Perspective on electrically active defects in $β$- $G a 2 O 3$ from deep-level transient spectroscopy and first-principles calculations

Gallium oxide (⁠ $G a 2 O 3$⁠) is a promising ultra-wide bandgap semiconductor with various desirable properties suitable for power electronics and UV optoelectronic applications. Among the five different $G a 2 O 3$ polymorphs, the monoclinic $β$-phase is the most stable one. Notable characteristics include a high breakdown field, highly tunable n-type conductivity, and the possibility for melt-based bulk growth.^1,2 However, achieving hole conductivity has proven a challenge, partly due to hole self-trapping,³ rendering conventional bipolar device prospects unlikely.

The $β$- $G a 2 O 3$ unit cell comprises five different lattice sites (two Ga sites and three O sites), resulting in a plethora of defects and potential configurations. Many of these defects are electrically active, providing electronic states within the bandgap, the occupation of which determines the charge state of the defect. A so-called charge-state transition level can be defined as the value of the Fermi level at which the energy of the defect in two different charge states is equal. Such charge-state transition levels are often (more ambiguously) referred to as energy levels or defect levels. Charge-state transition levels that are energetically deep in the bandgap, meaning at least a few tenths of an eV from the conduction band, are a source for recombination and trapping of charge carriers and are normally unwanted as they can be detrimental to device operation. Hence, further enhancement of devices based on $β$- $G a 2 O 3$ requires a comprehensive understanding of electrically active defects.

Electrically active defects and their charge-state transition levels can be probed by junction-spectroscopic techniques such as deep-level transient spectroscopy (DLTS), deep-level optical spectroscopy, and steady-state photo-capacitance measurements. These techniques are some of the most sensitive methods to study defects and are widely used for both traditional and novel semiconductors. The present Perspective focuses on the defects observed by DLTS. In the first DLTS study of electrically active defects in $β$- $G a 2 O 3$⁠,⁴ defect levels positioned 0.55, 0.74, and 1.04 eV below the conduction band edge (⁠ $E C$⁠) were observed, and electron traps were labeled $E 1$⁠, $E 2$⁠, and $E 3$⁠, respectively. Since then, a multitude of defect levels has been reported, and the labeling has been modified and extended but following a similar scheme.

Unfortunately, junction-spectroscopic techniques do not provide chemical or configurational information, and identifying the microscopic origin of the observed defect signatures is generally challenging. Here, first-principles defect calculations based on density functional theory are often a key ingredient, as many of the properties extracted by DLTS can now be readily calculated,^5–7 providing essential support in the identification and understanding of the defect behavior. Indeed, first-principles calculations played an influential role in the first identification of an electrically active defect level in $β$- $G a 2 O 3$⁠, namely, the $E 2$ center observed by DLTS at around 0.8 eV below $E C$⁠. Experiments indicated an origin related to iron impurities, which was explored from first-principles.⁸ These calculations revealed that Fe prefers to substitute for Ga and that both Ga sites yield an acceptor level in close agreement with the measured defect level. The prediction that Fe is expected to form in appreciable concentration on both Ga sites sparked further experiments, which verified that $E 2$ indeed consists of two levels.⁹ 

In the past, metal oxides such as $β$- $G a 2 O 3$⁠, exhibiting wide bandgaps and defects with a high degree of charge localization, have been challenging to describe. Indeed, conventional semilocal approximations suffer from so-called delocalization errors that result in underestimated bandgaps and a poor description of localized defect states.¹⁰ This is exemplified by the failure to describe hole polaron formation in $β$- $G a 2 O 3$⁠.³ However, with the emergence of hybrid functionals, the quantitative accuracy of calculated defect levels has increased considerably.^7,11 This has spurred the identification and understanding of defects in semiconductors in general.

The large bandgap is also a challenge for DLTS investigations in $β$- $G a 2 O 3$⁠, as only a part of the bandgap can be monitored using conventional techniques. For defects that are too deep (practically) for thermal emission of an electron from the defect level to the conduction band, photoionization can be utilized. Optical junction-spectroscopic techniques, including deep-level optical spectroscopy (DLOS) and steady-state photocapacitance (SSPC), have been widely employed to investigate defects in the lower part of the bandgap of $β$- $G a 2 O 3$⁠. Although the present Perspective is devoted to electrically active defects observed by DLTS in $β$- $G a 2 O 3$⁠, we have included a short discussion on optical techniques in Sec. VIII, considering the special importance of such measurements along with DLTS for this material.

In spite of being blind to a large part of the bandgap, DLTS has proven invaluable in uncovering crucial insights into the electrical activity within $β$- $G a 2 O 3$⁠, as will be detailed in this Perspective. The low symmetry and anisotropic behavior of the material foster the presence of metastability and other fascinating defect properties, including negative- $U$ behavior. While this presents challenges from an applicational standpoint, it renders $β$- $G a 2 O 3$ a captivating subject for scientific study. Indeed, metastability has been observed, and first-principles calculations suggest a significant presence of negative- $U$ behavior from some of the intrinsic defects. Thus, in order to advance $β$- $G a 2 O 3$⁠, a thorough understanding and control of electrically active defects is essential.

In the following, we will introduce the experimental technique DLTS, additional techniques related to DLTS, and the calculation of defect properties from first principles based on density functional theory and the supercell approach. Thereafter, the different reported defect signatures in $β$- $G a 2 O 3$ will be discussed, along with challenges and prospects for further identification and understanding of the defect behavior in this material.

$β$- $G a 2 O 3$ has a monoclinic crystal structure with space group C2/m,¹² as shown in Fig. 1. As mentioned above, there are several inequivalent sites for both oxygen and gallium, where the oxygen sites O1 and O2 have threefold local symmetry and O3 exhibits a fourfold symmetry. There are two inequivalent Ga sites, where Ga1 is tetrahedrally coordinated and Ga2 exhibits octahedral coordination. Considering the low symmetry of the monoclinic structure, it is perhaps not surprising that an abundance of defect configurations have been discovered,^13–20 resulting in a rich defect notation. In addition to the five different lattice sites, six interstitial sites have been pointed out in these studies: $ia$⁠, $ib$⁠, $ic$⁠, $id$⁠, $ie$⁠, and $if$⁠. Notably, several defects occur in the so-called split configurations, where sites are split between atoms or vice versa, and a combination of substitutional and interstitial sites can be involved. The configurations and notations of the defects that will be described in Secs. IV–VIII can be understood by referring to Fig. 1.

The electronic bandgap of $β$- $G a 2 O 3$ is in the range 4.7–4.9 eV.^21–23 The lowest bandgap is indirect, but the direct one is only slightly higher in energy. Due to the low symmetry, a pronounced directional dependence is found for the optical band gap.²⁴ The static dielectric constant is reported to be in the range 9.9–12.7 and is direction-dependent.^25,26 The effective mass of electrons is about 0.22–0.30 $m e$⁠,²² and nearly isotropic, while the effective mass of holes is substantially higher and anisotropic, leading to self-trapping of holes.³ As already mentioned, only $n$-type doping can be reliably achieved, with typical doping in the range from $10 16$ to $10 19 cm − 3$⁠.

