Abstract
Spin defects in wideband gap semiconductors are promising systems for the realization of quantum bits, or qubits, in solidstate environments. To date, defect qubits have only been realized in materials with strong covalent bonds. Here, we introduce a straindriven scheme to rationally design defect spins in functional ionic crystals, which may operate as potential qubits. In particular, using a combination of stateoftheart abinitio calculations based on hybrid density functional and manybody perturbation theory, we predicted that the negatively charged nitrogen vacancy center in piezoelectric aluminum nitride exhibits spintriplet ground states under realistic uni and biaxial strain conditions; such states may be harnessed for the realization of qubits. The straindriven strategy adopted here can be readily extended to a wide range of point defects in other wideband gap semiconductors, paving the way to controlling the spin properties of defects in ionic systems for potential spintronic technologies.
Introduction
The idea of realizing and harnessing coherent quantum bits in scalable solidstate environments has attracted widespread attention in the past decade^{1}. One of the milestones in the field has been the coherent manipulation of the single nitrogenvacancy (NV) defect spin in diamond^{2,3}, opening a new era for using atomlike point defects in crystals as solidstate qubits^{4,5}. However, inherent difficulties in growing and controlling the lattice of C diamond pose severe limitations to the use of the NV center for scalable quantum technologies, begging the question of whether analogs to this defect exist or can be engineered in other technologically mature materials^{6,7,8,9,10}, for examples IIIV crystals. Exploration of quantum defect spins in functional ionic crystals such as, e.g. wAlN, would potentially lead to new opportunities in scalable quantum technologies, including the realization of stronglycoupled spinmechanical resonator hybrid quantum systems^{11,12,13}. Indeed, the piezoelectric properties of wAlN make it an ideal system for measuring and controlling the vibrational motion of the lattice^{14,15} and may offer a variety of control schemes for quantum spins^{8,13}. The strong spontaneous polarization in wAlN combined with advanced heterostructuring and bandgap engineering techniques could add flexibility in designing device functionalities^{15}. Interestingly, wAlN has recently gained significant attention as an optomechanical system^{16,17}, making it attractive as a new host crystal for quantum defect spins.
A key step towards building hybrid quantum systems in ionic crystals, and in particular in aluminum nitride, is the identification of localized spintriplet states akin to the NV center in diamond^{6}. Within a crystalline environment, the spin sublevels of a spintriplet state may be split even in the absence of an external magnetic field and they can be chosen to function as a quantum bit^{3}. For optical addressability^{3}, the defect center should possess excited spintriplet states, which can lead to spinselective decays. These conditions are, however, challenging to realize in ionic crystals for several reasons. In an ionic solid, the energy of defect states derived from dangling bonds is usually close to either the conduction or the valence band edge and, in many cases it may be in strong resonance with that of the bulk band edges^{6}. In the case of wAlN, another critical constraint stems from doping: this material is naturally ntype, similar to several other nitrides^{18,19,20}, and attaining stable ptype doping is extremely challenging^{21,22,23,24,25}, effectively constraining the search for a defect qubit to ntype wAlN.
Here, using firstprinciples calculations^{26,27,28,29,30,31,32,33,34,35,36} we proposed a straindriven strategy to design stable spin defects in ionic crystals; in particular, we identified the strain conditions leading to the stabilization of localized spintriplet defect states in wAlN. We found that by applying moderate uni and biaxial strain to the host lattice in the presence of negatively charged N vacancies, we could stabilize spintriplet states, which are welllocalized within the fundamental gap of ntype wAlN. Our calculations also predicted that these defect spins have a number of spinconserved excited states, which could be used to optically address the spins^{3}, thus making them strong candidates for solidstate implementations of quantum bits in piezoelectric aluminum nitride.
