Spin Polarization and Magnetic Properties of V_{Ga}O_{N} and V_{Ga}O_{N}In_{Ga} in GaN: GGA+U Approach
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Abstract
Electronic structure of a defect center containing the gallium vacancy and substitutional oxygen atom at nitrogen site (V_{Ga}O_{N}) in zinc blende and wurtzite GaN was analyzed within GGA+U approach. The +U term was applied to d(Ga), p(N), p(O), and d(In). Neutral V_{Ga}O_{N} is in the stable high spin state with spin S = 1. The defect structure is strongly dependent on geometry of the defect and the charge state. Two spin structures, which arise due to two different configurations in V_{Ga}O_{N}, with O_{N} either along the c-axis or in one of three equivalent tetrahedral positions in wurtzite structure were analyzed. The weak ferromagnetic coupling between centers was found. The strength of magnetic coupling is increased when there is a complex containing V_{Ga}O_{N} with additional substitutional indium atom at the second neighbor to vacancy gallium site (V_{Ga}O_{N}In_{Ga}). Magnetic coupling between V_{Ga}O_{N}In_{Ga} is ferromagnetic due to strong spin polarization of p electrons of the nearest and distant nitrogen atoms.
Keywords
GaN In doping Complex V_{Ga}O_{N} Magnetic coupling LDA+U calculationsIII-nitride materials such as GaN and InGaN have found their applications in advanced solid-state lighting technologies [1, 2, 3] and optoelectronic devices including diodes or solar cells [4, 5, 6]. Moreover, the ferromagnetism (FM) in GaN or InN without doping by transition metal atoms was recently observed [7, 8, 9, 10, 11]. This FM was ascribed to the formation of native defects, such as cation vacancies or their complexes. For example in Ref. [9], analysis of the characteristics of hysteresis curves in irradiated GaN showed that the coercive field increases in line with the increase of concentration of gallium vacancy (V_{Ga}). The stable FM was discovered in n-type GaN, and it can be due to the presence of non-intentional donors, such as oxygen.
Measurements using positron annihilation spectroscopy of V_{Ga} and its complexes containing the gallium vacancy and substitutional oxygen atom at nitrogen site V_{Ga}O_{N} have found them to be dominating defects in as-grown n-GaN [12, 13, 14, 15, 16, 17, 18, 19, 20, 21]. Both defects were intensively studied in experiments for their optical properties [12, 13, 16, 17, 19, 21] indicating defects as possible sources of the green (GL) and yellow (YL) luminescence in GaN. In Ref. [12], Son et al. detect spin-polarized V_{Ga}O_{N} by electron paramagnetic resonance and suggest that YL and GL bands can be explained by 0/− and −/−2 V_{Ga}O_{N}–optical transition levels. Moreover, in Ref. [13], two electronic structures were observed, which arise due to two different configurations of V_{Ga}O_{N} in wurtzite (w) GaN, one with O_{N} either along the c-axis (axial configuration, referred below as a) and other one in one of three equivalent tetrahedral positions (basal configuration, referred below as b).
However, there is not unequivocal agreement on the structure of V_{Ga} and its complexes—both experimental [12, 13, 16, 17, 19, 21, 22] and theoretical [14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32] results are divergent and sometimes contradictory. For example, the prediction of multiband luminescence due to the difference in geometries of V_{Ga}O_{N} was demonstrated in Ref. [32]. Nevertheless, the magnetic properties for both a- and b-configuration of V_{Ga}O_{N} were not discussed there. Moreover, various aspects of magnetism in GaN are not yet fully elucidated. There are few works which analyze magnetic coupling of V_{Ga}-V_{Ga} [24, 25, 27, 31] or V_{Ga}O_{N}-V_{Ga}O_{N} [26].
In fact, the calculated defect structure strongly depends upon the used exchange-correlation functional [14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34]. V_{Ga}O_{N} in III-N nitrides were theoretically investigated by local density approximation (LDA)/generalized gradient approximation (GGA) [14, 15, 20] or the hybrid functional (HyF) calculations [16, 17, 18, 21, 26, 32, 33]. Finally, GGA+U approach with the +U term imposed on p(N) in an examination of V_{Ga} was also tried [30, 31]. Both the GGA+U [30, 31] and the HyF [16, 17, 18, 21, 22, 26, 27, 28, 32, 33, 34] methods enhance spin polarization V_{Ga} or its complexes and push defect levels deeper into the band gap in comparison with LDA/GGA [14, 15, 20, 29]. According GGA+U calculations, V_{Ga} demonstrates the U-induced splitting of partly occupied multiplets, and “negative-U_{eff}” behavior. The results for U(N) = 5 eV [30, 31] reproduce well the HyF calculations [17, 18, 23, 27, 28, 32, 33]. Both approaches give rise to stronger localization of the wave functions. This, in turn, alters the character of magnetic coupling between defects [31]. In fact LDA/GGA calculations are erroneous as they lead to overly long-ranged magnetic interactions [24, 25, 27, 31].
The increasing localization of the defect wave function has the opposite effect on the stability of the local magnetic moment and on the collective magnetization: in the former case, the increased localization stabilizes the high-spin (HS) state, while the coupling through the overlap of the wave functions of the neighboring defects is decreased [27, 30, 31, 35]. For both the V_{Ga} and the V_{Ga}O_{N} complexes, strong localization can lead to stable local spin moments [26, 27, 35] but it does not guarantee automatically a stable interaction between them [31, 35]. Partial delocalization of defect-induced bands may reduce the stability of HS state defect but also be responsible for the long-range magnetic interactions. This stabilization can be due to p-d exchange interaction of impurity like Mn [36] or the spin polarization of p electrons in low In-content InGaN.
Both In concentration and microscopic In distribution strongly influence the electronic structure and physical properties of InGaN [37]. The localization of the valence band maximum (VBM) states and the domination of the light emission of InGaN with low In content were observed [37]. Similar to GaN, V_{Ga}-complexes were also suggested to be the important non-radiative defects in InGaN quantum well [38, 39]. Strong effects in electronic structure of V_{Ga}O_{N}-hydrogen complex were found in p-type InGaN for high In content [33].
