Relation between Ga Vacancies, Photoluminescence, and Growth Conditions of MOVPE-Prepared GaN Layers

A set of GaN layers prepared by metalorganic vapor phase epitaxy under different technological conditions (growth temperature carrier gas type and Ga precursor) were investigated using variable energy positron annihilation spectroscopy (VEPAS) to find a link between technological conditions, GaN layer properties, and the concentration of gallium vacancies (VGa). Different correlations between technological parameters and VGa concentration were observed for layers grown from triethyl gallium (TEGa) and trimethyl gallium (TMGa) precursors. In case of TEGa, the formation of VGa was significantly influenced by the type of reactor atmosphere (N2 or H2), while no similar behaviour was observed for growth from TMGa. VGa formation was suppressed with increasing temperature for growth from TEGa. On the contrary, enhancement of VGa concentration was observed for growth from TMGa, with cluster formation for the highest temperature of 1100 °C. From the correlation of photoluminescence results with VGa concentration determined by VEPAS, it can be concluded that yellow band luminescence in GaN is likely not connected with VGa; additionally, increased VGa concentration enhances excitonic luminescence. The probable explanation is that VGa prevent the formation of some other highly efficient nonradiative defects. Possible types of such defects are suggested.


Introduction
Optimization of methods for high-quality GaN growth in the 1990s paved the way for GaN-based power-efficient optoelectronic devices [1]. Highly efficient InGaN blue and white LEDs are now commercially available [1]. Thanks to the high polarization coefficient and ability to form heterostructures, AlGaN/GaN high electron mobility transistors can be manufactured, and these can be found in many applications these days [2]. Other applications of GaN-based devices include UV detectors [3], photovoltaic cells [4], and scintillators [5]. Despite extensive research on both the application and basic properties of GaN and related materials in the past three decades, several fundamental questions remain unanswered. One of them is the role or even the presence of Ga vacancies in GaN.
Defects related to Ga vacancies (V Ga ) in GaN are extremely important for optoelectronic applications since they are believed to be the main cause for the nonradiative recombination of carriers [6]. Other scholars have proposed that V Ga are responsible for yellow band luminescence and form nonradiative centres if they are in a complex with donors [7][8][9]. V Ga -related defects deteriorate the frequency properties of high electron mobility transistors (HEMTs), since trapping and releasing a charge in deep levels is a rather slow process, decreasing the cut-off frequency. V Ga can also decrease electron channel mobility. Another problem of these defects is shortening of the device lifetime, since vacancies enhance the diffusion of atoms in the structure, which can lead to the decomposition of quantum wells (QWs) [10] or the back-diffusion of Mg atoms from the GaN capping layer through the AlGaN barrier [11,12] to the channel region in HEMT structures. Although V Ga and their complexes are very important defects in nitrides, there is a lack of knowledge about the link between formation of these defects and the technological parameters of the preparation of high-quality epitaxial layers by metalorganic vapor phase epitaxy (MOVPE), as well as the influence of vacancies on semiconductor properties.

Materials and Methods
The studied samples were prepared by Aixtron CCS 3 × 2 MOVPE apparatus with dynamic adjustment of showerhead height. The following precursors were used for epitaxy: trimethylgallium (TMGa), triethylgallium (TEGa), and ammonia (NH 3 ). The set of prepared samples is specified in Table 1. Samples were prepared with the aim of finding a link between vacancy formation and some MOVPE technological parameters, namely temperature, type of carrier gas, and type of precursor. Growth of samples was always repeated in both atmospheres, H 2 or N 2 . The growth temperature was changed from 850 • C to 1100 • C. TEGa was used for the growth in the temperature range of 850-950 • C; growth at higher temperatures with this precursor was not possible due to strong adduct formation. TMGa was used at the higher temperature range of 950-1100 • C. Growth at lower temperatures from TMGa was avoided due to strong carbon incorporation. Four samples, TEN3, TEH3, TMN1, and TMH1 were prepared at the same temperature (950 • C) to compare only the influence of precursor type. All samples had a 1 µm thick GaN layer prepared with relevant technological parameters. Their simple structure is shown in Figure 1a. Positron annihilation spectroscopy (PAS) [13,14] is a well-established technique with high sensitivity to vacancies in solids. Positrons