Results
Spinstate and stability of the native defects in wAlN
We determined the charge state (q) of a given defect D(q) as a function of the Fermi level (E_{F}) by computing the defect formation energy (E_{f}^{D(q)})^{37}:
where E^{D(q)} and E^{H} are the total energies computed for supercells with and without a defect, respectively; μ_{i} (i = Al, N, O) is the chemical potential and E_{F} is referred to the valence band edge (E_{V}). The last term on the right hand side of equation (1) corrects for artificial electrostatic interactions present in calculations with finitesize supercells and we used a correction scheme developed by Freysoldt, Neugebauer, and Van de Walle^{38,39}. Figure 1(a) reports E_{f}^{D(q)} as a function of the Fermi energy for the native defects of wAlN as obtained within PBE^{26}. The slope of E_{f}^{D(q)}(E_{F}) yields the charge state of the defect^{37}; a change in slope indicates a change in the charge state of the most stable defect, as a function of the energy of the Fermi level. The spin of each charge state is also indicated in the figure.
We first considered Alvacancy (V_{Al})related defects, including complexes with an N vacancy (V_{Al}V_{N}) and an O impurity at the N site (V_{Al}O_{N}). Paramagnetic states of cationvacancyrelated defects were previously investigated in IIInitrides, in particular GaN^{40,41}. The origin of the highspin states of these cationvacancyrelated defects was attributed to the strong electron localization of the N 2p states leading to a large exchange splitting^{40,41}. Furthermore, two recent theoretical papers suggested that the paramagnetic state of cation vacancy – oxygen impurity complexes in AlN and GaN may potentially operate as quantum bits, similar to the NV center in diamond^{42,43}. Consistent with previous calculations^{40,41,42,43}, we found that the V_{Al}related defects possess a variety of effective electron spins as shown in Fig. 1(a). However, our calculations showed that the paramagnetic ground states associated with V_{Al}related defects may not be suitable for quantum bit applications for several reasons: those defects with spin states higher than 1/2 are stable in pdoped AlN, which is extremely challenging to achieve^{20,21}; on the other hand, the defects stable in ntype AlN have spin zero. Additionally, the dangling bonds left by the presence of an Al vacancy have mainly N 2p character and their position in energy is very close to E_{V}^{40,41}. In some cases, e.g. for the V_{Al}O_{N} complex^{42,43}, the active occupied spinorbital states are in strong resonance with the valence band due to the large exchange splitting. Therefore, we excluded the V_{Al}related defects of wAlN from further consideration in the present work. However we do not exclude V_{Al}related defects from all possible potential qubit operations. For instance, there may exist excitedstate triplets, which may be suitable for quantum operations^{44}.
We now turn to the discussion of anion defects, i.e. V_{N} and O_{N}, from which the main conclusion of our work will be drawn. In this case, we also obtained results using a level of theory higher than PBE and, in particular, we adopted the PBE0 hybrid functional^{27,28} whose choice is extensively justified in the method section. In addition, we carried out calculations using manybody perturbation theory at the G_{0}W_{0} level^{33,34,35}. We first note that O_{N} has no effective spins in its ground state since substitutional oxygen either donates an electron to the host (q = +1) or traps an electron (q = −1), leading to a filled DX state^{45,46,47}. On the other hand, we found that V_{N} has S = 1/2 and 3/2 ground states for q = 0 and −2, respectively, in the ntype region, as shown in Fig. 1(a) (PBE results) and (b) (PBE0 results)^{25,27}. These findings indicate that the S = 3/2 state of V_{N}^{2−} might be of potential interest as a defect qubit, but it might be difficult to realize, as highly ndoped wAlN is difficult to obtain^{47,48}. Interestingly, the V_{N}^{0} S = 1/2 state has been recently detected using electron paramagnetic resonance^{18,19}. Based on these results, we considered the Fermi level range in the vicinity of the stability region of V_{N}^{0}, which might be accessible in experiments. Interestingly, we found that there is a metastable state of V_{N}^{−} with S = 1.
