Boron doping in gallium oxide from first principles

We study the feasibility of boron doping in gallium oxide ($\text{Ga}_2\text{O}_3$) for neutron detection. $\text{Ga}_2\text{O}_3$ is a wide band-gap, radiation hard material which has potential for neutron detection if it can be doped with a neutron active element. We investigate the boron-10 isotope (\B) as possible neutron active dopant. Intrinsic and boron induced defects in $\text{Ga}_2\text{O}_3$ are studied with semi-local and hybrid density-functional-theory calculations. We find that in growth conditions favourable for boron, boron substitutional defects are likely to form making boron doping of $\text{Ga}_2\text{O}_3$ feasible.


I. INTRODUCTION
Gallium oxide (Ga 2 O 3 ) is a wide gap semiconductor (band gap E g ∼ 4.9 eV [1]) with potential applications in ultraviolet optoelectronic devices, power electronics and laser lithography [2][3][4][5][6][7]. In this work, we are exploring further applications of Ga 2 O 3 for neutron detectors. There is a growing need for neutron detectors with low-power requirements, compact size and reasonable resolution for, e.g., non-invasive neutron imaging of organic materials, like human tissue or wood [8], safeguarding and nonproliferation of nuclear material [9], safety in the nuclear industry [10], space science [11] and autonomous radiation probes for hazardous environments [12] Most current neutron detectors use helium-3 gas ( 3 He), a non-radioactive isotope of helium, because of its extreme sensitivity in detecting neutron radiation [9,10,13]. However, innovation is greatly needed, since current neutron detectors are expensive, bulky and not radiation hard precisely because of their use of 3 He. The world's 3 He supply is extremely scarce and depleting rapidly. Moreover, the large size of 3 He detectors limits their portability and spatial resolution. Since 3 He detectors are not radiation hard they cannot be used in harsh environments like outer space, or fusion or nuclear reactors. For these reasons, semiconductor detectors have recently received increasing attention [9,10,[13][14][15][16][17][18]. However, the materials requirements for optimal energy, time and spatial resolution, detection efficiency, robustness and radiation hardness are daunting challenges [13], and there is currently no satisfying material choice nor commercially available semiconductor detectors. For this reason, we are here exploring Ga 2 O 3 as potential neutron detector material.
Solid state neutron detectors use neutron active elements, which convert neutrons to electronic excitation via a nuclear reaction. The ability of neutron active elements to capture neutrons is measured by the neutron cross section. The boron isotope 10 B has the largest neutron cross-section at 3840 barns, which is comparable to helium 3 He and larger than other candidates like lithium ( 6 Li) and beryllium ( 9 Be). Boron-based neutron detectors have recently been demonstrated experimentally [15,17,18], but are far from commercialisation. Wide band-gap materials have also been investigated in solid state neutron detectors, most notably gallium ni-tride (GaN) [14,16]. Here, we consider beta gallium oxide (β-Ga 2 O 3 ) as potential material for neutron detection, because β-Ga 2 O 3 is a radiation hard wide band-gap material, and gallium has similar chemical characteristics as boron which makes boron implantation on gallium sites favorable.
The electronic structure of β-Ga 2 O 3 and the behavior of defects in the material have attracted considerable interest and have been studied previously with density function theory (DFT) [1,19,20]. Defects have been investigated as a source of the observed intrinsic n-type conductivity and for the possibility of p-type doping of β-Ga 2 O 3 for opto-electronic applications. Boron-related defects have not been previously studied in β-Ga 2 O 3 .
We here investigate the possibility of boron doping with DFT. With the supercell approach, we calculate formation energies for simple point defects and complexes in β-Ga 2 O 3 in the diffuse doping limit. We study both intrinsic defects and boron defects to assess the feasibility of introducing boron to β-Ga 2 O 3 . Our work provides insight into the limits of boron doping and the potential of β-Ga 2 O 3 for neutron detection.