emitted by β+ radioisotopes have a continuous energy spectrum with a mean value in the order of hundreds keV. These energetic positrons are therefore called 'fast positrons'. The mean penetration depth of fast posi-  Positron annihilation spectroscopy (PAS) [13,14] is a well-established technique with high sensitivity to vacancies in solids. Positrons emitted by β+ radioisotopes have a continuous energy spectrum with a mean value in the order of hundreds keV. These energetic positrons are therefore called 'fast positrons'. The mean penetration depth of fast positrons into solids is in the order of 10 2 -10 3 µm depending on the material density. Consequently, fast positrons can be used for investigations of bulk samples only. For defect studies of thin films, it is necessary to decrease positron energy. This can be achieved by the moderation of fast positrons. A material with negative positron work function [15] can be used as a moderator. A fraction of fast positrons stopped in the moderator diffuses to its surface and escapes into the vacuum due to the negative positron work function. These so called 'slow positrons' are monoenergetic, with an energy of a few eV [16]. Slow positrons are collected in a slow positron beam and guided onto the sample. The energy of slow positrons in the beam can be increased using an electrostatic accelerator. Higher positron energy leads to higher positron penetration depth [16]. Hence, using a variable-energy slow positron beam one can probe the sample from the surface down to various depths. As an example, Figure 2 shows the implantation profiles of positrons with various energies implanted into GaN. The positron implantation profile is well-described by the Makhovian profile [17]. One can see in Figure 2 that varying the energy of slow positrons from 0.5 up to 12 keV enables variation in the mean positron penetration depth into GaN from to 2 to 347 nm. VEPAS is thus well-suited for defect studies of thin films or layered structure. The thickness of GaN films prepared for VEPAS studies is 1 µm (see Figure 1a). Hence, slow positrons are completely stopped inside the GaN layer even for the highest implantation energy of 12 keV used in measurement. the moderation of fast positrons. A material with nega be used as a moderator. A fraction of fast positrons st its surface and escapes into the vacuum due to the neg so called 'slow positrons' are monoenergetic, with an trons are collected in a slow positron beam and guid slow positrons in the beam can be increased using an e itron energy leads to higher positron penetration dep ergy slow positron beam one can probe the sample depths. As an example, Figure 2 shows the implantatio energies implanted into GaN. The positron implantat Makhovian profile [17]. One can see in Figure 2 that v from 0.5 up to 12 keV enables variation in the mean po from to 2 to 347 nm. VEPAS is thus well-suited for de structure. The thickness of GaN films prepared for VE Hence, slow positrons are completely stopped inside implantation energy of 12 keV used in measurement.  Measurement of a positron's lifetime requires a start signal providing a time stamp when a positron was born and a stop signal when the positron was annihilated. The stop signal is given by a 511 keV gamma ray emitted during positron annihilation. In order to obtain the start signal, the positron beam has to be pulsed, i.e., positrons are allowed to appear only in narrow time intervals, providing the start signal. The efficiency of positron moderation is very low (≤5 × 10 −3 ) [16] and pulsing of the slow positron beam leads to further reduction in intensity. As a consequence, a pulsed variable-energy slow positron beam requires a strong positron source.
VEPAS studies in the present paper were carried out on a LINAC-based pulsed variable energy slow positron beam MePS [18] operating at the ELBE facility [19] at Helmholtz-Zentrum Dresden-Rossendorf. Primary fast positrons were created by pair production induced by electron bremsstrahlung radiation. A tungsten moderator was used to obtain slow positrons. A high-intensity LINAC electron beam (average electron beam current 1.6 mA, electron energy 35 MeV) enabled achieving a high intensity of fast positrons necessary for a pulsed slow positron beam. Moreover, narrow electron pulses of ELBE LINAC (width smaller than 10 ps) were used as start signals for precise measurement of positron lifetimes. The annihilation gamma rays providing stop signals were measured using a CeBr 3 scintillation detector with digital signal processing. The time resolution of the spectrometer (FWHM of the resolution function) was 230 ps. The energy of slow positrons could be varied in the range from 0.5 to 12 keV.
Photoluminescence (PL) spectra of samples were collected using 325 nm excitation wavelength and a confocal microscope (LabRAM HR Evolution HORIBA, He-Cd laser, objective 74 CG, detector Synapse UV).
