The ultra-wide bandgap of gallium oxide provides a rich plethora of electrically active defects. Understanding and controlling such defects is of crucial importance in mature device processing. Deep-level transient spectroscopy is one of the most sensitive techniques for measuring electrically active defects in semiconductors and, hence, a key technique for progress toward gallium oxide-based components, including Schottky barrier diodes and field-effect transistors. However, deep-level transient spectroscopy does not provide chemical or configurational information about the defect signature and must, therefore, be combined with other experimental techniques or theoretical modeling to gain a deeper understanding of the defect physics. Here, we discuss the current status regarding the identification of electrically active defects in beta-phase gallium oxide, as observed by deep-level transient spectroscopy and supported by first-principles defect calculations based on the density functional theory. We also discuss the coordinated use of the experiment and theory as a powerful approach for studying electrically active defects and highlight some of the interesting but challenging issues related to the characterization and control of defects in this fascinating material.

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: i a, i b, i c, i d, i e, and i f. 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. IVVIII can be understood by referring to Fig. 1.

FIG. 1.

Top: Ball-and-stick model of the monoclinic crystal structure of β- G a 2 O 3, viewed along the b-axis. The two Ga sites (Ga1, Ga2), three O sites (O1, O2, O3), and six interstitial sites ( i a, i b, i c, i d, i e, and i f)18 are indicated. Bottom: Polyhedra showing the local symmetry of the five different lattice sites, with bond lengths and sites indicated.

FIG. 1.

Top: Ball-and-stick model of the monoclinic crystal structure of β- G a 2 O 3, viewed along the b-axis. The two Ga sites (Ga1, Ga2), three O sites (O1, O2, O3), and six interstitial sites ( i a, i b, i c, i d, i e, and i f)18 are indicated. Bottom: Polyhedra showing the local symmetry of the five different lattice sites, with bond lengths and sites indicated.

Close modal

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 capture rate of electrons for a trap is given by Ref. 28,
c n = n σ n v th, n ,
(1)
where n is the electron concentration in the conduction band, σ n is the electron capture cross section, and v th, n is the electron thermal velocity. The thermal emission rate can be derived from the condition of detailed balance and is given by Ref. 28 
e n ( T ) = γ σ na T 2 e E A / k B T ,
(2)
where γ is a material-dependent constant, σ na is the apparent capture cross section for electrons, which differs from σ n in Eq. (1) by being extrapolated to T = and modulated by an entropy factor, and E A is the activation energy for the thermal emission of an electron from the defect level to the conduction band edge. This activation energy corresponds to the thermodynamic charge-state transition, but can also include an energy barrier, as will be further discussed in Sec. IV B. It is evident from Eq. (2) that the emission rate is highly temperature-dependent and that the deeper the defect level, the slower the defect will re-emit the electron to the conduction band.

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)].

FIG. 2.

Schematic illustration of trap-level emission and capture processes during a DLTS measurement. The two leftmost schematics show the corresponding junction bias and capacitance. (a) The traps have emptied, and the reverse biased junction has reached a steady-state depletion width W ( V r ) with the corresponding junction capacitance C r. (b) Upon the application of the voltage pulse, the region with empty traps is flooded with electrons, turning on the capture process, and the traps are filled with electrons. Note that the depletion width is given as W ( V bi ), which is simply because the voltage pulse in this figure is equal in magnitude to the reverse bias, and thus the depletion-layer width is equal to the one resulting from the built-in voltage. (c) Upon returning to reverse bias, the depletion-region width is increased, and the Fermi level is shifted down. (d) The trap level emits electrons according to the temperature-dependent emission rate [Eq. (2)], which, as the depletion layer width decreases, is measured as a capacitance transient in the junction capacitance with the same time constant as the emission process.

FIG. 2.

Schematic illustration of trap-level emission and capture processes during a DLTS measurement. The two leftmost schematics show the corresponding junction bias and capacitance. (a) The traps have emptied, and the reverse biased junction has reached a steady-state depletion width W ( V r ) with the corresponding junction capacitance C r. (b) Upon the application of the voltage pulse, the region with empty traps is flooded with electrons, turning on the capture process, and the traps are filled with electrons. Note that the depletion width is given as W ( V bi ), which is simply because the voltage pulse in this figure is equal in magnitude to the reverse bias, and thus the depletion-layer width is equal to the one resulting from the built-in voltage. (c) Upon returning to reverse bias, the depletion-region width is increased, and the Fermi level is shifted down. (d) The trap level emits electrons according to the temperature-dependent emission rate [Eq. (2)], which, as the depletion layer width decreases, is measured as a capacitance transient in the junction capacitance with the same time constant as the emission process.

Close modal
FIG. 3.