In Fig. 1 we show the defectmolecule model^{49,50} for V_{N}^{−}, with the four active Al sp^{3} dangling bonds, {φ_{i,} i = 1 to 4}. The symmetryadapted molecular orbitals {a_{1}(1), a_{1}(2), e_{x}, e_{y}}^{51} present in an ideal C_{3v} environment are shown in Fig. 1(c). In such an environment, the ground state configuration of V_{N}^{−} is a_{1}(1)^{2} a_{1}(2)^{2}, which is a spinsinglet. Additionally, attaining the a_{1}(1)^{2} a_{1}(2)^{1} e^{1} spintriplet state is possible due to the Hund’s rule coupling. Interestingly, both the S = 0 and S = 1 configurations undergo strong JahnTeller (JT)^{52} distortions (see Fig. 1(c)). A tightbinding model^{53} with hopping matrix elements, t_{ij}, between states φ_{i} and φ_{j} provides a simple picture, and for the S=0 configuration it is easy to show that the a’(2) state is pushed down in energy due to the hybridization with the a’(3) state, thus giving rise to an energy gain. On the other hand, the JT distortion for the S = 1 state results in a different orbital ordering as , and the a′(3) state is pushed above a″, diminishing the hybridization between a′(3) and a′(2). Our calculations also showed that a″ is lowered in energy, thus reducing the energy gap between a′(2) and a″ and the S = 1 state is stabilized according to the Hund’s rule.
Straindriven design of defect spins
We found that in the absence of any strain the S = 1 state is higher in energy than the S = 0 one by only 55 meV and 7 meV, at the PBE and PBE0 levels of theory, respectively. Note that 7 meV is within the range of our numerical errors, estimated to be ~20 meV/defect at most (see supplementary information). This means that the two spin states (S = 0 and S = 1) of V_{N}^{−} are approximately degenerate in energy within PBE0, although they are associated with two distinct JTdistorted structures, as shown in Fig. 1(c). These results suggest that one may engineer the relative stability of the two spin states by straining the lattice. Strain effects have been extensively explored in nitrides and strain levels up to 4% are considered realistic^{54,55}. Figure 2(a) reports the energy difference between the S = 1 and S = 0 states of V_{N}^{−} as a function of compressive uniaxial strain applied to the host AlN lattice along the [] direction, and shows that the S = 1 state is significantly lower in energy than the S = 0 state. Even at a small compressive strain of −3%, the S = 1 state is lower in energy by about 0.25 eV than the S = 0 state within PBE0. This energy difference is well outside our numerical errors (see Supplementary Information) and we expect the energy difference to be representative of the free energy difference as well, since vibrational contributions to the free energy will be almost identical in the two spin configurations (see Supplementary Information). The uniaxial strain effect is twofold. The strained lattice environment is favorable for the S = 1 state as the lattice distortion is compatible with the a_{1} + e JT mode shown in Fig. 1(c). Furthermore the host lattice expands in the [0001] and [] directions according to the Poisson’s ratio effect^{56}, which is unfavorable for the S = 0 state because it tends to pull apart the Al_{3} and Al_{4} atoms, as well as the Al_{1} and Al_{2} atoms. Figure 2(b) reports the defect level diagram of the V_{N}^{−} S = 1 state under −1% uniaxial strain calculated at the G_{0}W_{0}@PBE level, showing the occupied a′(2) and a″ levels being located almost 2 eV below the conduction band minimum (CBM). We also note that in all cases a′(2) and a″ are almost degenerate in energy. As shown in Table 1, the band gap of AlN increases by 0.18 eV within G_{0}W_{0}@PBE, as the uniaxial strain changes from zero to −3%. However, the energy location of the occupied levels a′(2) and a″ does not vary and it remains 3.9 eV above E_{V} (ν in Table 1), indicating that the ^{3}A″ spintriplet state becomes more localized as the compressive uniaxial strain is increased, a beneficial effect for potential quantum information applications.
