling, these papers focused primarily on carrier scattering (especially polar optical phonon scattering), which acts to prevent charge carriers from accelerating to energies sufficient for impact ionization. However, they did not consider the possible effects of phonons on the ionization threshold energy itself. The first effect of strong electron–phonon coupling clearly would tend to increase E_cr, while the second is not easy to predict, although we speculate that avalanche may be dominated by local fluctuations to smaller threshold energy. We suggest that the effect of ionization threshold fluctuating in space and time is worthy of further investigation.

It is well-known that the optical absorption transitions in monoclinic β-Ga₂O₃ are anisotropic; the threshold energies for carrier generation using linearly polarized light depend on the crystallographic direction. In a perfect crystal, the onset of a direct optical transition with photon energy would be quite sharp. Disorder, any departure from perfect periodicity, causes a distribution of bandgaps; a missing, extra, or displaced atom at any point changes the local bandgap. Static disorder causes a time-independent, spatially varying bandgap distribution, while atoms’ random thermal motion due to phonons changes atom positions and, thus, bandgap dynamically in space and time. The same electron–phonon coupling responsible for the strong polar optical phonon scattering that limits carrier mobility^8–10 mediates this variation in the bandgap. Because optical transitions occur much faster than phonon vibrations, each photon absorption is like a snapshot in time of the dynamic disorder.

This phenomenon of phonon-induced exponential bandgap distributions was first observed by Urbach in silver halides and Martienssen in alkalides.^11,12 The static component of disorder is caused by distributions in bond geometry (non-uniform strain) and any non-periodicity of the crystal from random alloying, point defects, extended structural defects, and in the extreme case, amorphous structure.¹³ The nature and concentration of imperfections directly influence the slope and magnitude of the Urbach tails.^14,15 In contrast, dynamic disorder is temperature-dependent and increases with phonon populations as temperature increases. The electron–phonon interaction is the origin of this phenomenon, although the exact details of coupling to each phonon mode are rather complex.^16–19 

We measured optical transmission across a range of Fe-doped and Sn-doped β-Ga₂O₃ samples. Sn-doped samples >500 mm thick (data were normalized to the actual thickness for each sample) grown using edge-fed growth (EFG) were obtained from Novel Crystal Technologies (NCT) oriented in the (001), (010), and (−201) directions. These were polished on both sides by NCT except for the (010) for which we polished the back side to optical flatness. A (100)-oriented piece of a Fe-doped crystal grown by Synoptics using Czochralski (CZ) pulling along the (010) direction was meticulously cleaved using razor blades to a final thickness of 109 and 330 μm. All visible thickness steps were removed from both surfaces over an area larger than the measurement beam size and the 109 μm sample used for temperature-dependent measurements. Additionally, we measured a 36 mm long, 6 mm diameter rod cored from the same crystal and optically polished on both of its (010) ends. We also obtained a (010) unintentionally doped (UID) wafer from Synoptics and polished both sides using a 240-grit SiC pad followed by Al₂O₃ grits down to 0.3 μm and then final CMP using 0.050 μm colloidal silica. We measured the Fe-doped (100) β-Ga₂O₃ samples from 240 to 500 nm using a Hitachi U-4100 spectrophotometer equipped with an integrating sphere detector and a Rochon prism polarizer. We built a custom, short bore 1″ diameter furnace and measured the actual temperature of the sample using a thermocouple in contact with the sample before commencing measurements. To ensure accuracy, we averaged multiple measurements for a high signal-to-noise ratio at low transmitted intensity. Sn-doped and UID β-Ga₂O₃ samples were measured from 240 to 800 nm in a PerkinElmer Lambda 900 spectrophotometer with a broadband polarizer and an integrating sphere detector. Briefly, we determined α assuming an insignificant contribution of coherent reflections because of the large thickness compared to wavelengths as $α=− 1 dln⁡ ( T 1 − R )$ in which T is the measured transmission and R is the total reflection. Extensive details of the data processing used to convert transmission data to an absorption coefficient are given in the supplementary materials.

