3.1. Effect of Mg Concentration on SI AT-GaN:Mg Electrical Properties
The results of SIMS measurements, performed on the Ga-face of the samples, are presented in
Figure 1. The depth of the crater sputtered by O
− and Cs
+ bombarding ions was up to 8 µm. The depth dependence of the secondary ion count was stabilized after a possible uncontrolled surface contamination due to sample polishing or other surface effects. The in-depth profiles of [Mg], [H], [O], [Zn], [Mn], [Si], [Na], [C], and [Fe] are presented and it is seen that the concentrations of various chemical elements are approximately uniform throughout the whole probed depth. According to the expectations, the predominant chemical element in the two kinds of samples is Mg, whose concentration is 6 × 10
18 cm
−3 in sample #1 and 2 × 10
19 cm
−3 in sample #2. Because of using a getter, the oxygen concentration is nearly the same in both samples, being at the level of ~1 × 10
18 cm
−3. The concentrations of hydrogen in the samples #1 and #2 are 2 × 10
18 cm
−3 and 7 × 10
18 cm
−3, respectively. This low hydrogen concentration means that if all hydrogen would form Mg-H complexes, only about 30% of Mg would have been passivated. However, the role of hydrogen has not yet been clearly explained yet. It is generally assumed that hydrogen is an amphoteric impurity in GaN, i.e., it can act both as a donor or an acceptor, depending on the Fermi level position [
32]. In addition, hydrogen may also form complexes with intrinsic or extrinsic defects. For example, Fourier transform infrared spectroscopy (FTIR) studies of SI AT-GaN:Mg revealed the lines characteristic of the
VGaH
3 complex. In the first approximation, we will neglect the influence of hydrogen on the electrical properties of samples #1 and #2 as we assume that the electrically active Mg concentration is high enough to decide the Fermi level position and hydrogen acts only as a small additive to the total donor/acceptor concentration. The concentrations of other residual elements such as Si, Mg, Mn, and Fe do not exceed ~1 × 10
18 cm
−3. The background concentrations of these elements (in particular Mn) are too low to affect the electrical parameters of the material. The [Mg], [O], and [H] concentration values and resistivities for the samples used in this study are summarized in
Table 1.
The samples whose properties and resistivities are listed in
Table 1 represent two kinds of SI AT-GaN:Mg crystals with a different magnesium concentration ([Mg]), the same oxygen concentration ([O]), and a different hydrogen concentration ([H]). It is worth noting that the threefold increase in [Mg] by a factor of 3.3 results in a decrease in the material resistivity by 5 orders of magnitude. According to high-temperature Hall effect measurements, both materials are
p-type and it can be assumed that the enormous drop in resistivity is due to the huge rise in the hole concentration. In other words, there is an additional reason leading to the very high resistivity of the material with the lower [Mg]. The big difference in the resistivity between the materials with various [Mg] strongly suggests that in the material with lower [Mg], additional deep-level defects are present, which, apart from the Mg
Ga acceptors, efficiently compensate the shallow O
N donors. These defects are likely to be deep acceptor centers having an energy level near the middle of the bandgap.
The above-mentioned suggestion is experimentally confirmed by the results of TDDC and mobility-lifetime product (
μτ) measurements presented in
Figure 2, in which the temperature dependences of dark conductivity activation energies (
Figure 2a) and the
μτ product (
Figure 2b), established for the two materials with various [Mg], are compared. From a physical point of view, the
EADC values given in
Figure 2a represent the extrapolated-to-absolute zero Fermi level positions in both kinds of materials [
33]. Since the materials are of
p-type, the Fermi energy values are with respect to the VBM. The results indicate that in the material with the [Mg] = 6 × 10
18 cm
−3 (sample #1), the role of magnesium in the charge compensation is very small. This is because the ionized acceptor concentration formed by the electrically active magnesium is too small compared to the ionized donor concentration formed by the electrically active oxygen. In the case of the latter, the ionization level of O
N is at
Ec—33 meV and at 300 K all the oxygen atoms, whose concentration is 1 × 10
18 cm
−3, are positively ionized [
34]. In the case of magnesium, the ionization process is more complex, since a part of the [Mg] is passivated by hydrogen. Assuming that 30% of the [H] given in
Table 1 contributes to the Mg passivation, which equals 9 × 10
17 cm
−3, the Mg concentration that can be electrically active is 5.1 × 10
18 cm
−3. Moreover, the Mg
Ga ionization level is relatively deeply located in the bandgap, at around
Ev + 200 meV, and at RT typically only a few percent of the non-passivated Mg
Ga is negatively ionized [
34]. Taking that this part is 10% [
34], the concentration of the negatively ionized acceptors Mg
Ga− in the material with the total [Mg] = 6 × 10
18 cm
−3 (sample #1) is 5.1 × 10
17 cm
−3. Thus, the O
N shallow donors in this material are predominantly compensated by deep acceptors having the ionization level located closely to the Fermi level position given by the
EADC =
Ev + 1690 meV.