(a) DLTS signal processing using the simplest weighting function (boxcar). The peak is constructed by taking the difference in the capacitance measured at two different times for increasing temperatures. (b) The use of different rate windows results in a set of peaks from which one can extract pairs of T and e n that can be plotted according to Eq. (2) shown in the inset, providing E A and σ na.

FIG. 3.

(a) DLTS signal processing using the simplest weighting function (boxcar). The peak is constructed by taking the difference in the capacitance measured at two different times for increasing temperatures. (b) The use of different rate windows results in a set of peaks from which one can extract pairs of T and e n that can be plotted according to Eq. (2) shown in the inset, providing E A and σ na.

Close modal
A DLTS spectrum is created by applying a weighting function to the transients and plotting the resulting DLTS signal as a function of temperature. A trap level will then appear as a peak in the DLTS spectrum. In Fig. 3(a), this process is illustrated for the case of a simple boxcar weighting function. Here, the capacitance is measured at two different times, t 1 and t 2, and the DLTS signal is created by taking their difference. The DLTS signal peaks at one specific emission rate value, which is given by the times t 1 and t 2 (for the simple boxcar weighting function, this emission rate value is given by ln ( t 1 / t 2 ) / ( t 1 t 2 )). Changing the time between t 1 and t 2, referred to as the rate window, results in a set of peaks corresponding to different emission-rate values at specific temperatures [see Fig. 3(b)]. The simple boxcar weighting function is conceptually most straightforward but is not commonly used anymore. A widely used weighting function is the so-called lock-in function, i.e.,
w = { 1 : 0 < t < t i / 2 , 1 : t i / 2 < t < t i ,
(3)
which samples several points of the capacitance transients. Here, t i is the window length. More complex weighting functions also exist; examples are GS4 and GS6, which can provide higher energy resolutions, however, at the cost of a reduced signal-to-noise ratio.30,31 Independent of the weighting function, the principle remains the same, i.e., the DLTS signal peaks correspond to specific numerical emission rate values, which can be calculated for the different time windows.
Knowing the emission rate at specific temperatures allows for using Eq. (2) to determine E A and σ na. The equation has two unknowns, so the value of e n must be known for at least two temperatures. This is where the use of different rate windows comes into effect, providing several emission rate values at different temperatures. Equation (2) can be rearranged to
ln ( e n T 2 ) = ln ( γ σ na ) E A k B T .
(4)
Therefore, by having several pairs of ( e n, T) (in Fig. 3 four rate windows are shown), they can be plotted according to Eq. (4) in an Arrhenius plot, as shown in the inset in Fig. 3(b) and from a linear fit one can extract E A from the slope, and σ na from the intersect.
The concentration of the trap level is proportional to the peak height of the transients ( Δ C) and, as a consequence, to the peak amplitude of the DLTS signal. In the simplest approximation, it can thus be found from the following relation (assuming a constant defect and doping concentration, as well as a dilute limit with N t N D):
N t = 2 Δ C C rb N D = 2 S i, peak C rb F i N D ,
(5)
where C rb is the capacitance at reverse bias, N D is the donor concentration, S i, peak is the peak amplitude of the DLTS signal for window length i, and F i is a numerical factor, which depends on the weighting function and window length. A set of calculated values for F i can, for instance, be found in the appendix of Ref. 30.

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.

FIG. 4.

Schematic illustrations of additional techniques related to DLTS: (a) A depth-profile measurement. (b) A capture cross-section measurement. As the filling pulse duration increases, the DLTS signal intensity increases until it saturates. The capture cross section can be extracted from the shape of the curve. At higher (lower) temperatures, the capture process will be faster (slower), allowing the energy barrier for the thermally activated capture to be extracted from measurement at several temperatures.32 (c) Electric-field dependence of the emission rate in the n-type material. The logarithm of the normalized emission rate as a function of the square root of the electric field (bottom x-axis) yields a straight line if the field dependence of the emission process is dominated by the Poole–Frenkel effect, which is typical for donors. If the same quantity plotted against the electric field squared (top x-axis) yields a straight line, phonon-assisted tunneling is the dominating mechanism, which is typical for an acceptor. (d) A Laplace DLTS (LDLTS) spectrum showing two traps. The inset highlights how the same two traps can appear as a single peak ( E T) in a conventional DLTS spectrum.

FIG. 4.

