First, we obtained lattice constants a = 3.129 Å, c = 5.125 Å for bulk AlN, and a = 3.125 Å for monolayer, which are consistent with previous results [30, 31, 42].

We considered the doping of Si in bulk AlN. We placed a single Si atom into a large supercell with 288 atoms to simulate a dilute doping scenario in which the Si atom can be regarded as an isolated impurity (Figures 1A,B). We examined non-equivalent Si doping sites including the substitution of an Al atom (Si _Al ) or a N atom (Si _N ), the interstitial site with z coordination between two Al [labeled as Si _I,Al by Kroger–Vink notation] layers or N (Si _I,N ) layers (Figure 2A). Frenkel defects (Si _Al − Al _I and Si _N − N _I ) in which a Si atom occupies the Al or N site and repels the original Al or N atom to an adjacent interstitial site were also considered. We calculated the formation energies (E _f ) of these doping configurations under Al- and N-rich growth conditions. The growth conditions will affect the E _f of substituted Si, rather than interstitial Si. Both non-magnetic and magnetic states of the Si atom were considered.

The results of E _f are shown in Table 1. Si atoms at the Si _I,N site are not stable and transfer to the Si _I,Al site. In bulk AlN, Si atoms prefer to substitute Al atoms (Si _Al ), due to that the atomic radius and electronegativity of Si atoms are more close to those of Al atoms than N atoms. The ground magnetic state of Si _Al is an NM state. The E _f of Si _Al is smaller under N-rich than under Al-rich growth conditions, which is consistent with the experiment that Si incorporation in AlN films is suppressed under Al-rich growth conditions [43]. The magnetic state of the Si _Al site is not stable and will transition to the NM state. We further calculated the density of states (DOS), which indicates that the Si _Al site gives an impurity band very close to the conduction band, similar to the behavior of Si dopants in GaN [44]. Thus, the Si _Al site is non-magnetic in bulk AlN because the unpaired electron is quite delocalized. Instead, the metastable Si_N configuration prefers a magnetic state, with a magnetic moment of 1 μ _B . If enough N-vacancies exist in wurtzite AlN [45], external Si dopants can form Si _N configurations and exhibit magnetism in bulk AlN, consistent with Pan’s experiment [28].

The diffusion of Si atoms in bulk AlN can proceed through the interstitial Si_I,Al site. As bulk AlN has an anisotropic lattice, we considered Si diffusion along the pathway parallel (2 → 1) and vertical (1 → 3) to the [0001] direction, which was labeled as the c axis (Figure 2C). We estimated the diffusion barrier ΔE using the nudged elastic band (NEB) method [46]. The Si element has a larger ionic radius than Ga and N elements, so Si ions will extrude Al and N ions along the direction vertical to the diffusion pathway. We found that the energy of the unit cell of AlN increases more slowly under strain parallel to the c axis than strain vertical to the c axis (Figure 1D). Thus, the ΔE of path 1 → 3 is smaller than the ΔE of path 2 → 1 (Figure 2D) due to the smaller energy needed to extrude Al and N atoms. Thus, Si atoms prefer to diffuse vertical to the c axis in bulk AlN.

We introduced two Si _Al defects into the AlN supercell. The magnetic coupling between them can be NM, FM, or AFM. The total energy of AlN with two Si _Al defects is presented in Figure 2B. Both FM and AFM states will transition into NM states. The energy of AlN with two NM Si _Al defects decreases as the Si separation increases. Thus, Si atoms tend to act as NM dopants, distributed uniformly and diffuse vertically to the c axis in bulk AlN. That is consistent with experiments showing that Si incorporation in AlN is homogeneous under N-rich conditions [28, 43].