The article is structured as follows. First, we discuss the atomic structure of β-Ga 2 O 3 and present the computational methods used in this work (Section II). We then investigate the intrinsic defects found in β-Ga 2 O 3 (Section III B). Finally, we study boron defects and compare them to the intrinsic defects (Section III C) before concluding our work (Section IV). All defect calculations were carried out with the su- percell approach [21]. Point defects were introduced in a 160-atom supercell model of pristine where E(X q ) is the DFT total energy of the supercell containing a defect in charge state q, and E(0) the total energy of the defect-free crystal. µ i is the chemical potentials of the ith species whose number varies by ∆n i when defects are formed. ∆n i is negative for the removal of atoms (e.g., vacancies) and positive for the addition of atoms (e.g., interstitials). F is the Fermi energy of Ga 2 O 3 , defined with respect to the valance band maximum ( VBM ). The q( VBM + F ) term therefore accounts for the energy change upon removal or addition of electrons when charge defects are formed.
To remove spurious electrostatic interactions between supercells with charged defects, we include the Freysoldt-Neugebauer-Van de Walle (FNV) correction term E corr [22]. In the FNV scheme, we use a spatially averaged dielectric constant of 0 ∼ 10 [4,23] which includes ionic and electronic screening [24]. There has been some debate, if the electronic dielectric constant ∞ should be used instead for small supercells [25]. However, we observed that 0 is the correct choice by extrapolating supercells to the infinite supercell limit (see Appendix C). Our findings are in agreement with those of Ingebrigtsen et al. [24].
All DFT calculations in this work were performed with the all-electron numeric-atom-centered orbital code fhiaims [26][27][28][29]. We used the semi-local Perdew-Burke-Ernzerhof (PBE) functional [30] and the Heyd-Scuseria-Ernzerhof hybrid functional (HSE06) [31] to calculate the atomic and electronic structure of β-Ga 2 O 3 and defects therein. PBE calculations are employed as reference to previous work and to test the supercell dependence for charge corrections. For the final defect geometries, we always use the HSE06 functional to avoid spurious delocalization effects in PBE, as observed for, e.g., the oxygen vacancies in TiO 2 [32].
We set the fraction of Hartree-Fock exchange in HSE06 to 35%, a value which has been previously used for Ga 2 O 3 [23]. This yields a band gap of 4.95 eV for tight settings in FHI-aims and 4.76 eV for light settings (see below for these two settings), thus providing an acceptable compromise between accuracy and computational cost. Furthermore, we investigated the impact of long-range van der Waals (vdW) corrections based on the Tkatchenko-Scheffler (TS) method [33] on the atomic structure of bulk Ga 2 O 3 . Finally, scalar relativistic effects were included by means of the zero-order regular approximation (ZORA) [34].
Considering the computational cost of HSE06 calculations, we carried out most of our calculations with the cheaper "light" basis sets (which usually provide sufficiently converged energy differences) and used results with "tight" basis sets (which can better provide converged absolute energies) as reference. For light settings, we use the tier 1 basis set for oxygen and gallium, but exclude the f function for gallium. For tight settings, we use tier 2 for oxygen and the full tier 1 basis for gallium. Adding tier 2 for gallium did not improve the result with PBE thus the tier 1 basis set for gallium is enough to achieve convergence. A Γ-centered 2 × 8 × 4 k-point mesh was used for the 20-atom monoclinic unit-cell calculations, while for larger supercells (160-atom) we used a Γ-centered 2 × 2 × 2 k-point mesh. In pursuit of open materials science [35], we made the results of all relevant calculations available on the Novel Materials Discovery (NOMAD) repository [36].

