Ga vacancies in β-Ga2O3: split or not?

Ga vacancies and their unusual structure in β-Ga2O3 have been studied in recent years by several experimental techniques as well as theoretical calculations. Theory predicts that the so-called split Ga vacancy configuration is the one with the lowest formation energy. Positron annihilation spectroscopy faces challenges with β-Ga2O3 due to the colossal signal anisotropy and lack of a proper reference sample. Nevertheless, the majority of the data strongly suggests that all studied β-Ga2O3 samples contain high concentrations of split Ga vacancies in a wide distribution of configurations. EPR, IR spectroscopy and scanning transmission electron microscopy experiments all agree on the split Ga vacancy configuration as being the most likely interpretation of the data. However, the exact structure of the split Ga vacancies remains to be solved in terms of complexes with O vacancies and hydrogen.


Introduction
Interest in β-Ga 2 O 3 has seen a strong increase in the past decade thanks to its ultra-wide energy gap (4.6-4.9 eV) and high critical field (6-8 MV cm −1 ). 1) In addition, it is possible to grow large-area high quality bulk single crystals. 2) Unipolar devices have been developed, and controlled ntype doping is the key to achieving the desired properties. [3][4][5] Detailed identification of dominating lattice defects is an important goal in order to further improve the properties of β-Ga 2 O 3 devices. 6) Generally, the formation of acceptor-type native point defects is enhanced in n-type doped semiconductors but in most compound semiconductors this does not limit the carrier concentration. [7][8][9][10] In this review, the focus is on Ga vacancy related defects as they have acceptor character and hence ought to be efficient compensating centers, while O vacancy related defects have deep donor character. 6) Theoretical calculations of defect properties in β-Ga 2 O 3 predict unusually low-even negative-formation energies for Ga vacancy related defects in all of their negative charge state configurations at Fermi levels near the conduction band edge. [11][12][13][14] This suggests that Ga vacancy defects should be abundant and efficiently compensate for the n-type doping. Even complexes involving a Ga vacancy, O vacancy, and two hydrogen atoms are predicted to exhibit negative charge states at high Fermi levels. There is also abundant experimental evidence obtained by IR spectroscopy, EPR spectroscopy, positron annihilation spectroscopy and transmission electron microscopy that (split) Ga vacancies are present at very high concentrations in β-Ga 2 O 3 . [15][16][17][18] Theory also predicts that these split Ga vacancies have the lowest formation energies. [11][12][13][14] Three different split Ga vacancy configurations have been shown to form, where a next-nearest neighbor fourfold coordinated Ga(1) atom relaxes inwards into the interstitial space, as depicted in Fig. 1. The resulting split Ga vacancy has an open volume on both sides of the center interstitial, resulting in two "half-vacancies." The split Ga vacancy V Ga ia forms at Ga (1) and Ga (2) sites, while V Ga ib and V Ga ic are formed by two Ga(1) sites. It should be noted that the split Ga vacancies are monovacancies as they involve only one missing atom. Why these defects are not efficient in compensation of the donors, allowing for controllable n-type doping, remains an open question.
This progress review focuses on the question of the dominant form of existence of Ga vacancy related defects in β-Ga 2 O 3 : split or not? In addition, the possible formation of complexes with oxygen vacancies, hydrogen, and n-type dopants is discussed in the light of theoretical predictions and experimental evidence. Experimental results obtained with positron annihilation spectroscopy are given particular attention.

Positron annihilation spectroscopy
Positron annihilation spectroscopy has particular use in the identification and quantification of vacancy-type defects in semiconductors in the concentration range 10 15 -10 19 cm −3 . 19) It is possible to distinguish between different charge states of the vacancies and give information on their chemical environment, in addition to the size of the open volume and the concentrations. In the experiments, positrons are implanted into the sample material where they eventually annihilate with electrons emitting two 511 keV γ-photons. The positrons probe the interstitial space in a delocalized state in the crystal lattice and may get trapped at open volume defects thanks to the missing repulsive ion core. The trapping at vacancies causes changes in the positron-electron annihilation radiation: the positron lifetime and the Doppler broadening of the 511 keV annihilation line. The positron lifetime increases when the positron is trapped in a vacancy. Usually positron trapping at vacancies is detected as a narrowing of the Doppler broadened 511 keV line. Generally, the positron lifetime gives information about the open volume and concentration of the vacancy defects, while Doppler broadening gives information on the chemical surroundings of the vacancy, namely the electron configuration of the surrounding atoms. Interestingly, in β-Ga 2 O 3 the Doppler broadening signal anisotropy is colossal compared to other semiconductors, allowing for novel types of experiments. 17)

