Native defects and their complexes in spinel LiGa5O8: the puzzle of p-type doping

Recently, LiGa$_5$O$_8$ was identified as a cubic spinel type ultra-wide-band-gap semiconductor with a gap of about 5.36 eV and reported to be unintentionally p-type. Here we present first-principles calculations of the native defects and various of their complexes to try to explain the occurrence of p-type doping. Although we find Li-vacancies to be somewhat shallower acceptors than in LiGaO$_2$, and becoming slightly shallower in complexes with donors such as V$_{\rm O}$ and Ga$_{\rm Li}$ antisites, these $V_{\rm Li}$ based defects are not sufficiently shallow to explain p-type doping. The dominant defects are donors and in equilibrium the Fermi level would be determined by compensation between donors and acceptors, and pinned deep in the gap.


I. INTRODUCTION
Recently, LiGa 5 O 8 in the spinel structure was grown in thin-film form by mist-chemical vapor deposition and reported to have a gap of 5.39 eV and as grown p-type conductivity.[1] This is a quite surprising finding as many oxides are notoriously difficult to dope p-type.In β-Ga 2 O 3 for example, the formation of self-trapped hole polarons related to the heavy valence band mass is widely believed to preclude p-type doping.Although there are some reports of p-type doping via complexes, it is not clear whether these are robust or correspond truly to homogeneous doping and the hole mobilities were found to be rather low.[2,3] In closely related LiGaO 2 , with an even larger optical gap of 6.0 eV [4][5][6][7] native defects were studied [8] and n-type doping predicted by Si, and Ge.[9] However, p-type dopants were not yet identified.Substitutional N O was found to behave amphoteric and Zn doping suffers from site competition between Zn Li with donor behavior and Zn Ga acceptor behavior resulting in compensation.Diatomic molecules (N 2 , NO and O 2 in LiGaO 2 ) were also investigated as possible p-type dopants, but were all find to be deep acceptors.[10] Even in much lower gap ZnO, p-type doping has remained notoriously difficult.[11] It would greatly expand the opportunities for power-electronics to have a p-type ultrawide-band gap UWBG semiconductor but this surprising finding clearly deserves further scrutiny and requires first-principles calculations to identify the possible p-type dopants.
The spinel structure of LiGa 5 O 8 was established by Joubert et al. [12].It features Li in octahedral sites and Ga in both octahedral and tetrahedral sites.The com-pound has received some previous attention as a candidate phosphorescent materials by doping with Cr. [13][14][15].Its space group is P 4 3 32 or O 6 (No. 212) and its cubic lattice constant is 8.203 Å. [16][17][18] A figure of the structure, identifying the different types of Ga and O is given in Fig. S1 in Supplemental Material (SM).[19] Recently, one of us [20] calculated its electronic band structure using the quasiparticle self-consistent (QS) G Ŵ method with the screened Coulomb interaction Ŵ including ladder diagrams and its optical dielectric function using the Bethe Salpeter Equation (BSE) method, identifying the quasiparticle gap to be slightly indirect 5.72 eV with lowest direct gap 5.84 eV but a high exciton binding energy leading to an optical gap of 5.48 eV.These calculations did not include zero-point motion electron-phonon coupling effects which could reduce the gaps by a few 0.1 eV.To within the error bars this is consistent with the experimental data by Zhang et al. [1].In [20] we also showed that Si could provide a shallow n-type dopant.Here we study point defects using first-principles calculations.