Deep-level transient spectroscopy is an indispensable characterization technique for studying electrically active defects in semiconductors. The technique can establish the energy position in the bandgap of the defect charge-state transition level, and its capture cross section, which describes how efficient the defect is in capturing charge carriers. These two parameters can be seen as the electronic fingerprint of a defect.

Furthermore, the concentration of the defect and its depth distribution can be obtained. The sensitivity in DLTS measurements is higher than almost any other semiconductor diagnostic technique, with the possibility to detect defect concentrations as low as $1× 10 8 cm − 3$ in low-doped materials.^27,28

DLTS falls under the category of junction-spectroscopic techniques, where the junction capacitance is used as a probe to monitor the emission of charge carriers from a defect. Being a junction-spectroscopic technique, DLTS measurements require a rectifying junction, such as a $p$– $n$ junction or a Schottky diode. As $p$-type doping of $β$- $G a 2 O 3$ is unattainable at present, the most common device structures are $n$-type Schottky barrier diodes. Suitable metals for making Schottky diodes include, e.g., $Au$⁠, $Ni$⁠, and $Pt$⁠.²⁹ Unfortunately, the doping asymmetry in $β$- $G a 2 O 3$ makes it possible only to probe the upper part of the bandgap with DLTS. The rest of this discussion will, as such, deal with electron traps. These are defects with charge-state transition levels above mid-gap that interact mainly with the conduction band through the emission and capture of electrons.

The central quantity that is probed in a DLTS measurement is the capacitance arising from the depletion region in a rectifying junction. This capacitance can be modeled as that of a parallel-plate capacitor and is, as such, inversely proportional to the depletion-layer width, which again depends on the space-charge concentration in this region. The capacitance is, therefore, a suitable probe, as the emission of charge carriers from traps alters the effective space-charge concentration.

Figure 2 illustrates a band diagram during the DLTS measurement cycle for the case of one trap level. The diode of interest is kept at a reverse bias; this corresponds to Fig. 2(a). Then, a voltage pulse (⁠ $V \,p$⁠) is applied to the diode, which reduces the depletion-region width and thereby floods this area with free charge carriers. The defect level can now capture the charge [see Eq. (1): the capture rate strongly depends on the concentration of charge carriers], and the trap level below the Fermi level is soon filled with electrons [Fig. 2(b)]. When the pulse is removed again [Fig. 2(c)], the depletion-region width increases to $W ′( V r)$⁠, and the Fermi-level position in this region is lowered. The capture process is now suppressed as the concentration of free charge carriers is negligible in the depletion region. The trap will, however, thermally emit carriers according to Eq. (2), which are then swept out of the depletion region due to the electric field present therein [Fig. 2(c)]. The emission of electrons from the trap level leads to an increase in the concentration of positive space charge in the depletion region, gradually changing the depletion-layer width back to the steady-state value $W( V r)$⁠. When monitoring the capacitance, this emission process will be seen as an increase in the junction capacitance in the form of a transient with a time constant equal to that of the emission process. The DLTS measurement cycle of applying a reverse bias and then pulsing is repeated several times while slowly changing the temperature and collecting these capacitance transients, which will vary with temperature [see the leftmost plot in Fig. 3(a)].

The expression above [Eq. (5)] will often return an underestimation of the trap concentration as it overestimates the actual probing volume where trap occupancy changes. Indeed, the closest to the junction interface, the trap level always lies above the Fermi level, and the traps here will be unoccupied. On the other hand, the traps in the distance $λ$ from the depletion layer edge remain occupied as the trap level constantly lies beneath the Fermi level. These factors can be accounted for when calculating $N t$ by applying the so-called $λ$-correction and is often necessary to extract accurate values for $N t$⁠.^4,27,28

Depth profiling: From a regular DLTS measurement, the average trap concentration in the probed volume is extracted. However, it is also possible to perform a profiling measurement where the depth distribution of a trap level is determined, as illustrated in Fig. 4(a). There are several approaches to obtaining the depth distribution, where one such method is to vary the filling-pulse voltage while keeping the bias during the transient measurement constant. The temperature is also kept fixed at the peak of the DLTS signature. If the pulse voltage is small compared to the reverse bias, the probing volume is at the deep end of the probing range. Increasing the pulse voltage makes the probing volume gradually larger toward the junction interface. The ability to measure the depth profile of defect levels is useful, e.g., in the study of implantation-related defects and other cases where defect profiles are non-uniform.

Capture cross-section measurements: As explained in Sec. III A, the capture cross section extracted from a DLTS measurement is only an apparent one. However, it is possible to measure the capture cross section directly.³³ Typically, the filling-pulse duration for a standard DLTS measurement is set sufficiently long for all the traps to be occupied. However, for very short filling pulses, only a fraction of the traps will be filled. Thus, by measuring the DLTS signal intensity while systematically varying the filling-pulse duration, it is possible to monitor the filling process. Again, the temperature is kept fixed at the DLTS signature peak. Figure 4(b) illustrates the result of such a capture measurement. To first order, the DLTS signal intensity is exponentially dependent on the capture rate, which then can be determined by fitting the data. The capture cross section can then be extracted from the capture rate.²⁷ 

Capture barrier measurements: For defects showing a large difference in local-lattice geometry between the occupied and unoccupied trap state, a capture barrier can arise (see Sec. IV B). This barrier is included in the activation energy extracted by DLTS. The capture barrier can be measured by performing the capture measurements described above, but at several different temperatures (e.g., at the DLTS peak maxima for several different rate windows), as illustrated in Fig. 4(b). Any temperature-dependent change in the capture rate is normally ascribed to the capture barrier. The capture barrier can thus be found from the activation energy of the thermally activated capture process, which can be determined from the construction of an Arrhenius plot. As an example, such measurements revealed the capture barrier of the $E 4$ center in ZnO.^32,34

Electric-field dependence of the emission rate: The emission rate of carriers from deep-level defects can be enhanced by an electric field. The dominating mechanism for the enhancement is different depending on whether the defect is charged or neutral.³⁵ By performing DLTS measurements with different magnitudes of the reverse bias, which translates to different values of the electric-field strength in the depletion region, information about the mechanism can be deduced. The mechanism may provide information about the nature of the defect, e.g., whether it is a donor or an acceptor. The so-called Poole–Frenkel effect is recognized by a square-root dependence on the electric field for the logarithmic emission rate [see Fig. 4(c)], and this is normally interpreted as the defect being a donor-type in the $n$-type material or an acceptor-type in the $p$-type material.³⁶ Another mechanism for field-enhanced emission is via phonon-assisted tunneling, which also occurs for neutral defects. For phonon-assisted tunneling, the logarithmic emission rate is typically proportional to the electric field squared, as shown in Fig. 4(c).