Figure 1 presents the room temperature linear absorption coefficient (α) for E||b and E||c determined from a d = 330 μm thick sample cleaved on (100) faces from a Fe-doped boule grown by Synoptics. For comparison, we also show the absorption coefficient derived from the imaginary part of the dielectric tensor element for E||b using ellipsometry in Ref. 26 (ɛ_2,zz in the original notation). The fundamental and excitonic critical point energies (⁠ $C P 0 x b=5 .4 ± 0.6$ and $C P 0 b=5 .6 ± 0.4 eV$⁠) from Ref. 26 are indicated. It is immediately clear that the two experiments have measured very different regions of the optical absorption in both energy and α. By measuring many wafer-thick samples and incident polarizations in transmission, we have determined that very strong Urbach tails are present in β-Ga₂O₃, manifesting as linear regions on plots of log(α) vs photon energy. Through careful analysis of uncertainty (see the supplementary material), we determined that, although our data show two apparent slopes, the data for approximately α < 10 cm⁻¹ are significantly impacted by the precision and accuracy of both measurement and data processing (especially how it is corrected for reflectivity). As discussed below, we only place trust in and further analyze the steeper linear sections of the data for samples herein spanning approximately α > 10 cm⁻¹ (depending on sample thickness). We note that the Urbach tail for our E||b data extrapolates close to the CP values from Ref. 26; however, measurements were made on different samples; thus, we treat this as suggestive rather than conclusive.

There is by now widespread understanding that the dielectric or refractive index tensors for monoclinic β-Ga₂O₃ contain irreducible off-diagonal terms in the a–c plane containing the non-90° angle, while the b axis is decoupled. Because these ellipsometry experiments were done in reflection, they are capable of measuring both below and above bandgap, and thus of precisely determining critical point energies. However, because the reflected amplitude depends primarily on the real part of the dielectric constant, ellipsometry in this geometry cannot simultaneously accurately measure weak absorption from band tails far below the gap. On the other hand, transmission measurements like those herein can accurately measure light intensity from 100% to the noise floor (I_noise). Fundamentally, no matter what the origin of absorption, samples will extinguish light to intensities below the noise floor for all energies satisfying $I noise= I 0( 1 − R)$ $exp( − α d)$ in which all quantities besides d are understood to be energy dependent, I_o is the incident intensity, and R is the total reflection. Determination of the energy-dependent R either by measurement or modeling adds uncertainty. Using Beer's law for a generous case of I_noise/I₀ = 0.001–0.01 (two to three orders of magnitude measurable dynamic range), α can be measured in the range approximately 100–10⁵ cm⁻¹ for 500 nm layers (which would typically be mechanically supported on a wider-gap substrate) and in the range 0.1–100 cm⁻¹ for 500 μm thick wafers.

Tauc plot analysis to determine a bandgap²⁷ relies on measuring and extrapolating data for energies >E_g corresponding to absorption coefficients $α ≳ 10 4 c m − 1$ for a direct gap transition. This means that Tauc analysis of absorption data from samples possibly having band tails with d greater than a few μm should be suspect because optical thickness αd < 5 may occur in the band tail region at energies significantly below E_g. This can cause significant underestimation of E_g if this fact is not recognized. In Fig. 1(b), we illustrate an extreme case of this effect for β-Ga₂O₃ by applying Tauc analysis to data we deem reliable covering α = 0.1–1 cm⁻¹ from a 36 mm long (010) Fe-doped rod cored from the same Synoptics boule that supplied the Fe-doped sample in Fig. 1(a) and having polished ends. Despite the continuous curvature of the data in Fig. 1(b), if we apply the typical process of linear extrapolation from the highest energy data, this suggests a “bandgap” of 3.1–3.3 eV, which is obviously in error. Again, this is a result of applying Tauc analysis to a set of inappropriate absorption data not covering the range >E_g because for this very thick sample, the band tails extinguish the intensity for energies well below E_g. We speculate that some of the uncertainty and ranges of values for optical absorption thresholds previously published for β-Ga₂O₃ may arise from these underlying experimental issues. The polarized photocurrent onset of β-Ga₂O₃ is around 4.6 eV depending on the orientation of the crystal²⁸ and the applied bias.