A. Bulk Ga2O3 and chemical potentials
The optimized geometry of bulk β-Ga 2 O 3 is presented in Table I for HSE06 and semi-local functionals. Band gaps and formation enthalpies have been included for completeness. The PBE functional overestimates the lattice constants compared to experiment. Van der Waals effects slightly reduce the lattice constants towards better agreement with experiment. Conversely, the HSE06 functional reproduces the experimental geometry well and our results are consistent with those previously reported in the literature [19][20][21]23,25].
The HSE06 band structure β-Ga 2 O 3 is shown in Fig. 2. The band gap is indirect between the M-point and the Γ-point and has a value of 4.92 eV. The direct gap at the Γ-point is slightly larger (4.95 eV). The fact that indirect transitions are weak makes β-Ga 2 O 3 effectively a direct band-gap material. We reference the gallium chemical potential µ Ga to gallium metal and the oxygen chemical potential µ O to the oxygen molecule O 2 (see Appendix A for details). The chemical potentials need to be in equilibrium 2µ Ga +3µ O = H f (Ga 2 O 3 ), which defines the Ga-rich (µ Ga = 0) and O-rich (µ O = 0) limits. For the B chemical potential, we use diborane B 2 H 6 , which is used as precursor for boron doping e.g. in silicon [38]. The boron chemical potential is defined through the hydrogen chemical potential, which we reference against the hydrogen molecule: ). Note that we do not vary the hydrogen chemical potential in this study, but keep it fixed. A further constraint on the boron chemical potential is the formation of boron oxide B 2 O 3 . The upper bound of the boron chemical potential is therefore ) amounts to -1.12 eV in HSE06. Thus the boron and gallium chemical potentials are linked. Even in the most favorable boron-rich conditions the chemical potential of boron has to be lower than that of gallium.

B. Intrinsic defects
We first investigate intrinsic point defects. We do this not only to validate our calculations against previous studies, but also to study the competition between intrinsic defects and boron defects. We consider vacancy sites and simple interstitials, which we find to be energetically quite stable and thus enough for our assessment of boron defects [24,25]. Intrinsic defects for Ga-rich and O-rich cases are shown in Fig. 3.
The most important transition states of vacancy defects are listed in Table II. The charge transition levels of the oxygen vacancies (+2/0) are located deep below the conduction band minimum (CBM). Different coordinations yield slightly different transition states with the four-fold O(III) site being closest to the CBM. For n-type conditions (Fermi energy close to the CBM), the oxygen vacancies are therefore neutral while they would behave as donors for p-type conditions (Fermi energy close to the VBM). Conversely, gallium vacancies act as deep acceptors for most of the Fermi energy range. Here the (−2/−3) transition state for the lower coordinated Ga(I) is closer to the CBM than the octahedral Ga(II) state. We note in passing, that the Ga(I) vacancy in the -2 charge state requires a hybrid functional treatment. In the PBE functional the extra electrons do not localize, resulting in a formation energy that is too low.
The interstitial defects are more complex (see Fig. 3). We studied two oxygen interstitials, a split interstitial (O si ) on the O(I) site and a three-fold coordinated interstitial (O i ). For gallium interstitials, we considered two different configurations. In the V i Ga interstitial one gallium is removed from the Ga(I)-site and the second Ga(I) moves to an interstitial position with octahedral coordination. In the second configuration (Ga i ) we add one gallium atom with octahedral coordination into an interstitial position such that two nearby Ga(I) gallium atoms are pushed away from the interstitial gallium. The  Table III. From these defects only Ga i is donor-like near CBM while the gallium interstitial V i Ga is similar to simpler gallium vacancies and acts as a deep acceptor for most of the Fermi energy range. The interstitial configurations are shown in Appendix D. V i Ga can be considered as defect complex of a gallium vacancy and an interstitial but we have labeled it as an interstitial because the defect is more complex than the straightforward vacancy defects in Table II. Our results agree qualitatively and quantitatively with the existing literature for both intrinsic vacancy and interstitial defects. Our transition levels are consistently lower than those reported in Ref. 24, which is most likely due to the different amount of exact exchange in the HSE06 functional (32 % in Ref. 24 and 35 % in this work) and therefore a different bulk band gap of Ga 2 O 3 . On the experimental side, efforts are ongoing to identify point defects in Ga 2 O 3 [24,25]. However, thus far, no clear assignments have been possible.