We further explored the possible stability of the S = 1 V_{N}^{−} defect by considering biaxial strain. Figure 3(a) reports the energy difference between the S = 1 and S = 0 states of V_{N}^{−} as a function of biaxial strain applied in the (0001) plane, showing the stabilization of the S = 1 state above 3% strain, within the PBE0 approximation. Under 4% biaxial strain, the S = 1 state is lower in energy by 80 meV than the S = 0 state in PBE0, a result again outside the numerical errors and possible temperature effects (see Supplementary Information). We note that under the biaxial strain considered in Fig. 3(a), the lowestenergy geometry for the S = 0 state is again the C_{1h}^{(e)} JTdistorted structure, with the same orbital ordering shown in Fig. 1(c). However, an increase of biaxial strain to 2% induces a structural transition in the S = 1 state of V_{N}^{−}, and the defect symmetry changes from C_{1h} to C_{3v}. Moreover, the structural transition is accompanied by an a_{1}type displacement, in which Al_{4} (see Fig. 1(b)) is moved out of the (0001) plane and up in the zdirection, for example by 0.3 Å at 3% biaxial strain. Let us consider a tightbinding model in the C_{3v} symmetry. The orbital energy of the estates and the a_{1}(2) state are and , respectively; here, (= t_{23} = t_{34} = t_{24}) and (= t_{12} = t_{13} = t_{14}) are the inplane and outofplane hopping constants, which are decreased and increased, respectively, under tensile biaxial strain. Therefore, application of such strain could reverse the orbital order between e_{x,y} and a_{1}(2) in V_{N}^{−} beyond a certain critical strain level and lead to the formation of a ^{3}A_{2} S = 1 spintriplet state, according to Hund’s rule. Figure 3(b) reports the G_{0}W_{0} quasiparticle electronic structure of the S = 1 state of V_{N}^{−} under 3% biaxial strain, showing the doubly degenerate estates with two spinup electrons localized in the band gap of wAlN. We found that the band gap of wAlN is decreased from 5.94 eV to 5.35 and 5.10 eV under biaxial strain of 3% and 4%, respectively, within the G_{0}W_{0}@PBE approximation (see Table 1). However, in both cases, the estates are welllocalized and located deep in the gap, 1.8 eV (3% strain) and 1.7 eV (4% strain) below the CBM, within G_{0}W_{0}@PBE.
Hyperfine coupling between the V_{N} ^{−} defect spin and Al nuclear spins
Electron paramagnetic resonance (EPR)^{57} is a powerful experimental technique to detect and identify paramagnetic impurities in solids. EPR measurements yield hyperfine parameters, which are mainly determined by the interaction between an impurity’s electron spin and the surrounding nuclear spins, thus these measurements play a key role in the identification of pointlike paramagnetic impurities in solids^{57}. Son and coworkers resolved the detailed hyperfine structure of an electron spin1/2 in wAlN, mostly interacting with four ^{27}Al nuclei (nuclear spin I = 5/2, 100% natural abundance), and they identified its origin to be V_{N}^{0} with the help of abinitio density functional calculations^{18}. In Table 2, we report the computed principal values of the hyperfine tensor of V_{N}^{0} and we compare them with the previous theoretical and experimental data. Our results are in good agreement with all the previous results, supporting the interpretation of the resolved hyperfine structure being derived from V_{N}^{0}.
Similar to V_{N}^{0} in wAlN, the V_{N}^{−1} electron spin S = 1 may also interact with the nearest four Al atoms (Al_{14} shown in Fig. 1). We calculated the principal values of the hyperfine tensor of V_{N}^{−} (S = 1) as a function of strain and the results are reported in Fig. 4. Interestingly, the hyperfine parameters exhibit high sensitivity to the applied strain. For the uniaxial case, the hyperfine parameters (A_{xx}, A_{yy}, and A_{zz}) for Al_{2} decrease by −75 MHz as the compressive uniaxial strain is increased from 0 to −3%, while those of Al_{1,3,4} increase. To understand this trend, we show in Fig. 4 the isotropic Fermi contact term as a function of strain along with the hyperfine parameters. We found that the Fermi contact term, which is mainly determined by the electron spin density localized at the corresponding Al site, is mostly responsible for the change of the hyperfine parameters as a function of strain. Figure 4(a–c) indicate that as a result of uniaxial strain in the V_{N}^{−} S = 1 state the spin density is transferred from Al_{2} to the other Al atoms.
