Boron doping in gallium oxide from first principles
BBoron doping in gallium oxide from first principles
Jouko Lehtomäki , Jingrui Li and Patrick Rinke Department of Applied Physics, Aalto University, 00076 AALTO, Finland
Abstract.
We study the feasibility of boron doping in gallium oxide (Ga O ) forneutron detection. Ga O is a wide band gap, radiation-hard material with potentialfor neutron detection, if it can be doped with a neutron active element. We investigatethe boron-10 isotope as possible neutron active dopant. Intrinsic and boron induceddefects in Ga O are studied with semi-local and hybrid density-functional-theorycalculations. We find that it is possible to introduce boron into gallium sites atmoderate concentrations. High concentrations of boron, however, compete with theboron-oxide formation. a r X i v : . [ c ond - m a t . m t r l - s c i ] J u l oron doping in gallium oxide from first principles
1. Introduction
Gallium oxide (Ga O ) is a wide gap semiconductor (band gap E g ∼ O forneutron detectors. There is a growing need for neutron detectors with low-powerrequirements, compact size and reasonable resolution for, e.g., non-invasive neutronimaging of organic materials, like human tissue or wood [8], safeguarding and non-proliferation 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 ( He), a non-radioactive isotopeof 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 He. The world’s Hesupply is extremely scarce and depleting rapidly. Moreover, the large size of He-based detectors limits their portability and spatial resolution. Since He detectors arenot radiation-hard, they cannot be used in harsh environments like outer space, orfusion or nuclear reactors. For these reasons, semiconductor detectors have recentlyreceived increasing attention [9, 10, 13–18]. However, the materials requirements foroptimal energy, time and spatial resolution, detection efficiency, robustness and radiationhardness are daunting challenges [13], and there is currently no satisfying materialchoice nor commercially available semiconductor detectors. For this reason, we arehere exploring Ga O as potential neutron detector material.Solid state neutron detectors use neutron active elements, which convert neutronsto electronic excitation via a nuclear reaction. The ability of neutron active elements tocapture neutrons is measured by the neutron cross section. The boron isotope B hasthe largest neutron cross-section at 3840 barns, which is comparable to helium ( He)and larger than other candidates like lithium ( Li) and beryllium ( Be). Boron-basedneutron detectors have recently been demonstrated experimentally [15,17,18], but are farfrom commercialisation. Wide band-gap materials have also been investigated in solidstate neutron detectors, most notably gallium nitride (GaN) [14, 16]. Here, we considerbeta gallium oxide ( β -Ga O ) as a potential material for neutron detection, because β -Ga O is a radiation-hard wide band-gap material, and gallium has similar chemicalcharacteristics as boron which makes boron implantation on gallium sites favorable.The electronic structure of β -Ga O and the behavior of defects in the material haveattracted considerable interest and have been studied previously with density functiontheory (DFT) [1, 19–24]. Defects have been investigated as a source of the observedintrinsic n-type conductivity and for the possibility of p-type doping of β -Ga O foropto-electronic applications. Boron-related defects have not been previously studied in β -Ga O .In this work we investigated the possibility of boron doping with DFT. Withthe supercell approach, we calculated formation energies for simple point defects and oron doping in gallium oxide from first principles Figure 1.
Left: Conventional monoclinic unit cell of β -Ga O . All nonequivalentgallium and oxygen sites are color-coded. Right: Supercell of 160-atoms constructedfrom the conventional unit cell. complexes in β -Ga O in the diffuse doping limit. We studied both intrinsic defectsand boron defects to assess the feasibility of introducing boron into β -Ga O . Our workprovides insight into the limits of boron doping and the potential of β -Ga O for neutrondetection.The article is structured as follows. Section 2 reviews briefly the atomic structureof β -Ga O and outlines the computational details. In Section 3 we discuss the resultsof DFT calculations with a particular focus on boron doping in β -Ga O . Section 4concludes with a summary.