As discussed in the supplementary material, we carefully examined the effects of data processing, especially the numerical subtraction of reflectivity given by an index of refraction assumed constant or as a Cauchy function on the extracted values of α. The regions appearing as straight lines in Fig. 1(a) for α > 10–20 cm⁻¹ were insignificantly affected by various reflectivity subtraction methods, and thus, further analysis is done exclusively on these regions of data sets to extract E_u. The reasonable matching of data in Fig. 1(a) from the 36 mm rod with that from the cleaved wafer gives some confidence that the deep sub-gap region has been determined with fair accuracy, although the details of reflectivity subtraction do change the detailed shape and magnitude of the extracted α below the Urbach tail region. Also, it could be seen by eye that the Fe distribution was not completely homogeneous through the boule grown by Synoptics in one of their first growth campaigns. Although we refrain from detailed analysis, we speculate this deep sub-gap region reflects absorption by impurity (e.g., Fe) and native defect states as internal excitations and/or charge transfer excitations measurable by photocapacitance methods, such as DLOS.²⁹ 

In this investigation, we endeavored to investigate the contributions to E_u from structural defects and phonons, as well as their anisotropies. We note that phonon modes are excited even at 0 K by zero-point energy, as well as the thermal population, which increases with temperature. First, we attempted to separate the temperature-independent (zero-point phonon + structural defects) from temperature-dependent (thermally populated phonon) disorder by cooling to cryogenic temperatures. Figure 2 shows the Urbach tail region of α for an Sn (001) β-Ga₂O₃ wafer at room and liquid nitrogen temperature. This sample exhibited the smallest values of E_u we measured—still ∼60 meV at room temperature—but cooling to 77 K only reduced E_u by approximately 10 meV. The characteristic phonon energies in β-Ga₂O₃ range from approximately 19 to 93 meV;³⁰ thus, the phonon populations (which should be proportional to phonon-induced disorder via the mean-square displacement u²) for the various modes should be suppressed by factors ranging from ∼15 to 25 000 at 77 K. Cooling closer to 0 K would be expected to further reduce the measured E_u, although perhaps not dramatically as discussed below. This suggests that at room temperature and below, the majority contribution to E_u in these bulk crystal samples arises from temperature-independent zero-point phonons and structural defects, rather than temperature-dependent disorder from thermal occupation of phonon modes. The ratio of temperature-independent to temperature-dependent contributions to E_u appears to be near 5:1.

Similar data were measured at room temperature for a wide variety of Sn-doped samples and Fe doped (100) and UID (010) samples for different crystal orientations and polarizations. The data are summarized in Fig. 3 along with those shown in Fig. 2. Since two orthogonal in-plane incident light polarizations were investigated for each sample, Fig. 3 shows each pair of measurements from the same sample as two points connected by a line as a guide to the eye. Clearly, E_u varies quite radically with orientation within each sample, with doping, and to some degree with temperature.

First, we note that across all of the samples we measured, E_u was ∼60–140 meV at room temperature, which is a very large value compared to those typical for conventional single crystal semiconductors; typically, values of a few meV are expected for single crystalline, non-alloyed semiconductors, such as epitaxial III–V and II–VI semiconductors, while values of a few 10s of meV might be observed for high-quality randomly alloyed semiconductors.³² Thus, at least in these bulk crystals we have measured, the E_u values are quite significantly higher than those typical for more familiar semiconductor materials.

Next, we examined the possibility of underlying crystalline anisotropy in E_u across all the investigated samples; unfortunately, we cannot deduce any universal trends, although we cannot rule out the possibility of such effects in more perfect samples. For some samples, E_u is larger for E||a, others for E||b, etc. Thus, from this data set using melt-grown bulk crystals, we deduce that the extrinsic disorder from specific sets of defects present in different samples dominates E_u rather than intrinsic crystalline anisotropy. Samples with much higher crystalline perfection would need to be measured in order to accurately measure the intrinsic crystalline anisotropy in E_u. We note that at the small absorption coefficients associated with Urbach tails, surface layers of a few μm of highly damaged material, e.g., from cutting, lapping, and polishing could significantly contribute to the total sub-gap absorption.

In order to gain insight into the types of disorder primarily responsible for E_u, we examined the set of Sn-doped wafers of various orientations obtained from NCT since they are all produced from boules grown along the b-axis using EFG but cut on different crystal planes. We hypothesized that the degree of non-uniform strain and numbers of induced compensating native defects would scale with the concentration of Sn doping (which differed within the 10¹⁸/cm³ range for the samples we measured), and thus, E_u should scale with this variable. However, we found that E_u actually decreased with the concentration of Sn as shown in Fig. 4(a). For each wafer, we also had the on-axis x-ray rocking-curve FWHM specified by NCT. With the exception of one data point not shown, Fig. 4(b) indicates better correlation between E_u and the measured FWHM. On-axis x-ray rocking curves measure crystalline disorder in the same plane measured by incident polarized light (the width of θ–2θ scans measures disorder in the depth direction). The x-ray rocking curves will probe both the doping and point defect induced disorder and that arising from extended defects and near-surface polishing damage. Thus, we conclude that later two effects are dominant contributions to E_u in these samples. From our data, we cannot discern whether bulk structural defects or near-surface damage from wafer preparation (which can also be anisotropic) is the primary contributor.