Schematic illustrations of additional techniques related to DLTS: (a) A depth-profile measurement. (b) A capture cross-section measurement. As the filling pulse duration increases, the DLTS signal intensity increases until it saturates. The capture cross section can be extracted from the shape of the curve. At higher (lower) temperatures, the capture process will be faster (slower), allowing the energy barrier for the thermally activated capture to be extracted from measurement at several temperatures.32 (c) Electric-field dependence of the emission rate in the n-type material. The logarithm of the normalized emission rate as a function of the square root of the electric field (bottom x-axis) yields a straight line if the field dependence of the emission process is dominated by the Poole–Frenkel effect, which is typical for donors. If the same quantity plotted against the electric field squared (top x-axis) yields a straight line, phonon-assisted tunneling is the dominating mechanism, which is typical for an acceptor. (d) A Laplace DLTS (LDLTS) spectrum showing two traps. The inset highlights how the same two traps can appear as a single peak ( E T) in a conventional DLTS spectrum.

Close modal

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.

The defect formation energy E f is the energy cost for creating a defect in an otherwise perfect crystal. In the dilute regime, where no interactions occur between defects, it dictates the equilibrium defect concentration
c eq = N sites g exp ( E f k B T ) ,
(6)
where N sites is the density of sites available for defect formation in the crystal and g is a degeneracy factor.
As an example of how the defect formation energy is computed, we consider Fe Ga in β- G a 2 O 3. Defect calculations are performed using a periodically repeated supercell constructed from several bulk unit cells, usually consisting of a few hundred atoms. A total-energy ( E tot) calculation is carried out for the pristine β- G a 2 O 3 supercell, and for a supercell in which Fe Ga q in charge state q has been created by swapping atomic species i and electrons with reservoirs with chemical potentials μ i and E F, respectively. The latter is conventionally referenced to the valence band maximum ( E V). The defective supercell structure is relaxed to reach the equilibrium geometry, and the formation energy is calculated as
E f ( Fe Ga q ) = E tot ( Fe Ga q ) E tot ( β - Ga 2 O 3 ) + μ Ga μ Fe + q ( E V + E F ) + Δ q .
(7)
The chemical potentials of Fe and Ga are referenced to the calculated energy per atom of the elements in their standard state μ i 0. The value of μ i relative to the reference Δ μ i depends on the experimental condition but is subject to bounds. The lower bound on μ Ga is given by μ Ga 0, corresponding to Ga-rich conditions. The upper bound is imposed by the stability of the host crystal
2 Δ μ Ga + 3 Δ μ O = H f ( β -Ga 2 O 3 ) ,
(8)
where H f ( β -Ga 2 O 3 ) is the formation enthalpy per formula unit. The limits on μ Fe are similarly set by the formation of mixed secondary phases Ga 3Fe (Ga-rich) and Fe 2O 3 (O-rich), corresponding to the solubility limit.

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.

FIG. 5.

(a) Formation energy of substitutional and interstitial Fe defects in β -Ga 2 O 3 as a function of the Fermi-level position in the bandgap, (b) configuration coordinate diagram for the process of electron emission from Fe Ga 1 to the conduction band, (c) equilibrium structures of the neutral and negatively charged Fe Ga 1. The charge density of the trapped electron is shown by the yellow isosurface (5% of the maximum value).

FIG. 5.

(a) Formation energy of substitutional and interstitial Fe defects in β -Ga 2 O 3 as a function of the Fermi-level position in the bandgap, (b) configuration coordinate diagram for the process of electron emission from Fe Ga 1 to the conduction band, (c) equilibrium structures of the neutral and negatively charged Fe Ga 1. The charge density of the trapped electron is shown by the yellow isosurface (5% of the maximum value).

Close modal

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 ( Δ Q) 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 harmful54,55 one for device operation.

Metastability can also occur for the charge state of deep-level defects, resulting in so-called negative-U behavior. To understand this, it is useful to remember that a defect is deep, not just on account of the charge-state transition level being positioned deep in the bandgap, but also due to the high degree of spatial localization of the defect charge.41 Adding or removing charges can thus significantly affect the local lattice geometry and bonds at the defect. If two electrons are brought together into a localized defect state, there will be a significant Coulomb repulsion between them. Normally, this repulsion results in a positive correlation energy ( U) between the two successive charge-state transition levels5,
U = ε ( q 2 / q 3 ) ε ( q 1 / q 2 ) ,
(9)
where q 1, q 2, and q 3 are the charge states that correspond to having 0, 1, and 2 electrons in the defect state. In some cases, however, the occupation of a defect state by two electrons is accompanied by a large local lattice relaxation that overcomes the repulsion, resulting in a net positive interaction between the electrons, and thus a negative value of U in Eq. (9) (an Anderson negative-U system56). In a formation energy diagram, negative-U behavior can be easily spotted from a transition level skipping past a charge-state, that is, going directly from q 1 to q 3 in the above example. Then, there is no Fermi-level region where the q 2 state is the lowest in formation energy.

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.

FIG. 6.