Interestingly, the hyperfine parameters as functions of biaxial strain show a more drastic change. The hyperfine parameters for Al_{1} decrease by more than 250 MHz and become negative while those of the other basal Al nuclei (Al_{2–4}) significantly increase. This is due to the structural transition of the defect geometry from C_{1h} to C_{3v} beyond the 2% biaxial strain, as previously discussed. As a result of the transition, beyond 2% strain the Al_{2}, Al_{3}, and Al_{4} nuclei become symmetrically equivalent and hence their hyperfine parameters become equal, as shown in Fig. (e,f). The negative hyperfine parameters of Al_{1} are due to the negative spin density localized near the Al_{1} atom, which makes the Fermi contact term negative. Before the structural transition occurs, the spin density is distributed almost equally over the four nearest Al nuclei as it is derived from the a′(2) and a″ orbitals in the C_{1h} symmetry. The structural transition to C_{3v} induces, however, a change in the orbital ordering and the spin density is derived from the e_{x} and e_{y} orbitals, which are mainly localized at the basal plane of Al_{2}, Al_{3}, and Al_{4}, leading to increased Fermi contact terms as shown in Fig. 4 (e,f).
Spinconserved intradefect excitations in nitrogen vacancy spins
One of the most important properties of the NV center in diamond is the singlespin optical addressability, which relies on the presence of an excited ^{3}E spintriplet state and its spinselective decay^{3}. Similar to the diamond NV center, we found that in wAlN the atomic configuration in which the negatively charged N vacancy has a S = 1 ground state also exhibits S = 1 excited states; hence these states could be obtained by a spinconserving optical excitation from an occupied defect orbital to an empty defect orbital, as shown in Figs 2(b) and 3(b). Some of the empty defect levels are located slightly above the CBM. We note, however, that the lowestlying empty defectorbitals are not in resonance with the CBM; they remain localized, as shown by their dispersionless character in our computed band structure (not shown) due to the following reasons. The lowest conduction band of wAlN is a single parabolic band centered at Γ. The next available conduction states appear close the K and L points and they are located at 0.9 and 1.1 eV above the CBM, respectively, according to previous GW calculations^{58}, and consistent with our GW results. Furthermore we found that the CBM mainly exhibits a nitrogen p character, leading to a negligible hybridization with the aluminum dangling bonds created by the V_{N}^{−} defect.
The ZPL was obtained by carrying out calculations of total energy differences (ΔSCF calculations) within the PBE and PBE0 approximation^{59,60}. We first describe possible intradefect spinconserving excitation in the spindown channel, which may be similar to the excitation scheme of the NV center in diamond^{60}. In the case of uniaxial strain, we promoted an electron from a′(1) to a′(2) and the resulting configuration corresponds to the optically excited ^{3}A″ spintriplet state. The ZPL is rather insensitive to the amount of uniaxial strain, up to 3%, and it is calculated to be ~3.2 eV within PBE. A similar excitation for V_{N}^{−} under biaxial strain, shown in Fig. 3(b), corresponds to a transition to the excited ^{3}E spintriplet state; for the ZPL we found 3.35 and 3.25 eV under 3% and 4% biaxial strain, respectively within PBE. We note, however, that these near ultraviolet excitations may lead to photoionization of V_{N}^{−} to V_{N}^{0} considering the corresponding (0/−1) charge transition level is about 1.1 eV below the conduction band edge, as shown in Fig. 1(b). In the spinup channel, however, the ZPLs from a″ to a′(3) under uniaxial strain, and from e to a_{1}(2) under biaxial strain, are expected to be at much lower energies, in the near infrared range. For these cases, we used the PBE0 hybrid functional to better estimate the ZPLs^{60}. We find that the spinup ZPLs are 0.83 eV and 0.89 eV for −1% uniaxial strain and 3% biaxial strain, respectively.