2. Computational Details β -Ga O has a monoclinic crystal structure with space group C2/ m . The unit cellcontains two nonequivalent gallium sites and three nonequivalent oxygen sites. Themonoclinic cell with 4 Ga O units (i.e., 20 atoms) is shown in Fig. 1. The five differentsites are labeled as Ga(I), Ga(II), O(I), O(II) and O(III). The gallium sites Ga(I) andGa(II) are tetrahedrally and octahedrally coordinated by O ions, respectively. TheO(III) site is four-fold coordinated, while both O(I) and O(II) are three-fold coordinated.An O(I) site has two Ga(II) and one Ga(I) as neighbors, while an O(II) has two Ga(I)and one Ga(II) neighboring sites.All defect calculations were carried out with the supercell approach [25] in thiswork. Point defects were introduced in a 160-atom supercell model of pristine β -Ga O ,i.e., 32 Ga O units. Following Ref. [25], we calculated the defect formation energy oron doping in gallium oxide from first principles E f (X q ) = E (X q ) − E (0) + E corr + q ( (cid:15) VBM + (cid:15) F ) − X i ∆ n i µ i , (1)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 potential ofthe i th species whose number varies by ∆ n i when defects are formed. ∆ n i is negativefor the removal of atoms (e.g., vacancies) and positive for the addition of atoms (e.g.,interstitials). (cid:15) F is the Fermi energy of Ga O , defined with respect to the valance bandmaximum ( (cid:15) VBM ). The q ( (cid:15) VBM + (cid:15) F ) term therefore accounts for the energy change uponremoval or addition of electrons when charge defects are formed.To remove spurious electrostatic interactions between supercells with chargeddefects, we included the Freysoldt-Neugebauer-Van de Walle (FNV) correction term E corr [26]. In the FNV scheme, we used a spatially averaged dielectric constant of (cid:15) ∼
10 [4, 27] which includes ionic and electronic screening [28]. There has beensome debate, if the electronic dielectric constant (cid:15) ∞ should be used instead for smallsupercells [29]. However, we observed that (cid:15) is the correct choice by extrapolatingsupercells to the infinite supercell limit (see Appendix Appendix D). Our findings arein agreement with those of Ingebrigtsen et al. [28].The chemical potentials for species i can be written as µ i = µ i + ∆ µ i , where ∆ µ i ≤ µ i is acquired from T = 0 K DFT calculation of the appropriatephase, e.g. gas phase of the O molecule for O, and solid metal Ga with space groupCmce for gallium. We incorporated the external environment through the temperatureand partial pressure dependence of the chemical potentials of the gas-phase species, i.e.,here only oxygen∆ µ O ( T, p ) = 12 { [ H + ∆ H ( T )] − T [ S + ∆ S ( T )] } + 12 k B T ln pp ! . (2)Here H and S are enthalpy and entropy at zero temperature, respectively. All valueswere referenced to 1 atm pressure and obtained from thermodynamic tables [30].We estimated the boron doping concentration c in various conditions with theArrhenius relation [31] c (X q ) = N site N config exp ( − G f (X q ) /k B T ) , (3)where X q is the configuration of a boron dopant, N site the number of dopant sites perunit volume and N config their configurational degeneracy factor. The Gibbs free energyis approximated as G f (X q ) ≈ E (X q ) − E (0) + E corr + q ( (cid:15) VBM + (cid:15) F ) − X i ∆ n i µ i ( T, p ) , (4) oron doping in gallium oxide from first principles µ Ga ( T, p ) ≈ µ Ga , but for oxygen we use µ O ( T, p ) = µ + ∆ µ O ( T, p ) from eq. (2).With this approximation, we took into account only the pressure- and temperature-dependence of the oxygen chemical potential and discarding other entropy contributionsfrom the bulk phases. Note that this is very simplistic approximation for the Gibbs freeenergy as it is almost the same as the formation energy (eq. (1)) but still useful [21].With this approximation, the only difference between the Gibbs free energy and thezero temperature formation energy is that the gas-phase chemical potentials have atemperature- and pressure-dependence via the ideal gas relation.All DFT calculations in this work were performed with the all-electron numeric-atom-centered orbital code fhi-aims [32–35]. We used the semi-local Perdew-Burke-Ernzerhof (PBE) functional [36] and the Heyd-Scuseria-Ernzerhof hybrid functional(HSE06) [37] to calculate the atomic and electronic structure of β -Ga O and defectstherein. PBE calculations were employed as reference to previous work and to test thesupercell dependence for charge corrections. For the final defect geometries, we alwaysused the HSE06 functional to avoid spurious delocalization effects in PBE, as observedfor, e.g., the oxygen vacancies in TiO [38]. We set the fraction of Hartree-Fock exchangein HSE06 to 35%, a value which has been previously used for Ga O [27]. This yieldsa 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 betweenaccuracy and computational cost. Scalar relativistic effects were included by means ofthe zero-order regular approximation (ZORA) [39].Considering the computational cost of HSE06 calculations, we carried out most ofour calculations with the cheaper “light” basis sets (which usually provide sufficientlyconverged energy differences) and used results with “tight” basis sets (which can betterprovide converged absolute energies) as reference. For light settings, we used the tier-1basis set for oxygen and gallium, but exclude the f function for gallium. For tightsettings, we use tier-2 for oxygen and the full tier-1 basis for gallium. Adding tier 2 forgallium did not improve the result for PBE. The tier-1 basis set for gallium is thereforeenough to achieve convergence. A Γ-centered 2 × × k -point mesh was used for the20-atom monoclinic unit-cell calculations, while for larger supercells (160-atom) we useda Γ-centered 2 × × k -point mesh. In pursuit of open materials science [40], we made theresults of all relevant calculations available on the Novel Materials Discovery (NOMAD)repository [41].