Laplace DLTS: In conventional DLTS analysis, defect levels with similar emission rates can be hard to distinguish from one another in the spectrum due to the inherent broadness of the peaks. One route to overcome this challenge is to use the more refined weighting functions, such as GS4 or GS6. However, this is not always sufficient. A technique that produces an even greater resolution is Laplace DLTS, which makes use of an algorithm that effectively performs an inverse Laplace transform of the transients.³⁷ A simple schematic showing what a Laplace DLTS spectrum can look like is given in Fig. 4(d). The spectrum is composed of much sharper peaks than a typical DLTS spectrum and can be very helpful in resolving whether a DLTS signature is, in fact, composed of several levels. However, there is a trade-off between resolution and noise, and the concentration of the defect(s) under scrutiny needs to be sufficiently high for Laplace DLTS to be applicable.

The activation energy found by DLTS correlates with the measurement temperature, where the deepest traps are found at the highest temperatures. Thus, for wide-bandgap materials, only a part of the bandgap can be practically measured by DLTS. For $β$- $G a 2 O 3$⁠, this limit is roughly 1.5–1.8 eV below $E C$ (1.8 eV is the highest activation energy that has been reported³⁸). Hence, a large part of the bandgap is inaccessible by DLTS, which is further limited by the lack of $p$-type doping. However, by using light to excite charge carriers to or from the traps, it is possible to probe larger parts of the bandgap. Two related junction-spectroscopic techniques that rely on light excitation are steady-state photo-capacitance (SSPC) and deep-level optical spectroscopy (DLOS), but there are a number of related techniques based on similar principles. There are several practical approaches to perform such measurements, but in general, SSPC records the steady-state capacitance of a rectifying junction kept at reverse bias in the dark and after illumination to optically ionize traps in the depletion region. Here, the sample temperature is kept sufficiently low to suppress the thermal emission of carriers. The steady-state reverse-bias capacitance of the junction is monitored as the sample is illuminated by photons with increasing energy, and the optical emission of carriers from traps appears as a step in the capacitance. DLOS, on the other hand, relates the initial change in trap occupancy to the transient capacitance after optical excitation, although other versions using electrical injection or thermal excitation exist as well.³⁹ From the recordings, one can, e.g., plot the optical cross section as a function of energy, where the energy level position is found by fitting to analytical models. This allows for probing a wide range of the bandgap. However, the DLOS signatures of defects can be challenging to interpret, particularly if multiple defect levels exist within the same energy range. Indeed, this holds true in $β$- $G a 2 O 3$⁠, where, e.g., both the O and Ga vacancies are expected to be in the same energy range.⁴⁰ 

Over the last few decades, first-principles calculations based on density functional theory and the supercell method have come into prominence as an effective and powerful approach to study defects in semiconductors. Such calculations are now routinely used both to support experimental investigations, for instance, DLTS studies and to provide new physical insights into defect physics. The calculation of total energies and forces allow for the prediction of stable atomic structures of defects in different charge states, and thus investigation into their thermodynamic and electrical properties. An established formalism revolving around the defect formation energy as the key quantity has been developed,⁵ allowing the stability of the defect and its charge-state transition levels in the bandgap to be quickly assessed. On this foundation, more advanced methodologies based on one-dimensional configuration coordinate (CC) diagrams have been developed to study the dynamics of defect charge-state transitions and even predict the corresponding defect signatures observed in real experiments,⁴¹ including the main defect quantities obtained from DLTS measurements.⁷ 

In the following, we shall briefly explain the methodology, focusing on the parts relevant for DLTS studies. For more details, see reviews and papers elsewhere.^{5–7,41–43} In particular, methodology papers by Wickramaratne et al.⁷ and Coutinho et al.⁴⁴ have focus on DLTS.

A correction $Δ q$ is also included in Eq. (7) due to the finite size of the employed supercell. This accounts for the electrostatic interaction between the defect charge and its images under periodic boundary conditions.^45–48 

The formation energy can be conveniently plotted as a function of the Fermi-level position in the bandgap, as shown for $Fe Ga$ and $Fe i$ in Fig. 5. The slope of the linear segments correspond to the charge state, and kinks occur at Fermi-level positions for which the formation energy of the defect in two different charge states is equal. The positions of these thermodynamic charge-state transition levels relative to the band edge corresponds approximately to the $E A$ value in a DLTS measurement.

The formation energy diagram already provides us with a lot of useful information. First, when the Fermi level is located in the upper part of the bandgap, it will be energetically preferable for Fe impurities to substitute for Ga rather than residing at an O or interstitial site. Second, $Fe Ga$ can be stabilized in positive, neutral, and negative charge states, and there is a thermodynamic (⁠ $0$/ $−$⁠) transition level located 0.7 and 0.8 eV below $E C$ for the tetrahedral $Fe Ga 1$ and octahedral $Fe Ga 2$ configurations, respectively. These levels are both likely to form when Fe impurities are present and located within the energy range that can be probed by DLTS.

To study the kinetics of the carrier emission and capture processes that is actually measured in DLTS, we must consider the coupling between the electronic and ionic structure. This concept can be described by using a one-dimensional CC diagram,^6,7,43 as depicted for the process of electron emission and capture between the (⁠ $0$/ $−$⁠) level of $Fe Ga 1$ and $E C$ in Fig. 5(b). The ground (g) state is $Fe Ga 1 −$⁠, and the excited (e) state is $Fe Ga 1 0$ with the emitted electron located at $E C$⁠. The configuration coordinate (⁠ $Q$⁠) along the horizontal axis is a one-dimensional parametrization of the collective motion of all atoms in the supercell between the equilibrium structure of $Fe Ga 1$ in the neutral and negative charge states.⁴² Calculating the total energy as a function of $Q$ results in a harmonic potential energy curve for each of the states, where the energy is minimized at the corresponding equilibrium structure (⁠ $Q g$ and $Q e$⁠). Their difference (⁠ $ΔQ$⁠) involves mainly the Fe impurity and the four adjacent O ions. Indeed, as shown in Fig. 5(c), the negative charge state corresponds to an electron trapped in a localized defect state displaying Fe 3 $d$ character. When this electron is emitted, the Fe impurity relaxes into the ideal Ga1 site and the Fe–O bonds contract by $∼$5%.

The two potential-energy curves in the CC diagram are displaced along the vertical axis by the ionization energy (⁠ $E i$⁠), that is, the energy difference between the ground and excited states in their equilibrium configurations. This corresponds to the Fermi-level position of the thermodynamic (⁠ $0$/ $−$⁠) transition level relative to $E C$⁠, as also indicated in Fig. 5(b). In the classical picture, electron emission occurs when $Fe Ga 1 −$ traverses the ground-state potential-energy curve to reach the crossing point with the $Fe Ga 1 0+ e C −$ curve, after which the defect relaxes to its new equilibrium configuration. The activation energy for carrier emission is then given by the sum of the ionization energy and the capture barrier (⁠ $E b$⁠), as indicated in Fig. 5(c). This should be kept in mind when comparing $E A$ values from DLTS with charge-state transition levels in a formation energy diagram. If the capture barrier is large, $E A$ can deviate significantly from $E i$⁠. As explained in Sec. III C, an experimental value of $E b$ can be estimated from the temperature dependence of the electron capture rate of a DLTS signature.