Experiment
The standard approach for studying bulk crystals is sandwiching a low-activity 22 Na positron source between two identical pieces of the sample material. The fast positrons emitted by the source probe the sample up to about 200 μm providing averaged data over these depths. Thin films can be studied with a slow positron beam where the positrons emitted by the source are slowed down and implanted in a single sample at kinetic energies up to 40 keV, probing the near-surface region up to 2 μm below the surface. For a detailed account on the experimental techniques, see Ref. 19).
The positron lifetime experiment represents the probability distribution of positron annihilation at time t and is analyzed as a superposition of exponential decay components: is typically used as an experimental parameter as it coincides with the center of mass of the lifetime spectrum and has very high statistical accuracy thanks to the >10 6 collected annihilation events. Typically, the lifetime is 20%-50% longer for a positron trapped at a vacancy than for a positron in the delocalized state in the lattice (e.g. 160 ps for the lattice and 235 ps for the Ga vacancy in GaN 7) ).
The motion of the annihilating electron-positron pair causes a Doppler shift in each annihilation event ΔE = cp L /2, where p L is the longitudinal momentum component of the pair in the direction of the annihilation photon emission. The shape of the 511 keV peak gives thus the one-dimensional momentum distribution of the annihilating electron-positron pairs as shown schematically in Fig. 2. A Doppler shift of 1 keV corresponds to a momentum value of p L = 0.54 a.u. Due to limited resolution and a relatively low peak-to-background ratio (∼10 2 on the right hand side of the peak) in conventional experiments, integrated parameters are used to describe the shape of the Doppler-broadened annihilation peak. The low electronmomentum parameter S is defined as the ratio of the counts in the central region (typically p L < 0.4 a.u.) of the annihilation line to the total number of counts in the line. The high electron-momentum parameter W is the fraction of the counts in the wing regions of the line (typically p L > 1.5-2.0 a.u.). Typically, the S parameter increases by 3%-5% when positrons annihilate as trapped at vacancies, while the decrease of the W parameter is 15%-50%.
As discussed in detail in Refs. 17 and 20, it is important to note that the data obtained in Doppler broadening experiments are by definition dependent on the measurement direction relative to the crystal orientation. Typically the "longitudinal" measurement direction mentioned above is along a crystal orientation within the sample surface or perpendicular to it (see Fig. 2). When performing defect studies in "conventional" semiconductors with Doppler broadening, the measurement orientation is usually disregarded as the relative differences in the signals in the different directions are at most of the order of 0.05%-0.1% and hence negligible. However, β-Ga 2 O 3 is colossally different in this respect. Relative changes of the S parameter as high as 4% have been measured by rotating the samples. The root cause lies in the quasi-one-dimensional positron state in the β-Ga 2 O 3 lattice, 17) but the detailed mechanisms giving rise to the unusually high anisotropy are still to be elucidated. While this phenomenon causes severe challenges to the interpretation of the results as the magnitude is similar to typical vacancy-induced changes, it also allows for novel approaches for defect identification with positron annihilation.
For defect identification, the measured (S, W) parameters are typically plotted in a plane as illustrated in Fig. 3. This figure highlights the anisotropy issue faced when studying β-Ga 2 O 3 . In "conventional" semiconductors such as GaAs or GaN, the annihilations in the defect-free lattice (bulk) and at cation vacancies produce clearly different signals and the data are straightforward to interpret. 19) Changing the measurement orientation for these materials produces changes in the parameters that are smaller than the symbol size in the figure. However, in β-Ga 2 O 3 , the full extent of the measured anisotropy is larger than the largest predicted distance between bulk and a mono-vacancy (unrelaxed V Ga1 ) along a single orientation, in this case [100]. 17)

Theory
An important advantage of positron annihilation spectroscopies is the possibility of predicting the annihilation signals from first principles. 19) Delocalized and trapped positron states and their associated lifetimes are modeled using twocomponent density functional theory for electron-positron systems. The local density approximation employed for the correlation potential and enhancement factor typically predicts too high annihilation rates with 3d electrons in comparison with the experiments, but differences and ratios in positron lifetimes and Doppler spectra can be compared with high confidence. 21,22) For comparison with experiments, the three-dimensional momentum density is first projected into the desired crystal direction and then convoluted with the experimental detector resolution function. A detailed description of the annihilation parameter modeling applied to β-Ga 2 O 3 is found in Ref. 17).