Since many samples were measured under many conditions, we introduce Table I to assist the reader.

We next examine the temperature-dependent contributions to E_u through measurements at elevated temperatures. Figure 5 shows the extracted absorption coefficients for temperature-dependent data on a Fe-doped (100) sample for (a) E||b and (b) E||c incidences and global fitting (data from all temperatures fit together) to Eq. (1). Separately for the two incident linear polarizations, the Urbach focus (α₀, E₀) is determined from all temperature data, while E_u (and, thus, σ₀) change with temperature. We estimate uncertainties of ±2 meV for all Urbach energies, arising from both noise in the data and the variation induced in numerical data processing. The values of E₀ and α₀ are indicated in the figure for each incident polarization, and the extracted E_u(T) values up to 633 K are plotted as Fig. 6. Data obtained at higher temperatures were less reliable and repeatable because of the onset of blackbody emission from the furnace and the more significant convection-driven turbulence in the open-air optical path and, thus, is not included in further analysis. The onset of significant changes in native defect concentrations in n-type Ga₂O₃ is known to be above ∼900 °C in air, although it is possible that redistributions of impurities may occur. The fact that these wafers were grown from the melt and cooled through temperature ranges much greater than those used for measurements argues against this being significant.

In the first expression, K is the proportionality constant, $⟨ u 2 ⟩ X , 0$ is the temperature-independent contribution of the structural disorder, $⟨ u 2 ⟩ P h , 0$ represents the temperature-independent zero-point excitation of phonons, and $⟨ u 2 ⟩ P h , T$ is the temperature-dependent contribution of thermal phonon occupation. In the second expression, K is defined as $Θ σ 0$⁠, the ½ in the second term represents the zero-point phonon displacements, and X characterizes displacements from structural defects such that X = 1 would be equivalent to the zero-point phonon displacements. The zero-point excitation of the lattice produces finite Urbach energy at X = 0 and T = 0, $E u( 0 , 0)= Θ 2 σ 0$⁠. The mean-square thermal displacements are proportional to the population of phonons, given by the Einstein model in which $Θ$ can be recognized as $ℏ ω k b$⁠. Cody points out that the Debye temperature modeled for acoustic modes $Θ D= 4 3Θ$⁠. Because of the large number of phonon modes within a small energy range for β-Ga₂O₃, it is not possible to separate their contributions from temperature-dependent E_u measurements alone. Thus, we take $Θ$ to represent a properly averaged representative phonon mode.

Figure 6 shows the results of fitting of our measured E_u data for Fe-doped (100) β-Ga₂O₃ with Eq. (3). The uncertainty estimates were derived from sensitivity analysis of the fitting (thus the different uncertainties for the two data sets). The best fit parameters are Θ = 606 ± 145 K for E||b and 588 ± 71 K for E||c corresponding to Debye temperatures of 808 ± 193 and 784 ± 93 K for E||b and for E||c, respectively. These values are in reasonable agreement with the experimental value of 738 K³⁶ and the predicted value from first principles calculations of 872 K.³⁷ We find the steepness parameter σ₀ = 4.60 ± 0.34 (5.69 ± 0.19) and structural disorder X = 0.60 ± 0.40 (0.93 ± 0.26) for E||b (E||c). Examining the static structural disorder, we note that for E||b its mean-square displacement is about half that of the zero-point phonons, while for E||c, the two effects are nearly equal. Thus, the zero-point phonon contributions are larger for both directions than either the structural defect or finite-temperature phonon contributions; despite X being larger for E||c, it is offset by the proportionality constant $σ 0$ also being higher, which ultimately results in measured E_u values for E||b being larger overall. Using the fitting values and Eq. (3), we determine that E_u at 0 K is predicted to be 106 meV for E||b and 100 meV for E||c for this Fe-doped sample. For E||b, 69 meV would be from zero-point phonons and 37 meV from structural disorder, while for E||c, 52 meV would be from zero-point phonons and 48 meV from structural disorder. Thus, we find evidence for intrinsic anisotropy in static structural E_u of order 17 meV and minimum possible values at room temperature of 88 meV for E||b and 68 meV for E||c. The breakdown of the three different contributions to E_u at different temperatures and for different polarizations is summarized in Table II.