In the material with the [Mg] = 2 × 10
19 cm
−3 (sample #2), the compensation mechanism is entirely different. This fact is indicated by the Fermi level position given by the
EADC =
Ev + 397 meV, which is much closer to the VBM. The significant shift of the Fermi level towards the VBM with increasing [Mg] seems to be due to three reasons. Firstly, the compensation is not affected by the very deep acceptors with the ionization level located in the middle of the bandgap. Secondly, the concentration of negatively ionized acceptors
exceeds the concentration of positively ionized shallow donors
. This fact can be confirmed by a rough calculation, taking into account that the [H] = 7 × 10
18 cm
−3 (see
Table 1), and assuming that 30% of the [H] takes part in the Mg passivation. In this way, the concentration of the non-passivated Mg that can be electrically active is 1.79 × 10
19 cm
−3, and taking into account that only 10% of these Mg atoms are ionized, the concentration of negatively ionized acceptors
is 1.79 × 10
18 cm
−3. Thirdly, the Fermi level in the material (sample #2) is located clearly above the Mg
Ga level and this fact indicates that other acceptors, with deeper ionization levels, are likely to be also involved in the compensation of the
. shallow donors.
The temperature dependences of the
μτ product presented in
Figure 2b are consistent with the Fermi energies given in
Figure 2a for both kinds of materials with the various [Mg]. It is seen that increasing the [Mg] from 6 × 10
18 cm
−3 (sample #1) to 2 × 10
19 cm
−3 (sample #2) and keeping the [O] = 1 × 10
18 cm
−3 leads to a substantial rise in the
μτ values determined in the temperature range of 300–700 K. However, for the latter material, the measurements were stopped at 600 K due to a high dark current. It is worth noting that at 300 K, the ratio of the
μτ values for samples #2 and #1 is around 4, but at 600 K, this ratio is around 20. In other words, the temperature increase in the
μτ is stronger for sample #2 than for sample #1. Since the
μτ product determines the quality of semiconductor material in terms of its application for the fabrication of various kinds of sensors, used as photodetectors, X-ray detectors, and radiation detectors, the results shown in
Figure 2b are of practical importance [
31,
35]. They demonstrate that by increasing the doping level in SI AT-GaN:Mg crystals, it is possible to obtain the material suitable for making unique detectors operating at 600 K. In view of the fact that above 300 K the mobility of charge carriers decreases with temperature due to the lattice scattering, the observed in
Figure 2b rise in the
μτ values as a function of temperature can only result from the strong increase in the excess charge carriers lifetime [
35]. The lifetime strongly depends on the properties and concentrations of deep-level defects acting as the recombination centers and the ionization levels of the most efficient recombination centers are located in the vicinity of the middle of the bandgap [
35]. According to the results shown in
Figure 2a, the efficient recombination centers are likely to be present only in the material with the lower [Mg] (sample #1). The observed changes in the lifetime against temperature reflect the changes in the charge carrier recombination rate, being equal to the lifetime reciprocal. In other words, for both kinds of materials, the recombination rate decreases with temperature, and in the case of deeper recombination centers (sample #1), the decrease is slower than for the centers whose ionization levels are located closer to the VBM or CBM [
35]. In view of the SRH recombination model, proposed by Shockley, Read, and Hall [
36,
37], the recombination takes place when an electron from the conduction band is captured by the empty defect level and then a hole from the valence band is captured by an electron occupying this level. With increasing temperature, the thermal emission rate of charge carriers rises exponentially and the probability of releasing the electron or hole from a defect center to the conduction or valence band becomes much higher than that of their capture by this defect center [
35].