Positron experiments in β-Ga 2 O 3
Reports of positron experiments in β-Ga 2 O 3 can be divided into two categories. The first category (also chronologically) consists of experiments where the anisotropy of the Doppler broadening signals is not considered, either due to performing only lifetime experiments or due to the lack of knowledge of the unusual properties of β-Ga 2 O 3 . The second category consists of experiments where the colossal anisotropy of the Doppler broadening signals is acknowledged, leading also to the realization that the split Ga vacancy configuration is the key to interpreting most of the experimental results. The early work on β-Ga 2 O 3 with positron annihilation spectroscopy consists of positron lifetime experiments in sintered polycrystalline material, with a single crystal as a reference point. 23,24) Large open volume defects were found in the polycrystalline samples, but the single crystal produced a single positron lifetime of roughly 180 ps that was interpreted as the "bulk" lifetime, or the lifetime of delocalized positrons in the β-Ga 2 O 3 lattice. A similar single lifetime (176 ps) was found also in the single crystal substrate in the work performed after the interest in β-Ga 2 O 3 was revived, still interpreted as the bulk lifetime. 25) Large vacancy clusters have also been observed by positron lifetime experiments recently, both in sintered bulk samples and in annealed hetero-epitaxial thin films. [26][27][28] Doppler broadening experiments were the main approach used to study the possible involvement of Ga vacancies in the electrical compensation of n-type doping in hetero-and homo-epitaxial β-Ga 2 O 3 thin films in Ref. 25). With the lack of earlier positron annihilation studies on defects in β-Ga 2 O 3 , the interpretations were based on comparison to well-known phenomena in GaN and ZnO. The anisotropy of the Doppler broadening signals was not considered, and also the possible strong relaxation of the Ga vacancies was ignored. The data strongly suggested that the (unrelaxed) Ga vacancy defects would be present at high concentrations in semi-insulating Si-doped and undoped thin films, providing an explanation for the electrical compensation. This interpretation was later reinforced by experiments in Sndoped n-type thin films where the Ga vacancy concentrations were found to be very low. 29) However, the controversy in relation to theoretical predictions remained: Ga vacancy concentrations were found to be low in n-type material and high in semi-insulating material, while theory pointed towards the opposite.
The work on the effect of alloy composition and Si doping on vacancy defects in (In x Ga 1−x ) 2 O 3 thin films was also performed without knowledge of the unusual anisotropy effects exhibited by the Doppler broadening signals in β-Ga 2 O 3 . 30) Detailed investigations of these effects could shed more light on the delicate balance between the different phases in the continuous-composition spread samples used in the studies.
The colossal anisotropy of the Doppler broadening signals in β-Ga 2 O 3 was revealed only very recently. 17) This finding   17,19) The data for different materials are shown in arbitrary units and shifted with respect to one another to show the relative differences. The anisotropy or measurement orientation dependence of the (S, W) parameters in GaAs and GaN is smaller than the symbol size. In β-Ga 2 O 3 , the largest predicted bulk-vacancy difference that is predicted to occur for the V Ga1 in measurement along [100] is smaller than the full measured anisotropy.