From Fig. 6 and Table II, we have measured anisotropy in E_u between b and c directions at elevated temperatures. Using the fitting values from Eq. (3), we can deduce the various causes of E_u and its anisotropy. The fitting from the sample used for elevated temperature measurements, with X = 0, gives an estimate for the phonon-only minimum possible values of E_u at 293 K (85 and 68 meV) and at 0 K (66 and 52 meV) for b and c directions, respectively. As discussed before, these values are two to ten times larger than values for more traditional semiconductors. We do note that the (001) Sn-doped sample we measured, which showed E_u(293 K) slightly larger than 60 meV, demonstrates that slightly smaller values are possible; we suspect but cannot prove from the present data sets that very thin but highly damaged surface layers related to polishing may contribute on the order of ∼10 meV, which would bring the uncertainties between samples close to alignment. Looking at the breakdown of contributions to E_u at 0 K, first, we define the bottom three rows of Table II. E_u(X, 0) denotes the modeled Eu using T = 0 and the fit parameters from Fig. 6, while Eu(0, 0) is only the zero-point phonon component and the final row is computed as $Θ X 2 σ 0$⁠. We see that anisotropy in the modeled zero-point phonon component E_u(0, 0) is just below the threshold for being significant, while both the total E_u and the isolated component for static structural defects are isotropic within our uncertainties. Thus, since anisotropy in X (from structural defects) would be the same at 0 K and elevated temperatures, we conclude that the anisotropy in E_u observed at 293 K is caused by phonon anisotropy. At the highest temperature of 633 K, we measure 187 and 167 meV for E||b and E||c, respectively, yielding 20 meV anisotropy with uncertainty estimated as 2–3 meV. Evaluating Eq. (3) with the best fit parameter values and taking into account the parameter sensitivity of the fitting gives the same values but with much higher uncertainty. Thus, we conclude experimentally that there is some minor anisotropy on the order of 5% of the total E_u at room temperature induced by the anisotropic phonons in β-Ga₂O₃.

Thus, we can extract this measure of the electron–phonon coupling by combining temperature-dependent measurements of both E_g and E_u. As per the discussion of Fig. 1, ellipsometric measurements of E_g are more reliable unless few-mm thick samples are available for transmission, while transmission measurements of E_u are more reliable. First, we observe that small amounts of static structural disorder should have negligible effect on E_g; thus, we assume that $E g( 0 , X)≈ E g( 0 , 0)$—this is also justified by the dominance of zero-point phonons in the static disorder. Making this assumption, the numerator of Eq. (6) is calculated at each of our measured temperatures from the Bose–Einstein parameterization of the $E g( T , X)− E g( 0 , X)$ given in Ref. 26. $E u( T , X)$ is our measured temperature-dependent Urbach energy data and $E u( 0 , 0)$ is given in Table II. The extracted values of $⟨ u 2 ⟩ 0D$ are given in Fig. 7. We note that Ga₂O₃ has many modes of different energies; thus, the quantity we extract represents some weighted average over our measured temperature range, including any subtle differences between phonons’ effects on E_u and E_g. The values we extract consistently increase with temperature, which we interpret as arising from this population-weighting of phonon modes that would favor lower-energy modes at lower temperatures but include all modes at very high temperatures.

The product $⟨ u 2 ⟩ 0D$ gives a measure of the electron–phonon coupling as an energy, which we find to have some temperature dependence. Because the zero-point displacements are by definition temperature independent, this must arise from the population weighting of phonon modes, which have different effective D values. Phonons in β-Ga₂O₃ are plentiful, complex, and anisotropic, with energies spanning approximately 10–100 meV.³⁰ An interesting open question is how exactly to quantify the coupling between the phonons’ displacements and or induced dipoles and the distribution of bandgaps at each temperature that generate the Urbach edges. The simplest but most computationally expensive method would appear to be direct calculation of bandgaps in a large number of supercells with different superpositions of the phonon displacements applied and fitting the Urbach formula to the resulting distribution of bandgaps.