The calculation of formation energy and CC diagrams is already a highly useful tool to interpret experimental DLTS data, providing potential defect origins for further experimental study. However, a quantum treatment is also possible within this model, considering the process as the nonradiative capture of a carrier via multiphonon emission, which can yield the capture cross section as a function of temperature.⁷ From such calculations, one can see how the capture rate is limited by quantum tunneling in the low-temperature limit, and thermally activated crossing of a capture barrier in the high-temperature limit. Combined with the temperature dependence of $E i$ (through calculations of the temperature dependence of the bandgap and the vibrational free energy difference between the defect in the two charge states), the temperature dependence of $E A$ can be extracted from an Arrhenius plot, akin to how real DLTS data is analyzed.⁷ Calculated capture cross sections can also be compared with experimental measurements, although care should be taken when comparing them to the apparent capture cross section obtained from a DLTS measurement (see Sec. III C). The apparent values can also be affected by factors such as the magnitude of the reverse-bias and the doping level of the material.^49–51 

Previous studies have shown that many defects in $β$- $G a 2 O 3$ exhibit metastability. First of all, considering the low symmetry of the monoclinic crystal structure, defects can in general assume a large number of crystallographically inequivalent configurations. As mentioned, impurities can occupy different lattice sites, and many intrinsic defects undergo striking lattice distortions to form split configurations. First-principles calculations have proven quite beneficial in this regard, as the relative stability of different configurations can be assessed, guiding experiments to the most pertinent segments of the configuration space. In some cases, different defect configurations will be favored depending on the conditions, and these configurations can exhibit different electrical behavior. Such effects have been invoked to explain certain experimental observations on DLTS signatures in $β$- $G a 2 O 3$⁠.^{9,19,38,52,53} There are also practical implications of the metastability; external stimuli such as temperature or electrical field can transform the defect from a passive configuration into a useful or harmful^54,55 one for device operation.

The negative-U behavior and occurrence of several different geometries is well-exemplified by the Ga–O divacancy (⁠ $V Ga V O$⁠) in $β$- $G a 2 O 3$⁠. Figure 6 shows the formation energy of the two most favorable configurations, namely, $V Ga 2 V O 2$ and $V Ga ib V O 1$ (19 inequivalent close-associate configurations are shown in Ref. 19). For Fermi-level positions above $E cross$ in Fig. 6, $V Ga 2 V O 2$ is the lowest in formation energy. This configuration displays a triple negative charge state, where the two Ga1 ions associated with $V O 2$ are brought together in a dimer by two electrons occupying a state in the bandgap (the corresponding charge density is shown by the yellow isosurface in Fig. 6). When an electron is removed from this state, the dimer becomes unstable, resulting in negative-U behavior. Indeed, it is more favorable for the defect to lose both electrons, fully breaking the bond. The charge-state transition then occurs directly from triple negative to single negative. It should be emphasized that the negative U only occurs when the charge-state transition is accompanied by a significant separation of the Ga1 ions from each other as they break up. This is demonstrated in Fig. 6 by showing the formation energy when the defect is frozen in the equilibrium geometry of the $2−$ charge state, such that the electronic correlation energy (⁠ $U el$⁠) is isolated from the relaxation.⁵ In this case, there is a large separation between $ε$(⁠ $−/2−$⁠) and $ε$(⁠ $2−/3−$⁠) due to the Coulomb repulsion arising upon electron addition.

For most positions of the Fermi level in Fig. 6 (below $E cross$⁠), the divacancy will however prefer the $V Ga ib V O 1$ configuration. The electrical properties of this divacancy configuration are different, as it does not exhibit the negative-U $ε$(⁠ $−/3−$⁠) level. This is because the Ga2–Ga2 dimer found in this configuration is too weak.¹⁹ 

There are several ways to reveal negative-U behavior experimentally.⁴⁴ We shall mention one such method here which is based on DLTS. First, consider the sequential thermal emission of two electrons from a negative-U level to $E C$⁠. The question then arises: which of the two corresponding charge-state transition levels will be observed? In a conventional DLTS measurement, only the deepest of the two charge-state transitions, corresponding to the emission of the first electron, will be observed. Since the second electron is bound less strongly, its emission will proceed immediately after the first electron. Hence, one peak will be observed in the spectrum, but two electrons will be emitted per defect (the “two-electron” peak). By making two modifications to the DLTS measurement, a peak associated with the second electron emission can also be observed. This “one-electron” peak will occur lower in temperature and be approximately halved in amplitude. First, the duration of the filling pulse must be made short enough to allow only one electron to be recaptured by the defect. Second, in the event that any defects manage to capture a second electron, all defects must be optically emptied before each pulse. This is because any defects capturing a second electron will be frozen out in the temperature range where the one-electron peak is observed.⁵⁷ Without photoionization, the peak would decrease to zero over time due to the repetitive pulses required by DLTS. Such measurements previously revealed the one- and two-electron peaks of the negative-U centers $Z 1 / 2$ in 4H-SiC.⁵⁷ 

Before discussing the most commonly reported DLTS signatures, we provide a brief overview of the defect structure in $β$- $G a 2 O 3$⁠. This will be based on previous defect calculations^{13,14,16,19,20,38,58} employing the Heyd–Scuzeria–Ernzerhof^59,60 hybrid functional. We consider native defects, relevant impurities, and notable complexes and discuss which defects are unlikely to be observed experimentally (and can be ruled out from the discussion on DLTS signatures in Sec. VII).

Figure 7 shows the formation energy of native defects in $β$- $G a 2 O 3$ under O-rich and Ga-rich conditions. For each defect, only the lowest-energy configuration under n-type conditions (seeing as $β$- $G a 2 O 3$ is primarily n-type or semi-insulating²) is shown. Results for other configurations can be found in previous reports.^{13,14,16,19,20,38,58}

Ga interstitial: In its most favorable configuration, $Ga i$ forms a split configuration in which two Ga ions share a Ga2 lattice site (⁠ $Ga iad 2$⁠). $Ga iad 2$ behaves as a deep triple donor displaying a negative- $U$ (⁠ $3+/+$⁠) level near $E C$⁠. The (⁠ $2+/+$⁠) level, which would be probed in a conventional DLTS measurement, is 0.91 eV below $E C$⁠. However, $Ga i$ shows a high formation energy in excess of 6 eV under n-type conditions, making it highly unlikely to form under equilibrium conditions. Furthermore, if $Ga i$ is formed (e.g., by irradiation), its thermal stability is expected to be low. Barriers of 0.7 and 0.9 eV have been calculated for migration of $Ga i 3 +$ and $Ga i +$⁠, respectively, along the [010] direction, which would make it mobile even at room temperature.²⁰ 

O interstitial: $O i$ can assume a split configuration, which behaves as an exceedingly deep single donor,³⁸ but the preferred configuration under n-type conditions has the O ion sitting in the large eight-sided channel, acting as a deep double acceptor. The (⁠ $−/2−$⁠) level is located 1.47 eV below $E C$⁠, which is on the higher end of activation energies that can be probed by DLTS. The formation energy of $O i$ is relatively low, especially in the O-rich limit. However, $O i$ has an even lower migration barrier of $∼$0.1 eV,³⁸ making it extremely mobile.