Discussion
We proposed a straindriven defect design scheme to obtain point defects with localized spintriplet ground states for implementation of spin qubits in piezoelectric aluminum nitride. We found that negatively charged nitrogen vacancies exhibit localized spintriplet states in ntype aluminum nitride under realistic strain conditions. Nitrogen vacancies are naturally incorporated in aluminum nitride during crystal growth and they are known to be the main source of the intrinsic ntype behavior of aluminum nitride^{18,19,20,21}, thus making the nitrogen vacancy spins easily amenable to experimental investigations^{18,19}. Extra ntype doping control might be achievable by introducing substitutional oxygen impurities (O_{N}) during crystal growth^{24}. As shown in Fig. 1(a,b), O_{N} can donate electrons in the region where V_{N}^{0} is stable and provide the extra charge necessary to form V_{N}^{−}.
In addition to nitrogen vacancies, V_{Al}related defects are also commonly observed in experiments^{22,23}. Our calculations showed that such defects could in principle introduce a variety of electron spins in the host lattice, which would make it difficult to isolate a single defect spin for qubit applications. However, as shown in Fig. 1(a), the V_{Al}related defects are magnetically passivated (i.e. S = 0) in the stability region of V_{N}^{−}, thus allowing for the V_{N}^{−} spins to be easily isolated for qubit applications.
We showed that both the S = 0 and S = 1 configurations of V_{N}^{−} undergo static JahnTeller (JT)^{52} distortions as shown Fig. 1(c) and the relative stability of the S = 0 and S = 1 states can be controlled by applying a uniaxial or biaxial strain to the host lattice. Regarding possible temperature effects on the relative stability under strain, dynamic JahnTeller effect^{52} might be an important factor to be considered in addition to the vibrational entropy effect discussed in the Supplementary Material. We also note that potential dynamic JahnTeller effects at elevated temperature might be straindependent, as we found the defect structure and its lattice environment change significantly in response to external strain perturbations. An investigation of potential dynamic JahnTeller effects will be the subject of future studies.
We also showed that excited spintriplet states are present for negatively charged nitrogen vacancies and these states may play an important role in potential optical manipulations of the vacancy spins. We found that the zero phonon lines for a spinconserving intradefect excitation in the spindown and spinup channels of the V_{N}^{−} S = 1 state are in the near ultraviolet and in the near infrared range, respectively. We pointed out that the near ultraviolet excitation could lead to photoionization of the V_{N}^{−} defect, thus the near infrared excitation may be more suitable for potential optical manipulation of the V_{N}^{−} spin.
The excited triplet may couple to the ground state triplet nonradiatively, similar to the NV center in diamond. We note, however, that additional important issues remain to be considered in order to address the potential optical manipulation of the V_{N}^{−} spin. The optical manipulation of the NV center is based on the spinorbitinduced spinselective decay through dark singlet states^{3}. For the V_{N}^{−} system studied here there are openshell singlet states such as ^{1}A_{1} or ^{1}E states in the C_{3v} symmetry case and ^{1}A″ in C_{1h}, which may be close in energy to the ^{3}A_{2} triplet in C_{3v} and the ^{3}A″ triplet in C_{1h}, respectively. These single states may play a role in the optical manipulation of the V_{N}^{−} spin. However, whether these singlet states are positioned in energy between the ground spin triplet and the excited triplet remains to be seen^{61}, as well as whether the spinorbit interaction between the excited triplet states and the singlet states is sufficiently strong to couple them. We further note that the C_{3v} symmetry of the NV center in diamond is responsible for specific selection rules for the spinselective decay^{62}. The V_{N}^{−} defect in wAlN has the C_{3v} symmetry only under the biaxial strain, while it has C_{1h} symmetry under uniaxial strain; such symmetry difference may be an important factor in establishing the potential optical manipulation of the V_{N}^{−} spin.