3. Results O and chemical potentials The optimized geometry of bulk β -Ga O is presented in Table 1 for the HSE06and PBE functionals. Band gaps and formation enthalpies have been included forcompleteness. The PBE functional overestimates the lattice constants compared toexperiment. Conversely, the HSE06 functional reproduces the experimental geometry oron doping in gallium oxide from first principles X Y N X M I LF Y Z F Z I E n e r g y F ( e V ) Figure 2.
HSE06 band structure of β -Ga O along the path defined in Ref. [43]. well and our results are consistent with those previously reported in the literature[19, 20, 25, 27, 29].The HSE06 band structure of β -Ga O is shown in Fig. 2. The band gap of 4.92 eVis indirect between a point in the I-L line for the VBM and the Γ-point for the conductionband minimum (CBM). The direct gap at the Γ-point is slightly larger (4.95 eV). Thefact that indirect transitions are weak makes β -Ga O effectively a direct band-gapmaterial. Table 1.
Lattice parameters ( a , b , c and β ) of bulk β -Ga O , as well as the bandgap ( E g ) and formation energy ( H f ) calculated with different DFT functionals. H f isgiven in eV per Ga O unit. Also listed are experimental (Exp.) results for the latticeparameters [42] and band gap [1] as reference. PBE HSE06 Exp. a [Å] 12 .
46 12 .
23 12 .
23 [42] b [Å] 3 .
08 3 .
05 3 .
04 [42] c [Å] 5 .
88 5 .
81 5 .
80 [42] β [°] 103 . . . E g [eV] 1.95 4.95 4.9 [1] H f [eV] -10.6 -10.1 -11.3 [42]We reference the gallium chemical potential µ to gallium metal and the oxygenchemical potential µ to the oxygen molecule O (see Appendix A for details). Thechemical potentials need to be in equilibrium (i.e, 2 µ Ga + 3 µ O = E (Ga O )), whichdefines the Ga-rich (∆ µ Ga = 0) and O-rich (∆ µ O = 0) limits. An important constrainton the boron chemical potential is the formation of boron oxide B O . The upper boundof the boron chemical potential is therefore 2 µ B + 3 µ O ≤ E (B O ). We use solid boron oron doping in gallium oxide from first principles µ B . We first investigate intrinsic point defects. We do this not only to validate ourcalculations against previous studies, but also to study the competition between intrinsicdefects and boron defects. Here we present only vacancy sites while in Appendix B weprovide calculations for other relevant intrinsic defects [28, 29].The most important transition states of vacancy defects are listed in Table 2. Thecharge transition levels of the oxygen vacancies (cid:15) (+2 /
0) are located deep below theCBM. Different coordinations yield slightly different transition states with the four-foldO(III) site being closest to the CBM. For n-type conditions (Fermi energy close to theCBM), the oxygen vacancies are therefore neutral while they would behave as donorsfor p-type conditions (Fermi energy close to the VBM). Conversely, gallium vacanciesact as deep acceptors for most of the Fermi energy range. Here the (cid:15) ( − / −
3) transitionstate 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 ahybrid functional treatment. In the PBE functional the extra electrons do not localize,resulting in a formation energy that is too low.