Antisites: $O Ga$ will be disregarded as they are unstable with respect to the constituent $V Ga$ and $O i$ under n-type conditions.³⁸ From Fig. 7, $Ga O$ is similar to $Ga i$⁠, acting as a deep triple donor with a negative- $U$ (⁠ $3+/+$⁠) transition level near $E C$⁠, and a high formation energy under n-type conditions. The preferred $Ga O 1$ configuration has the (⁠ $2+/+$⁠) level at 1.05 eV below $E C$⁠. It has been suggested that mobile $Ga i$ produced by irradiation can become trapped at $V O$⁠, forming thermally stable $Ga O$ that could be observed by DLTS.³⁸ For instance, estimating the dissociation energy of $Ga O 1$ as the sum of the 1.7 eV binding energy (with respect to $V O 1$ and $Ga iad 2$ under n-type conditions) and the 0.9 eV $Ga i +$ migration barrier, a value of 2.6 eV is obtained.

O vacancy: $V O$ behaves as a deep double donor with negative- $U$ behavior. Under Ga-rich and n-type conditions, the formation energy is about 1 eV, which is lowest among the native defects. Nonetheless, a high concentration of $V O$ is not expected in the as-grown material. Among the three different sites, the favored $V O 2$ displays the deepest (⁠ $2+/0$⁠) level at 2.2 eV below $E C$⁠. The (⁠ $+/0$⁠) level is located at 2.48 eV, which is practically inaccessible to DLTS. For the $V O 1$ and $V O 3$ configurations, however, the corresponding level is about 1.7 eV below $E C$⁠. As such, the possibility of observing a DLTS signal from these defects cannot be ruled out, provided that the barriers to transform into $V O 2$ are large.¹⁵ 

Ga vacancy: $V Ga$ is a deep polaronic triple acceptor that can trap up to four holes, resulting in charge states ranging from $1+$ to $3−$⁠. It has a low formation energy in the n-type material, particularly toward O-rich conditions, acting as a compensating acceptor. Indeed, Son et al.⁶¹ have reported that annealing as-grown and unintentionally n-type materials in $O 2$ ambient at 1450  $°$C results in the semi-insulating material in which a magnetic resonance signal from $V Ga 2 −$ (S = 1/2) can be observed. For $V Ga$⁠, the simple vacancies on the Ga1 and Ga2 sites (⁠ $V Ga 1$ and $V Ga 2$⁠) are only metastable. Rather, a neighboring Ga ion will jump into an interstitial position, such that the vacancy is split between two Ga sites connected by the resulting $Ga i$⁠. There are three favorable split configurations, labeled as $V Ga ia$⁠, $V Ga ib$ and $V Ga ic$⁠, where the latter is the lowest in energy under n-type conditions.^13,15,38 The (⁠ $2−/3−$⁠) acceptor levels of the different $V Ga$ configurations fall in the range 1.8–2.7 eV below $E C$⁠, which is too deep for DLTS. Furthermore, $V Ga 3 −$ can migrate along [001] with a low barrier of 1.0 eV.²⁰ 

Here, we shall focus on impurities that are both commonly found in $β -Ga 2 O 3$ and have been discussed previously as potential origins of DLTS signatures.

Hydrogen: Despite the large bandgap, $H i$ acts as a shallow donor in $β -Ga 2 O 3$⁠. When the Fermi level is at $E C$⁠, two $H i$ can form interstitial molecular hydrogen $( H 2 ) i$ with a binding energy of 0.85 eV. However, both forms of interstitial H are highly mobile.^58,62,63 H is, therefore, more likely to be found in a trapped configuration. H trapped by $V O$ acts as a shallow donor on the O1 and O3 sites, but $H O 2$ displays a (⁠ $+/−$⁠) level 0.2 eV below $E C$⁠.⁶³ The (⁠ $0/−$⁠) level that might be observed as a two-electron peak in a DLTS measurement is 0.68 eV below $E C$⁠. However, the thermal stability of this center is also rather low, with a maximum binding energy of 0.7 eV.^63, $V Ga$ can be decorated with several H, forming highly stable complexes^13,38,62 (most notably the $V Ga ib$2H complex¹⁷). However, as mentioned above, the acceptor levels of $V Ga$ are quite deep, and complexing with H tends to shift the levels to even lower Fermi-level positions.^13,19,63 Relevant H complexes with impurities and higher-order intrinsic defects are discussed below.

Group-13 donors: Si, Ge, and Sn substitute for Ga and act as shallow donors in $β -Ga 2 O 3$⁠. Although these defects would not be observed by DLTS, it has been found that anionic $H i −$ can be stabilized in their vicinity, forming charge-neutral complexes with a (⁠ $2+/0$⁠) level 0.5–0.7 eV below $E C$ (e.g., $Si Ga 1$H, $Ga Ga 1$H, and $Sn Ga 2$H complexes, as shown in Fig. 8). However, these complexes exhibit modest binding energies in the range 0.5–1.0 eV (for $E F= E C$⁠) and thus a low thermal stability.⁶³ 

Iron: Fe prefers to substitute for Ga in $β -Ga 2 O 3$⁠, acting as an acceptor with the (⁠ $0/−$⁠) level located 0.68 and 0.78 eV below $E C$ for $Fe Ga 1$ and $Fe Ga 2$⁠, respectively. Similar to other single acceptor impurities,^64,65 $Fe Ga$ can act as a trap for H, forming a deep donor complex with a binding energy of 0.74 eV under n-type conditions. $Fe Ga 2$H exhibits a (⁠ $+/0$⁠) level at $E C−$1.44 eV.

Titanium: Ti prefers to substitute for Ga as well, acting as a single donor with a (⁠ $+/0$⁠) level at 1.15 and 0.62 eV below $E C$ for $Ti Ga 1$ and $Ti Ga 2$⁠, respectively. In contrast to $Fe Ga 1$ and $Fe Ga 2$⁠, the DLTS signatures of $Ti Ga 1$ and $Ti Ga 2$ are thus not expected to overlap.