It is worth mentioning that a number of alternative straindriven spin manipulation and readout schemes are being actively developed in the literature^{63,64}, and the V_{N}^{−} spins proposed in our study may lead to excellent platforms for the implementation of straindriven spin control schemes due to the strong piezoelectricity of the host lattice^{8}.
Methods
Abinitio charged defect calculations
We carried out density functional theory (DFT) calculations with semilocal (PBE^{26}) and hybrid (PBE0^{27,28}) functionals, using plane wave basis sets (with a cutoff energy of 75 Ry), normconserving pseudopotentials^{29}, and the Quantum Espresso code^{30}. In the case of wAlN, the selfconsistent HartreeFock mixing parameter, as derived using the method of Ref. [28], is 24%^{28}, which justifies the use of the PBE0 hybrid functional, whose mixing parameter is defined to be 25% (see below). We examined the most common native defects, which may be easily accessible in experiment (including Al^{22,23} and N vacancies^{18,19}, O impurities^{24}, and cationanion defect complexes^{23,25}). To mimic the presence of isolated defects, we employed supercells with 480 and 96atoms, when using the PBE and PBE0 approximations, respectively, and full geometry optimizations were performed with both functionals. We sampled the Brillouin zone by a 2 × 2 × 2 kpoint mesh. Numerical errors in terms of supercell size, kpoint sampling and the plane wave cutoff energy were examined and are summarized in the supplementary information. We considered 7 different types of defects and 6 to 7 different charge states for each of them, in addition to 2 to 3 spin multiplicities for each case. For all defect states, total energy minimizations were started from three different initial geometries (with symmetry C_{3v} and C_{1h}) and the state with the lowest energy was selected as the ground state. In total, we explored approximately 300 different defect states.
The use of PBE0 to describe AlN is appropriate, based on our recent work^{28}, as the average electronic dielectric constant of AlN is ~4.1^{28} and we expect the optimal amount of the HartreeFock mixing to be 1/4.1 = 24%, which is almost identical to the 25% mixing parameter entering the definition of the PBE0 functional. The accuracy of the PBE0 functional to describe pristine AlN was checked for several properties, which are summarized in Table 3 and are all in excellent agreement with experiment. In particular, using the PBE0 functional we calculated the electronic and static dielectric constants of AlN with a combined finite Efield and Berry phase method^{65,66}. We calculated the electronic dielectric constants to be 4.06 and 4.22 for and , respectively, to be compared with experimental values of 4.13 ± 0.02 for and 4.27 ± 0.05 for ^{67}. For the static dielectric constants, we calculated and to be 7.71 and 8.95, respectively, to be compared with experimental values of 9.18 for^{68} and the result of 8.5 obtained from polycrystalline AlN^{69}. Note that PBE0 accurately describes the anisotropic nature of the dielectric constants of AlN. The accuracy of the PBE0 functional to describe the native defects in AlN was checked by calculating defect formation energies and charge transition levels of the N vacancy and O impurity as shown in Fig. 1(b). Recently, similar results using another hybrid functional (HSE) were reported in Ref. [25]; the authors were able to successfully explain the experimentally observed optical absorption and emission of the defects and we verified that our PBE0 results are in almost perfect agreement with those of Ref. [25].
The zerophonon line (ZPL) was obtained by carrying out calculations of total energy differences (ΔSCF calculations) within the PBE and PBE0 approximation^{59,60}. We employed 480atom and 288atom supercells along with 75 Ry and 65 Ry planewave cutoff energies for the PBE and PBE0 calculations, respectively. By using the PBE functional, we checked the numerical error induced by reducing the supercell size from 480 atoms to 288 atoms and the planewave cutoff energy from 75Ry to 65Ry to be around 0.05 eV.