To check the hypothesis that magnetic coupling between V_{Ga}O_{N}-complexes can be more stable in InGaN, in the present paper, we study V_{Ga}O_{N}-structure in zb (zinc blende)– and w-GaN and InGaN using GGA+U approach. After presenting the details of calculations in Sect. 1, the justification of the chosen approach is underlined. Next, we present results of calculations of the formation energies of defects (Sect. 1.3). Sections 2.1–2.3 contain the results of calculations of electronic structure of V_{Ga}O_{N} in GaN and transition levels (Sect. 2.2) and electronic structure of InGaN (Sect. 2.3). The magnetic coupling between defects is analyzed in Sect. 2.4. Next, the structural relaxation is discussed in Sect. 3. Finally, Sect. 4 summarizes the results.
1 Method of Calculations
1.1 Details of Calculations
Calculations based on the density-functional theory were performed using the ultrasoft pseudopotentials [40], the Perdew-Burke-Ernzerhof GGA exchange-correlation potential [41], including the +U term implemented in the QUANTUM-ESPRESSO code [42] along the theoretical framework developed in Ref. [43]. We employed ultrasoft atomic pseudopotentials and chose 3d, 4s, and 4p orbitals for Ga; 4d, 5s, and 5p for In; 2s and 2p for N; and O as valence orbitals. The plane wave basis with the kinetic energy cutoff (E_{cut}) of 40 Ry provided a convergent description of the analyzed properties. The Brillouin zone summations were performed using the Monkhorst-Pack scheme with a 2 × 2 × 2 k-point mesh [44]. Methfessel-Paxton smearing method with the smearing width of 0.136 eV was employed for obtaining partial occupancies. The zb 216- and 512-atoms and w 128-, 192-atoms supercells were considered, and ionic positions were optimized until the forces acting on ions were smaller than 0.02 eV/Å. The spin-orbit coupling was neglected. Formation energies were calculated according to Ref. [45] for N-rich conditions [30]. The +U corrections were imposed on d(Ga) and p(N) [30, 31]; d(Ga), p(N), and p(O); and d(Ga), p(N), p(O), and d(In) for pure GaN, GaN:V_{Ga}O_{N}, and InGaN:V_{Ga}O_{N}, respectively.
1.2 Justification for the Chosen Method
1.2.1 Impact +U Correct on Band and Crystal Structure of GaN
We found that U(Ga) = 3.0 eV along with U(N) = 5 eV reproduces the experimental E_{gap} of 3.2 and 3.4 eV for both zb- and w-GaN [49] (Fig. 1, (3, 3′)) and the binding energy of Ga 3d level centered about 15.5 eV below the VBM—in agreement with Ref. [50]. These values are also in agreement with HyF results [51]. Such an underestimation of the band gap and band structure follows from the sublinear dependence of the LDA/GGA total energy on the occupation [43]. Moreover, the sensitivity of E_{gap} in both U(Ga) and U(N) is explained by the orbital compositions of both the VBM and the minimum of the conduction band (CBM).
GGA+U calculations give the lattice constants a_{zb} = 4.57 Å and (a_{w} = 3.19 and c_{w} = 5.2 Å) for zb- and w-GaN, respectively. These values are very close to the experimental data of a_{zb}^{exp} = 4.55 Å [52, 53] and (a_{w}^{exp} = 3.18 and c_{w}^{exp} = 5.18 Å [52]) for zb- and w-structure, respectively.
1.2.2 Impact of +U Correction on Electronic Structure of GaN with Defect
The +U corrections were imposed on d(Ga), p(N), p(O), and d(In). U(O) = U(N) = 5 eV, and U(In) = U(Ga) = 3 eV for GaN. However, including U(O) and U(In) terms have a small effect on V_{Ga}O_{N} structure due to small contributions of p(O) and d(In) orbitals to the defect states (see Sects 2.1 and 2.3 for the detailed discussion of electronic structure of both GaN:V_{Ga}O_{N} and GaN:V_{Ga}O_{N}In_{Ga}).
1.2.3 GGA+U vs HSE Results
It was noted above that HyF [16, 17, 18, 21, 22, 26, 27, 28, 32, 33] calculated V_{Ga}-structure is in agreement with GGA+U results for U(N) = 5 eV [30, 31]. Here, in order to verify the agreement further, we perform calculations for Heyd, Scusseria, and Ernzherof functional, based on the PBE functional where parameter α is a fraction of the exchange that is replaced by Hatree-Fook exchange [54]. Calculations were done for isolated V_{Ga}O_{N} and V_{Ga}O_{N}-V_{Ga}O_{N} (3NNs axial configuration in notation of Sect. 2.4) for w-GaN. Screening parameter α = 0.25 was set to reproduce band gap of ~ 3.0 eV. Nevertheless, the results indicate that energies of spin polarization (ΔE^{PM−FM}) (defined in Sect. 2.1) agree to within 0.05 eV or less, and energies of magnetization (ΔE^{AFM−FM}) (defined in Sect. 2.2) agree to within 0.005 or less. That shows the good agreement for calculations of magnetic properties within these two approaches.
We note that the problem of choosing the α parameter for getting accurate defect levels is still an open issue [55, 56], as well choosing the U parameter in GGA+U approach [30].
1.3 Formation Energy of Defects
Formation energy of charged V_{Ga}O_{N} was calculated. One geometry of a-V_{Ga}O_{N} was considered for cubic GaN and two a- and b-V_{Ga}O_{N} configurations were analyzed for w-GaN. Because the aim was to understand the influence of In doping on spin-polarized properties, the number of configurations of complex V_{Ga}O_{N} was chosen: In_{Ga} as second nearest neighbor to V_{Ga} when forming O_{N}-In_{Ga} (referred below as o) or N-In_{Ga} (referred later as n) chains where these O_{N} and N are the nearest possible positions to V_{Ga} neighbors. Hence, the two geometries as referred as a-o- and a-n-V_{Ga}O_{N}In_{Ga} were considered for zb-crystal. And four geometry configurations referred as a-o-, a-n-, b-o-, and b-n-V_{Ga}O_{N}In_{Ga} were analyzed for w-GaN. In this work, the formation enthalpy of InGaN was calculated and the binding energy, E_{bind}, defined as a difference in the total energies of compounds that contain V_{Ga}O_{N}In_{Ga} or (V_{Ga}O_{N} and In_{Ga}) were taken into consideration.