Left: Formation energy diagram of the two most favorable configurations of the Ga–O divacancy in β -Ga 2 O 3 ( V Ga 2 V O 2 and V Ga ib V O 1) for intermediate conditions (halfway between O-rich and Ga-rich conditions). The V Ga 2 V O 2 configuration is energetically preferred only under n-type conditions. The negative effective correlation energy ( U eff) associated with the / 3 transition of V Ga 2 V O 2 is indicated. The dark grey line shows the formation energy of V Ga 2 V O 2 frozen in the relaxed, which demonstrates that the purely electronic correlation ( U el) energy is positive. Right: Relaxed structures of the two different V Ga V O configurations. For V Ga 2 V O 2, the Ga1–Ga1 dimer formation associated with the negative-U transition from q = to 3. The yellow isosurface (set at 10% of the maximum value) shows the charge density of the two trapped electrons.

FIG. 6.

Left: Formation energy diagram of the two most favorable configurations of the Ga–O divacancy in β -Ga 2 O 3 ( V Ga 2 V O 2 and V Ga ib V O 1) for intermediate conditions (halfway between O-rich and Ga-rich conditions). The V Ga 2 V O 2 configuration is energetically preferred only under n-type conditions. The negative effective correlation energy ( U eff) associated with the / 3 transition of V Ga 2 V O 2 is indicated. The dark grey line shows the formation energy of V Ga 2 V O 2 frozen in the relaxed, which demonstrates that the purely electronic correlation ( U el) energy is positive. Right: Relaxed structures of the two different V Ga V O configurations. For V Ga 2 V O 2, the Ga1–Ga1 dimer formation associated with the negative-U transition from q = to 3. The yellow isosurface (set at 10% of the maximum value) shows the charge density of the two trapped electrons.

Close modal

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 calculations13,14,16,19,20,38,58 employing the Heyd–Scuzeria–Ernzerhof59,60 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

FIG. 7.

Formation energy of native defects in β -Ga 2 O 3 as a function of the Fermi-level position in the bandgap under O-rich (left) and Ga-rich (right) conditions. For each defect, only the lowest-energy configuration under n-type conditions is shown. The O antisite is unstable and thus not included. The shaded rectangle indicates the Fermi-level region accessible to DLTS (highest reported E A is 1.8 eV below E C38).

FIG. 7.

Formation energy of native defects in β -Ga 2 O 3 as a function of the Fermi-level position in the bandgap under O-rich (left) and Ga-rich (right) conditions. For each defect, only the lowest-energy configuration under n-type conditions is shown. The O antisite is unstable and thus not included. The shaded rectangle indicates the Fermi-level region accessible to DLTS (highest reported E A is 1.8 eV below E C38).

Close modal

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 complexes13,38,62 (most notably the V Ga ib2H 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 1H, Ga Ga 1H, and Sn Ga 2H 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 

FIG. 8.

Formation energy of relevant impurities in β -Ga 2 O 3 as a function of the Fermi-level position in the bandgap under intermediate conditions (halfway between O- and Ga-rich). Hydrogen, group-13 donors and transition metals iron and titanium are shown on the left, middle, and right, respectively. The shaded rectangle indicates the Fermi-level region expected to be accessible to DLTS.

FIG. 8.

Formation energy of relevant impurities in β -Ga 2 O 3 as a function of the Fermi-level position in the bandgap under intermediate conditions (halfway between O- and Ga-rich). Hydrogen, group-13 donors and transition metals iron and titanium are shown on the left, middle, and right, respectively. The shaded rectangle indicates the Fermi-level region expected to be accessible to DLTS.

Close modal

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 2H 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

FIG. 9.

(a) Formation energy of V Ga V O as a function of the Fermi-level position in the bandgap under intermediate conditions. One divacancy is shown for of each type of dimerization (Ga1–Ga1, Ga1–Ga2, and Ga2–Ga2). The vertical dotted lines highlight the Fermi-level positions of the charge-state transition levels below E C. Formation energies of the same configurations with one and two H trapped at V Ga are also shown. The shaded rectangle indicates the Fermi-level region expected to be accessible to DLTS. (b) Relaxed structures of the corresponding structures, indicating the different types of dimerization. The yellow isosurface shows the charge density associated with the Ga dimer state and is set at 10% of the maximum value. When the Fermi level crosses E cross, V Ga ib2H- V O 1 will transform into V Ga ibH–H O 1 through an H jump indicated by the arrow.

FIG. 9.