Two samples were also prepared for secondary ion mass spectroscopy (SIMS) measurement to study the connection between the contamination of layers and the technology of their preparation (see Figure 1b). These samples contained GaN layers with relevant technological parameters, with a thickness of 50 nm separated by 10 nm thick AlGaN layers that served as markers. All examined layers of the first and second sample were grown in H 2 and N 2 atmospheres, respectively. The concentrations of common contaminants (C, O, H, Si) were measured in EAG laboratories [20].

Variable Energy Positron Annihilation Spectroscopy
A positron lifetime spectrum is in general a sum of exponential components convoluted with the response function of the spectrometer. Each component corresponds to a certain positron state and is characterized by its lifetime τ i and relative intensity I i . positrons are annihilated in the GaN layer. The mean positron lifetime values for energies above 8 keV were, therefore, supposed to be relevant for determination of defect type and concentration.
From inspection of Figure 4, one can conclude that the mean positron lifetime at high energies (E > 8 keV) approaches values located between the calculated bulk positron lifetime (i.e., lifetime of free positrons in a defect-free lattice) and the calculated lifetime of positrons trapped in V Ga . This indicates that GaN films contain V Ga , and some positrons are trapped and annihilated in V Ga while the remaining positrons are annihilated in the free state, i.e., not trapped at defects. The only exception comprises TMN3 and TMH3 samples grown at the highest temperature of 1100 • C. In TMN3 and TMH3, the mean positron lifetime at high energies approached a value that was higher than the calculated lifetime for V Ga . This indicates that TMN3 and TMH3 contain defects with a higher open volume than V Ga .     as for TEH1 were also obtained for other GaN films deposited at temperatures below 1100 • C. The results of the decomposition of positron lifetime spectra for the TMH3 sample deposited at 1100 • C are plotted in Figure 6. In contrast to the films deposited at lower temperatures, the lifetime τ 2 for the TMH3 sample at high energies approaches a significantly higher value, corresponding to the calculated lifetime of positrons trapped in a complex consisting of two Ga and two N missing ions (2V Ga +2V N complex). A similar result was obtained for TMN3 film deposited at 1100 • C using N 2 carrying gas (see Supplementary Materials). The development of the lifetime τ 2 measured at the highest positron energy of 12 keV on different GaN film deposition temperatures is plotted in Figure 7a. For deposition temperatures lower than 1100 • C, the lifetime τ 2 remains close to the calculated value for V Ga . However, for films deposited at 1100 • C the lifetime τ 2 increases to the value corresponding to a 2V Ga + 2V N complex. Thus, one can conclude that GaN films grown at temperatures below 1100 • C contain V Ga . However, at higher temperatures the nature of the defects changes and films grown at 1100 • C contain 2V Ga + 2V N complexes.         Table 1. Labels indicate notation of sample and dominant type of defects in the sample. Solid lines are plotted to guide eyes only.
The V Ga concentration was calculated from the lifetimes and intensities of both components using the two-state trapping model [21]. Data measured for the highest positron energy of 12 keV corresponding to the highest positron penetration depth were used for determination of the vacancy concentration for all samples. Results for all samples are summarized in Figure 7b. V Ga formation is not only influenced by the temperature, but it is also strongly dependent on the choices of precursor and carrier gas. The temperature dependence of V Ga concentration for samples grown from TEGa and TMGa precursors have opposite characters. Unfortunately, there was not sufficient overlap of temperature intervals in which both precursors could be used for epitaxy; only around 950 • C could both types of precursors be used. TEGa is preferred for temperatures below 950 • C, since at this temperature range TMGa is less decomposed and too much carbon would be incorporated into the layers. At temperatures above 950 • C, TEGa cannot be used due to adduct formation, which impacts the growth rate.
Very characteristic of layers grown from TEGa is that the measured V Ga concentration is significantly higher in a nitrogen atmosphere compared to samples grown in hydrogen. In the case of TEGa grown in nitrogen, V Ga concentration decreases with temperature. This is unexpected from a thermodynamic point of view, since the vacancy concentration should be increasing with temperature as exp (-E f /kT), where E f is the vacancy formation energy [22]. Several possible hypotheses explaining this behaviour can be suggested. Since the TEGa precursor should be fully decomposed in the studied temperature range [23], limited Ga precursor decomposition can be ruled out. One of the possible explanations is the influence of surface morphology on vacancy formation, because in a N 2 atmosphere the epitaxial surface is rather rough (V-shape defects are formed). A lower V Ga concentration in a H 2 atmosphere can be due to passivation or filling of V Ga by hydrogen incorporated in the GaN lattice. In complexes of V Ga associated with H atoms positrons cannot be trapped or their lifetime is much shorter. For comparison, the lifetime of a positron trapped in a V Ga is 200-211 ps, while positron lifetime for V Ga -H complexes is shortened to 180 ps and for V Ga -2H it is further shortened to 150 ps, so it is even shorter than 155 ps, which is the bulk positron lifetime for GaN, i.e., lifetime of free positrons in a perfect (defect-free) GaN crystal [24].