The first two terms on the right-hand side are the total energies of the supercell with and without the complex, respectively. n_{i} is number with the +(−) sign corresponding to the removal (addition) of atoms. E_{VBM} is the energy of the VBM of bulk GaN, and ε_{F} is the Fermi energy referenced to this E_{VBM}. The energy E_{VBM} is determined from the total energy difference between the pure crystal with and without a hole at the VBM in the dilute limit by algorithm from Ref. [45]. μ_{i} are the variable chemical potentials of atoms in the solid, which in general are different from the chemical potentials μ_{i}(bulk) of the ground state of elements (Ga bulk, In bulk, and N_{2}, O_{2}). Chemical potentials of the components in the standard phase are given by total energies per atom of the elemental solids: μ(Ga bulk) = E_{tot}(Ga bulk), μ(In bulk) = E_{tot}(In bulk), while μ(N bulk) = E_{tot}(N_{2})/2 and μ(O bulk) = E_{tot}(O_{2})/2 (ΔH_{f}(NO) was neglected. In N-rich condition, μ(Ga) = E_{tot} (Ga bulk) + ΔH_{f}(GaN) and μ(In) = E_{tot} (In bulk) + ΔH_{f}(InN) are taken, where ΔH_{f} is the enthalpy of formation per formula unit, and it is negative for stable compounds. ΔH_{f} at T = 0 K is obtained by considering the reaction to form or decompose a crystalline GaN and InN from its components and dependent on an cohesive energy, E_{coh}, of Ga, In, N, and O. The obtained results for E_{coh} of Ga, N, and O were shown in Refs. [30, 57]. Calculated E_{coh} (In) and ΔH_{f}(zb-GaN), ΔH_{f}(InN) are 2.56 (2.5 [58]) and − 1.24 (−1.27), −0.36 (−0.32) eV [59]), (experimental values presented in brackets).
The last term, E_{correct}, includes two corrections. The first one, ΔE_{PA}, is the potential alignment correction of the VBM. The VBM in the ideal supercell and in the supercell with a (charged) defect differs by the electrostatic potential and is obtained by comparing the potential at two reference points far from the defect in the respective supercells with (P[D^{q}]) and without (P[0]) the defect, ΔE_{PA} = q(P[D^{q}] − P[0]). Second correction is an image charge correction as expressed by 2-order Makov-Payne form: \( {E}_{\mathrm{M}\mathrm{P}}=\frac{q^2{\alpha}_{\mathrm{M}}}{2{\varepsilon}_d{W}^{1/3}} \), where α_{M} is the lattice-dependent Madelung constant, which for hexagonal structure is 3.5, W is the supercell volume, and ε is the static dielectric constant. E_{MP} was calculated to be 0.2, 0.4 eV for charged defects (q = − 1, − 2). Results of calculations are presented in Sect. 2.2.
2 Results
This section summarizes the obtained results for the defect structure and formation energy and discusses magnetic interaction between defects.
2.1 Electronic Structure and Spin Polarization of V _{Ga}O_{N}
In the case of non-vanishing spin polarization, the exchange coupling splits e_{2} into spin-up e_{2↑} and spin-down e_{2↓} states by the splitting exchange energy defined as Δε_{ex} = ε(e_{2↓}) − ε(e_{2↑}), where ε is the energy of the defect level, and a_{1} into a_{1↑} and a_{1↓} (Fig. 3, the blue lines). The Δε_{ex}, in general, depends on the symmetry of defect, the charge state and U. The e_{2↓} and a_{1↓} of neutral V_{Ga}O_{N} in zb-GaN are localized in the band gap at 2.6 and 1.9 eV above the VBM, respectively. The e_{2↑} and a_{1↑} are resonances with the VBM (Fig. 3a). According to this point of view V_{Ga}O_{N} is a deep acceptor containing two holes.
As presented above in Section 2.3, in w-crystal, the defect can exists in two different geometries tagged as a- and b-V_{Ga}O_{N}. According to GGA+U calculations, single-electron level representations of electronic structures are different for a- and b-V_{Ga}O_{N} (Fig. 3 b, c, left panels). Structure of a-V_{Ga}O_{N} in w-GaN is similar to the one of zb-GaN, e_{2↓} and a_{1↓} are 2.45 and 1.65 eV with the respect to the VBM, and e_{2↑} and a_{1↑} are hybridized with the valence bands (Fig. 3b). Introducing O atom into basal plane of defect leads to strong symmetry perturbation, e_{2↓} is split by 0.5 eV into two a_{2↓(1)} and a_{2↓(2)} singlet states. Thus, “t_{2↓}” in this case is a composite band containing a_{1↓}, a_{2↓(1)}, and a_{2↓(2)} levels located about 1.4, 2.3, and 2.8 eV above the VBM, respectively (Fig. 3c).
The energies of V_{Ga}O_{N} levels strongly depend on the charge state q. Single-electron energy levels of V_{Ga}O_{N}^{q} for q = 0, − 1, and − 2 with their respective charge states are shown in Fig. 3. The physics behind the calculated electronic structure of charged defects is determined by the following counteracting effects [30]: (i) the intracenter Coulomb repulsion is dominant in the non-spin-polarized calculations. Without spin polarization, the energy of e_{2}, a_{1} increases by ~ 0.5–0.6 eV with the q changing from 0 to − 2 (the levels are shown in green color in Fig. 3); (ii) The effect of the value exchange splitting, for example, in zb-GaN with q changing from 0 to − 2, Δε_{ex} decreases from 2.6 to 0 eV; (iii) the U-induced potential which is attractive (repulsive) for occupied (unoccupied) orbitals [43]. This effect is clearly seen for e_{2↓} that decreases with q changing from 0 to − 1 (Figs.3 a, b). These results are in agreement with HyF calculations from Ref. [17] where negative-U_{eff} behavior of V_{Ga}O_{N} was observed.
