On the feasibility of p-type Ga₂O₃

Gallium oxide (Ga₂O₃) occurs in five different phases, commonly referred to as α, β, γ, δ, and ε. Among these, the monoclinic β-Ga₂O₃ is the most stable and technologically relevant phase.¹ Thus, we will refer to this phase hereinafter. Ga₂O₃ has an ultrawide bandgap, making it suitable for applications as a deep UV transparent conducting oxide (TCO). Typical values of the bandgap range from 4.4 to 4.9 eV depending on the polarization of the incident light with respect to the b and c axes of the crystal.^2–7 The n-type conductivity of Ga₂O₃ is easily tunable over many orders of magnitude ranging from 10⁻¹² to 10² S/cm.^3,8–10 In order to further extend its utility and applications, it is necessary to realize p–type Ga₂O₃. However, such a task has been proven challenging. The various theoretical studies have been devoted to investigating the p–type conductivity in wide bandgap oxides, pointing out the difficulty in achieving such a task.^11–14 

Regarding Ga₂O₃, Liu et al.¹⁵ and Feng et al.¹⁶ reported the fabrication of p–type Ga₂O₃ nanowires doped with nitrogen and zinc, respectively. In both cases, the authors reported that the homojunction of the doped nanowires and the n–type Ga₂O₃ substrate showed rectifying behavior. Zinc and nitrogen have been used as acceptor dopants in other materials, and they are potential candidates for p–type doping in Ga₂O₃. Zinc has been used as a cation substituting dopant in various III-V semiconductors,^17,18 while nitrogen has been used as an acceptor dopant in other TCOs such as SnO₂. However, the hole concentration was too low for useful p–n junction fabrication.¹⁹ The limited number of reports on the existence of p–type Ga₂O₃ indicates that there may be issues with the reliability and reproducibility of these results. Additionally, an extensive study of possible p-type dopants in Ga₂O₃ is still missing.

In the present letter, we employed density functional theory (DFT)^20,21 to investigate the various cation substitutional impurities as potential dopants for effective p–type doping in Ga₂O₃. Specifically, the studied impurities include X_Ga, where X = Li, Na, K, Be, Mg, Ca, Cu, Au, and Zn. These elements are representative of groups 1, 2, 11, and 12 of the periodic table which are likely to result in p–type conductivity.

We employed DFT calculations using the projector augmented wave (PAW)^22,23 method as implemented in the Vienna Ab-initio Simulation Package (vasp).²⁴ We used both the generalized gradient approximation (GGA) in the parameterization by Perdew, Burke, and Ernzerhof (PBE)²⁵ and the hybrid functional by Heyd, Scuseria, and Ernzerhof (HSE).^26,27 The amount of mixing of the Hartree-Fock exchange was set at a = 0.32 in order to achieve a bandgap of 4.8 eV, consistent with the experimental bandgap. All the calculations were performed in a 120-atom supercell which is found to produce converged results with respect to the formation energies of impurities in Ga₂O₃.²⁸ The cutoff energy of the plane-wave expansion was set at 450 eV for all dopants except for Li in which the cutoff energy was set at 500 eV. The sampling of the Brillouin zone was carried out using a 2 × 2 × 2 Monkhorst-Pack k–mesh, and the force convergence criterion for the lattice relaxations was set at 5 × 10⁻³eV/Å and 5 × 10⁻² eV/Å for the PBE and HSE calculations, respectively.

The formation energy of a defect in charge state q is given by

where $Etotdef$ is the total energy of the supercell containing the defect and $Etotbulk$ is the total energy of the bulk supercell. The chemical potential μ_i of the species i added (n_i > 0) or removed (n_i < 0) for the creation of the defect affects the formation energy of the defect but not the transition levels. The Fermi energy, E_F, is referenced at the valence band maximum, E_VBM, in the bulk such that E_F ≥ 0. Finally, the term Q_c accounts for the electrostatic corrections that are necessary to be considered due to the long range electrostatic interaction of the charged defects of the neighboring supercells.²⁹ Based on the formation energies, the ionization level (thermodynamic transition level), ϵ(q₁/q₂), between the charge states q₁ and q₂ is defined as

where $Efq|EF=0$ is the formation energy of the defect in charge state q evaluated at E_F = 0. The ionization levels can be observed experimentally using deep level transient spectroscopy (DLTS) in which the final charge state is able to relax to its equilibrium configuration.