When TMGa is used as the precursor, V Ga concentration is not influenced by the choice of atmosphere in the MOVPE reactor, but it is strongly enhanced with increasing temperature. The lifetime corresponding to a cluster 2V Ga -2V N was measured for the highest growth temperature of 1100 • C, which means that the actual V Ga concentration was twice the concentration of 2V Ga -2V N complexes shown in Figure 7b, since each complex contains two V Ga . Significantly different V Ga concentration when TMGa or TEGa was used for the GaN growth suggests that the reaction pathway has important influence on the vacancy formation. This could be a reason why theoretical predictions of V Ga concentration based on formation energy [25] are underestimated. In the future, layers grown from novel precursors and the vacancy formation in them might help to understand the growth kinetics, as suggested in [26,27].

Photoluminescence
Knowing the V Ga concentration in particular GaN layers, we investigated its correlation with PL properties. It could bring valuable insight into the quest for yellow band (YB) luminescence origin. In previous works, YB luminescence was attributed to a C N defect [28], C N -O N complex [29], which is energetically favourable, or C N -Si Ga [30] in n-type GaN layers. Other sources of YB luminescence that were proposed in the literature were V Ga -3H and V Ga -O N -2H complexes [31] or V Ga -O N [32]. V Ga were also supposed to be responsible for nonradiative recombination [6].
Typical room-temperature PL spectra for MOVPE-prepared GaN layers (samples TEN1 and TEH1) are shown in Figure 8 to illustrate the position of PL bands. Contrary to expectations, for sample TEN1, with a higher V Ga concentration, neither an increase in YB intensity nor a decrease of PL efficiency was observed.  Figure 9a,b, correlation between excitonic and YB intens concentration is shown, summarizing the data obtained from al lation between VGa concentration and YB luminescence intens suggests that VGa are not the dominant reason for YB luminesce In Figure 9a,b, correlation between excitonic and YB intensity with respect to the V Ga concentration is shown, summarizing the data obtained from all studied layers. No correlation between V Ga concentration and YB luminescence intensity was observed, which suggests that V Ga are not the dominant reason for YB luminescence.

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In Figure 9a,b, correlation between excitonic and YB intensity with respect to the VGa concentration is shown, summarizing the data obtained from all studied layers. No correlation between VGa concentration and YB luminescence intensity was observed, which suggests that VGa are not the dominant reason for YB luminescence. However, there is unexpected and clear positive correlation between VGa concentration and excitonic luminescence. It means that VGa cannot be efficient nonradiative centre. Since there is no known mechanism in which VGa presence would enhance free-exciton emission in GaN, we suggest the following explanation for the correlation observed in Figure 9a: higher exciton intensity is caused by the suppression of other recombination channels, especially the nonradiative ones (because enhancement of yellow luminescence was not observed with increasing VGa concentration, Figure 9b). Correlation between VGa concentration and GaN exciton intensity means that there exists an anticorrelation between nonradiative centre concentration and VGa concentration. In other words, the formation of an unknown nonradiative centrum is suppressed when the Ga vacancies are present. However, there is unexpected and clear positive correlation between V Ga concentration and excitonic luminescence. It means that V Ga cannot be efficient nonradiative centre. Since there is no known mechanism in which V Ga presence would enhance free-exciton emission in GaN, we suggest the following explanation for the correlation observed in Figure 9a: higher exciton intensity is caused by the suppression of other recombination channels, especially the nonradiative ones (because enhancement of yellow luminescence was not observed with increasing V Ga concentration, Figure 9b). Correlation between V Ga concentration and GaN exciton intensity means that there exists an anticorrelation between nonradiative centre concentration and V Ga concentration. In other words, the formation of an unknown nonradiative centrum is suppressed when the Ga vacancies are present.