Spin polarization energy, ΔE^{PM−FM} (in eV) and total local magnetic moment (in μ_{B}) of charged V_{Ga}O_{N} (a-, b-configurations) and V_{Ga}O_{N}In_{Ga} (a-o-, a-n-, b-o-, and b-n-configurations) in zb- and w-GaN
zb | w | zb | w | |||||||
---|---|---|---|---|---|---|---|---|---|---|
Defect | V_{Ga}O_{N} | V_{Ga}O_{N}In_{Ga} | ||||||||
q | 0 | − 1 | − 2 | 0 | − 1 | − 2 | 0 | − 1 | 0 | − 1 |
ΔE ^{ PM−FM} | 1.70^{a} | 0.38^{a} | 0^{a} | 1.62^{a} | 0.33^{a} | 0^{a} | 1.78^{a-o} | 0.50 ^{a-o} | 1.69 ^{a-o} | 0.50 ^{a-o} |
– | – | – | – | – | – | 1.8 ^{a-n} | 0.52 ^{a-n} | 1.73 ^{a-n} | 0.52 ^{a-n} | |
– | – | – | 1.86^{b} | 0.40^{b} | 0^{b} | – | – | 1.91 ^{b-o} | 0.41 ^{b-o} | |
– | – | – | – | – | – | – | – | 2.02 ^{b-n} | 0.48 ^{b-n} | |
μ _{ tot} | 2 | 1 | 0 | 2 | 1 | 0 | 2 | 1 | 2 | 1 |
Every considered geometry of the neutral V_{Ga}O_{N} is the magnetic centrum in HS state with the local magnetic moment μ_{tot} of 2 μ_{B}. For example, ΔE^{PM−FM}(V_{Ga}O_{N}^{0}) are 1.7, 1.62, and 1.86 eV for defects in zb-GaN, and a- and b-geometry in w-GaN, respectively (Table 1). V_{Ga}O_{N}^{−1} is the paramagnetic centrum with μ_{tot} = 1 μ_{B}. ΔE^{PM−FM}(V_{Ga}O_{N}^{−1}) are 0.38 and 0.3, 0.4 eV for defects in zb-GaN, and a- and b-geometry in w-GaN, respectively (Table 1). Finally, V_{Ga}O_{N}^{−2} is the non-spin-polarized centrum (in this case ΔE^{PM−FM} = 0). According to our calculations, b-configuration stabilizes the HS state in neutral V_{Ga}O_{N} by about 0.24 eV in comparison with a-geometry. This value is close to the energy gain of formation energies in b-V_{Ga}O_{N} and a-V_{Ga}O_{N} (see Sect. 2.2).
The geometry of the defect affects also the localization of the V_{Ga}O_{N} states. The effect is displayed in the plots of the density of spin polarization, Fig. 2a,e–f. Figure 2 indicates that the V_{Ga}O_{N} states are dominated in all cases by the localized and spin-polarized contribution of the three sp^{3} orbitals of the N nearest neighbors because O-atom is more electronegative than N-atom and two electrons with the opposite spins are located on sp^{3} oxygen orbital. In contrast to GGA method in which electrons that occupy for example, “t_{2↓}” are spread over four p orbitals of the nearest neighbors [31, 60], GGA+U calculations showed that the partial occupancy is avoided [31, 60]. Moreover, in the case of b-V_{Ga}O_{N}, one can observe the anisotropy for three sp^{3} of N orbitals (Fig. 2d, f). The U-induced symmetry breaking of e_{2} level, i.e., the quasi-Jahn-Teller effect, was observed in b-V_{Ga}O_{N}. In a-V_{Ga}O_{N} contributions of the three N neighbors to the V_{Ga} states of the vacancy, wave function are almost equal, whereas for b-V_{Ga}O_{N} the wave function is dominated by the two basal N ions located in the (x,y) plane, and the contribution of the remaining N ion is strongly reduced, see Fig. 2d–f.
2.2 Formation Energy and Transition Levels
The calculated E_{form} of V_{Ga}O_{N} assuming ε_{F} at the VBM (N-rich condition)
q | 0 | − 1 | − 2 |
---|---|---|---|
a-V_{Ga}O_{N}, zb-GaN | 1.35 | 3.1 | 5.2 |
a-V_{Ga}O_{N}, w-GaN | 1.64^{a} | 3.54^{a} | 5.73^{a} |
b-V_{Ga}O_{N}, w-GaN | 1.4^{b} | 3.5^{b} | 5.9^{b} |
2.3 Spin Polarization of V _{Ga}O_{N}In_{Ga}
In Sect. 1.3 the geometrical configurations of V_{Ga}O_{N}In_{Ga} were discussed. Although our calculations show that binding energy in such a complex is low, ~ 0.1–0.3 eV, the formation energy is 1.9 eV which is a little higher than in the case of V_{Ga}O_{N} and considerably lower than in the case of V_{Ga} [30, 34]. Because the formation energy for different geometries (a- or b-) is similar, both V_{Ga}O_{N} and V_{Ga}O_{N}In_{Ga} can exist in non-equivalent atomic configurations.
In this section, the results for calculated energy of spin polarization, local magnetic moment, spin density, and density of charge of V_{Ga}O_{N}In_{Ga} are presented. Moreover, the effect of addition of In impurity on the spin-polarized properties is analyzed by comparing these results with similar results for V_{Ga}O_{N}. Next, in order to get a clearer picture of the influence of In on electronic structure, the complex V_{Ga}O_{N}nIn_{Ga}, where n = 2, 3, and 4, is investigated in zb-GaN.
The defect states of V_{Ga}O_{N}In_{Ga} stem from the result of the interaction between the vacancy orbitals and the O- and In- impurity states.
According to GGA+U calculations, in zb-GaN, the electronic structure of V_{Ga}O_{N}In_{Ga} is similar to V_{Ga}O_{N}. There is only a difference in the defect energy levels of about 0.1–0.2 eV. With the increasing number of In content from 1 to 4, the level of e_{2} increases no more than 0.5 eV. It demonstrates the stability of defect symmetry in zb-structure.
Strong effect on the electron structure was observed for wurtzite crystal (see Fig. 3b, c, right panels). As with O-atom in basal plane, also in this case for both a- and b-V_{Ga}O_{N}In_{Ga}, “t_{2}” is split into three singlet states a_{1↓}, a_{2↓(1)}, and a_{2↓(2)} due to strong tetragonal perturbation generated by In atom in the crystal structure. a_{2↓(2)} of b-n-V_{Ga}O_{N}In_{Ga} is higher in energy, and it is a resonance state with the conduction bands (Fig. 3c).