(a) Formation energy of V Ga V O as a function of the Fermi-level position in the bandgap under intermediate conditions. One divacancy is shown for of each type of dimerization (Ga1–Ga1, Ga1–Ga2, and Ga2–Ga2). The vertical dotted lines highlight the Fermi-level positions of the charge-state transition levels below E C. Formation energies of the same configurations with one and two H trapped at V Ga are also shown. The shaded rectangle indicates the Fermi-level region expected to be accessible to DLTS. (b) Relaxed structures of the corresponding structures, indicating the different types of dimerization. The yellow isosurface shows the charge density associated with the Ga dimer state and is set at 10% of the maximum value. When the Fermi level crosses E cross, V Ga ib2H- V O 1 will transform into V Ga ibH–H O 1 through an H jump indicated by the arrow.

Close modal

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 GaH), 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 ibH-H O 1, which is closely related to V Ga ib2H (the lowest-energy V Ga2H configuration, and the dominant O–H infrared absorption line in β -Ga 2 O 3 annealed in H 217). This configuration shows no negative-U level, but if the Fermi level is lowered below E cross in Fig. 9(b), V Ga ib2H- 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 I.

List of defect levels in β- G a 2 O 3 with calculated ionization energies (Ei) in the range accessible to DLTS. Nonradiative capture barriers (Eb) are also included. The upper estimates for SiGa1H, GeGa1H, and SnGa2H assume a (2 + / +) level at EC, as explained in Ref. 63. For divacancies, the type of Ga dimerization is given in parenthesis.

DefectLevelEi (eV)Eb (eV)
Gaiad2 2 + / + 0.91 0.01 
GaO2 2 + / + 1.05 0.00 
VO1 +/0 1.72 0.02 
VO3 +/0 1.68 0.03 
HO2 0/ − 0.68 0.11 
SiGa1+/0 ≤0.92 … 
GeGa1+/0 ≤1.44 … 
SnGa2+/0 ≤1.08 … 
FeGa1 0/ − 0.68 0.04 
FeGa2 0/ − 0.78 0.02 
TiGa1 +/0 0.62 0.11 
TiGa2 +/0 1.15 0.00 
VGa2VO1 (Ga1–Ga2) 2−/3− 0.63 0.16 
VGa1VO2 (Ga1–Ga2) 2−/3− 0.65 0.17 
VGa2VO2 (Ga1–Ga1) 2−/3− 1.63 0.00 
V Ga ibH-VO1 (Ga2–Ga2) −/2− 0.53 0.18 
VGa1H-VO1 (Ga2–Ga2) −/2− 0.50 0.20 
VGa1H-VO2 (Ga1–Ga2) −/2− 0.93 0.08 
VGa2H-VO1 (Ga1–Ga2) −/2− 0.89 0.08 
V Ga ib2H-VO1 (Ga2–Ga2) 0/ − 0.71 0.10 
VGa12H-VO1 (Ga2–Ga2) 0/ − 0.64 0.13 
VGa12H-VO2 (Ga1–Ga2) 0/ − 1.16 0.03 
VGa22H-VO1 (Ga1–Ga2) 0/ − 1.19 0.02 
DefectLevelEi (eV)Eb (eV)
Gaiad2 2 + / + 0.91 0.01 
GaO2 2 + / + 1.05 0.00 
VO1 +/0 1.72 0.02 
VO3 +/0 1.68 0.03 
HO2 0/ − 0.68 0.11 
SiGa1+/0 ≤0.92 … 
GeGa1+/0 ≤1.44 … 
SnGa2+/0 ≤1.08 … 
FeGa1 0/ − 0.68 0.04 
FeGa2 0/ − 0.78 0.02 
TiGa1 +/0 0.62 0.11 
TiGa2 +/0 1.15 0.00 
VGa2VO1 (Ga1–Ga2) 2−/3− 0.63 0.16 
VGa1VO2 (Ga1–Ga2) 2−/3− 0.65 0.17 
VGa2VO2 (Ga1–Ga1) 2−/3− 1.63 0.00 
V Ga ibH-VO1 (Ga2–Ga2) −/2− 0.53 0.18 
VGa1H-VO1 (Ga2–Ga2) −/2− 0.50 0.20 
VGa1H-VO2 (Ga1–Ga2) −/2− 0.93 0.08 
VGa2H-VO1 (Ga1–Ga2) −/2− 0.89 0.08 
V Ga ib2H-VO1 (Ga2–Ga2) 0/ − 0.71 0.10 
VGa12H-VO1 (Ga2–Ga2) 0/ − 0.64 0.13 
VGa12H-VO2 (Ga1–Ga2) 0/ − 1.16 0.03 
VGa22H-VO1 (Ga1–Ga2) 0/ − 1.19 0.02 

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 

TABLE II.