Isolated divacancy: As discussed in Sec. V, Ga and O vacancies can form divacancy complexes in a number of different configurations, some of which show a (⁠ $−/3−$⁠) level and thus retain the negative-U behavior of the O vacancy. As indicated in Fig. 9, three groups of divacancies can be identified, depending on the type of Ga dimer formed in the $3−$ charge state: Ga1–Ga1, Ga1–Ga2, or Ga2–Ga2, where the two former display (⁠ $2−/3−$⁠) levels at around 1.6 and 0.6 eV below $E C$⁠, respectively, and the latter cannot be stabilized in the $3−$ charge state. These complexes are thermally stable with large binding energies of 1.6–3.0 eV depending on the Fermi-level position. As discussed in Sec. V, a change in Fermi-level can induce a change in divacancy configuration if the temperature is also sufficiently high to surmount the required transformation barriers. These barriers were computed and discussed in Ref. 19 but jumps involving Ga ions generally show lower barriers than O jumps.^15,19

Hydrogenated divacancy: Similar to the isolated vacancies, divacancies are strong traps for H. The H can terminate a dangling O bond at the $V Ga$ (⁠ $V Ga$H), or occupy the $V O$ (⁠ $H O$⁠) in the complex. The former retains the negative-U behavior of the isolated $V Ga V O$ but shifts the corresponding charge-state transition level down in Fermi-level position by about 0.25 eV per H atom in the complex.¹⁹ Conversely, if H substitutes for $V O$⁠, the negative-U behavior is lost. Notably, the preferred configuration of the doubly hydrogenated divacancy is $V Ga ib$H-H $O 1$⁠, which is closely related to $V Ga ib$2H (the lowest-energy $V Ga$2H configuration, and the dominant O–H infrared absorption line in $β -Ga 2 O 3$ annealed in $H 2$¹⁷). This configuration shows no negative-U level, but if the Fermi level is lowered below $E cross$ in Fig. 9(b), $V Ga ib$2H- $V O 1$ becomes favored with a (⁠ $0/−$⁠) level 0.7 eV below $E C$⁠. This transformation can occur through a single H jump, as indicated by the arrow in Fig. 9(a). Hydrogenated divacancies are also stable with binding energies of 1.4–2.5 eV under n-type conditions.¹⁹ 

For convenience when comparing theory and experiments below, Table I lists the ionization energy (difference between $E C$ and the thermodynamic charge-state transition level) and capture barrier for the defects that are potentially observable by DLTS, as discussed above. Specifically, we have included defect levels with $E i$ values up to 1.7 eV. Here, it can also be pointed out that, at elevated temperatures, the bandgap shrinkage becomes considerable,⁶⁶ which can increase the range of accessible levels. In the following discussion, the $E b$ value is mentioned only in cases where it is larger than 0.1 eV.

Table II lists the extracted activation energy, apparent capture cross section and proposed defect origin for the most commonly observed DLTS signatures in $β$- $G a 2 O 3$⁠, to be discussed below. Previously reported values scatter somewhat in the literature due to, e.g., differences in electron concentration in the studied material,⁵⁰ the magnitude of the employed reverse-bias voltage,⁴⁹ or overlap between different signatures. The values provided in Table II were taken from Refs. 8, 38, and 53. In cases where the electric-field dependence of the parameters has been studied, we also list the low-field value.⁴⁹ 

Among the DLTS signatures that are commonly reported across different $β$- $G a 2 O 3$ sample types, the $E 1$ level (⁠ $E A∼0.6 eV$⁠) has the lowest activation energy, see Fig. 10. The $E 1$ center was first observed and labeled by Irmscher et al.⁴ The investigated samples therein were as-received bulk crystals grown by the Czochralski (CZ) method. The level has also been observed in bulk crystals grown by the edge-defined film-fed (EFG) method^68–70 and in epitaxial films grown by molecular beam epitaxy (MBE),⁸ halide vapor phase epitaxy (HVPE),⁷¹ and pulsed laser deposition (PLD).⁷² The reported concentrations of $E 1$ in bulk samples are typically one to two orders of magnitude lower than those of the dominant defect level $E 2$ (discussed in Sec. VII B).

Irradiation studies on $β$- $G a 2 O 3$ have been performed to gauge whether the microscopic defect structure of $E 1$ is related to intrinsic defects. However, there have been conflicting reports, with some claiming that the $E 1$ concentration is unaffected,^70,71 while others see a small increase in the trap concentration in response to irradiation.^49,73 In addition to irradiation, Polyakov et al.^74,75 observed the emergence of the $E 1$ level in bulk EFG-grown $β$- $G a 2 O 3$ samples exposed to H plasma. However, one cannot conclude from this that $E 1$ is H-related, as the plasma treatment causes considerable crystal damage.

An attempt at revealing the microscopic origin of the $E 1$ center was made by subjecting $β$- $G a 2 O 3$ to heat treatments in $Ar$ and $H 2$ ambient.⁶³ It was shown that the concentration of $E 1$ increased considerably following annealing in an $H 2$ ambient (Fig. 10), in contrast to what was observed in samples that were heat-treated in an $Ar$ flow. Consequently, $E 1$ was proposed to, indeed, be associated with a $H$-related defect.

Assuming a H-related origin for $E 1$⁠, hybrid functional calculations were performed to identify potential defects. As discussed in Secs. VI B and VI C, isolated H behaves as a shallow donor but is more likely to be trapped at defects due to its high mobility. Calculations have revealed numerous complexes involving H, but among these, only a small subset exhibits charge-state transition levels in the vicinity of $E C$⁠. In Ref. 63, three types of defects were found to display a charge-state transition level compatible with $E 1$⁠: (i) the (⁠ $−/2−$⁠) level of singly hydrogenated $V Ga V O$ with Ga2–Ga2 dimerization (e.g., $V Ga ib$H- $V O 1$ with $E i=0.53$ eV and $E b=0.18$ eV), (ii) the (⁠ $0/−$⁠) level of $H O 2$ (⁠ $E i=0.68$ eV and $E b=0.11$ eV), and (iii) the (⁠ $+/0$⁠) level of $Si Ga 1$H, Ge $Ga 1$H, or $Sn Ga 2$H (estimated to be located within 1.44 eV of $E C$⁠). Since the two latter exhibit rather low thermal stability, a singly hydrogenated $V Ga V O$ was proposed as the most probable candidate.

This assignment was later contested by Polyakov et al.⁵¹ based on measurements of the electric-field dependence of the $E 1$ emission rate. The field dependence was found to fit a Poole–Frenkel-type emission, which is characteristic of a donor in $n$-type materials. This result points to $Si Ga 1$H or $Sn Ga 2$H, as these are the only donors among the candidates listed above. This defect type is, on the other hand, hard to reconcile with the thermal stability observed for the $E 1$ center.⁶³ 

Based on comparison with predicted capture cross sections, a $Ti Ga 1$ origin has also been suggested.⁷⁶ The donor level of $Ti Ga 1$ (⁠ $E i=0.62$ eV with $E b=0.11$ eV) is indeed close to the $E 1$ level and consistent with the observed Poole–Frenkel emission.⁵¹ However, this model does not explain the observed response to heat treatments in $H 2$⁠.⁶³ Furthermore, the concentration of $E 1$ does not correlate with the Ti-content measured by secondary-ion mass spectrometry (SIMS).⁹ To conclude, a firm identification of this level is still lacking.