Table 2.
Transition levels of vacancy defects. All energies (in eV) are given withrespect to the conduction band minimum (CBM). The transition level is the energy atwhich two defect charge states, q and q , are in equilibrium. Reference [28] uses 32%fraction of exact exchange in HSE06 while in Ref. [29] 26% exact exchange is used withno range separation. Vacancy q/q Transition levelsite This work Ref. [28] Ref. [29]Ga(I) (-2/-3) -1.65 -1.76 -1.64Ga(I) (-1/-2) -2.21 -2.32 -Ga(II) (-2/-3) -2.06 -2.17 -2.12Ga(II) (-1/-2) -2.39 -2.50 -O(I) (+2/0) -1.38 -1.50 -1.71O(II) (+2/0) -2.11 -2.23 -2.29O(III) (+2/0) -1.24 -1.36 -1.56Our results agree qualitatively and quantitatively with the existing literature forsimple vacancy defects. Our transition levels are consistently lower than those reportedin Ref. [28], which is most likely due to the different amount of exact exchange in theHSE06 functional (32 % in Ref. [28] and 35 % in this work) and therefore a differentbulk band gap of Ga O . On the experimental side, efforts are ongoing to identify pointdefects in Ga O [28, 29]. However, thus far, no clear assignments have been possible. oron doping in gallium oxide from first principles Figure 3.
Structure of the boron defect sites in Ga O supercell: (a) Boron on Ga(II)-site with three-fold coordination, (b) boron on Ga(I)-site with four-fold coordinationc) two 4-fold coordinated boron atoms on the Ga(II) site. Ga, O and B atoms arecolored in light green, red and blue, respectively. Next we turn to boron point defects. We did initial calculations for neutral defects withthe PBE functional, which are shown in Appendix C. PBE and HSE06 give the sameformation energy ordering for neutral defects. We therefore scanned a variety of neutraldefects with PBE. A clear picture emerges: 4-fold coordinated boron defects are thelowest in energy. We then picked three substitutional defects on Ga-sites with one ortwo borons and further investigated them with HSE06.The boron defect geometries are shown in Fig. 3 and the corresponding formationenergies in Fig. 4 for three different chemical environments (O-rich, Ga-rich andintermediate conditions µ Ga = µ O = H f (Ga O )). Boron preferably incorporates intothe tetrahedrally coordinated Ga(I) site. The neutral B Ga(I) substitutional defect is verystable 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 galliumdue to its much smaller ionic size. This leads to a larger relaxation of the surroundingatoms such that B
Ga(II) becomes 3-fold coordinated and introduces a dangling bond onone of the neighboring oxygen atoms. In this site, boron can therefore act as donor witha 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. 3. Each boron is 4-fold coordinated, which makes the formationenergy competitive to the other two boron defects we discussed. Similar two boronstructures were constructed on the Ga(I) and interstitial sites but they were not 4-foldcoordinated thus resulting in considerably higher formation energies.Next, we address the range of boron chemical potential, in which boron defectsform preferentially. By combining the equilibrium condition of β -Ga O and therestriction of B O formation on the boron and oxygen chemical potentials, we arriveat ∆ µ B − ∆ µ Ga ≤ [ H f (B O ) − H f (Ga O )] = − .
28 eV, where H f is the heat offormation. The implication is that to prevent the formation of B O , the chemicalpotential of boron must always be lower than that of gallium µ B ≤ µ Ga . Thus the mostboron rich environment is ∆ µ B = − .
28 eV + ∆ µ Ga . oron doping in gallium oxide from first principles − F o r m a t i o n e n e r g y ( e V ) Ga-rich
Fermi energy ε F (eV) − Intermediate − V O(I) V O(II) V O(III) O si V icGa(I) B Ga(I) B Ga(II) B Ga(II)
O-rich
Figure 4.
Defect formation energies E f for multiple intrinsic defects and the mostimportant boron defects. The chemical potential of Gallium is µ Ga = H f (Ga O ) forthe intermediate case. The boron chemical environment is set to ∆ µ B = − .