Secondary Ion Mass Spectroscopy
To find possible candidates for nonradiative defects that are anticorrelated with V Ga and to elucidate whether V Ga form complexes with atoms from impurities, we prepared multilayer samples designed for SIMS characterization according to the scheme in Figure 1b. Hydrogen, oxygen, carbon, and silicon concentration profiles were measured by SIMS. There was one significant feature observed in all GaN layers-contamination from all four elements was higher when H 2 was used for the reactor atmosphere. Interesting correlations between V Ga and carbon for samples grown from TMGa, and between V Ga and hydrogen for samples grown from TEGa were observed. The obtained hydrogen and carbon concentrations for GaN layers prepared according to the above-defined technological parameters are summarized in Figure 10a,b.
In layers grown in a H 2 atmosphere at lower temperatures, hydrogen contamination was approximately three times higher than for samples grown in a N 2 atmosphere. We suppose that NH 3 is less efficiently decomposed in a H 2 atmosphere and hydrogen could be incorporated into the epitaxial layer. Hydrogen can form complexes with V Ga that are invisible to PAS methods, as was explained above. Since in samples grown from TEGa in a H 2 atmosphere the oxygen concentration measured by SIMS was in the order of 10 17 cm −3 (not shown here), we can also suppose the formation of V Ga -O N -2H, which is energetically favourable. In fact, if H and O are abundant, formation of V Ga -3H, V Ga -2H, or V Ga -O N -2H complexes is much more probable than formation of isolated V Ga . The smallest formation energy of V Ga is found when the Fermi level is near the conduction band minimum but still reaches a value around 4.5 eV. On the other hand, at the same Fermi level position, the formation energy of V Ga -3H, V Ga -2H, and V Ga -O N -2H is around 3 eV, 2.6 eV, and 0.5 eV, respectively [33]. Since none of these defects can be detected by VEPAS due to the small open volume of the defects, their presence may explain the significantly lower detected V Ga concentration in layers grown from TEGa in a H 2 atmosphere. Some of these defects probably also act as nonradiative centres [33]. and to elucidate whether VGa form complexes with atoms from impurities, we prepare multilayer samples designed for SIMS characterization according to the scheme in Figu 1b. Hydrogen, oxygen, carbon, and silicon concentration profiles were measured by SIM There was one significant feature observed in all GaN layers-contamination from all fou elements was higher when H2 was used for the reactor atmosphere. Interesting correl tions between VGa and carbon for samples grown from TMGa, and between VGa and hy drogen for samples grown from TEGa were observed. The obtained hydrogen and carbo concentrations for GaN layers prepared according to the above-defined technological p rameters are summarized in Figure 10a,b.
(a) (b) Figure 10. Concentration of (a) hydrogen and (b) carbon contamination measured by SIMS on laye prepared on multi-layered samples with technological parameters as defined in Table 1.
In layers grown in a H2 atmosphere at lower temperatures, hydrogen contaminatio was approximately three times higher than for samples grown in a N2 atmosphere. W suppose that NH3 is less efficiently decomposed in a H2 atmosphere and hydrogen coul be incorporated into the epitaxial layer. Hydrogen can form complexes with VGa that ar invisible to PAS methods, as was explained above. Since in samples grown from TEGa a H2 atmosphere the oxygen concentration measured by SIMS was in the order of 10 17 cm (not shown here), we can also suppose the formation of VGa-ON-2H, which is energetical favourable. In fact, if H and O are abundant, formation of VGa-3H, VGa-2H, or VGa-ON-2 complexes is much more probable than formation of isolated VGa. The smallest formatio energy of VGa is found when the Fermi level is near the conduction band minimum bu still reaches a value around 4.5 eV. On the other hand, at the same Fermi level positio the formation energy of VGa-3H, VGa-2H, and VGa-ON-2H is around 3 eV, 2.6 eV, and 0 eV, respectively [33]. Since none of these defects can be detected by VEPAS due to th small open volume of the defects, their presence may explain the significantly lower d tected VGa concentration in layers grown from TEGa in a H2 atmosphere. Some of thes defects probably also act as nonradiative centres [33].