3.2. Effect of Mg Concentration on the Properties and Concentrations of Deep Traps
In the LPITS method, the occupation of defect levels is changed by the capture of excess charge carriers excited by optical pulses and after the excitation is switched off, the photocurrent relaxation waveforms (PRWs) induced by the thermal emission of excess electrons or holes from the defect levels are produced [
31]. At a given temperature, the PRW is observed as a bottom part of the photocurrent decay, occurring in the photocurrent transient when the optical excitation is switched off. Usually, the PRW represents the slower part of the photocurrent decay that appears for times longer than ~1 μs from the termination of the optical pulse. The faster decay rate, which is observed for the shorter times, is related to the excess charge carrier recombination. Thus, the PRWs are the distinctive parts of the photocurrent transients and their analysis as a function of temperature enables the properties and concentrations of charge carrier traps to be determined [
31]. Most frequently, a PRW observed in the photocurrent transient recorded at a given temperature is composed of a few exponential signals related to the thermal emission of charge carriers, and an advanced numerical procedure is used to extract the time constant of each exponential signal [
31]. This procedure, based on the inverse Laplace transformation algorithm (ILT), uses the CONTIN code [
38] adapted to the commercially available MATLAB computational environment. Assuming that retrapping excess charge carriers is neglected, the time dependence of a PRW exponential component at a given temperature
T can be written as [
31]
where
t denotes time,
eT is the charge carrier thermal emission rate characteristic of the defect center that captured the carriers when the sample was illuminated, and
I(0) is the amplitude of the exponential signal when the optical excitation pulse is terminated. The
eT is equal to the reciprocal of the exponential signal time constant and the
I(0) can be expressed in the form [
31]
where
nT (0) is the concentration of electrons or holes trapped by the defect center at
t = 0 when the optical excitation pulse is switched off,
q is the elementary charge,
E is the electric field between two planar contacts dependent on the applied voltage and
C is the geometrical parameter equal to the area of the cross-section of a sample region through which the excess charge carriers emitted from the defect center flow to the electrodes on the sample surface. The temperature dependence of the excess charge carrier thermal emission rate is given by the Arrhenius equation [
31]
where
Ea is the activation energy,
kB is the Boltzmann constant,
A =
γσa is the pre-exponential factor, equal to the product of the material constant γ, dependent on the effective mass, and the apparent capture cross-section for electrons or holes
σa. Thus, each defect center is characterized by the activation energy
Ea for electron or hole thermal emission and pre-exponential factor
A. For GaN with wurtzite structure, the values of
γn and
γp, necessary for obtaining the apparent capture cross-sections for electrons or holes, are 6.48 × 10
20 and 6.33 × 10
21 K
−2 cm
−2 s
−1, respectively.
To extract the parameters
Ea and
A, characterizing defect center properties, the PRWs recorded in the time domain at temperatures
Tj (j = 1, 2, 3, ……) within the range of 300–700 K, were transformed into the one-dimensional (1D) Laplace spectra
SLj in the domain of the thermal emission rate in which the sharp peaks indicated the
eT values characteristic of the defect centers detected at each temperature. Next, all the 1D Laplace spectra were assembled to create the (2D) Laplace spectrum which visualized in the 3D space contained the sharp folds whose ridgelines depicted the temperature dependences of the emission rate for the defect centers that had trapped the charge carriers during the sample illumination [
31]. The
T and
eT data derived from these ridgelines were used to draw the Arrhenius plots in coordinates ln(
T2/
eT) = f(1/
kBT), being the signatures of defect centers. For each trap, the activation energy
Ea and the pre-exponential factor A in the Arrhenius equation were determined from the slope and intercept of the Arrhenius plot by means of linear regression. The concentration (
NT) of a defect center was determined from the amplitude of the exponential component of the PRW observed at a given temperature. According to Equation (2), the amplitude
I(0) is proportional to the concentration
nT(0) of charge carriers trapped by the defect center when the optical excitation pulse is switched off. To find the experimental values of
I(0) for the exponential signals resulting from the analysis of the PRW measured at a given temperature, the waveform was fitted with the sum of exponential signals in the form of Equation (1), whose number and time constants were derived from the 1D Laplace spectrum. More details on extracting the trap concentration from LPITS measurements is given elsewhere [
31].
The characteristics of charge carrier traps determined by the LPITS for two kinds of SI AT-GaN:Mg crystals with the [Mg] = 6 × 10
18 cm
−3 (sample #1) and [Mg] = 2 × 10
19 cm
−3 are shown in
Figure 3. The results were obtained by applying the CONTIN procedure to the analysis of the sets of the photocurrent transients generated in the samples of these crystals at temperatures 300–700 K.
Figure 3a,b shows the one-dimensional Laplace spectra indicating the presence of three exponential components related to the thermal emission of charge carriers in the photocurrent relaxation waveforms observed for the samples #1 and #2 at temperatures 309 and 325 K, respectively. In addition,
Figure 3a contains the one-dimensional Laplace spectrum indicating the thermal emission of charge carriers from deep-level defects at 675 K. It is worth noting that the three traps, which manifest themselves through the Laplace peaks labeled as T1, T2, and T3, are present in both kinds of crystals. The fourth trap, observed at 675 K through the Laplace peak labeled as T4, is only detected in the material with the [Mg] = 6 × 10
18 cm
−3.