II. COMPUTATIONAL METHODS
We use the Vienna Ab initio Simulation Package (VASP) [21,22] using the projector augmented wave (PAW) approach [23] and the Heyd-Scuseria-Ernzerhof (HSE) type hybrid exchange correlation functional [24,25] with adjusted inverse screening length µ and fraction of non-local exchange α.We found that α = 0.372 and µ = 0.2 Å−1 gave a direct gap at Γ gap of 5.845 eV, and indirect S − Γ gap of 5.717 eV, in excellent agreement with the QSG Ŵ calculations of [20].The band structures are shown for reference in Fig. S2 in SM. [19] We also tested the fulfillment of Koopman's theorem for V Li and Li Ga defects as shown in Fig. S3 in SM. [19] We find the cubic lattice constant a = 8.171 Å, slightly smaller than the experimental value of 8.203 Å.
Because the cubic unit cell (shown in Fig. S1 in SM) already contains 56 atoms or four formula units of LiGa 5 O 8 it is rather challenging to perform calculations for symmetric larger supercells.Even a 2 × 2 × 2 cell corresponds to 448 atoms.While for higher convergence, this may be necessary in the future, we here find it more important to study more types of defects and stick to the 56 atom cell.We use the standard approach of calculating defect energies of formation as where D q indicates the defect D in charge state q, C : D q means the crystal with defect an C without defect, ϵ v is the valence band maximum (VBM) with respect to the average electrostatic potential which is aligned far away from the defect with that of the perfect crystal by means of the V align correction and E corr is a Madelung correction for the electrostatic energy of the periodic array of localized defect charges in the homogeneous background used to compensate the defect charge.This and the alignment term for charge defects are obtained using the Freysoldt-Neugebauer-Van de Walle (FNV) procedure [26].The µ i are the chemical potentials of the atoms added to or removed from the perfect crystal go create the defects.

III. RESULTS
First we determine the chemical potential range in which LiGa 5 O 8 is stable.As can be seen in Fig. 1 this is a rather narrow range.The points A,B correspond to Orich conditions (Ga-poor) and C,D to O-poor (Ga-rich) conditions.The points E and F correspond to realistic O-chemical potentials corresponding including the pressure and temperature dependent term (k B T ln p(O 2 )/p 0 ) [27] and at the growth temperature (T=900 • C) reported in [1] and can be seen to be close to O-rich.The points A, E and C all correspond to relatively Li rich compared to Ga, while B, F and D are Li-poor.The actual values of the chemical potentials at these points is given in Table S1 in SM. [19] Our results for simple native point defects are shown in Fig. 2 for the chemical potential conditions F, which corresponds to realistic O-chemical potentials and somewhat Li-poor.Additional figures for other chemical potential conditions are given in Fig. S4 in SM.We can see that Ga Li acts as a shallow donor, V O are deep donors on both O types.The V Li is a relatively deep acceptor with acceptor binding energy or 0/− transition level at 0.74 eV.Ga vacancies have higher energy and the Li Ga acceptors have higher formation energy and higher 0/− transition level for both the tetrahedral Ga (Ga-tet) and octahedral Ga (Ga-oct) sites.Consistent with point F being less Li rich, we find (see Fig. S4 in SM [19]) that the Li-vacancy formation energy is increased under condition E compared to F by about 1 eV but Li Ga are lowered in condition E and come closer to the V Li .This shifts the intersection points with Ga Li and V O which will be shown later to be important but makes no overall qualitative change.As also shown in difference and Ga-vacancies are even further increased by about 6 eV.These high-energy of formation defects thus play little role in the charge neutrality and can safely be ignored.These Ga-vacancies can only be expected to be formed under high-energy particle irradiation conditions, which distort the equilibrium.