Next, we investigated the Si doping in 2D AlN. We put a single Si in a 6 × 6 × 1 supercell of monolayer AlN. We considered the substituted Si _Al and Si _N in buckling and planar structures, the adsorption of Si atoms on top of Al (Si _T,Al ), N atoms (Si _T,N ), hexagonal site (Si_H), and bridge site (Si_B) (Figure 3A). As shown in Table 2, the Si atom at the (Si_B) site is not stable and will transfer to the Si _T,N site. If we consider the Si in the NM state, the buckling Si _Al structure is not stable will transition to the planar Si _Al structure. We calculated the energy of Si _Al configuration at different buckling heights in the NM state while keeping all atoms fixed. The energy profile in Figure 3B shows that the NM Si _Al in the planar structure has the lowest energy.

However, if we consider Si in a spin-polarized state, the buckling Si _Al structure becomes the most stable doping configuration (Table 2; Figure 3B). The optimized Si–N bond lengths are 1.697 Å and 1.749 Å in the plane and buckling Si_Al structures, respectively. As the ionic radius of the Si dopant is larger than that of the Al ion in Si-doped 2D AlN, the Si dopant prefers a buckling structure to release the strain and make the system energetically lower. That is similar to alkali metals which also form buckling Si_Al structures in monolayer AlN [47]. The stable Si _Al structure has a buckling height of 0.49 Å and a magnetic moment of 1 μ _B per Si atom.

Si_N and other adsorption configurations, which can induce a magnetic moment of 1 or 2 μ _B per Si atom, have much higher E _f (Table 2). Thus, the Si atom prefers to substitute the Al atom with a non-zero magnetic moment in the monolayer AlN.

Based on the HSE06 hybrid functional [40], the band structure and partial density of states (PDOS) of single-Si-doped monolayer AlN are displayed in Figures 3C, D. A diluted Si-doped AlN is still a semiconductor. The pristine AlN monolayer has D_3h symmetry, and the 2p states of the N atom split into a singlet p _z state and doubly degenerate states p _x and p _y with lower energy. The induced Si dopants in the Si_Al site deform the D_3h symmetry of the AlN lattice, resulting in small splitting between p _x and p _y states (Figure 3D). The PDOS of the Si-p and N-p orbitals overlapped at an energy of around −6 eV in the deep valence band (Figure 3D). The Si atom contributes three valence electrons to participate in the bonding with neighboring N-p orbitals, lifting an unpaired electron to form a dispersion-less impurity band in the spin-up channel within the band gap (Figure 3C). The asymmetric PDOS creates a magnetic moment of 1 μ _B per Si atom. Si-p _z and Si-s orbitals dominate the electronic states of the impurity band. The impurity band is a deep donor level, which is consistent with the n-type conductivity of Si-doped AlN in the experiment [28].

The spin density of Si_Al structures is displayed in Figures 3E,F. The negative spin density is insignificant compared to the positive spin density (Figure 3F). Spin density exhibits localization at the Si site. The nearest N neighbors around the Si dopant had a small contribution to the magnetic moment and exhibited an FM coupling with the Si dopant (Figure 3E). The Al atoms display virtually no spin polarization. We can conclude that the additional charge carriers due to Si doping promote ferromagnetism in monolayer AlN. The situation is similar to that of Cr-doped AlN nanowires with the ferromagnetic coupling of the Cr atoms [48].

We further decomposed magnetic moment origin. The p orbitals of Si dopants and the nearest-neighboring N atoms dominate the magnetic moments. The Si-s orbital and nearest-neighboring N-s orbital have a small contribution to the magnetic moments. That is different from Mg- or Ga-doped monolayer AlN where spin-polarized 2p orbitals of N neighbors around dopants dominate the magnetic moment [26, 27].

The diffusion of Si atoms in monolayer AlN can proceed along the direction vertical to or parallel to the AlN sheet. The former one can proceed between two equivalent Si_Al sites on the different sides of the AlN sheet (Figure 4A). We performed NEB calculations and allowed the atomic positions to be relaxed. The diffusion barrier △E is estimated at 0.11 eV (Figure 4B). The accurate energy profile of Si diffusion indicates that the planar Si_Al configuration is a metastable state (Figure 4B). If the initial buckling height was set to be small in the simulation of structure relaxation, one may finally predict a planar Si_Al configuration in spin-polarized calculations. That may be the reason for the difference between our results and previous work [32, 33]. The other diffusion pattern is through low-energy adsorption sites on the same side of the AlN sheet, which is Si_T,N − Si _H − Si_T,N pathway (Figure 4C). Through the NEB calculations, the optimized saddle point was still close to the H site. The diffusion barrier △E was estimated at 1.11 eV (Figure 4D), which is much larger than △E crossing the AlN sheet.