According to the data shown in
Figure 3a,b, the thermal emission rates at 309 K for the T1, T2, and T3 traps are 2512, 479, and 145 s
−1, respectively, and at 325 K are 5129, 1072, and 339 s
−1, respectively. For trap T4, present only in the material with the lower [Mg], the thermal emission rate of charge carriers at 675 K is 19 s
−1. The dependence of the thermal emission rate reciprocal as a function of the thermal energy reciprocal for this trap is shown in
Figure 3c. The dependences of the thermal emission rate reciprocals plotted against the thermal energy reciprocal for the T1, T2, and T3 traps are shown in
Figure 3d.
Each Arrhenius plot is the trap signature and is used to determine the parameters of the Arrhenius equation characterizing the temperature dependences of the thermal emission rate of electrons or holes. The slope of the plot allows finding the activation energy
Ea for the thermal emission of charge carriers from the trap and the intercept gives the pre-exponential factor A in the Arrhenius equation. The
Ea value for the T4 trap is given in
Figure 3c and the
Ea values for the T1, T2, and T3 traps are shown in
Figure 3d. All the deep trap parameters extracted from the Arrhenius plots presented in
Figure 3c,d as well as the trap concentrations estimated from the amplitudes of the exponential signals found in the photocurrent relaxation waveforms recorded for both kinds of samples of SI AT-GaN:Mg crystals are listed in
Table 2.
The LPITS results summarized in
Table 2 reveal the qualitative and quantitative changes in the defect structure of SI AT-GaN:Mg crystals induced by increasing the [Mg] from 6 × 10
18 to 2 × 10
19 cm
−3. The qualitative result is that the T4 trap is present only in the material with the lower [Mg]. Moreover, the activation energy value of this trap is close to the Fermi energy value of
Ev + 1690 meV derived from the TDDC measurements. Quantitatively, the results in
Table 2 indicate that the T4 trap predominates in the material with the lower [Mg] and its concentration is near that of the T1 trap in the material with the higher [Mg] in which this trap is predominant. On the other hand, the T1 trap concentration in the former is the lowest compared to the concentrations of traps T2 and T3. It is worth noting that the T2 and T3 trap concentrations in the latter material are by the order of magnitude higher than that in the former, but in both materials, the inequality [T2] > [T3] is fulfilled. Moreover, a fact worth emphasizing is that the activation energy of the T1 trap, which is predominant in the material with the higher [Mg], is close to the Fermi energy value of
Ev + 397 meV obtained for the material by the TDDC measurements. It should be added that the LPITS measurements do not allow us to distinguish directly between the electron or hole traps. However, by using both parameters
Ea and
A, the properties of traps detected by LPITS with those found by DLTS can be indirectly compared.
The parameters determined from the Arrhenius plot for the T1 trap well match those for the hole trap DP1, with the activation energy and pre-exponential factor of
Ev + 484 meV and 1.77 × 10
4 K
−2s
−1, respectively, detected by the current transient method in the
p-type GaN film playing role a gate in a high-electron-mobility transistor (HEMT) [
39]. This trap, similarly to the other hole traps found in a
p-GaN HEMT and
p-GaN Schottky diodes with the
Ea of 480–490 meV and
A in the range of 10
3–10
4 K
−2s
−1, was suggested to be attributed to nitrogen vacancies [
40]. A trap with parameters close to those of the T3 trap is the hole trap EHa revealed by DLTS in homoepitaxial
p-type GaN with the Mg concentration of 8 × 10
15 cm
−3 after the irradiation with 137-keV electrons to produce nitrogen atoms displacement [
24]. The
p+/
p−/
n+ GaN epitaxial structures exposed to the electron bombardment were grown by metalorganic vapor phase epitaxy (MOVPE) on freestanding GaN substrates. The activation energy and pre-exponential factor derived from the Arrhenius plot for the EHa trap are
Ev + 520 meV and 1.9 × 10
6 K
−2s
−1, respectively [
24]. On the grounds of the first-principle calculations based on a hybrid functional, this trap was proposed to be attributed to
VN (3+/+) or N
i(2+/+). It is worth adding that with increasing the electron fluence from 4.6 × 10
15 to 1.9 × 10
16 cm
−2, the EHa trap concentration went up from 4.0 × 10
14 to 2.3 × 10
15 cm
−3, respectively [
24]. In turn, the T2 trap parameter values are comparable with the
Ea =
Ev + 480 meV and
A = 2.3 × 10
5 K
−2s
−1 determined by DLTS for a Mg-doped
p-type GaN epitaxial layer exposed to the irradiation with 1.8-MeV protons [
41]. The layer with a [Mg] = 2 × 10
17 cm
−3 was grown by ammonia-based molecular beam epitaxy [
41]. Before the irradiation, the
Ev + 480-meV trap concentration in the material was ~10
13 cm
−3, and after irradiations with proton fluences of 1 × 10
13 and 3 × 10
13 cm
−2, its concentration increased to 3.8 × 10
14 and 6.4 × 10
14 cm
−3, respectively [
41]. Finally, the T4 trap with parameters listed in