C. Boron defects
Next we turn to boron point defects. We did initial calculation for neutral defects with the PBE functional, which are shown in Appendix B. PBE and HSE06 give the same formation energy ordering for neutral defects. We therefore scanned a variety of neutral defects with PBE. A clear picture emerges: 4-fold coordinated boron defects are the lowest in energy. We then picked three substitutional defects on Ga-sites with one or two borons and further investigated them with HSE06.
The boron defect geometries are shown in Fig. 4 and the corresponding formation energies in Fig. 5 for three Table II. Transition levels of vacancy defects. All energies (in eV) are given with respect to the conduction band minimum (CBM). The transition level is the energy at which two defect charge states, q and q , are in equilibrium. Reference (Ga 2 O 3 )). Such intermediate conditions are likely closer to the experimental growth conditions than either of the two extreme limits.
Boron preferably incorporates into the tetrahedrally coordinated Ga(I) site. The neutral B Ga(I) substitutional defect is very stable and does not introduce charge states into the band gap. Boron on the Ga(II) site, B Ga(II) , is not able to maintain the 6-fold coordination of the substituted gallium due to its much smaller ionic size. This leads to a larger relaxation of the surrounding atoms such that B Ga(II) becomes 3-fold coordinated and introduces a dangling bond on one of the neighboring oxygen atoms. In this site, boron can therefore act as donor with a ε(+1/0) transition state at 1.29 eV above the VBM.
Another interesting boron defect is the two-boron complex on the Ga(II) site (2B Ga(II) ) shown in Fig. 4. Each boron is 4-fold coordinated, which makes the formation energy competitive to the other two boron defects we discussed. However, since two boron atoms replace one gallium atom in this defect, the constraint on the boron chemical potential (µ B < µ Ga ) makes it hard for this defect to compete with gallium vacancies. For this reason, it is unlikely that this two-boron defect complex forms without forming B 2 O 3 first. Similar two boron structures were constructed on the Ga(I) and interstitial sites but they were not 4-fold coordinated thus resulting in considerably higher formation energies.
Next, we address the range of boron chemical potential, in which boron defects form preferentially. In Fig. 5, the boron chemical potential is as boron rich as possible without forming ). The range of feasible boron chemical potential depends on the competition between gallium vacancies and boron substitutionals, which in turn depends on the Fermi-level position during growth.
Clearly the incorporation of neutral borons on gallium sites, especially Ga(I), is the most preferable way of doping. Neutral boron defects are preferable as we are not interested in making electronically active defects but incorporating boron as a neutron active material.
In experimental crystal growth, the Fermi level is suspected to be pinned deep below the CBM [24,25] (e.g. 0.8 ∼ 2.6 eV below CBM in [24]). Under such conditions, boron incorporation is relatively likely in the intermediate chemical potential conditions depicted in the middle panel of Fig. 5. Also in oxygen-rich conditions, boron can be incorporated, but it starts to compete with gallium vacancies. Gallium-rich growth conditions are not favorable for B implantation. To make a more refined assessment on B incorporation in Ga 2 O 3 one would also have to consider the role of hydrogen [24,39], with which B competes [40].

IV. CONCLUSION
We have investigated boron related point defects in β-Ga 2 O 3 with DFT for a possible use of the material in solid-state neutron detectors. We found that boron preferably incorporates onto 4-fold coordinated gallium sites. Such boron defects are electronically neutral and do not introduce trap states in the band gap. Larger boron complexes have similar formation energies, but are unlikely to form due to competition with the B 2 O 3 formation. Intermediate growth regimes between the Ga-rich and O-rich limits are most conducive to boron incorporation. ACKNOWLEDGMENT We thank F. Tuomisto, V. Havu, S. Kokott and D. Golze for fruitful discussions. The generous allocation of computing resources by the CSC-IT Center for Science (via Project No. ay6311) and the Aalto Science-IT project are gratefully acknowledged. This work was supported by the Academy of Finland through its Centres of Excellence Programme under project number 284621, as well as its Key Project Funding scheme under project number 305632.

Appendix A: Chemical potentials
For completeness, we report the total energies used for calculating the chemical potentials in Table IV. For gallium, we used gallium metal in the orthorhombic structure with 8 atoms per unit cell as reference. The calculations were performed with a 8 × 8 × 8 Γ-centered kpoint mesh. Oxygen is referenced to the oxygen molecule