Ga Li
The V Li appears to be the only somewhat viable acceptor but p-type doping is not predicted from it because its binding energy is rather deep and secondly it would be compensated by O-vacancies and Ga Li .Considering that under conditions E and F the system is rather O-rich and Ga-poor, the Ga Li antisite is apparently a low energy of formation donor.Nonetheless, it is remarkable that the 0/− transition level for the V Li is significantly lower than in LiGaO 2 , where it is 1.03 eV.[8].An important difference is that in that material Li occurs in a tetrahedral environment and its wave function localizes on one of the O neighbors.[29] Whereas here it appears to be delocalized over several of the neighboring O atoms as can be seen in Fig. 3.An important conclusion is that the V Li does not seem to be a polaronic acceptor in the present material.
Next, we study various defect pair complexes.Our reason for doing so is that we found in our recent study of donor acceptor pairs in LiGaO 2 [30] that the acceptor level of the complex tends to be pushed closer to the VBM and the donor level closer to the CBM.This results from bonding anti-bonding interactions between their defect wavefunctions.We might thus obtain shallower acceptor levels.In Fig. 4, we focus again on the chemical potential condition F. A more complete set of figures for other chemical conditions is given in Fig. S5 in SM.The subpanels give Li Ga−tet , Li Ga−oct , V Ga−tet and V Ga−oct and V Li each paired with three different donors, V O1 , V O2 and Ga Li along with the corresponding constitutive point defects.The pairs are chosen as nearest Formation energy (eV) Fermi level (eV) Formation energy (eV) Fermi level (eV) Formation energy (eV) Fermi level (eV) Formation energy (eV) Fermi level (eV) Formation energy (eV) Fermi level (eV)