Table 2 is likely to be related to the same defect as the hole trap H5 detected by minority carrier transient spectroscopy (MCTS) in samples of
n-type GaN grown by MOCVD on free-standing
n+-GaN substrates [
42]. The values of the
Ea and
A parameters for this trap derived from the Arrhenius plot are
Ev + 1760 meV and 7.6 × 10
9 K
−2s
−1, respectively [
42]. The lower activation energy for the hole thermal emission from the trap H5 and higher pre-exponential factor
A, proportional to the hole capture cross-section, can result from the substantially higher electric field strength under the MCTS experiment than that under the LPITS measurements [
42]. In the case of the former, the trap was probed in the space charge region of a
p+-
n junction where the electric field is of the order of 10
6 V/cm. In the case of the latter, the hole thermal emission from the trap T4 was induced in the homogeneous SI GaN sample at the electric field of ~286 V/cm. It is worth adding that the red PL bands, with peak positions around 1.8 eV, have been observed in both
p- and
n-type GaN, but their origins remain unknown [
19,
43].
3.3. Identification of Detected Traps Based on HSE-Hybrid Functional Calculations
The comparison of the T1–T4 trap parameters given in
Table 2 with those of derived from the Arrhenius plots determined by other methods used to observe and analyze the thermal emission of charge carriers in
p-type GaN allows us to conclude that the traps detected by LPITS are hole traps and their activation energies refer to the VBM. The next step in their identification is to propose the atomic configurations, as well as the charge state changes associated with the thermal emission of holes, and to establish whether they are donors or acceptors. All these facts can be found from the results of simulating the formation energy and properties of native defects in GaN obtained by the ab initio calculations made using the Heyd, Scusseria, and Ernzherof (HSE) range-separated hybrid functional as well as the projector-augmented wave (PAW) formalism implemented in the Vienna Ab-initio Simulation Package (VASP) code [
15,
16]. The recent results of these calculations are sufficiently unambiguous to be used for the comparison of the activation energies of experimentally detected deep traps with the known transition levels of native defects in GaN. The electronic properties of native point defects identified in both kinds of samples of SI AT-GaN:Mg crystals on the grounds of the experimental results obtained by LPITS measurements are summarized in
Table 3. The comparison given in
Table 3 indicates that the experimentally determined activation energies for hole emission from the T1, T2, and T3 defect states related to the nitrogen displacement from the substitutional sites are within 40 meV lower than the energies calculated by using the HSE hybrid functional corresponding to the defects charge state transitions. This fact can be accounted for by assuming that the defects, whose activation energies with an accuracy of about ±10 meV are derived from the LPITS measurements, are influenced by a deformation potential induced by the local strain introduced into the lattice of SI AT-GaN:Mg crystals due to the size mismatch between Mg and Ga atoms [
44]. The atomic radii of Mg and Ga are 140 and 123 pm, respectively, and the large difference between them results in two effects: a size effect, related to the lattice deformation and an electronic effect, related to the creation of the deformation potential [
44,
45]. As far as the size effect is concerned, the calculations performed by Van de Walle [
44] indicate that the nitrogen atoms in the first coordination sphere surrounding the
relax outwards by 6.1% of the bulk Ga-N bond length in the direction parallel to the
c axis. On the other hand, in the second coordination sphere, the nitrogen atoms are compressed due to a small atomic radius (71 pm) and can be easily removed from the substitutional to interstitial positions by the elastic force [
44,
45]. In other words, a change in the strain state from compressive to tensile is likely to be accompanied by the generation of nitrogen interstitials and nitrogen vacancies [
43,
44]. There are also two experimental facts confirming this change [
43]. The first is obtained by Raman spectroscopy and shows that the strain-sensitive E
2(high) mode of the Raman shift is monotonically reduced for Mg-concentrations higher than 7 × 10
18 cm
−3 [
43]. With increasing the [Mg] in the material, the compressive strain goes down, and for [Mg] as high as 2 × 10
19 cm
−3 no symmetry-forbidden modes are observed [
43]. The second is derived from the photoluminescence measurements indicating that the change in strain state is correlated with the change in optical spectra involving a saturation of the DAP luminescence and enhancing the signals related to the transitions induced by deep-level defects [
43]. The results allow us to conclude that the change in the strain state is accompanied by the change in the optical electron transitions, for the predominant contribution of the shallow states to the photon emission is replaced by the contribution of deep states [
43].