It has been established that the dominant intrinsic carrier scattering mechanism (i.e., excluding ionized-impurity scattering) in β-Ga₂O₃ at room temperature is polar-optical phonon (POP) scattering. In compound semiconductors with polar bonds, Frölich coupling is typically assumed in which carriers scatter from the fluctuating dipole field induced by transverse-optical phonons. The second order deformation potential extracted here is not identical to the Frolich coupling energy responsible for POP phonon scattering.⁶ While E_u, mobility, and breakdown all depend on electron–phonon coupling, they each have different dependencies and must be carefully interpreted.

Despite its seeming straightforwardness, electrical breakdown in inorganic crystalline materials, such as β-Ga₂O₃, is a complex topic. The fact that Ga₂O₃ exhibits strong Urbach tails predicts qualitatively that the mobility and, thus, the mean free path for carrier acceleration in a high field are curtailed, making it less likely for carriers to reach kinetic energies sufficient to impact ionize. However, an unexamined topic (to our knowledge) is how the energy required for impact ionization varies in the presence of significant time-and-space bandgap disorder associated with Urbach tails. For example, if we use the empirical scaling that breakdown field is proportional to $E g 2$⁠, we would expect a variation in E_cr with the position in Ga₂O₃. These two physical effects associated with phonon-induced bandgap disorder push the critical breakdown field in different directions; thus, the net effect is difficult to predict. Breakdown in a-Se and a-Si:H—extreme examples of static structural disorder having Urbach tails on-par with what is observed for Ga₂O₃—has been studied, and in general, it is found that avalanche multiplication in these materials occurs at higher fields than for ordered semiconductors with similar bandgap materials.³⁸ This argues that in the case of static structural disorder, the lower mobility prevents carriers from easily accelerating to high energies in a field. The effects of phonon-related disorder—bandgap fluctuations and carrier scattering from phonons especially in the case of large electron–phonon coupling as in β-Ga₂O₃—have not been elucidated. If the effects are similar to structural disorder, which may be likely since the root cause of both is mean-squared displacement from ideal sites, we may expect that materials with higher electron–phonon coupling might have even higher breakdown fields than their counterparts of the same bandgap. It has been established that POP phonon scattering should reduce the empirically predicted breakdown field in perfectly crystalline Ga₂O₃ of ∼8 to ∼5–6 MV/cm,5 but the effects of the combination of thermal and structural disorder have not been accounted to date. This issue will have a profound effect across all materials because it has not been incorporated into theoretical frameworks and, thus, has priority for future investigations.

We investigated the sub-gap absorption of a number of bulk β-Ga₂O₃ wafers using linearly polarized transmission. Overall, we find significant Urbach tails that range from 60 to nearly 140 meV at room temperature depending on doping, crystallographic direction, and individual sample. Static structural defects and zero-point phonons account for the majority of the effect, but a non-negligible component arises from finite temperature phonons. There is some evidence for the phonons inducing some anisotropy into the Urbach energies, although this is at the 10% level so not dominant compared to the mean isotropic values. We predict based on comparison to results from amorphous materials that UWBG materials with high electron–phonon coupling may have even larger breakdown fields than predicted using scaling from conventional semiconductors where this is weaker. Detailed theory will be needed to resolve whether or not the impact ionization threshold also varies as a result of static structural disorder and whether this changes E_cr.

The supplementary material includes extensive details of the data processing used to convert transmission data into absorption coefficients, along with the associated error estimates.

The authors would like to acknowledge the funding provided by the Air Force Office of Scientific Research under Award No. FA9550-21-0078 (Program Manager: Dr. Ali Sayir). We thank Alyssa Mock for sharing the ellipsometry data used for Fig. 1 and acknowledge fruitful discussions with Professor Zlatan Aksamija, Professor Elif Ertekin, Professor Feliciano Giustino, and Professor Dragica Vasileska.

The authors have no conflicts to disclose.

Ariful Islam: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Writing – original draft (lead); Writing – review & editing (equal). Nathan David Rock: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Writing – review & editing (supporting). Michael A. Scarpulla: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Supervision (lead); Validation (equal); Writing – review & editing (lead).

The data that support the findings of this study are available within the article. Additional data are available from the corresponding author upon reasonable request.