However, in samples grown from TMGa, hydrogen does not play a significant rol Instead, carbon contamination seems to be anticorrelated with VGa concentration. It seem that for the lowest temperature of 950 °C, the carbon concentration is so high that it filling not only VN, but also VGa, thus forming n-type CGa defects. On the other hand, at th highest temperature, when there is low carbon contamination, both types of vacancie remain unfilled and their concentration is so high that they can merge into clusters, as wa observed at 1100 °C. According to the PL results, at the highest carbon concentration P efficiency drops significantly, suggesting the formation of strong nonradiative defect Figure 10. Concentration of (a) hydrogen and (b) carbon contamination measured by SIMS on layers prepared on multi-layered samples with technological parameters as defined in Table 1. However, in samples grown from TMGa, hydrogen does not play a significant role. Instead, carbon contamination seems to be anticorrelated with V Ga concentration. It seems that for the lowest temperature of 950 • C, the carbon concentration is so high that it is filling not only V N , but also V Ga , thus forming n-type C Ga defects. On the other hand, at the highest temperature, when there is low carbon contamination, both types of vacancies remain unfilled and their concentration is so high that they can merge into clusters, as was observed at 1100 • C. According to the PL results, at the highest carbon concentration PL efficiency drops significantly, suggesting the formation of strong nonradiative defects, which could be C Ga according to [34], or it may be due to some complex related to C Ga such as C Ga -C N or C Ga -H i .

Conclusions
Twelve samples with GaN layers prepared by MOVPE under different technological conditions were investigated by a unique VEPAS method to find a link between technological conditions and V Ga formation. Different correlations between V Ga concentration and technology parameters were observed for layers grown from TEGa and TMGa precursors. In the case of TEGa, V Ga formation was significantly influenced by the type of reactor atmosphere (N 2 or H 2 ), while no similar behaviour was observed for samples grown from TMGa. V Ga formation was suppressed with increasing temperature for growth from TEGa in a N 2 atmosphere. On the contrary, opposite temperature dependence for V Ga formation was observed for growth from TMGa, with cluster formation for the highest temperature of 1100 • C. Significantly different V Ga formation when TMGa or TEGa was used for the GaN growth suggests that the reaction mechanism has an important influence on formation of vacancies. This could be a reason why V Ga concentrations theoretically predicted from formation energies are underestimated. According to the correlation of PL results with V Ga concentrations determined by VEPAS, it can be concluded that YB luminescence is not generally connected with V Ga in MOVPE-prepared GaN layers. Surprisingly, the presence of V Ga was correlated with intensity of excitonic luminescence. The probable explanation is that the V Ga concentration in GaN layers is anticorrelated with some efficient nonradiative defects. According to SIMS results and correlation with detected V Ga concentrations, we have suggested several candidates for such nonradiative defects, which could be C Ga, C Ga -C N , or C Ga -H i in samples grown from TMGa and V Ga -3H, V Ga -2H, or V Ga -O N -2H complexes in samples grown from TEGa.
Supplementary Materials: The following supporting information can be downloaded at: https: //www.mdpi.com/article/10.3390/ma15196916/s1, Figure S1: Results of decompositions of positron lifetime spectra for the sample TEN1 plotted as a function of the energy of incident positrons. Figure S2: Results of decompositions of positron lifetime spectra for the sample TEN2 plotted as a function of the energy of incident positrons. Figure S3: Results of decompositions of positron lifetime spectra for the sample TEN3 plotted as a function of the energy of incident positrons. Figure S4: Results of decompositions of positron lifetime spectra for the sample TEH1 plotted as a function of the energy of incident positrons. Figure S5: Results of decompositions of positron lifetime spectra for the sample TEH2 plotted as a function of the energy of incident positrons. Figure S6: Results of decompositions of positron lifetime spectra for the sample TEH3 plotted as a function of the energy of incident positrons. Figure S7: Results of decompositions of positron lifetime spectra for the sample TMN1 plotted as a function of the energy of incident positrons. Figure S8: Results of decompositions of positron lifetime spectra for the sample TMN2 plotted as a function of the energy of incident positrons. Figure S9: Results of decompositions of positron lifetime spectra for the sample TMN3 plotted as a function of the energy of incident positrons. Figure S10: Results of decompositions of positron lifetime spectra for the sample TMH1 plotted as a function of the energy of incident positrons. Figure S11: Results of decompositions of positron lifetime spectra for the sample TMH2 plotted as a function of the energy of incident positrons. Figure S12: Results of decompositions of positron lifetime spectra for the sample TMH3 plotted as a function of the energy of incident positrons. References [35,36] are cited in the supplementary materials.