The T1 trap activation energy of 433 meV compares well with the
VN (3+/+) donor level, which according to the HSE calculations made by Lyons and Van de Walle [
16], is located at 450 meV above the VBM. It is worth adding that by means of HSE calculations with finite-size corrections, Yan et al. found the
VN (3+/+) level position at 470 meV above the VBM [
19]. In turn, the HSE calculations performed by Diallo and Demchenko [
15] gave the
VN (3+/+) level position at 540 meV above the VBM. These data illustrate that there is a spread in the values of the HSE-calculated transition energies. The discrepancies in the calculated transition levels can be due to the differences in the lattice relaxations of a defect, various supercells sizes, and the
k-point sampling methods, as well as due to the use of different electrostatic correction scheme [
15]. The possibility of the (3+/+) charge state change for the nitrogen vacancy is predicted by most HSE calculations [
15,
16]. This means that
VN exhibits the properties of a negative-
U center that being singly positively ionized needs to capture two holes to be transferred to the (3+) charge state and the binding energy of these holes is greater than in the case of the one-hole capture [
15,
16]. The experimental fact that the T1 trap concentration in the material with the [Mg] = 2 × 10
19 cm
−3 (
Table 2) is two times higher than that of the T2 trap, identified with the
VN (2+/+), supports assigning to the T1 trap the negative-U defect properties. Attributing the T1 trap to
VN (3+/+) means that during the sample illumination, this trap captures two holes and the charge released by the thermal emission when the illumination is switched off is two times larger than in the case of the T2 trap [
31]. Actually, both trap concentrations are the same; however, the amplitude of the exponential component of the photocurrent relaxation waveform, which is used to determine the trap concentrations, is doubled in the case of the thermal emission from the T1 trap [
31]. The values of the capture cross-section for holes (
Table 3) are also in line with the proposed identifications of T1 and T2 traps. For the former, the
σp value is nearly by the order of magnitude lower due to the stronger Coulomb repulsive force while capturing two holes simultaneously.
The trap T3 is proposed to be attributed to the N split interstitial N
i–N
i. The trap activation energy of 460 meV is close to the donor level related to the (2+/+) charge state change in this defect whose location in the bandgap calculated by Kyrtsos et al. is at
Ev + 500 meV [
17] and by Diallo and Demchenko at
Ev + 510 meV [
15]. It should be noted that the activation energy for the hole emission is likely to be diminished due to the deformation potential in the defect’s vicinity [
15,
44]. In the calculations, the initial N
i–N
i bond length is assumed to be 0.113 nm, which well matches the bond length of a free N
2 molecule (0.11 nm) [
15]. The HSE calculations show that the N split interstitial can capture one electron, becoming singly negatively ionized, or up to three holes, being in the (+), (2+), and (3+) charge states [
15]. When illuminating the SI AT-GaN:Mg sample during the LPITS experiment, the N
i–N
i (+) captures an excess hole and goes to the (2+) charge state. After switching off the optical excitation, the hole is thermally released to the valence band and the defect returns to the initial charge state. It is worth adding that the N
i–N
i bond distance is dependent on the charge state and for the extreme cases (−) and (3+) are 0.141 and 0.111 nm, respectively [
15].
The results listed in
Table 3 indicate that during the growth of SI AT-GaN:Mg crystals in N-rich conditions, the formation of nitrogen vacancies is correlated with the formation of nitrogen interstitials. In other words, the compressive strain induced by the large Mg atomic radius (0.140 nm) results in the formation of Frenkel pairs, which are two separated defects: a nitrogen vacancy and a nitrogen interstitial. Because of the small nitrogen atomic radius (0.071 nm), the latter is very mobile at the growth temperature of ~500 °C [
46]. Using density-functional total energy calculations, Wixom and Wright [
46] have shown that the N interstitial migration is very effective both perpendicular and parallel to the
c-axis, since there are low barriers for the defect migration in the all-positive charge states: (+), (2+), and (3+). According to the calculated results, the N interstitial diffusion length strongly increases with temperature and at ~500 °C is of the order of 1000 nm. On the other hand, the results shown in
Table 2 and
Table 3 indicate that in the SI AT-GaN:Mg samples with the Mg concentrations of 6 × 10
18 and 2 × 10
19 cm
−3 the concentrations of N interstitials are 3.6 × 10
17 and 3.6 × 10
18 cm
−3, respectively. These values allow us to conclude that the average distance between the defects in the materials with the lower and higher Mg doping level are 14 and 6.5 nm, respectively. Thus, the probability of the N
i–N
i split interstitials formation during the growth of SI AT-GaN:Mg crystals is very high. It should be stressed that for the material of each kind the [N
i] and [
VN] values are comparable and this fact supports the view that the Frenkel pairs are created during the growth of SI AT-GaN:Mg crystals.