28 + ∆ µ Ga which gives the lowest possible formation energies for boron related defects whilepreventing formation of B O . See text for more details. In Fig. 4 we show intrinsic defects and boron defects in different chemicalenvironments, for which the boron chemical potential obeys ∆ µ B = − .
28 eV + ∆ µ Ga .Clearly the incorporation of neutral borons on gallium sites, especially Ga(I), is themost preferable way of doping. Boron complexes with multiple boron atoms are notfavored, since the penalty term of not forming B O suppresses them. Furthermore,neutral boron defects are preferable as we are not interested in making electronicallyactive defects, but incorporating boron as a neutron active material. We now perform a semi-quantitative analysis of boron doping based on the borondefects on gallium sites. Our main goal is to ascertain, if we can introduce significantconcentrations of boron for neutron detection. We are aiming for a boron concentrationof 10 cm − . This number is estimated based on experiments performed for GaN [15],which demonstrated neutron detection in boron-doped GaN for a boron density of5 . × cm − . The density of B is 10 × cm − considering a B abundanceof ∼ N config in the Arrhenius relation in eq. (3) isequal to 1 and the site density N site for both gallium sites is 1 . × cm − . Inserting theformation energies of the boron defects shown in Fig. 4 into the Arrhenius relation revealsthat the concentration ratio B Ga(II) / B Ga(I) = exp (cid:16)h E (cid:16) B Ga(I) (cid:17) − E (cid:16) B Ga(II) (cid:17)i /k B T (cid:17) isbetween 2 . × − and 6 . × − for temperatures between 600 K and 2100 K, whichis the relevant range for doping and Ga O crystal growth. We therefore only consider oron doping in gallium oxide from first principles 7 H P S H U D W X U H . % R U R Q F R Q F H Q W U D W L R Q F P % 2 U H J L P H * D 2 U H J L P H H 9 H 9 H 9 H 9 H 9 H 9 Figure 5.
Concentration of boron defect B
Ga(I) as a function of growth temperaturewhere different lines have different chemical environments described in terms ofdifference ∆ µ Ga − ∆ µ B . Concentrations are calculated with the Arrhenius relation(eq. (3)). Boron oxide is a limiting factor ∆ µ Ga − ∆ µ B ≥ .
28 eV marked with adashed line. B Ga(I) in the following. Similarly, 2B
Ga(II) is also excluded from further consideration asit has a considerably higher formation energy than B
Ga(II) in all chemical environmentswhere the formation of B O is unfavorable.First we investigate the boron concentrations as a function of temperature ina chemical environment optimal for boron implantation. The formation energy ofB Ga(I) depends on the chemical environment through the difference in the galliumand boron chemical potential µ Ga − µ B . This is further constrained by the formationof the competing B O phase, which results in the inequality ∆ µ Ga − ∆ µ B ≥ [ H f (Ga O ) − H f (B O )] = 1 .
28 eV that guarantees that the formation of B O isunfavorable.In Fig. 5, we plot the boron concentrations for different chemical environmentsas a function of temperature for growth temperatures from 600 K up to 2100 K.Higher temperatures favor boron incorporation and the boron concentration increaseswith growth temperature. Furthermore, boron rich conditions (i.e. small values of∆ µ Ga − ∆ µ B ) are more conducive to boron incorporation than gallium rich (high values).Unfortunately, the divider line of ∆ µ Ga − ∆ µ B = 1 . eV implies that in reality the Bdopability might be quite low. Even at the highest crystal growth temperatures we arelimited to a boron concentration of 2 . × cm − ( ∼ . cm − . Growth methods that extendinto the B O regime, but suppress the formation of boron oxide, would be beneficial.Finally, we make a connection between the boron chemical potential and the oxygenenvironment. In Fig. 6, we plot the boron chemical potential as a function of theoxygen partial pressure. We do not convert ∆ µ B into a partial pressure, since boron oron doping in gallium oxide from first principles µ B has to reduce withincreasing oxygen partial pressure. The relation arises from the fact that the galliumchemical potential is tied to the oxygen chemical potential via equilibrium conditions.The boron concentration depends on Gibbs free energy (4) via Arrhenius relation (3)where the chemical potentials are ∆ µ Ga − ∆ µ B which can be then transformed intoexpression H f (Ga O ) − ∆ µ O − ∆ µ B via equilibrium condition