In order for a defect to contribute to the p–type conductivity of the material, the defect should be stable in the neutral charge state and the transition from the neutral to the negative charge state should occur close to or below the valence band maximum (VBM). In other words, the determination of the ε(0/–) transition level with respect to the VBM is essential for the p–type conductivity of the material. Typically, shallow defect levels either exhibit ionization energies of the order of a few tenths of an eV from the band edges or they resonate with them. Hence, the accurate determination of the ionization levels is imperative for the study of possible p–type dopants.

The ionization levels can be obtained using a semilocal (LDA or GGA) approach. This approach suffers from the artificial electron self-interaction, and also lacks the derivative discontinuities of the exchange-correlation potential with respect to the occupation number. This leads to the known bandgap error, i.e., the fact that the calculated bandgap is much smaller than the real bandgap of the material.^30,31 Hence, while GGA calculations produce reasonable results for the atomic geometries, they usually fail to reproduce the experimentally observed ionization levels. In some cases, corrections for on-site coulombic interactions (GGA + U) can improve the results of GGA calculations in the study of defect levels.³² On the other hand, in hybrid functionals, the introduction of a portion of a non-local Hartree-Fock exchange corrects the self-interaction error which is observed in LDA/GGA. As a result, the bandgap is increased and the agreement of the ionization levels with experiments is greatly improved, making hybrid functionals more suitable for such calculations. However, their merit comes with increased computational cost.

Even though the transition levels obtained by GGA are not always accurate, GGA can still give valuable insights into the existence of possible acceptor levels for certain dopants, having the advantage of being computationally less expensive. This approach can be justified by how the band edges are aligned between PBE and HSE. The alignment is performed with respect to a common reference level. A typical way to determine the reference level is by performing a surface calculation in both schemes and to define the reference level as the vacuum level at a large distance from the surface. Another way is to define the average electrostatic potential in the bulk as the common reference level. Aligning the VBM with respect to the average electrostatic potential is equivalent to the alignment with respect to an external vacuum level, when the charge density in the two theoretical schemes is identical.^33,34 In this work, the average electrostatic potential in the bulk is used as the common reference level. Typically, aligning the band edges obtained by PBE and HSE results in the VBM of the hybrid approach to lie lower than the VBM of the semilocal approach. Additionally, the conduction band minimum (CBM) of the HSE calculations lies above the CBM of the PBE calculations. A schematic representation is shown in Fig. 1. In general, we may distinguish two discrete cases of defect levels. In the case of a deep level, like (a) in Fig. 1, the widening of the bandgap does not affect its position significantly. On the other hand, shallow levels, like (b) in Fig. 1, which originate from hydrogenic-like impurities, tend to follow the band edge to which they are bound.^33,34 Hence, GGA is useful for determining the possible existence of relevant levels since the deep levels are correctly identified. In our studies, we calculated the transition levels of different impurities using predominantly GGA and performed HSE calculations for some representative cases.

Due to the low symmetry of the Ga₂O₃ crystal, there are two inequivalent gallium and three inequivalent oxygen sites. Hence, there are two substitutional sites for Ga and three for O. One of the gallium atoms is tetrahedrally coordinated with respect to its oxygen neighbors, while the other is octahedrally coordinated. Details of the structure can be found elsewhere.²⁸ We refer to them as Ga⁽¹⁾ and Ga⁽²⁾, respectively. Each of the dopants is used as a substitute to the gallium site, and both coordinations are investigated.

Figure 2 illustrates the obtained transition levels using GGA. In the case of Cu_Ga and Au_Ga, the introduced levels are very deep. Hence, copper and gold are not expected to introduce any relevant acceptor levels. This behavior is consistent with group 11 dopants in other wide bandgap oxides.¹¹ The transition levels introduced by the rest of the dopants lie bellow the midgap, still quite far from the VBM. Regarding the two different gallium sites, the substitutional impurity which resides in the Ga⁽²⁾ site introduces the transition level which is closer to the VBM. Specifically, the levels obtained by GGA lie 0.36/0.21, 0.42/0.26, 0.55/0.34, 0.26/0.21, 0.26/0.22, 0.34/0.24, and 0.35/0.27 eV above the VBM for Li, Na, K, Be, Mg, Ca, and Zn, respectively. All of them suggest deep levels, but further investigation was carried out using HSE.