To characterize the impact of the impurity concentration on the size effect, the coefficient
βsize which relates the fractional change in the lattice parameter (Δ
a/
a) to the impurity concentration was calculated [
44]. Thus, for Mg-doped GaN, the Δ
a/
a =
βsize × [Mg
Ga], where the
βsize = 1.5 × 10
−24 cm
3 [
44]. Consequently, for the Mg concentrations in the SI AT-GaN:Mg, crystals equal to 6 × 10
18 and 2 × 10
19 cm
−3, the Δ
a/
a values are 9 × 10
−6 and 3 × 10
−5, respectively. These positive changes in the lattice parameter should be compensated by the negative changes induced by the formation of N vacancies in the N-sublattice [
24,
43]. The -Δ
a/
a can be defined as [
VN]/[N
N], where [
VN] is the nitrogen-vacancy concentration and the [N
N] is the concentration of nitrogen atoms in the N-sublattice which is known to be equal to 4.45 × 10
22 cm
−3. In the materials with the Mg concentrations of 6 × 10
18 and 2 × 10
19 cm
−3, the N vacancy concentrations determined from the LPITS measurements are 2.8 × 10
17 and 4.0 × 10
18 cm
−3, respectively, which gives the −Δ
a/
a values of 6.3 × 10
−6 and 9.0 × 10
−5, respectively. The fact that the results obtained by the theoretical calculations and by the LPITS studies are of the same order confirms the strain involvement in the formation of the Frenkel pairs.
The T4 trap activation energy of 1870 meV exactly matches the HSE-calculated acceptor level of gallium vacancy (
VGa) located at
Ev + 1870 meV [
15]. The level is related to the hole thermal emission from the
VGa (−), resulting in the (−/2−) charge state transition. When the sample is illuminated during the LPITS measurements, the
VGa (2−) captures an excess hole from the valence band, becoming
VGa (−), and after switching off the illumination, the hole is thermally emitted back to the valence band. In other words, in the equilibrium conditions, the
VGa is doubly negatively ionized and effectively compensates the singly positively ionized oxygen-related shallow donors. This is the case for the material with the [Mg] = 6 × 10
18 cm
−3 in which the
VGa (−/2−) is experimentally detected through the T4 trap whose concentration is found to be ~1 × 10
19 cm
−3. It should be emphasized that this trap has not been detected in the material with the [Mg] = 2 × 10
19 cm
−3. This result is of great importance since it indicates that under the N-rich conditions during the crystal growth, the
VGa are formed as non-stoichiometric defects due to the presence of an excess of nitrogen [
47]. A similar phenomenon occurs under the growth of undoped GaAs crystals when
VGa are formed due to a small deviation from stoichiometry under the As-rich conditions [
48]. In other words, Mg-doping consumes the Ga vacancies present in the material by the reaction Mg
i +
VGa → Mg
Ga. Therefore, it can be assumed that in the material with the [Mg] = 6 × 10
18 cm
−3, the initial [
VGa] had been 1.6 × 10
19 cm
−3, and during the crystal growth the part of this concentration was filled with the Mg interstitials. On the other hand, in the material with the [Mg] = 2 × 10
19 cm
−3, the initial [
VGa] had been presumably the same; however, all the Ga vacancies were filled with the Mg interstitials, for the [Mg] is higher than [
VGa]. Thus, the T4 trap, identified with the
VGa (−/2−), has not been detected in this material by the LPITS measurements. The proposed interaction of Mg with
VGa is consistent with the positron annihilation spectroscopy results obtained by the studies of grown-in vacancy defects in bulk GaN crystals grown by the ammonothermal method [
6]. A high concentration of
VGa-related defects was observed only in the undoped with Mg
n-type samples. In the Mg-doped samples, with the [Mg] of the order of 10
19 cm
−3, no positron trapping at vacancy defects was observed [
6].
3.4. Compensation Mechanism
In order to further verify if the observed deep defects act as an electron or hole trap and confirm the data in a quantitative manner, Hall effect measurements at elevated temperatures were performed. It was found that in high temperatures, regime
p-type conductivity of the investigated samples was revealed.