of gallium and oxygenchemical potential. Higher partial pressures imply higher oxygen chemical potential,and in order to keep the boron concentration constant, the boron chemical potentialhas to be lowered.As a side note, there is a distinct possibility that low O pressures are not accessibledue to formation of gallium suboxide (Ga O) which makes β -Ga O unstable [21, 44].A possible formation of gallium suboxide would depend on the growth method and wedo not explore this phenomenon further in this context.In Fig. 6 we also marked the B O growth regime. It is apparent that meaningfulboron concentrations fall into this B O regime at lower growth temperatures. Onlyat 1200 K and above we can obtain reasonable concentrations near the B O limit.Acquiring even boron concentrations of 1 . × cm − ( ∼
1% of Ga(I) sites) wouldrequire going above the B O limit even for high temperatures. To stress the limitation,we calculated the required partial pressures with boron gas B as the boron reference.For a temperature of 1200 K and oxygen partial pressures p (O ) above 10 − bar, thepartial pressure of B would have to be below 10 − bar: the boron environment wouldhave to be extremely poor even in oxygen poor conditions, which are also limited dueto stability of gallium oxide. Such low amounts of boron or oxygen would also limit thegrowth/doping rate.From these results, it is apparent, that it is challenging to introduce highconcentrations of boron into β -Ga O without formation of B O . For neutron detectorsit is possible to enhance the neutron activity by constructing thicker layers of thematerial to obtain a higher number of neutron active atoms, but here we do not exploretechnical device details. Compared to previous experimental results (5.12 × cm − ),achievable boron concentrations appear to be quite moderate in Ga O according to ourcalculations.
4. Conclusion
We have investigated boron related point defects in β -Ga O with DFT for a possibleuse of the material in solid-state neutron detectors. We found that boron preferablyincorporates onto 4-fold coordinated gallium sites. Such boron defects are electronicallyneutral and do not introduce trap states in the band gap. Larger boron complexeshave similar formation energies, but are unlikely due to their competition with B O formation. The Ga-rich growth regime turns out to be the most conducive to boronincorporation. oron doping in gallium oxide from first principles p O2 E D U % R U R Q F K H P L F D O S R W H Q W L D O B H 9 7 . % 2 / L P L W H H H H p O2 E D U % R U R Q F K H P L F D O S R W H Q W L D O B H 9 % 2 U H J L P H 7 . p O2 E D U % R U R Q F K H P L F D O S R W H Q W L D O B H 9 7 . Figure 6.
Contours of B
Ga(I) concentration as a function of oxygen partial pressuregiven by eq. (2). Limiting boron oxide is marked as a dashed green line and the areafavorable for B O formation is marked with light grey. Boron can be introduced as a substitutional defect to gallium sites in meaningfulconcentrations, but the concentrations are still modest compared to previous boron-based neutron active materials, mostly due to the limitations imposed by B O . Thelimitation might likely inhibit introducing boron also to other oxide materials such asIn O . The situation would be improved, if growth methods could be extended into theB O stability region. Acknowledgment
We thank F. Tuomisto, V. Havu, S. Kokott and D. Golze for fruitful discussions. Thegenerous allocation of computing resources by the CSC-IT Center for Science (viaProject No. ay6311) and the Aalto Science-IT project are gratefully acknowledged.This work was supported by the Academy of Finland through its Centres of ExcellenceProgramme under project number 284621, as well as its Key Project Funding schemeunder project number 305632.
Appendix A. Chemical potentials
For completeness, Table A1 lists the DFT-calculated energies of several relavant systemswhich were used for calculating the chemical potentials. For gallium, we used Ga metalin the orthorhombic structure with 8 atoms per unit cell as reference. The referencefor oxygen is the O molecule. Boron is referenced to its α phase with a rhombohedralcrystal structure with 12 atoms in a unit cell. For boron oxide (B O ), we took the α -phase with 15 atoms per unit cell [45]. The calculations for Ga, B and B O werecarried out using a 8 × ×
8, 2 × × × × k -point mesh. oron doping in gallium oxide from first principles Table A1.