Reported activation energies (EA), apparent capture cross sections (σna), proposed defect origins, reported response to irradiation and correlation with impurity concentration for commonly observed DLTS signatures in β − Ga2O3. The EA and σna values for E1, E2, E 2 and E3 are taken from Refs. 8 and 38, and the values for E2a/2b are based on the simulated and Laplace DLTS results from Ref. 9. The values for E3* and E4 (not observed by Ingebrigtsen et al.8,38) are taken from Refs. 53 and 67. Low-field values for EA and σna are given in parenthesis where available.49,51

SignatureEA (eV)σna (cm2)Proposed origin(s)Irradiation-inducedImpurity
E1 0.56 (0.64) 0.3–5 × 10−13 (7 × 10−13 V Ga ibH-VO⟩, HO2No  H anneal time 
   (Si/Sn)GaH, TiGa1   
E2 0.78 (0.82) 7 × 10−15 (1 × 10−14FeGa No ∝ [Fe] 
E2a 0.6  FeGa1 
E2b 0.7  FeGa2 
E 2  0.75 3–7 × 10−14 VGa2VO1⟩, V Ga ib2H-VO1⟩ Yes  
E3 1.01 (1.05) 0.02–1 × 10−12 (4 × 10−13TiGa2 No ∝ [Ti] 
E31.05 1.5-4.5 × 10−14  Yes  
E4 1.17 2 × 10−14 VGa12H-VO2⟩ Yes  
SignatureEA (eV)σna (cm2)Proposed origin(s)Irradiation-inducedImpurity
E1 0.56 (0.64) 0.3–5 × 10−13 (7 × 10−13 V Ga ibH-VO⟩, HO2No  H anneal time 
   (Si/Sn)GaH, TiGa1   
E2 0.78 (0.82) 7 × 10−15 (1 × 10−14FeGa No ∝ [Fe] 
E2a 0.6  FeGa1 
E2b 0.7  FeGa2 
E 2  0.75 3–7 × 10−14 VGa2VO1⟩, V Ga ib2H-VO1⟩ Yes  
E3 1.01 (1.05) 0.02–1 × 10−12 (4 × 10−13TiGa2 No ∝ [Ti] 
E31.05 1.5-4.5 × 10−14  Yes  
E4 1.17 2 × 10−14 VGa12H-VO2⟩ Yes  

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) method68–70 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).

FIG. 10.

DLTS spectra from Ref. 63 recorded on an as-received and H 2-annealed (900  °C for 30 min) bulk β- G a 2 O 3 crystal. The inset shows the mean and standard deviation of the E 1 concentration ([ E 1]), as extracted from DLTS measurements conducted on several samples annealed in H 2 at increasing annealing times.

FIG. 10.

DLTS spectra from Ref. 63 recorded on an as-received and H 2-annealed (900  °C for 30 min) bulk β- G a 2 O 3 crystal. The inset shows the mean and standard deviation of the E 1 concentration ([ E 1]), as extracted from DLTS measurements conducted on several samples annealed in H 2 at increasing annealing times.

Close modal

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 ibH- 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 1H, Ge Ga 1H, or Sn Ga 2H (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 1H or Sn Ga 2H, 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.

FIG. 11.

(a) DLTS spectrum of a ( 201) oriented EFG-grown β- G a 2 O 3 sample, showing the E 2 and E 3 signatures, where the former consists of overlapping peaks E 2 a and E 2 b.9 (b) This is evidenced in the Laplace DLTS spectrum, showing three distinct peaks.9 (c) Linear relationship between Ti and Fe impurity concentrations from SIMS, and E 2 and E 3 trap concentrations from DLTS for various different samples.8,9

FIG. 11.

(a) DLTS spectrum of a ( 201) oriented EFG-grown β- G a 2 O 3 sample, showing the E 2 and E 3 signatures, where the former consists of overlapping peaks E 2 a and E 2 b.9 (b) This is evidenced in the Laplace DLTS spectrum, showing three distinct peaks.9 (c) Linear relationship between Ti and Fe impurity concentrations from SIMS, and E 2 and E 3 trap concentrations from DLTS for various different samples.8,9

Close modal

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 GaH). 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 H74,75 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 Ga68,82,83) 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 ib2H- 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 ibH-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 

FIG. 12.

DLTS spectra of a HVPE-grown β- G a 2 O 3 sample from Ref. 52, showing the E 2 and E 2 signatures after He and H implantations followed by three different zero- and reverse-bias annealing steps at 680 K.

FIG. 12.

DLTS spectra of a HVPE-grown β- G a 2 O 3 sample from Ref. 52, showing the E 2 and E 2 signatures after He and H implantations followed by three different zero- and reverse-bias annealing steps at 680 K.

Close modal

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 12H- 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 12H- V O 2 complex is envisioned to transform into the V Ga ibH- V O 1 complex, through rearrangement and dissociation of a single H, leading to the appearance of E 1.