Figure 4a,b presents the hole concentration (
Figure 4a) and resistivity (
Figure 4b) vs. inverse temperature for both samples. In the case of sample #2, the slope of both curves indicates the generation of holes with a conductivity activation energy of about
EA = 500 meV, reflecting an approximate Fermi level position pinned at deep donor level (
EDD) located at 400–500 meV above VBM. The data are derived from Equation (4), being the charge neutrality equation taking into account Mg acceptors, shallow and deep donors, and, as well as from Equation (5), making use of the changes in the occupation of the deep donors charge state. The charge neutrality is given by
where
p and
n are concentrations of holes and electrons (in our case
n is negligible, since all shallow donors act as compensating centers and do not provide electrons to the conduction band) in the valence and conduction bands, respectively,
concentration of ionized oxygen donors (equal to concentration of neutral donors
NDO, since all donors compensate acceptor states),
NMg—total concentration of Mg,
—concentration of ionized Mg acceptors,
gMg is the degeneracy factor of the Mg shallow acceptor level,
EMg = 150 meV—ionization energy of Mg acceptor,
NDD and
, —concentrations of neutral, singly, and triply ionized deep donors, respectively,
gDD—degeneracy factor of the deep donor. In the first approximation and for the sake of simplicity, we neglect the influence of the N
i–N
i defect and introduce into the charge neutrality equation only multivalent N vacancies with
donor level and
= 3000 meV below the CBM acceptor level. These N vacancies are dominant deep defects according to LPITS results and we assume they all undergo the negative-U effect. Further, we take advantage of the fact that the ratio
between the concentrations of centers of the subsequent electron charge states
q and
q − 1 undergo the equation:
where all degeneracy factors
gq,
gq−1 were assumed to be unity. We keep in mind that the total concentration of N vacancies
. The combination of Equations (4) and (5) adopted for multivalent defect centers [
49] is used for the description of experimental data with input parameters from SIMS or LPITS:
NMg, NDO,
NDD,
EMg, and
EDD above VBM (
Figure 4).
For this paper, a detailed analysis of the electron transport measurements was performed only for sample #2 due to a large concentration of defects in the N-sublattice. In sample #1 (treated as a reference sample with a small concentration of these defects), the defect structure is very complex. At this stage, we state that
VGa defects (identified earlier as the T4 trap) play the most important role in its electrical properties. We see in
Figure 4 that for this highly compensated sample, resistivity is much higher than in sample #2, whereas the slope of the hole concentration and resistivity corresponds to the activation energy of conductivity equal to 1.4 eV, which agrees quite well with experimentally derived
EADC = 1.7 eV (
Figure 2a) and activation energy of T4 trap (
Figure 3c). Thus, the Fermi level is close to the (−/2−) and (0/−) transition level of
VGa and related defects [
15] (of concentration even 1 × 10
19 cm
−3), making presented experimental data of sample #1 consistent. The concentration of N
i–N
i, and
VN defects seems to be small (at the order of 10
17 cm
−3), and therefore their participation in compensation of this sample is limited. A detailed quantitative description of experimental electrical data and a discussion of the compensation mechanism are beyond the scope of this paper due to the complex defect structure of this sample. At this stage, we state that
VGa plays a key role in explaining the experimental Hall effect data because a smaller [Mg]/[O] ratio leads to a high compensation resulting in a mid-gap position of
EF, closer to the acceptor level of
VGa. In sample #2, where compensation is smaller and Mg concentration is larger, substantial total concentration of
VN (8 × 10
18 cm
−3) and also
Ni–Ni (concentration of about 2 × 10
18 cm
−3) are detected at LPITS measurements. In this case, the Fermi level is lower and pinned to donor levels of
VN and N
i–N
i at about 450 meV above VBM. In the temperature-dependent Hall effect, experiment hole transition from donor levels of
VN, mainly (3+/+), located at about
EV + 0.45 eV [
15], are mostly responsible for the observed transport properties. Consequently, the
p-type conductivity at high temperatures was observed. The Hall effect data vs. temperature of sample #2 can be well described by the charge neutrality equation main parameters taken from SIMS, LPITS (
NMg = 2 × 10
19 cm
−3,
NDO = 1 × 10
18 cm
−3), including the compensating deep donor concentration
NDD = 1.1 × 10
19 cm
−3 of ionization energy
EV + 450 meV, which is very close to estimated total concentration of
VN and N
i–N
i centers (6 × 10
18 cm
−3). We note that some content of detected N
i–N
i centers (concentration 2 × 10
18 cm
−3) of similar ionization energy is also present. Thus, the donor defects in N-sublattice play a significant role in the compensation mechanism. Our final remark is that the presented results are consistent also from the thermodynamic point of view, as N vacancies and N
i–N
i are expected as compensating donors if
EF is low.