The $E 2$ level (⁠ $E A∼0.8$ eV) is the most significant DLTS signature in $β$- $G a 2 O 3$ in terms of prevalence in bulk samples. The first reports of $E 2$ were in CZ- and EFG-grown crystals with a concentration in excess of $10 16 cm − 3$⁠.^4,68,69 The level has been suggested to be the main compensating acceptor in bulk crystals.⁴ 

$V Ga 1$ was initially proposed as a possible microscopic origin for the $E 2$ level.¹⁶ This assignment was based on the agreement between their predicted (⁠ $2−/3−$⁠) level and the $E 2$ activation energy, as well as the low formation energy and compensating acceptor behavior of $V Ga$⁠. However, theoretical studies taking into account the energetically preferred split configurations of $V Ga$ show that the relevant levels are at least 1 eV deeper in the bandgap, making this an unlikely candidate for $E 2$⁠.^{13,15,20,38,77}

A convincing link between the $E 2$ level and $Fe$ was later presented.⁸ First, H irradiation was performed and found to induce no significant change of the $E 2$ concentration, making an extrinsic origin more likely than an intrinsic one. Second, the concentration of the $E 2$ level across several types of samples (EFG, HVPE, and MBE) was compared with impurity concentrations obtained from SIMS analysis. This comparison revealed an almost linear relation between the $Fe$ and $E 2$ concentrations in the samples, although not in a 1:1 ratio [Fig. 11(c)]. Calculated formation energies for Fe-related defects in $β$- $G a 2 O 3$ further corroborated the experimental findings.⁸ Indeed, the formation energy of $Fe Ga$ is relatively low, and this configuration is preferred for relevant Fermi-level values.⁸ Both $Fe$ $Ga 1$ and $Fe$ $Ga 2$ are predicted to act as deep acceptors with calculated (⁠ $0/−$⁠) levels at $E C−0.68$ and 0.78 eV, respectively. These predicted values are in good agreement with the extracted $E A$ value of $E 2$⁠.

Further insight into the microscopic origin of the $E 2$ level was gained from Laplace DLTS measurements.⁹ As mentioned in Sec. III C, Laplace DLTS allows for the separation of overlapping signatures in conventional DLTS spectra, and the $E 2$ level was found to consist of two levels, $E 2a$ (⁠ $E A∼0.6$ eV) and $E 2b$ (⁠ $E A∼0.7$ eV), see Figs. 11(a) and 11(b). The existence of the two closely spaced levels is in agreement with theory, which predicts similar (⁠ $0/−$⁠) transition levels for the two $Fe$ $Ga$ defect configurations. The concentration ratio of $E 2a$ to $E 2b$ was further found to be approximately 1:5, which was the same ratio previously found for the magnetic resonance signals of $Fe Ga$ on the sixfold and fourfold coordinated site.⁷⁸ The ratio is also consistent with theory, considering the small difference in formation energy predicted for $Fe Ga1$ to $Fe Ga2$⁠.

The electric-field dependence of the $E 2$ emission rate has also been investigated.⁴⁹ The emission rate is found to increase with higher electric fields via phonon-assisted tunneling, which is characteristic for an acceptor type defect, in accordance with the assignment to $Fe Ga$⁠.

Thus, the $E 2$ level is an exception among the DLTS signatures in $β$- $G a 2 O 3$ as the microscopic origin is rather well established. However, the deviation from a 1:1 ratio between the concentration of $E 2$ from DLTS and Fe from SIMS is not fully understood. One possible explanation is that Fe exists in other forms, e.g., in various complexes between $Fe Ga$ and other defects (such as $Fe Ga$H). However, further studies are required to clarify this.

The $E 2 ∗$ level (⁠ $E A∼0.75$ eV) is close in activation energy and thus overlaps strongly with the $E 2$ level, which has resulted in a shared signature label. The $E 2 ∗$ center was first recognized by Ingebrigtsen et al.⁸ as a distinct defect that could be introduced by H irradiation, hinting at an intrinsic origin. It has, however, also been observed in the as-grown material. Notably, McGlone et al.^54,55 observed $E 2 ∗$ in Si-doped $β -Ga 2 O 3$ layers grown by MBE on semi-insulating Fe-doped substrates to make field-effect transistors and found it to be the dominant cause of threshold voltage and on-resistance instability.

As the dominant DLTS signature generated by irradiation damage, there are numerous irradiation studies on $E 2 ∗$ in the literature.^{8,38,71,79–81} Polyakov et al. have reported donor compensation and $E 2 ∗$ generation in samples irradiated with H (both MeV⁷¹ and GeV⁸⁰), $α$-particles,⁷³ and neutrons,⁷⁹ (⁠ $E 2 ∗$ was also observed in samples exposed to Ar⁶⁷ or H^74,75 plasma). Here, $E 2 ∗$ was introduced at concentrations much too low to account for the observed donor compensation,⁷³ which might indicate a higher-order rather than a primary intrinsic defect origin. Significantly higher concentrations of compensating acceptors are generated with optical charge-state transition levels in the lower half of the bandgap (believed to be related to $V Ga$^68,82,83) as revealed by current–voltage profiling under light illumination.⁸⁴ 

In Refs. 8 and 38, H irradiation caused near-complete removal of charge carriers from the depletion region, which could be largely recovered during the first DLTS scan up to 650 K, after which $E 2 ∗$ grew as a low-temperature shoulder to $E 2$⁠. The centers were introduced with a near linear irradiation-dose dependence and remained stable in subsequent DLTS measurements.

The introduction of $E 2 ∗$ was further explored in a H/He implantation study by Zimmermann et al.,⁵² revealing that the thermally activated introduction was contingent on the application of a reverse-bias voltage during the heat treatment up to 680 K. Furthermore, the concentration of $E 2 ∗$ could be increased and decreased by performing zero-bias anneals (ZBA) and reverse-bias anneals (RBA), and applying a larger reverse-bias resulted in a larger concentration of $E 2 ∗$⁠. The effect was particularly pronounced for implantations with H, as shown in Fig. 12. Note that when an implantation is performed (as opposed to irradiation), the implanted species is present in the region probed by DLTS, and can interact with defects therein. Based on these observations, it was suggested that the main effect of the reverse bias is an effective lowering of the Fermi-level position within the space-charge region. Hence, $E 2 ∗$ was ascribed to a defect complex involving intrinsic defects (and likely H) that can occur in several different configurations, where the configuration responsible for $E 2 ∗$ is more likely to form if the Fermi-level is lowered from $E C$⁠. Zimmermann et al.⁵² pointed to $V Ga V O$ as a defect fulfilling several of the requirements imposed by the experimental results.