Calculated ΔE^{PM−FM} values for different configurations of V_{Ga}O_{N}In_{Ga} are given in Table 1. According to our calculations, the inclusion of In_{Ga} stabilizes HS state of the defect by 0.1–0.16 eV in comparison to V_{Ga}O_{N} (see Table 1). For example, in zb-GaN, the differences in the spin polarization energies of V_{Ga}O_{N} and V_{Ga}O_{N}In_{Ga} are 0.08 and 0.14 eV, for q = 0 and − 1, respectively. The same energy gain takes place in w-structure, for example, the differences between ΔE^{PM−FM} (b-n-V_{Ga}O_{N}In_{Ga}) and (b-V_{Ga}O_{N}) are 0.16 and 0.08 eV, in cases q = 0 and − 1, respectively. Generally, the increase of ΔE^{PM−FM} is in agreement with the changes of electron structure. Shifting up the single-electron levels (a_{2↓(2)} in particular) to the CBM leads to the increase in the energy of exchange splitting, and therefore in the energy of ΔE^{PM−FM}. The above results obtained for V_{Ga}O_{N}In_{Ga} are typical: the stability of the HS state predicted for V_{Ga}O_{N}In_{Ga} is 10–12% higher than that predicted for V_{Ga}O_{N.}
The increase in discrepancy of ΔE^{PM−FM} was observed with the rising of In content. ΔE^{PM−FM} is 1.7, 1.78, 1.88, 1.9, and 1.8 for n = 0, 1, 2, 3, and 4. For n = 4, ΔE^{PM−FM} is smaller than for n = 2 and n = 3.
The contour (Fig. 2k, l) is plotted in the (100) plane and shows that a large contribution to the electron density comes from the nitrogen atoms and strong ionic bonds resulting from sp^{3} hybridization. The spherical symmetry around anions is observed, which indicates that the bonds in GaN:V_{Ga}O_{N}In_{Ga} are dominated by the ionic component.
By calculating the contributions of individual atoms projected onto relevant atomic orbitals to the total DOS (Fig. 4a), one finds that the main contribution comes from the defects states p(N) of the N nearest neighbors of V_{Ga} (Fig. 4b)—in agreement with Fig. 2. Both p(N) and p(O) orbitals also build the VBM of GaN (see Fig. 4b–d). The contribution of the d(In) orbitals to the spin density is non-negligible due to the substantial contribution of d(In) to the VBM and defect states (see Figs. 2 and 4f).
2.4 Magnetic Interaction of V _{Ga}O_{N} -V _{Ga}O_{N} and V _{Ga}O_{N}In_{Ga} -V _{Ga}O_{N}In_{Ga}
In the present section, we study V_{Ga}O_{N}-V_{Ga}O_{N} and V_{Ga}O_{N}In_{Ga}-V_{Ga}O_{N}In_{Ga} defect pairs and analyze the impact of crystal distortion on their properties by comparing the results of magnetic interaction calculations. The magnetic coupling between vacancy complexes is discussed as the possible origin of the experimentally observed FM in GaN. Electronic structure of a defect pair is determined by three factors: (i) the distance between vacancy complex, (ii) the relative orientations of complexes with respect to each other and to the crystal axes, and (iii) the charge state. V_{Ga}O_{N}-V_{Ga}O_{N} and V_{Ga}O_{N}In_{Ga}-V_{Ga}O_{N}In_{Ga} configurations were considered, in which the defects are the third nearest neighbors (3NNs) and fourth nearest neighbors (4NNs) with respective spatial separation of about ~ 5.2 and ~ 6.5 Å (it is a distance between V_{Ga} in the relaxed structure). In w-GaN, the defects can be located either in the same (x,y) basal plane perpendicular to the c-axis, which is referred to as the xy-case, or they can be oriented along the c-axis, which is denoted here as the c-case. Finally, we mention that the defect pair in such configurations has eight nearest neighbor atoms (two O and six N atoms). Because the goal of work was to analyze the FM in GaN, we do not consider here 1NNs configurations with the seven nearest neighbor atoms.
Energy of magnetic coupling ΔE^{AFM(FiM)-FM} together with total magnetic moment (in μ_{B}) of the complex in the charge state q (it is a sum q_{1} and q_{2,} where) obtained for zb- and w-GaN calculations. For simplicity, both ΔE^{AFM−FM} and ΔE^{FiM-FM} are denoted by ΔE^{AFM−FM} and the actual spin configurations and values of magnetic moment are given in the columns “μ”
q | Configuration | ΔE ^{ AFM−FM} | μ | Configuration | ΔE ^{ AFM−FM} | Μ |
---|---|---|---|---|---|---|
V_{Ga}O_{N}-V_{Ga}O_{N} | V_{Ga}O_{N}In_{Ga}-V_{Ga}O_{N}In_{Ga} | |||||
zb-GaN | ||||||
0 | a-a (3NNs) | 0.014 | 4 (FM) | a-o-a-o (3NNs) | 0.1 | 4 (FM) |
a-a (4NNs) | 0.0045 | 4 (FM) | a-o-a-o (4NNs) | 0.01 | 4 (FM) | |
– | – | – | a-n-a-n (3NNs) | 0.1 | 4 (FM) | |
– | – | – | a-n-a-n (4NNs) | 0.014 | 4 (FM) | |
− 2 | a-a (4NNs) | 0.001 | 2 (FM) | a-o-a-o (4NNs) | 0.006 | 2 (FM) |
– | – | – | a-n-a-n (4NNs) | 0.008 | 2 (FM) | |
w-GaN | ||||||
0 | xy-a-a (3NNs) | 0.01 | 4 (FM) | xy-a-n-a-n (3NNs) | 0.05 | 4 (FM) |
xy-a-a (4NNs) | 0.006 | 4 (FM) | xy-a-n-a-n (4NNs) | 0.008 | 4 (FM) | |
c-a-a 3 (3NNs) | − 0.016 | 0 (AFM) | c- a-n-a-n (3NNs) | − 0.1 | 0 (AFM) | |
c-a-a (4NNs) | − 0.002 | 0 (AFM) | c- a-n-a-n (4NNs) | − 0.007 | 0 (AFM) | |
xy-b-b (3NNs) | 0.09 | 4 (FM) | xy-b-n-b-n (3NNs) | 0.15 | 4 (FM) | |
xy-b-b (4NNs) | 0.003 | 4 (FM) | xy-b-n-b-n (4NNs) | 0.012 | 4 (FM) | |
c-b-b (3NNs) | 0.008 | 4 (FM) | c-b-n-b-n (3NNs) | 0.015 | 4 (FM) | |
c-b-b (4NNs) | 0.00015 | 4(FM) | c-b-n-b-n (4NNs) | 0.009 | 4 (FM) |
- (i)
V_{Ga}O_{N}-V_{Ga}O_{N} are spin-polarized and the coupling between defect spins is FM in both charge states. Magnetic moment is 4 and 2 μ_{B}, for q = 0 and − 2, respectively. The observed FM is weak with ΔE^{AFM−FM} energy of about 0.0045 and 0.001 eV, for q = 0 and − 2, respectively (for 4NNs pair). The obtained values are a little larger than those obtained by HSE06 approach in Ref. [26] due to shorter spatial separation between defects in our work.