Reference systems used in the calculations of the chemical potentials. Foreach system the energy is given per formula unit except for gallium and oxygen whereit is given per atom.
System Energy (eV) System Energy (eV)Ga -53183.059 B -676.609O -2046.547 B O -7505.521Ga O -112515.856 Appendix B. Intrinsic defects
The interstitial defects in Ga O are more complex than the single vacancies (seeFig. B1). We studied two oxygen interstitials, a split interstitial (O si ) on the O(I) siteand a three-fold coordinated interstitial (O i ). For gallium interstitials, we considered twodifferent configurations. In the V iGa 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 coordinationinto an interstitial position such that two nearby Ga(I) gallium atoms are pushed awayfrom the interstitial gallium. The chosen transition levels are listed in Table B1. Fromthese defects only Ga i is donor-like near CBM while the gallium interstitial V iGa is similarto simpler gallium vacancies and acts as a deep acceptor for most of the Fermi energyrange. The interstitial configurations are shown in Appendix E. V iGa can be consideredas defect complex of a gallium vacancy and an interstitial but we have labeled it as aninterstitial because the defect is more complex than the straightforward vacancy defectsin Table 2. Our results agree qualitatively and quantitatively with the existing literaturefor both intrinsic vacancy and interstitial defects, see Ref. [28]. Table B1.
Transition levels of interstitial defects. All energies (in eV) are given withrespect to the CBM.
Defect q/q Transition levelThis work [28]O si (+1/0) -3.08 -3.26O i (-1/-2) -1.20 -1.23V i Ga (-2/-3) -2.46 -2.55V i Ga (-1/-2) -2.73 -2.82V i Ga (0/-1) -3.00 -3.29Ga i (+3/+1) -0.69 -0.60 oron doping in gallium oxide from first principles Figure B1.
Formation energies for intrinsic vacancy and interstitial defects in β -Ga O for Ga-rich (left) and O-rich (middle) conditions as a function of the Fermienergy. The interstitial locations are shown on the right. Appendix C. Boron defects with the PBE functional
In Table C1 we tabulate neutral defects calculated with the PBE and HSE06 functional.The formation energies are given for the Ga-rich ( µ Ga = 0 eV) and boron rich( µ B = − .
17 eV) limit. The gallium and oxygen vacancies are listed for referenceto demonstrate that they have the same energetic ordering as neutral vacancies withthe HSE06 functional.Boron defects B
Ga(I) and B
Ga(II) are substitutional defects on Ga-sites. Morecomplex substitutional defects are (2B)
Ga(II) , (2B)
Ga(I) and (3B) − (2Ga(II)), in whichtwo or three boron atoms replace Ga atoms. The B i interstitial has a lower formationenergy than the (2B) − Ga i interstitial, in which a gallium atom moves to an interstitialsite and the vacant Ga-site is filled with two substitutional borons. Appendix D. Electrostatic corrections
We verified the FNV corrections for the Ga(II) vacancy in two charge states by anexplicit supercell convergence with the PBE functional. The results are shown inFig. D1. The structures are multiples of the unit cell, which have been relaxed after theremoval of one gallium in the Ga(II)-site. For The FNV correction we use a dielectricconstant ε of 10. Applying the FNV correction results in horizontal lines with formationenergies that are independent of the supercell size. Appendix E. Interstitial defects in β -Ga O In Fig. E1 we show the atomic configurations for the interstitial defects. The structureof vacancies is straightforward and therefore not shown for brevity. oron doping in gallium oxide from first principles Table C1.
The formation energies (eV) of neutral Boron defects and vacanciescomputed with PBE and HSE06 functional. See text for details.
Defect E f (PBE) E f (HSE06)V Ga(I)
Ga(II)
O(I)
O(II)
O(III)
Ga(I) -0.196 1.240B
Ga(II)
Ga(II) i Ga(I) − (2Ga(II)) 4.344 -(2B) − Ga i Ga(II), -3Ga(II), -2
Figure D1.
Defect formation energies for V
Ga(II) in the − − References [1] Mohamed M, Janowitz C, Unger I, Manzke R, Galazka Z, Uecker R, Fornari R, Weber J R, VarleyJ B and Van de Walle C G 2010
Applied Physics Letters oron doping in gallium oxide from first principles Figure E1.
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