SF0802-3 © 2023 The Author(s). Published on behalf of
The Japan Society of Applied Physics by IOP Publishing Ltd was accompanied by the observation that all bulk crystals and homo-epitaxial thin films exhibit this anisotropy. 31) In addition, the experimental anisotropy was found to be significantly larger than that predicted by theoretical calculations for the β-Ga 2 O 3 lattice. Instead, the anisotropy predicted for various split Ga vacancy configurations [11][12][13][14] was found to be comparable to the experiments. This is another highly unusual property of β-Ga 2 O 3 : in other semiconductors the vacancy related Doppler broadening signals are more isotropic than those of the lattice.
The colossal anisotropy of the positron Doppler broadening signals measured in all β-Ga 2 O 3 crystals does not depend on the doping. It is found in undoped, Mg-doped, Fedoped, Sn-doped, indicating that all these crystals contain high concentrations of split Ga vacancies. 31) However, the configuration distributions are different in the different crystals, as revealed by temperature-dependent positron lifetime experiments. 31) It is remarkable that the measured lifetimes in the as-grown crystals are in the range 175-185 ps, coinciding with the earlier interpretation as "bulk." It is, however, now evident that this lifetime corresponds to the split Ga vacancies. Theoretical calculations predict 10-30 ps longer lifetimes for the split Ga vacancies (in various configurations, including O vacancies and H decoration) than in the lattice, suggesting that the true lattice positron lifetime in β-Ga 2 O 3 is clearly shorter than 170 ps. 17,31,32) Hydrogen implantation experiments and subsequent annealings reveal a complex interplay between hydrogen, split Ga vacancies, and potentially also unrelaxed Ga vacancies. 32) In particular, the latter observation requires further studies for a more detailed picture as theory predicts the unrelaxed Ga vacancy to be a metastable configuration with a relatively low (less than 1 eV) barrier for relaxation. 11) Figure 4 shows the average positron lifetime measured in a Fe-doped β-Ga 2 O 3 crystal grown by the edge-defined filmfed method. 31) This crystal produces the shortest positron lifetimes measured so far. The grey shadow behind the markers shows the variation in the early experimental assignments of the bulk lifetime. The temperature dependence of the average positron lifetime in the Fe-doped crystal is a clear indication of the data being dominated by (negatively charged) vacancies, 19) even if the exponential decay profile contains only a single component. This alone demonstrates that the vacancy concentrations are high even in this sample, although the lifetime is shorter than the early assignments of the bulk lifetime. The blue shadow shows the range of values of the average positron lifetime measured in polycrystalline material, corresponding to relatively large vacancy clusters. 24,[26][27][28] Preliminary experiments after subjecting the Fe-doped crystal to ion irradiation give a range of average lifetime values given by the red shadow in Fig. 4. Theoretical calculations suggest that the positron lifetime should be 15-20 ps longer in unrelaxed than in split Ga vacancies. Hence, if the average positron lifetime in the Fe-doped crystal corresponds to the split Ga vacancies it is possible that irradiated samples contain unrelaxed Ga vacancies.
The colossal anisotropy of the Doppler broadening signals makes it extremely important to pay close attention to the measurement direction and give a detailed account of the measurement geometry in any Doppler broadening experiment performed in β-Ga 2 O 3 . Together with the lack of a proper "defect-free" reference sample, making useful interpretations concerning defect identities or concentrations requires the use of unusual approaches. Analyzing the anisotropy itself provides access to the detailed identities. 17,31,32) Another approach is to make a detailed analysis of the positron diffusion lengths in slow positron experiments. [33][34][35] This gives the possibility to interpret the changes in terms of defect concentrations, but also there the anisotropy needs to be acknowledged. It is not yet known whether the quasi-one-dimensional positron state in the β-Ga 2 O 3 lattice causes anisotropy in positron diffusion as well.