The divacancy model was further explored from hybrid functional calculations in Ref. 19. As already discussed in Secs. V and VI C, isolated and hydrogenated $V Ga V O$ display: (i) several different configurations where the most favorable one depends on the Fermi-level position and (ii) thermodynamic charge-state transition levels near $E C$ with positions largely governed by the type of Ga dimerization and the number of trapped H. Based on the experimental observations on $E 2 ∗$ and theoretical results, which also included calculated energy barriers for transformations between different configurations, specific candidates were proposed.¹⁹ Among the isolated divacancies, configurations with Ga1–Ga2 dimers show a (⁠ $2−/3−$⁠) level compatible with $E 2 ∗$⁠, as indicated in Fig. 9(a), e.g., $V Ga 2 V O 1$ with $E i=0.63$ eV and $E b=0.16$ eV. However, these are not the global-minimum configurations for any Fermi-level position in the bandgap, and a triple acceptor level is perhaps inconsistent with the measured $σ na$ value of about $10 − 14 cm 2$⁠. Doubly hydrogenated divacancies with Ga2–Ga2 dimerization were also proposed, with a particularly interesting candidate being the (⁠ $0/−$⁠) level of $V Ga ib$2H- $V O 1$ (⁠ $E i=0.71$ eV and $E b=0.10$ eV). This is the global-minimum configuration for Fermi-level positions below $E cross$ in Fig. 9(a). When the Fermi-level is above $E cross$⁠, however, $V Ga ib$H-H $O 1$ becomes the preferred configuration, which can be reached through a single H jump into the $V O 1$⁠, removing its negative-U level.

The $E 3$ center (⁠ $E A∼1.0$ eV) has followed a similar story to that of $E 2$⁠, but its origin remains a subject of some debate. The first reports on the center were from measurements on CZ-grown crystals, where $E 3$ was sporadically observed at concentrations up to $4× 10 16 cm − 3$⁠, and an extrinsic origin was tentatively suggested.^4,69

By measuring on various CZ-, EFG- and HVPE-grown $β$- $G a 2 O 3$ samples, Zimmermann et al.⁵² found that the concentration of $E 3$ from DLTS scaled linearly with the concentration of Ti impurities from SIMS, this time approximately in a 1:1 ratio, as shown in Fig. 11(c). In this case, the relation was demonstrated for a narrower range of Ti concentrations, comprising solely bulk samples; in HVPE samples, the concentration of Ti was below the SIMS detection limit and the DLTS spectra showed no $E 3$ signature. These results were also corroborated by hybrid functional calculations showing a $Ti Ga 2$ donor level compatible with $E 3$ (⁠ $E i=1.13$ eV).⁵² A donor origin is also consistent with the electric-field dependence of the $E 3$ emission rate, which follows Poole–Frenkel emission.⁴⁹ 

The $Ti Ga 2$ model is, however, not completely established. First, there are conflicting reports concerning the response of $E 3$ to irradiation, where some observe no introduction of $E 3$⁠,^8,38,81 and others see an increase in concentration.^{70,71,73,79,80} Polyakov et al.⁶⁷ observed an increase in the concentration of $E 3$ also in the near-surface region of $β$- $G a 2 O 3$ exposed to Ar plasma. One explanation could be that these are overlapping signatures related to different defects, i.e., an $E 3 ∗$ signature,⁸⁵ with additional contributions from the partially overlapping $E 2 ∗$ and $E 4$ levels. Here, it should be pointed out that starting from the temperature range where $E 3$ is observed and up to 650 K, irradiation produces several overlapping signatures that display larger uncertainties in the extracted activation energies and can disappear after multiple DLTS cycles up to 650 K.³⁸ The second observation that has been challenging to explain within the $Ti Ga 2$ model comes from a recent study on the thermal stability of different Schottky contacts (Au, Pt, and Ni) on CZ-grown $β$- $G a 2 O 3$⁠.⁵³ Here, the $E 3$ center was present already in the first measurement from 100 to 650 K (ramp-up) in freshly processed samples with Au and Pt contacts but appeared in the sample with Ni contacts only after being subjected to the thermal load, i.e., in the measurement from 650 to 100 K (ramp-down).⁵³ Further studies are required to explain these observations.

Before discussing $E 4$ (⁠ $E A∼1.2$ eV), we note that this DLTS signature is sometimes interchanged with others in the literature (such as $E 3$⁠). As noted in the Sec. VII D, there is also a larger uncertainty in the extracted parameters of traps in this temperature range. $E 4$ is not a prominent signature in the as-grown material, but Polyakov et al.^71,73,79,80 have shown that it responds to irradiation. In the H-irradiation studies by Ingebrigtsen et al.,^8,38 a signature labeled $E 4 ∗$ was observed with a higher activation energy of $1.4±0.15$ eV (as part of a broad DLTS signature that likely consists of several overlapping levels), and it is not clear whether this is actually $E 4$ or a distinct level. Interestingly, the $E 4 ∗$ center could be mostly removed following several DLTS cycles up to 650 K.³⁸ Here, it should also be pointed out that the DLTS spectra in Refs. 8 and 38 do not correspond to as-irradiated samples, since the irradiation resulted in a near-complete removal of charge carriers in the depletion region. Rather, the measurements were performed after an initial DLTS scan up to 650 K to thermally recover the charge carriers. Hence, it is entirely plausible that other DLTS signatures could have been present in the as-irradiated state (such as the $E 4$ center observed in Refs. 71, 73, and 79), but that these were removed during the initial DLTS scan. Regardless, the irradiation studies suggest an intrinsic-related defect origin for the $E 4$ (and $E 4 ∗$⁠) center.

A specific defect model for $E 4$ was recently proposed by Palvan et al.⁵³ in their study on the thermal stability of different Schottky contacts on CZ-grown $β$- $G a 2 O 3$⁠. The $E 4$ center was present in the temperature ramp-up DLTS spectrum for all freshly processed diodes but disappeared after the thermal load and did not recover over time. Thus, $E 4$ is formed during CZ-growth but is unstable against DLTS measurements up to 650 K (note that this observation reinforces our suspicions regarding the thermal stability of irradiation-induced $E 4$ centers). In addition, following the disappearance of $E 4$⁠, the $E 1$ center appears at a similar concentration level, revealing a possible link between $E 1$ and $E 4$⁠. Considering that a singly hydrogenated $V Ga V O$ model had been suggested for $E 1$⁠, other hydrogenated divacancy complexes were considered as a possible model for $E 4$⁠. This led to the proposal of a $V Ga 1$2H- $V O 2$ origin (⁠ $E i=1.16$⁠). As indicated in Fig. 9(a), such doubly hydrogenated configurations showing Ga1–Ga2 dimerization display (⁠ $0/−$⁠) levels falling $∼$1.2 eV below $E C$⁠. During the thermal load, the $V Ga 1$2H- $V O 2$ complex is envisioned to transform into the $V Ga ib$H- $V O 1$ complex, through rearrangement and dissociation of a single H, leading to the appearance of $E 1$⁠.