To investigate the magnetic coupling between the Si dopants in monolayer AlN, we introduced two Si_Al defects into a large rectangle supercell with 112 atoms with different Si–Si distances (Figure 1C). We further considered that two Si_Al defects can be on the same side or opposite sides of the AlN sheet. The energies of these structures in the NM, FM, and AFM states are shown in Figure 5C. We found that the magnetic coupling between Si atoms prefers an AFM state. At a large Si–Si separation, the structures containing two Si atoms on the same and opposite sides of the AlN sheet have similar energies. However, two Si_Al defects are likely to occur on the opposite sides of the AlN sheet at a small Si–Si distance. That structure can decrease the Coulomb repulsion between the unpaired electrons located on Si dopants. A Si atom had to overcome an energy barrier to get close to the other one. The energy barrier is about 0.03 and 0.015 eV for two Si atoms on the opposite sides and same sides of AlN sheet, respectively (Figure 5C). Moreover, considering the large diffusion barrier of Si along the AlN sheet (Figure 4D), Si atoms might keep a uniform distribution at a low doping concentration. Moreover, we calculated the electronic structure which indicates that monolayer AlN with two Si_Al defects is a semiconductor. 2D AlN has been grown on Ag (111) [9] or SiN substrate [49] which has a large influence on the properties of the AlN layer near the substrate. The Si doping on the top surfaces of AlN layers should produce the magnetic behaviors as the basal planes of multilayer AlN are bonded by a weak van der Waals force.

Si dopants should also exhibit magnetic moments in a few-layer AlN. We calculated the doping of Si in bilayer AlN with a stable AA’ stacking pattern [50]. We found that Si atoms also tend to substitute Al atoms in N-rich conditions and exhibit a magnetic moment of 1 μ _B per Si atom (Table 3). Two Si_Al defects on opposite sides of the bilayer AlN have lower energy (Figure 5D). Si atoms also tend to get close to each other in bilayer AlN, similar to Si dopants in monolayer AlN (Figure 5D).

We noticed that the intrinsic defects of monolayer AlN, such as Al-vacancy and N-vacancy, can induce a non-zero magnetic monument [51]. We further found that the Al vacancy seems to have an AFM coupling with the Si dopants, thereby influencing the magnetic moment and magnetic coupling between the Si dopants. We will consider the effect of intrinsic defects on the magnetic properties of Si-doped AlN in later work.

By fully substituting the Al of monolayer AlN by Si, we obtained monolayer SiN which adopts a hexagonal lattice with a lattice constant of 2.904 Å and a buckling height of 0.578 Å. We compared different spin configurations of monolayer SiN (Figure 8A) and identified the FM state as its ground state. Thus, in Si-doped monolayer AlN, there should be a phase transition from an AFM to an FM state as the Si content increases.

We adopted the Dalpian’s band structure model to explain such phase transition [52]. There are four valence electrons per Si atom. As shown in Figure 3D, up-spin Si-2p states have higher energy than down-spin Si-2p states. An unpaired electron of Si produces a magnetic moment of 1 μ _B in the monolayer AlN. We used the ɛ _↑ , ɛ _↓ , △ _↑↑ , and △ _↑↓ to represent the energy of spin-up and spin-down Si-2p states, hopping matrix elements among the Si-2p states with the same spins and different spins, respectively. When the magnetic coupling between Si dopants is FM, as shown in Figure 6A. From a molecular mode, the hybridization between the down-spin Si-2p states of the neighboring Si dopant produces molecular levels: ε ↓ + = ε ↓ + | Δ ↓ ↓ | , ε ↓ − = ε ↓ − | Δ ↓ ↓ | . (3) As the down-spin Si-2p states are fully filled, there is no energy gain from such hybridization. On the other hand, hybridization between the partially filled up-spin Si-2p states leads to an energy gain of −2|Δ _↑↑ |.