- (ii)
The inclusion of In impurity modifies the coupling. The absolute values of the coupling strength are few times higher than they are for 4NNs V_{Ga}O_{N} and amount to 0.01–0.014 and 0.006–0.008 eV, for q = 0 and − 2, respectively.
- (i)
The dependence on geometry configuration was observed. For example, in xy- basal plane, the ground state of (V_{Ga}O_{N}-V_{Ga}O_{N}) ^{0} is FM with ΔE^{AFM−FM} of about 0.01, 0.006 eV (for 3NNs and 4NNs) and 0.09, 0.005 eV (for 3NNs and 4NNs) in a- and b-V_{Ga}O_{N} geometry, respectively.
- (ii)
Unlike zb-crystal, in wurtzite, the dependence of the coupling on the orientation in crystal of the pairs is rather strong: the total energy differences between the xy- and c-orientations are larger than the corresponding values of ΔE^{AFM−FM}. When defects are oriented along the c-axis, the magnetic interaction is AFM.
- (iii)
Similar to the results for zb-crystal, magnetic coupling of V_{Ga}O_{N}In_{Ga}-V_{Ga}O_{N}In_{Ga} is larger than of V_{Ga}O_{N}-V_{Ga}O_{N}. Generally, for all configurations, In doping facilitates the stability of FM phase in GaN by order of magnitude (Table 4).
The dominant mechanism of magnetic interaction between V_{Ga}O_{N} is determined by the interplay between the counteraction of the bonding-antibonding (BA) and the exchange of spin-up–spin-down (Δe_{ex}) splittings.
2.5 Crystal Structure zb- and w-GaN and Relaxation
But we note that structural distortions are more complex. Although, the displacements of the second and third neighbors are an order of magnitude smaller, the effect of atomic relaxations around defects, involving not only the nearest but also more distant neighbors, cannot be neglected. The states of the defect complexes are determined by the overlap of the N and O dangling bonds given by the N-N (N-O) distances. In ideal structure (after relaxation without defect), N-N is equal to 3.18 Å. In V_{Ga}O_{N}, the N-N and N-O dangling bonds are 3.55 and 3.48 Å, respectively. But in V_{Ga}O_{N}In_{Ga}, these values are shorter; three N-N are 3. 58, 3.52, and 3.48 Å, respectively, and N-O is 3.42 Å. It implies that the defect states are more localized, and the energy of spin polarization of such defects is higher.
3 Summary and Conclusions
In summary, spin states of V_{Ga}O_{N} and V_{Ga}O_{N}In_{Ga} complexes in both zb- and w-GaN, and magnetic coupling between them, were studied within GGA+U calculations. The U(Ga) =3 eV and U(N) = 5 eV terms were imposed on d(Ga) and p(N) leading to the correct band gap of GaN.
Charge states q from 0 to − 2 of V_{Ga}O_{N} were considered. In both crystal structures for neutral V_{Ga}O_{N} with S = 1, high-spin configuration is stable. Wave functions of V_{Ga}O_{N} have a multi-orbital character, being composed of three p(N) and one p(O) orbitals of vacancy neighbors. But the main contribution to the spin density comes from sp^{3} N orbitals.
Two different electronic structures, which arise due to two different geometry configurations of V_{Ga}O_{N}, with O_{N} either along the c-axis and in one of three equivalent tetrahedral positions in w-GaN, were analyzed. The latter geometry configuration assumes stronger stability of HS state and more delocalization of defect state.
Introducing In_{Ga} as second neighbor to V_{Ga} on the one hand imposes changes to the electronic structure, on the other gives rise to the delocalized wave function of the defect as the crystal structure is perturbed, and finally contributes to long-tail spin density distribution. Magnetic moments originate mainly from sp^{3} N orbitals, and the contribution of p orbitals of distant N, d(Ga), and d(In) states is about 20%.
Various relative orientations of the defects and several charge states (q = 0 and q = − 2) were considered, and consequences regarding the observed FM in GaN were pointed out. Using a relation predicted from mean field model, T_{c} = 2zS(S + 1)J/3k_{B}, the room temperature of FM implies that (assuming z = 6 neighbors and S = 1) the coupling constant J(V_{Ga}O_{N}In_{Ga}) = 0.01 and J(V_{Ga}O_{N}) = 0.0045 eV, i.e., 920 K and 415 K, respectively. These values have a limited reliability as the distribution of defects is random and the coupling depends on the distance between defects.
Comparing the obtained results with experiments, we note that, according to the results, the observed collective ferromagnetism in GaN systems [7, 8, 9, 10, 11] can originate from magnetic interaction between V_{Ga}O_{N} defects. And in Ga-rich InGaN alloys, we predict even stronger FM.
Notes
Funding Information
The work was supported by the National Science Centre (Poland), Grant No. 2015/17/D/ST3/00971. Calculations were done at Interdisciplinary Center for Mathematical and Computational Modeling, University of Warsaw (Grant No. GA65-27).