Discussion
Positron annihilation spectroscopy is generally considered a powerful method for detailed identification and quantification of vacancy-type defects in semiconductors. However, this is based on two conditions: (1) the existence of a suitable reference sample that can be considered defect-free from the perspective of positrons, hence producing annihilation signals of the lattice (usually called "bulk"), and (2) the dominance of a single type of defect in the annihilation signals, at least in some externally controllable state of the sample(s). When these conditions are fulfilled, it is reasonably straightforward to interpret both the positron lifetime and Doppler broadening signals through comparison with state-of-the-art theoretical calculations. 19) In particular, the Doppler broadening data are typically presented as normalized by the data in the reference sample, both when presenting (S, W) plots or coincidence Doppler data as ratio curves for identifying the fingerprints of the various defects.  Figure 3 illustrates the concept of an (S, W) plot. For example, if GaN samples with varying concentrations of Ga vacancies are analyzed, the data points will fall on the line connecting the "bulk" and vacancy points (black circle markers) and the positioning on the line is directly dependent on the defect concentration.
As is evident from the results reported so far in β-Ga 2 O 3 , neither of the two conditions is fulfilled. Theoretical predictions for the typical fingerprints have been published, 17,31,32) with the hope that future research efforts leading to having suitable samples. In the meantime, the colossal anisotropy of the Doppler broadening signals can be utilized for making rather strong-although somewhat qualitative-arguments concerning the structure of the Ga vacancy defects as well as their concentrations. Importantly, the general form and span of the anisotropy indicate that trapping at split Ga vacancies is responsible for a significant fraction of positron annihilation signals in all studied β-Ga 2 O 3 samples. 31) This means that their concentrations are well above 10 18 cm −3 . However, the relative differences between the differently doped materials and the nontrivial behavior of the positron lifetime as a function of temperature [see Ref. 31)] indicate that all of the samples contain more than just one type of split Ga vacancy defect at significant concentrations. Based on the positron data, it is highly likely that the in-grown split Ga vacancies are present in the form of different types of complexes with O vacancies and hydrogen, in addition to the different configurations of the "clean" split Ga vacancy. [11][12][13][14] Figure 5 shows the (S, W) parameters in selected samples measured along different crystal orientations: Fe-doped and Sn-doped single crystals grown by EFG (edge-defined filmfed growth) and a Mg-doped single crystal grown by the Czochralski method. 31) The surface orientation of the EFG samples is (001) and the Czochralski crystal is (100). The data have been adjusted for the different detector resolutions and measurement geometries (by measuring some of the crystals with several setups) so that they can be shown in the same figure. To highlight the evolution of the anisotropy in the (S, W) data, points measured between the high-symmetry crystal orientations in the Mg-doped Czochralski crystal are also shown. 17) As pointed out above, the data cannot be normalized to a well-defined reference point. Hence Fig. 5 shows the "raw" S and W parameters. The purple, black and green bars in the figure represent the calculated overall anisotropy of a typical split Ga vacancy, the β-Ga 2 O 3 lattice, and an unrelaxed Ga vacancy, respectively. 17) It is clear from the figure that the experimental anisotropy is significantly larger than that calculated for the β-Ga 2 O 3 lattice (or the unrelaxed Ga vacancies), irrespective of the sample. This is a strong indication of the experimental signals being dominated by positrons annihilating as trapped at split Ga vacancies. Interestingly, preliminary results in the ion irradiated Fedoped EFG crystal suggest that the overall anisotropy is strongly reduced. In combination with the positron lifetime obtained in the crystal (see Fig. 3) it is likely that this crystal contains unrelaxed Ga vacancies.
The fact that EPR experiments have only been reported for irradiation or high temperature annealing induced defects in β-Ga 2 O 3 supports the interpretation that the in-grown Ga vacancies are present as complexes rather than isolated. 16,[36][37][38][39] The EPR experiments appear to agree that these defects are in the doubly negative charge state, hence acting as compensating centers for n-type conductivity. There is an ongoing debate on whether the EPR signals originate from unrelaxed or split Ga vacancies, but comparing experiments with state-of-the-art theoretical calculations suggests that the irradiation-induced Ga vacancies are in the split configuration. 37) IR absorption spectroscopy is extremely powerful in identifying hydrogen-related defect centers, and has been applied to β-Ga 2 O 3 in recent years. 15,[40][41][42][43] The experiments are typically performed on samples with hydrogen and/or deuterium introduced by diffusion or ion implantation. The observed vibrational modes are consistent with split Ga vacancy configurations complexed with 1-3 hydrogen or deuterium atoms, demonstrating the strong propensity of these defects to form complexes. Also split Ga vacancy-O vacancy complexes have been observed. 41) The IR experiments fully support the interpretation that in-grown Ga vacancy defects are predominantly present in the split configuration. We note that the split Ga vacancies have also been observed by scanning transmission electron microscopy (STEM), also supporting their presence at high concentrations. 18) Combining the findings of positron annihilation spectroscopy, STEM, EPR, and IR absorption strongly suggests that in-grown Ga vacancies in β-Ga 2 O 3 are present in the split Ga vacancy configuration at high concentrations, well above 10 18 cm −3 . The experimental evidence also suggests that these split Ga vacancies are present as a distribution of various configurations, most likely complexed with O vacancies and hydrogen. This observation also offers a possible solution to the question of compensation, that is, why the n-type doping of β-Ga 2 O 3 works so well even if the Ga vacancy concentrations are extremely high. Many of the split Ga vacancy-O vacancy-(multi-)hydrogen configurations are predicted to be neutral for a large part of the band-gap, 14) and these will not cause compensation. However, the exact structure of the dominant native defects remains to be solved and is the The Japan Society of Applied Physics by IOP Publishing Ltd subject of future work. It should be noted that it is still uncertain whether irradiation-induced Ga vacancies could also be in the unrelaxed configuration, as suggested by EPR and positron experiments. 32,36)

Summary
Point defects in β-Ga 2 O 3 have been studied in recent years by several experimental techniques. Positron annihilation spectroscopy, a powerful method for identifying and quantifying vacancy-type defects, faces challenges with β-Ga 2 O 3 due to the colossal signal anisotropy and lack of a proper reference sample. Nevertheless, the data can be interpreted as the signals being dominated by split Ga vacancy defects in all studied β-Ga 2 O 3 samples. Theory predicts that the split Ga vacancy configurations have the lowest formation energies, and EPR, IR and STEM experiments all agree on the split Ga vacancy configuration as being the most likely interpretation. The exact structure of the split Ga vacancies remains to be solved in terms of complexes with O vacancies and hydrogen.