When the coupling between Si dopants is AFM, as shown in Figure 6B. The hybridization between the down-spin Si-2p states and the up-spin Si-2p states of the neighboring Si dopant produces molecular levels: ε ± = 1 2 ε ↑ + ε ↓ ± ε ↑ − ε ↓ 2 2 + | Δ ↑ ↓ | 2 . (4) Similarly, the hybridization between the up-spin Si-2p states and the down-spin Si-2p states of the neighboring Si dopant will produce the same molecular levels. Moreover | Δ ↑ ↓ | ≪ ε ↑ − ε ↓ 2 , such hybridizations lead to an energy gain of 2 3 ε − + ε + − 3 ε ↓ + ε ↑ ≅ − 4 | Δ ↑ ↓ | 2 ε ↑ − ε ↓ . (5) Thus, the energy difference between the FM and AFM states is given as Δ E F M − A F M = − 2 | Δ ↑ ↑ | + 4 | Δ ↑ ↓ | 2 ε ↑ − ε ↓ .

At low Si concentration, the localized unpair p orbital of Si dopants leads to a small |△ _↑↑ |. Adding the small exchange splitting (ɛ _↑ − ɛ _↓ ) (Figure 3D), ΔE _FM−AFM is positive. Si-doped monolayer AlN exhibits the AFM state at low concentration. However, the lattice constant and the Si–Si distance of Si-doped AlN decrease as Si content increases. The unpaired p orbital of Si dopants becomes more delocalized and exchange splitting becomes large (Figure 7D), which results in the ground state of heavy Si-doped monolayer AlN changing to FM.

The HSE band structure of FM monolayer AlN is shown in Figure 7A, indicating that it is a 2D FM semiconductor with an indirect band gap of 1.556 eV, consistent with previous work [53]. We considered four possible magnetic configurations, including FM, AFM1, AFM2, and AFM3 configurations (Figures 8A–D). We expanded their energy by the Heisenberg Hamiltonian: E F M = E 0 − 3 J 1 S 2 − 3 J 2 S 2 − 3 J 3 S 2 (6) E A F M 1 = E 0 + J 1 S 2 + J 2 S 2 − 3 J 3 S 2 (7) E A F M 2 = E 0 + J 1 S 2 − J 2 S 2 + J 3 S 2 (8) E A F M 3 = E 0 − J 1 S 2 + J 2 S 2 + J 3 S 2 , (9) where S is the spin per Si atom which was normalized to 1. J ₁, J ₂, and J ₃ are the nearest-neighboring, next-nearest-neighboring and thirdly-nearest-neighboring exchange interaction constants, respectively. We calculated these interaction constants through the relations between energy and spin configuration, the results are given in Table 4. Through the Monte Carlo simulation, the Curie temperature T _c of monolayer SiN was estimated to be about 105 K (Figure 8E).

We expanded current calculations to Ge- and Sn-doped monolayer AlN. First, we found that single Ge and Sn dopants also prefer a magnetic buckling Si_Al structure in monolayer AlN in the dilute doping scenario. The corresponding buckling height is 0.73 Å and 1.28 Å, respectively. Different from Si dopants, the Ge and Sn dopants in the planar Si_Al structure are a saddle point, rather than a metastable state through the NEB calculations (Figures 4E,F). The diffusion barriers vertical to the AlN sheet are 0.6 and 1.0 eV for Ge and Sn dopants, respectively. Second, monolayers GeN and SnN are also 2D ferromagnetic semiconductors, with a band gap of 1.628 and 1.076 eV, respectively (Figures 7B,C). Their exchange constants are given in Table 4. Their T _c is estimated as 431 and 594 K through MC simulation, respectively (Figures 8F,G).