eV
Fig. 5: Summary of all transition levels of point defects and their complexes calculated.
As the most relevant example, let's consider V Li + V O1 .When the Fermi level is close to the VBM, the V O1 is in a 2+ state while the V Li is neutral.Increasing the Fermi level, the first transition of the complex is 2 + /+.However, we can see that the 2+/+ transition level of the complex is indeed closer to the VBM than the 0/− level of the isolated V Li .Thus, in some sense the acceptor level becomes shallower, and this corresponds indeed to the V Li emitting a hole to the VBM, which one might think of as contributing to p-type conductivity.However, overall the defect is still a donor and is then still in a positive charge state, which requires it to be compensated either by free electrons or by a negative defect.Thus by itself, this complex cannot lead to p-type doping because we combined a double donor with a single acceptor.Similar considerations apply to the V Li -Ga Li pair.Eventually, at higher Fermi level position in the gap, we can see a narrow range of 0 charge state and a transition to the −1 charge state.This point again lies higher than the corresponding donor 2 + /0 transition.Unexpectedly, for the V Li -V O2 complex the 2 + /+ transition is higher than for the isolated V Li .
The Li Ga acceptor is a double acceptor and when it combines with a double donor such as V O1 , we obtain a 2+/0, transition and a 0/2− transition at high Fermi energy.Thus all the complexes essentially display ampho-teric behavior, combining donor and acceptor like properties.Close to the VBM, they are dominated by the donor-like behavior and close to the CBM by the acceptor like behavior.For the double acceptor double donor pairs a large region of neutral charge state occurs in the center of the gap.Finally, for the V Ga based complexes, we combine a triple acceptor with a double donor.However, none of them show promising shallow acceptor like behavior.
An overview of all the transition levels is given in Fig. 5.We can see that the levels closest to the VBM are the 2 + /+ level of the V Li + V O1 and V Li + Ga Li at 0.58 eV and 0.65 eV above the VBM.First of all, these are still not shallow enough to consider shallow acceptors and secondly, they correspond to 2 + /+ levels, rather than 0/− levels.A shallow level is not sufficient to explain p-type doping.Basically, these are deep donors instead.We also need to consider the overall charge neutrality and compensation issues.Because the lowest energy defects close to the VBM are the Ga Li donor and V O type donors, compensation is expected.
The concentration of defects in various charge states and the net free electron and hole concentrations in the bands are governed by the charge neutrality condition where N i is the density of sites available for given fect, q i its charge and i a degeneracy factor.For example if the charged defect features a with a single electron, (such as the V 0 Li ) it can be in spin up or spin down state and the degeneracy is 2 but if it is filled or empty, the degeneracy is 1.Its formation energy in a given charge state depends on the electron chemical potential, ∆E f or (D qi i ) = ∆E f or (D qi i , µ = 0) + q i µ.The band electron and hole concentrations n and p also depend on the electron chemical potential and temperature, As long as the chemical potential is not too close to the band edge, with m n the electron effective mass and similar equations hold for the hole concentration.Together, these equations can be solved for a given temperature, say the growth temperature, where we assume the system was in thermodynamic equilibrium before the defects concentrations were frozen in by quenching the system to room temperature.They then determine the position of the chemical potential (i.e. the Fermi level) as well the defect concentrations and free carrier concentrations.In practice because of the exponential dependence on the energies of formation only the lowest energy for formation defects play a significant role and if the Fermi level stays deep in the gap, the n and p are negligible.Keeping only the V −1 Li , V +2 O and for simplicity equating the formation energy of the two types of O, and Ga 2+ Li charged defects in the neutrality equation, and using a growth temperature of 1200K, we can solve for µ graphically, using Mathematica and find µ = 3.25 eV.This is close to the crossing point of the V −1 Li formation energy with the Ga 2+  Li , indicating that even the V O may be neglected.In that case, the equilibrium Fermi energy is set by because the density of available sites for V Li and Ga Li is the same and both their degeneracies are 1, which can be simplified to 3µ = kT ln 2 + ∆E f or (V Li ) − ∆E f or (Ga Li ).
Even at T = 1200 K, kT ln 2 = 0.072 eV is pretty small compared to the difference in formation energies of 9.74 eV, but there is no problem in keeping it and we find µ = 3.25 in almost perfect agreement with the calculation including the O-vacancies.Thus we conclude that considering only the native defects, the Fermi level is pinned at the intersection of the dominant donor Ga Li and the shallowest acceptor-type defect V Li .This is under condition F which is most Li-poor and hence the likeliest to give Li-vacancies and Ga on Li antisites.But the Fermi level is then pinned deep in the gap and insulating behavior is expected.
Considering now the complexes, these are less likely because the density per unit volume of ij pairs is N i N j and of those only the nearest neighbor ones correspond to the pair considered.In spite of this pre-factor disadvantage of the complex, its energy is lowered by the bonding between donor and acceptor and as this goes in the exponential, it has a more important effect.Numerical solution of the neutrality equation, shows that µ now shifts to 3.54 eV, in other words, it shifts close to the intersection of the V −1 Li and the [V Li − Ga Li ] +1 , which now becomes the lowest energy donor.So, even though the 2 + /+ level moved closer to the VBM, the lower energy of these complexes ultimately shifts the Fermi level even higher in the gap.
We also consider what happens to the chemical potential at room temperature if we assume the defect concentrations from the growth temperature are frozen in but their charge state might change and including the free carrier concentrations.When we only include the native defects, we find that the chemical potential then drops to 0.82 eV above the VBM but the hole concentration is still negligible ( ∼ 10 6 cm −3 ) and the neutrality is still produced by the equilibrium between Ga 2+ Li and V −1 Li defects which have a concentration of 2.1×10 22 cm −3 and 4.2×10 22 cm −3 respectively.We indeed need twice the concentration of single negative acceptors as 2+ donors to obtain neutrality.When we include the lowest energy of formation defect pairs, we find the room temperature chemical potential at 0.83 eV, the hole and electron concentrations are negligible (p≈ 1.4 × 10 6 cm −3 ) and the neutrality then comes from the balance of V −1 Li with [V Li − Ga Li ] +1 complexes, both of which are found around 7.2×10 23 cm −3 with a smaller concentration of Ga +2 Li of or 7×10 19 cm −3 .
Thus in thermal equilibrium, even if we include complexes, our calculations unambiguously predict insulating behavior because the Fermi level stays too far away from the valence band edge.
We also considered some impurity doping candidates for p-type doping such as N substituting for O and Zn substituting for Ga.The results are shown in Fig. 6.We can see that substitutional N on both O 1 or O 2 sites behaves as an amphoteric defect.Although its lowest level is fairly close to the VBM, it is a 2 + /+ transition, typical of a deep donor.For Zn Ga , we find acceptor behavior for both tetrahedral and octahedral site with slightly shallower transition level 0/− for the tetrahedral site but both are about 1 eV or larger above the VBM.Furthermore, there is site competition with Zn Li which acts as a shallow donor, in the +1 charge state throughout the gap.Formation energy (eV) Fig. 6: Energies of formation for N and Zn dopants.