HSE calculations were performed for Li, Mg, and Zn. The choice of these three dopants for further investigation lies in the fact that they demonstrated promising results with GGA and because each one represents groups 1, 2, and 12 of the periodic table, respectively. In all three cases, the results obtained by GGA indicate deep acceptor levels, and as mentioned earlier, such levels are expected to appear even deeper using HSE due to the widening of the bandgap.

Figure 3 shows the ε(0/–) transition levels for Li, Mg, and Zn. The levels obtained by HSE occur at 1.67/1.84, 1.25/1.05, and 1.39/1.22 eV above the VBM for Li, Mg, and Zn, respectively. Table I summarizes our results. Such deep levels are not expected to contribute to the p–type conductivity of the material. In addition to the ε(0/–) level, we observe also the binding of a second hole in a very deep ε(+/0) donor level. This level occurs at 1.54/0.97, 0.96/0.98, and 0.92/0.86 eV for Li, Mg, and Zn at the Ga⁽¹⁾ and Ga⁽²⁾ sites, respectively. The existence of a very deep donor level is consistent with previous results in wide bandgap oxides.¹¹ 

The difficulty in identifying shallow acceptor dopants in Ga₂O₃ is not surprising. Other TCO materials such as SnO₂, In₂O₃, TiO₂, and ZnO face similar issues.^13,35 Robertson and Clark¹² found that the dopability of oxides can be viewed in terms of the VBM and the CBM on an absolute energy scale. Common n–type oxides, including Ga₂O₃, exhibit very deep VBM with respect to the vacuum level.^36,37 On the other hand, in semiconductors where the VBM lies higher, the defect levels are more likely to remain close to the valence band edge, leading to p–type materials.

A projected density of states calculation reveals that the VBM of Ga₂O₃ is dominated by O 2p states. The O 2p orbitals exhibit small dispersion, large effective masses, and high density of states. This is a common feature among wide bandgap oxides.¹⁴ Materials with this feature favor the formation of localized oxygen hole polarons.^11,38 Additionally, Varley et al.¹⁴ showed that Ga₂O₃ exhibits self-trapped holes, in agreement with experimental observations.³⁹ Thus, materials with a very low energy VBM which is dominated by O 2p states are not optimal for p–type conductivity. On the other hand, the success of copper based oxides to achieve p–type conductivity arises from the 3d electrons of copper. The copper 3d orbitals are close in energy to the oxygen 2p orbitals.³⁵ Thus, the covalency of these orbitals results in a more dispersive valence band which allows for delocalized hole states with smaller effective masses.

Furthermore, the position of a defect level is not the only factor affecting the effectiveness of certain doping. Other factors include the solubility of the dopant and the presence of compensating native defects. The solubility of the dopant is directly related to the formation energy of the defect. The major limitation, however, is compensation by native defects.¹² In the case of Ga₂O₃, oxygen vacancies act as compensating donors, hindering the p–type conductivity of the material.^28,40 Hence, the aforementioned factors portrait the difficulty of the realization of p–type Ga₂O₃.

In conclusion, we utilized density functional theory both in the PBE and HSE formalism to investigate potential cation substitutional dopants from groups 1, 2, and 12 for producing p–type Ga₂O₃. Our results demonstrate that these dopants introduce deep acceptor levels with ionization energies of more than 1 eV. In addition, similar to other wide bandgap oxides, these dopants can trap an extra hole at a very deep donor level which hinders p–type conductivity even more. We also reviewed the challenges in realizing p–type conductivity in wide bandgap oxides and specifically in Ga₂O₃. Evidently, traditional doping techniques are not expected to produce fruitful results. In view of these findings, we hope to stimulate further experimental work towards the realization of p–type Ga₂O₃, aiming at appropriately engineering the valence band.

The authors gratefully acknowledge financial support from the U.S. Army Research Laboratory through the Collaborative Research Alliance (CRA) Grant No. W911NF-12-2-0023 for Multi-Scale multidisciplinary Modeling of Electronic Materials (MSME). The computational resources were provided by the DoD HPC Open Research Systems and the 2014 Army Research Office DURIP Award Grant No. W911NF-14-1-0432 made to Dr. E. Bellotti.