In addition to the commonly reported defect centers discussed in Secs. VII AVII E, many other DLTS signatures have been reported in the literature. Here, we shall point out two types of signatures not discussed above: (i) Farzana et al.86 observed some relatively shallow levels at low concentrations in the Ge-doped material grown by plasma-assisted MBE. The levels displayed similar E A values of 0.18 ± 0.02, and 0.21 ± 0.02 eV and distinct σ na values of 1 7 × 10 19 and 0.2 2 × 10 15 cm 2, respectively. Similarly, Polyakov et al.73 observed the introduction of a level at 0.28 eV below E C with a σ na value of 6 × 10 18 cm 2 following α-particle irradiation, albeit with a very low concentration. (ii) Ingebrigtsen et al.85 reported two exceedingly deep signatures labeled E 5 and E 6 displaying activation energies of 1.5 ± 0.15 and 1.8 ± 0.2 eV, respectively. Interestingly, these traps were generated in as-received epitaxial films of (010)-oriented β- G a 2 O 3 grown by MBE following an initial heat treatment up to 675 K with an applied reverse-bias voltage. After generation, the defects were thermally stable. Further characterization of these signatures is required.

Considering the large portion of the β -Ga 2 O 3 bandgap that is inaccessible to DLTS measurements, the utilization of complementary optical junction-spectroscopic techniques such as SSPC and DLOS that can probe defect levels deeper in the bandgap becomes particularly important for this material. Numerous studies have reported results from such measurements on β -Ga 2 O 3,18,40,68,70,71,73,81–83,86–91 revealing several defect levels in the range between 0.8 and 4.5 eV below E C. Although a comprehensive discussion on these results is beyond the scope of the present Perspective, we shall provide a short overview and discuss how the optical and thermal defect signatures relate to each other and to results from first-principles calculations, highlighting some of the opportunities and challenges.

For most of the optical levels referenced above, a clear defect identification is lacking. However, several signatures have been reported to respond to irradiation.54,68,70,73,82,83,86 As mentioned in Sec. VII C, irradiation-induced optical levels are introduced at much higher rates overall compared to those observed by DLTS and are thus believed to be responsible for most of the observed charge-carrier removal.88,90 As such, primary intrinsic defect origins have been suggested for many of these levels. Indeed, based on the hybrid functional calculations discussed in Sec. VI, one would expect to observe defect signatures from V O and V Ga in the middle to lower part of the bandgap,83 and the latter is likely to play an important role as a compensating acceptor.38,55,88 Certain levels have also been correlated with impurities, such as Fe,40 Ti,40 Mg,92 and N.91 

As an illustrative example, we again discuss the cases of Fe and Ti in β -Ga 2 O 3.40 As explained in Secs. VII B and VII D, the E 2 and E 3 centers have been assigned to the ( 0 / ) levels of Fe Ga 1 and Fe Ga 2, and the ( + / 0) level of Ti Ga 2, respectively.8,9 Zimmermann et al.40 performed SSPC measurements on samples with known concentrations of these defects from DLTS to identify the corresponding optical levels of Fe and Ti. To understand this, we first consider the difference between thermal and optical ionization of deep-level defects.

Optical charge-state transition levels of defects can be understood from the CC model explained in Sec. IV B.41 In the classical formulation of the Franck–Condon (FC) principle, optical transitions occur instantaneously with no change in geometry. The optical absorption energy of Fe Ga 1 in Fig. 5(b) can thus be drawn as a vertical line from the minimum of the ground-state curve to the excited-state curve in the CC diagram.41 The resulting absorption energy will be larger than E i by the so-called FC shift in the excited state d e FC. The FC shift is shown for the ground state in Fig. 5(b). We obtain d e FC values of 1.17 eV for Fe Ga 1, 1.09 eV for Fe Ga 2, and 1.15 eV for Ti Ga 2, resulting in corresponding absorption energies of 1.85, 1.87, and 2.30 eV. When comparing these predictions with experimental data, it should be kept in mind that, for such large FC shifts, a significant vibrational broadening of the onset of the photoionization profile can be expected. As explained in Refs. 40, 41, 83 and 93, the shape of the absorption onset can be simulated by going beyond the classical FC picture, treating the ground- and excited-state curves in the CC diagram as quantum harmonic oscillators and considering vibronic transitions between their vibrational sub-levels. For a photoionization process bringing the electron into the conduction band, the model of Kopylov and Phiktin can be used.94 

The SSPC measurements performed by Zimmermann et al.40 revealed an SSPC signature labeled T 1 EFG with a photon energy position and trap concentration close to the calculated absorption energies of Fe Ga 1, Fe Ga 2, and Ti Ga 2 and concentrations of E 2 and E 3 from DLTS. Using the CC diagrams of Fe Ga 1, Fe Ga 2, and Ti Ga 2 as input (together with the recorded photon flux for the optical excitation in the SSPC measurements, and the concentrations of E 2 a, E 2 b, and E 3) to simulate their SSPC spectra, the T 1 EFG signature was ascribed to overlapping signatures from these three defects. Thus, a unified energy-level scheme was found for Fe and Ti in β -Ga 2 O 3.