Firstprinciples calculations of hyperfine tensors
We carried out calculations of hyperfine tensors between V_{N} spins and nuclear spins in wAlN at the PBE level of theory. The calculations were performed in two steps. First, we calculated the groundstate wavefunctions for V_{N} using the 480atom supercell as described in the previous section. Then, we used the gaugeincluding projectoraugmented wave method (GIPAW) of Ref. [70] to calculate the hyperfine tensor, which is comprised of the isotropic Fermi contact term and the anisotropic dipolar coupling term. Numerical convergence in terms of the energy cutoff, the kpoint sampling, and the supercell size was checked, with a numerical error in the hyperfine tensor less than 10 MHz. To verify the accuracy of the PBE functional, we calculated the hyperfine parameters of the neutral V_{N} spin (S = 1/2) and compared them to previous theoretical and experimental results^{18} as shown in Table 2. We found an excellent agreement between our and previous results. In addition, we found that adding core polarization effects^{71,72} improves the agreement between our results and experiment, thus we included core polarization effects throughout all of our calculations.
Largescale manybody GW calculations
In order to check the robustness of our predictions, we carried out calculations of quasiparticle energies within the G_{0}W_{0} approximation^{31,32}, using the Γ point only and 480atom supercells with defect geometries optimized at the PBE level of theory. GW calculations were performed utilizing a spectral decomposition technique for the dielectric matrix^{33,34} and an efficient contour deformation technique for frequency integration^{35}, as implemented in the WEST code (www.westcode.org), where the evaluation of virtual electronic states is not required. Calculations carried out with the WEST code started from the results obtained with the semilocal PBE functional and perturbative corrections to the KohnSham eigenvalues were obtained. The massively parallel implementation of the GW method in WEST takes advantage of separable expressions for both the Green’s function (G) and the screened Coulomb interaction (W). The newly developed technique for largescale GW calculations allowed us to explore defective AlN systems of unprecedented size, containing ~2000 electrons. For the band gap of wAlN, we obtained 5.94 eV within the G_{0}W_{0}@PBE approximation using our 480atom supercell with a point defect, which is in very good agreement with the previous G_{0}W_{0}@LDA results of 5.8 eV^{58} and 6.08 eV^{73} obtained for the pristine bulk.
Additional Information
How to cite this article: Seo, H. et al. Design of defect spins in piezoelectric aluminum nitride for solidstate hybrid quantum technologies. Sci. Rep. 6, 20803; doi: 10.1038/srep20803 (2016).
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Acknowledgements
We thank David Awschalom for suggesting wAlN as a new potential host material for quantum defect spins. We also thank William Koehl, Jonathan Skone, Matthew Goldey, Márton Vörös, and EunGook Moon for helpful discussions. HS is primarily supported by the National Science Foundation through the University of Chicago MRSEC under award number DMR1420709. MG and GG are supported by DOE grant No. DEFG0206ER46262. Part of this work (MG) was done at Argonne National Laboratory, supported under U.S. Department of Energy contract DEAC0206CH1135. This research used resources of the National Energy Research Scientific Computing Center (NERSC), a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DEAC0205CH11231, resources of the Argonne Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DEAC0206CH11357, and resources of the University of Chicago Research Computing Center.
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The Institute for Molecular Engineering, The University of Chicago, Chicago, IL, USA
 Hosung Seo
 , Marco Govoni
 & Giulia Galli
Materials Science Division, Argonne National Laboratory, Argonne, IL, USA
 Marco Govoni
 & Giulia Galli
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Contributions
H.S. and G.G. designed the research. Most of the calculations were performed by H.S., with contributions from all authors. M.G. implemented the GW manybody perturbation theory in the WEST code. All authors contributed to the analysis and discussion of the data and the writing of the manuscript.
Competing interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to Hosung Seo.
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