IV. DISCUSSION AND CONCLUSION
In spite of a rather exhaustive study of the possibilities, our calculations are unable to explain p-type doping in LiGa 5 O 8 .In equilibrium we clearly find donors like Ga Li to compensate the potential V Li acceptor.The Li vacancy acceptor itself is somewhat shallower than in LiGaO 2 where it occurs in tetrahedral coordination as opposed to octahedral coordination found here.This different environment does avoid the polaronic distortion and we find a more spread out vacancy wave function, but, despite this being somewhat encouraging, the level is still too deep in the gap to provide p-type behavior.Complex formation with V O or with Ga Li can push the lowest transition levels closer to the VBM.Such complexes have less available sites but may nonetheless become slightly favored by the binding energy lowering of the energy of formation.However, these complexes are net amphoteric or donors rather than acceptors.Thus, they play the role of compensating the isolated V Li acceptors but do not become acceptors themselves.The lowest energy complex found here is the V Li -Ga Li pair and because it is lower in energy and becomes a single donor it shifts the Fermi level even deeper in the gap.Even if we consider that, after cooling to room temperature, the defects may change their charge state and thereby affect the free carrier concentrations, we find the Fermi level to still be at about 0.8 eV above the VBM, which is too high to give p-type doping.The maximum p-type concentration we obtain in this way is of order 10 6 cm −3 .
The only way to explain the observation of p-type conduction is then a non-equilibrium situation.One might envision, regions of the sample where V Li occur or their complexes occur and emit holes but these p-type regions would then have to be compensated by opposite charge space charge regions.This non-equilibrium could possibly occur near grain boundaries, or other extended defects or near surfaces.
In [1] p-type doping is reported for various growth conditions and in that sense labeled robust.Notably they report it to occur for both Li-rich and Li-poor conditions.However, as we noted the Li-chemical potential range in which LiGa 5 O 8 is stable is rather narrow, so this is perhaps not too surprising.These authors also report that annealing in O 2 could render samples which were initially p-type to become insulating.This might lead one to the assumption that oxygen vacancies play a role in the p-type conductivity.However, O-vacancies are clearly donors and not acceptors.Even if they could play a role in complex formation with V Li and thereby push the 2 + /+ transition of the complex closer to the VBM than the 0/− transition of the acceptor, this in itself cannot explain p-type doping.The O-annealing effects could perhaps rather point to a surface region annealing effect and restore the native insulating behavior we predict from our defect calculations.Although our results at present did not find any plausible explanation for p-type doping, we hope that the results on defect levels in this system will prove useful to its further characterization.

Fig. 1 :
Fig. 1: Chemical potential ranges of stability of LiGa 5 O 8 .Point E and F correspond to realistic O-chemical potentials at the growth condition.This figure was visualized by the Chesta code.[28]

Fig. 2 :
Fig. 2: Native defect formation energies for realistic O chemical potential conditions F.

Fig. 3 :
Fig. 3: V Li wave function (teal colored isosurface ) Dark green spheres: Ga-tetahedral, light green sphere: Ga-octahedral, yellow sphere: O1 which is the O bonded with 4 Ga, red sphere: O2 which is the O bonded with 1 Li and 3 Ga, blue spheres: Li.

Fig. 4 :
Fig.4: Energies of formation for different charge states for various donor-acceptor pair defect complexes, including Li Ga−tet , Li Ga−oct , V Ga−tet and V Ga−oct and V Li each paired with three different donors, V O1 , V O2 and Ga Li , under chemical potential condition F.