References
- 1.Nakamura, S., Mukai, T., Senoh, M.: Appl. Phys. Lett. 64, 1687 (1994)ADSCrossRefGoogle Scholar
- 2.Arif, R.A., Ee, Y.K., Tansu, N.: Appl. Phys. Lett. 91, 091110 (2007)ADSCrossRefGoogle Scholar
- 3.Li, Y., You, S., Zhu, M., Zhao, L., Hou, W., Detchprohm, T., Taniguchi, Y., Tamura, N., Tanaka, S., Wetzel, C.: Appl. Phys. Lett. 98, 151102 (2011)ADSCrossRefGoogle Scholar
- 4.Yamamoto, A., Islam, M.R., Kang, T.T., Hashimoto, A.: Phys. Stat. Sol. (c). 9, 1309 (2010)CrossRefGoogle Scholar
- 5.Dahal, R., Li, J., Aryal, K., Lin, Y.J., Jiang, H.X.: Appl. Phys. Lett. 97, 073115 (2010)ADSCrossRefGoogle Scholar
- 6.Neufeld, C.J., Toledo, N.G., Cruz, S.C., Iza, M., DenBaars, S.P., Mishra, U.K.: Appl. Phys. Lett. 93, 143502 (2008)ADSCrossRefGoogle Scholar
- 7.Roul, B., Rajpalke, M.K., Bhat, T.N., Kumar, M., Kalghatgi, A.T., Krupanidhi, S.B., Kumar, N., Sundaresan, A.: Appl. Phys. Lett. 99, 162512 (2011)ADSCrossRefGoogle Scholar
- 8.Jeganathan, K., Purushothaman, V., Debnath, R., Arumugam, S.: AIP Advances. 4, 057116 (2014)ADSCrossRefGoogle Scholar
- 9.Kilanski, L., Tuomisto, F., Szymczak, R., Kruszka, R.: Appl. Phys. Lett. 101, 072102 (2012)ADSCrossRefGoogle Scholar
- 10.Xu, J., Li, Q., Zhang, W., Liu, J., Du, H., Ye, B.: Chem. Phys. Lett. 616, 161 (2014)ADSCrossRefGoogle Scholar
- 11.Xie, Q.Y., Gu, M.Q., Huang, L., Zhang, F.M., Wu, X.S.: AIP Advances. 2, 012185 (2012)ADSCrossRefGoogle Scholar
- 12.Son, N.T., Hemmingsson, C.G., Paskova, T., Evans, K.R., Usui, A., Morishita, N., Ohshima, T., Isoya, J., Monemar, B., Janzén, E.: Phys. Rev. B. 80, 153202 (2009)ADSCrossRefGoogle Scholar
- 13.Sedhain, A., Li, J., Lin, J.Y., Jiang, H.X.: Appl. Phys. Lett. 96, 151902 (2010)ADSCrossRefGoogle Scholar
- 14.Mattila, T., Nieminen, R.M.: Phys. Rev. B. 55, 9571 (1997)ADSCrossRefGoogle Scholar
- 15.Neugebauer, J., Van de Walle, C.G.: Appl. Phys. Lett. 69, 503 (1996)ADSCrossRefGoogle Scholar
- 16.Reshchnikov, M.A., Morkoç, H., Park, S.S., Lee, K.Y.: Appl. Phys. Lett. 81, 4970 (2002)ADSCrossRefGoogle Scholar
- 17.Reshchikov, M.A., Demchenko, D.O., Usikov, A., Helava, H., Makarov, Y.: Phys. Rev. B. 90, 235203 (2014)ADSCrossRefGoogle Scholar
- 18.Lyons, J., Alkauskas, A., Jannoti, A., Van de Waale, C.G.: Phys. Status Solidi (b). 252, 900 (2015)ADSCrossRefGoogle Scholar
- 19.Saarinen, K., Laine, T., Kuisma, S., Nissilä, J., Hautojärvi, P., Dobrzynski, L., Baranowski, J.M., Pakula, K., Stepniewski, R., Wojdak, M., Wysmolek, A., Suski, T., Leszczynski, M., Grzegory, I., Porowski, S.: Phys. Rev. Lett. 79, 3030 (1997)ADSCrossRefGoogle Scholar
- 20.Van de Walle, C.G., Neugebauer, J.: J. Appl. Phys. 95, 3851 (2004)ADSCrossRefGoogle Scholar
- 21.Reshchikov, M.A., Morkoç, H.: J. Appl. Phys. 97, 061301 (2005)ADSCrossRefGoogle Scholar
- 22.Demchenko, D.O., Diallo, I.C., Reshchikov, M.A.: Phys. Rev. Lett. 110, 087404 (2013)ADSCrossRefGoogle Scholar
- 23.Diallo, I.C., Demchenko, D.O.: Phys. Rev. Appl. 6, 064002 (2016)ADSCrossRefGoogle Scholar
- 24.Dev, P., Xue, Y., Zhang, P.: Phys. Rev. Lett. 100, 117204 (2008)ADSCrossRefGoogle Scholar
- 25.Dev, P., Zhang, P.: Phys. Rev. B. 81, 085207 (2010)ADSCrossRefGoogle Scholar
- 26.Wang, X., Zhao, M., Wang, Z., He, X., Xi, Y., Yan, S.: Appl. Phys. Lett. 100, 192401 (2012)ADSCrossRefGoogle Scholar
- 27.Wang, X., Zhao, M., He, T., Wang, Z., Liu, X.: Can cation vacancy defects induce room temperature ferromagnetism in GaN? Appl. Phys. Lett. 102, 062411 (2013)ADSCrossRefGoogle Scholar
- 28.Gillen, R., Robertson, J.: J. Phys. Condens. Matter. 25, 405501 (2013)CrossRefGoogle Scholar
- 29.Volnianska, O., Boguslawski, P.: High-spin states of cation vacancies in GaP, GaN, AlN, BN, ZnO, and BeO: a first-principles study. Phys. Rev. B. 83, 205205 (2011)ADSCrossRefGoogle Scholar
- 30.Volnianska, O., Zakrzewski, T., Boguslawski, P.: Point defects as a test ground for the local density approximation +U theory: Mn, Fe, and VGa in GaN. J. Chem. Phys. 141, 114703 (2014)ADSCrossRefGoogle Scholar
- 31.Volnianska, O., Boguslawski, P.: Local and collective magnetism of gallium vacancies in GaN studied by GGA+U approach. J. Magn. Magn. Mater. 401, 310–319 (2016)ADSCrossRefGoogle Scholar
- 32.Xie, Z., Sui, Y., Buckeridge, J., Sokol, A.A., Keal, T.W., Walsh, A.: Appl. Phys. Lett. 112, 262104 (2018)ADSCrossRefGoogle Scholar