The strong overlap between the SSPC signatures of Fe and Ti impurities highlights what is arguably the main challenge when interpreting results from optical junction-spectroscopic measurements on β -Ga 2 O 3, i.e., defects exhibiting strong electron–phonon coupling show broad optical signatures. Considering the even stronger electron-phonon coupling predicted for intrinsic defects,83 and the large number of levels that are expected to occur in the same energy range, the resulting signatures are challenging to interpret and distinguish.83 Nevertheless, future application of this approach, as well as its extension to DLOS,91 might help elucidate the origins of other optical junction-spectroscopic defect signatures in β -Ga 2 O 3.

β- G a 2 O 3 has proven to be a fascinating material with a rich defect structure. Since the first DLTS study on β- G a 2 O 3 reported in 2011,4 a wide range of defect levels have been observed and investigated by a broad international community. Although more than a decade has passed since then, we believe the development in understanding and identifying defects has been more efficient relative to comparable semiconductor materials, including other oxides and carbides. This can to a large extent be attributed to the development of first-principles defect calculations and their increased use in combination with experiments, i.e., in concerted efforts to unravel the properties and microscopic origin of defects. However, β- G a 2 O 3 is a challenging material from a defect-physics point of view, with its ultra-wide bandgap, low crystal symmetry, multiple inequivalent lattice sites, and the tendency to form split configurations.

The experimental and theoretical studies discussed in this Perspective have revealed many interesting phenomena related to metastability of defects in β- G a 2 O 3. One property that keeps reoccurring in the defect assignments proposed in theoretical studies is negative-U behavior, including H O 2, isolated and hydrogenated divacancies, and complexes between donor impurities and anionic H. Experiments, however, have thus far provided no firm evidence of such behavior. Proving negative-U behavior experimentally is notoriously difficult,57,95 as the charge-state under scrutiny is metastable or even unstable.

As discussed in Sec. V, the direct observation and investigation of negative-U behavior rely to a large extent on controlling the capture process,33 i.e., through appropriate reduction of the filling-pulse length, causing incomplete filling of the trap.57 However, to successfully conduct measurements of the capture process, the process needs to be sufficiently slow for the measurement instrumentation. In practical terms, this means that the capture cross section of interest must be sufficiently low, and, more importantly, the doping in the sample must be sufficiently low as the capture rate increases with free carrier concentration. Unfortunately, obtaining sufficiently low-doped β- G a 2 O 3 crystals pose a challenge for these types of measurements. This is probably the main obstacle for further investigations into negative-U at present. However, if the material quality improves further, and in particular with improved control of the charge carrier concentration down to the 10 13 and 10 14 cm 3 range, this can become feasible.

Accurate measurements of the capture cross sections of DLTS centers in β- G a 2 O 3 will also allow additional comparison with theory. As discussed in Sec. IV B, methodological developments allow nonradiative capture coefficients to be calculated from first-principles based on the one-dimensional CC model. This method includes the evaluation of the Sommerfeld parameter, which describes the enhancement or suppression of the carrier-capture rate by charged defects due to the Coulomb interaction.6,7

Metastability of defect configurations is an additional, although related, topic that needs further investigation. Uncontrolled introduction and removal of deep-level defects can be detrimental to device performance and must be understood if β- G a 2 O 3 is to live up to its potential as the next material for high-power and harsh-environment device applications. Another aspect that needs to be investigated further is the dependence of defect stability on crystal orientation. For example, recent calculations using the nudged elastic band method have revealed a large anisotropy in the diffusivity of V Ga, with barriers of 1.0 and 2.1 eV for migration along [001] and the remaining crystal directions, respectively.20 Such effects could have implications, e.g., for the evolution of elementary defects generated by irradiation and measured using DLTS.

Financial support is acknowledged from the Research Council of Norway through the GO2DEVICE and GO-POW research projects (Grant Nos. 314017 and 301740) and the Norwegian Micro- and Nano-Fabrication Facility, NorFab, Project No. 295864. Computations were performed on resources provided by UNINETT Sigma2— the National Infrastructure for High Performance Computing and Data Storage in Norway.

The authors have no conflicts to disclose.

Amanda Langørgen: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Project administration (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Lasse Vines: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Project administration (equal); Supervision (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). Ymir Kalmann Frodason: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Project administration (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available within the article.

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