- 33.Dreyer, C.E., Alkauskas, A., Lyons, J.L., Speck, J.S., Van de Waale, C.G.: Appl. Phys. Lett. 108, 141101 (2016)ADSCrossRefGoogle Scholar
- 34.Lyons, L., Van de Waale, C.G.: Comp. Mat. 3, 12 (2017)Google Scholar
- 35.Volnianska, O., Boguslawski, P.: J. Phys.: Condens. Matter. 22, 073202 (2010)ADSGoogle Scholar
- 36.Zhang, L., Li, J., Du, Y., Wang, J., Wei, X., Zhou, J., Cheng, J., Chu, W., Jiang, Z., Huang, Y., Yan, C., Zhang, S., Wu, Z.: New J. Phys. 14, 013033 (2012)CrossRefGoogle Scholar
- 37.Zhu, S.-G., Shi, J.-J., Zhang, S., Yang, M., Bao, Z.-Q., Zhang, M.: Appl. Phys. B. 104, 105 (2011)ADSCrossRefGoogle Scholar
- 38.Chichibu, S.F., et al.: Nature Mater. 5, 810 (2006)ADSCrossRefGoogle Scholar
- 39.Lozac’h, M., Nakano, Y., Sang, L., Sakoda, K., Sumiya, M.: Jpn. J. Appl. Phys. 51, 121001 (2012)ADSCrossRefGoogle Scholar
- 40.Vanderbilt, D.: Phys. Rev. B. 41, 7892(R) (1990)ADSCrossRefGoogle Scholar
- 41.Perdew, J.P., Burke, K., Ernzerhof, M.: Phys. Rev. Lett. 77, 3865 (1996)ADSCrossRefGoogle Scholar
- 42.Giannozzi, P., Baroni, S., Bonini, N., Calandra, M., Car, R., Cavazzoni, C., Ceresoli, D., Chiarotti, G.L., Cococcioni, M., Dabo, I., Dal Corso, A., de Gironcoli, S., Fabris, S., Fratesi, G., Gebauer, R., Gerstmann, U., Gougoussis, C., Kokalj, A., Lazzeri, M., Martin-Samos, L., Marzari, N., Mauri, F., Mazzarello, R., Paolini, S., Pasquarello, A., Paulatto, L., Sbraccia, C., Scandolo, S., Sclauzero, G., Seitsonen, A.P., Smogunov, A., Umari, P., Wentzcovitch, R.M.: J. Phys. Condens. Matter. 21, 395502 (2009)CrossRefGoogle Scholar
- 43.Cococcioni, M., de Gironcoli, S.: Phys. Rev. B. 71, 035105 (2005)ADSCrossRefGoogle Scholar
- 44.Monkhorst, H.J., Pack, J.D.: Special points for Brillouin-zone integrations. Phys. Rev. B. 13, 5188–5192 (1976)ADSMathSciNetCrossRefGoogle Scholar
- 45.Lany, S., Zunger, A.: Modelling Simul. Mat. Sci. Eng. 17, 084002 (2009)ADSCrossRefGoogle Scholar
- 46.Magnuson, M., Mattesini, M., Höglund, C., Birch, J., Hultman, L.: Phys. Rev. B. 81, 085125 (2010)ADSCrossRefGoogle Scholar
- 47.Lambrecht, W.R.L., Segall, B., Strite, S., Martin, G., Agarwal, A., Morkoç, H., Rockett, A.: Phys. Rev. B. 50, 14155 (1994)ADSCrossRefGoogle Scholar
- 48.Ding, S.A., Neuhold, G., Weaver, J.H., Häberle, P., Horn, K., Brandt, O., Yang, H., Ploog, K.: J. Vac. Sci. Thechnol. A. 14, 819 (1996)ADSCrossRefGoogle Scholar
- 49.Bougrov, V., Levinshtein, M.E., Rumyantsev, S.L., Zubrilov, A.: In: Levinshtein, M.E., Rumyantsev, S.L., Shur, M.S. (eds.) Properties of Advanced Semiconductor Materials GaN, AlN, InN, BN, SiC, SiGe, pp. 1–30. John Wiley & Sons, Inc., New York (2001)Google Scholar
- 50.Maruyama, T., Miyajima, Y., Hata, K., Cho, S.H., Akimoto, K., Okumura, H., Yoshida, S., Kato, H.: J. Electron. Mater. 27, 200 (1998)ADSCrossRefGoogle Scholar
- 51.Stroppa, A., Kresse, G.: Phys. Rev. B. 79, 201201 R (2009)ADSCrossRefGoogle Scholar
- 52.Chandrasekhar, D., Smith, D.J., Strite, S., Lin, M.E., Morkoc, H.: Characterization of group III-nitride semiconductors by high-resolution electron microscopy. J. Cryst. Growth. 152, 135–142 (1995)ADSCrossRefGoogle Scholar
- 53.Strite, S., Ruan, J., Li, Z., Manning, N., Salvador, A., Chen, H., Smith, D.J., Choyke, W.J., Morkoc, H.: J. Vacuum Sci. Technol. B. 9, 1924 (1991)ADSCrossRefGoogle Scholar
- 54.Heyd, J., Scuseria, G.E., Ernzerhof, M.: J. Chem. Phys. 118, 8207 (2003)ADSCrossRefGoogle Scholar
- 55.Lany, S., Zunger, A.: Phys. Rev. B. 81, 205209 (2010)ADSCrossRefGoogle Scholar
- 56.Ramprasad, R., Zhu, H., Rinke, P., Scheffler, M.: Phys. Rev. Lett. 108, 066404 (2012)ADSCrossRefGoogle Scholar
- 57.Volnianska, O.: Magnetic properties of isolated Re ion and Re-Re complex in ZnO studied by GGA+U approach. J. Magn. Magn. Mater. 441, 436–442 (2017)ADSCrossRefGoogle Scholar
- 58.Kittel, C.: Introduction to Solid State Physics, 8th edn. John Wiley & Sons, Inc, Hoboken, NJ (2005)zbMATHGoogle Scholar
- 59.Harrison, W.A.: Electronic Structure and the Properties of Solids Dover, New York, p. 176 (1989)Google Scholar
- 60.Adeagbo, W.A., Fisher, G., Ernst, A., Hergert, W.: Magnetic effects of defect pair formation in ZnO. J. Phys. Condens. Matter. 22, 436002 (2010)ADSCrossRefGoogle Scholar
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