Towards identification of silicon vacancy-related electron paramagnetic resonance centers in $4H$-SiC

The negatively charged silicon vacancy (${\mathrm{V}}_{\text{Si}}^{-}$) in silicon carbide (SiC) is a paramagnetic and optically active defect in hexagonal SiC. The ${\mathrm{V}}_{\text{Si}}^{-}$ defect possesses $S=3/2$ spin with long spin coherence time, and it can be optically manipulated even at room temperature. Recently, electron spin resonance signals have been observed besides those associated with the ${\mathrm{V}}_{\text{Si}}^{-}$ defects in the $4H$ polytype of SiC. The corresponding centers share properties akin to those of the ${\mathrm{V}}_{\text{Si}}^{-}$ defects and thus they may be promising candidates for quantum technology applications. However, the exact origin of the new signals is unknown. In this paper, we report ${\mathrm{V}}_{\text{Si}}^{-}$-related pair defect models as possible candidates for the unknown centers. We determine the corresponding electronic structures and magneto-optical properties as obtained by density functional theory calculations. We propose models for the recently observed electron paramagnetic resonance centers, and we predict their optical signals for identification in future experiments.

Figure 1

Kohn-Sham levels introduced by antisites to the band gap of $4H$ SiC situated at both $h$ and $k$ sites as calculated by PBE functional. Energy levels for ${\mathrm{Si}}_{\text{C}}^0$ are slightly lower for ${\mathrm{Si}}_{\text{C}}^0-k$ than for ${\mathrm{Si}}_{\text{C}}^0-h$, while no states are located in the band gap either for ${\mathrm{C}}_{\text{Si}}^0-h$ or for ${\mathrm{C}}_{\text{Si}}^0-k$.

Figure 2

Defect models of axial ${\mathrm{V}}_{\text{Si}}^{-}-X$ defect complexes, where $X=\{{\mathrm{C}}_{\text{Si}}^0,{\mathrm{Si}}_{\text{C}}^0\}$. Neighboring order between ${\mathrm{V}}_{\text{Si}}^{-}$ and $X$ are indicated in brackets. The possible (a) ${\mathrm{V}}_{\text{Si}}^{-}-X-k$ and (b) ${\mathrm{V}}_{\text{Si}}^{-}-X-h$ pair defects along with the (c) perfect lattice are illustrated. Labels of atoms and the $c$-axis are indicated.

Figure 3

Electronic structure of all investigated defect models. Notations of defects are aligned with those in Fig. 2. Green arrows represent the electrons of ${\mathrm{V}}_{\text{Si}}^{-}$ defects residing at open orbitals, i.e., the active space of the electronic structure. Black arrows stand for electrons at closed energy levels introduced by ${\mathrm{Si}}_{\text{C}}^0$. Valence and conduction bands of $4H$ SiC are also indicated.

Figure 4

A combination of the orbitals of ${\mathrm{V}}_{\text{Si}}^{-}-k$ and ${\mathrm{Si}}_{\text{C}}^0-k$ yielding mixed states, particularly fully occupied $e$ and $a_1$ states, where the $e$ level falls into the valence band (VB). The fully occupied $a_1$ state and, in addition, the half-occupied $a_1$ and $e$ states lie in the band gap. Green arrows stand for the unpaired electrons establishing the spin density [orange lobes in Fig. 5], whereas black arrows represent the paired electrons.

Figure 5

Defect structure of (a) ${\mathrm{V}}_{\text{Si}}^{-}-k$ and (b) ${\mathrm{V}}_{\text{Si}}^{-}-{\mathrm{Si}}_{\text{C}}^0-k$ and the corresponding charge densities established by electrons in the majority spin channel (orange lobes) generated by using the same isovalues, (c) and (d), respectively. The supercell structure is shown in orthographic view and the lattice is represented by a wire structure. In the core of the defects, Si and C atoms are represented by yellow and cyan balls, respectively, while dashed balls stand for the ${\mathrm{V}}_{\text{Si}}$.

Figure 6

Formation energies of isolated ${\mathrm{V}}_{\text{Si}}, {\mathrm{C}}_{\text{Si}}$, and ${\mathrm{Si}}_{\text{C}}$ defects under stoichiometric conditions calculated by the HSE06 functional in the Fermi-level region between the $(0/-)$ and $(-/2-)$ charge transition levels of ${\mathrm{V}}_{\text{Si}}-h/k$.

Figure 7

Defect structures of the investigated basal (a)–(c) ${\mathrm{V}}_{\text{Si}}^{-}-{\mathrm{C}}_{\text{Si}}-h{k}^{\text{(a)}}$ and (b)–(d) ${\mathrm{V}}_{\text{Si}}^{-}-{\mathrm{C}}_{\text{Si}}-h{k}^{\text{(b)}}$ defect configurations. The supercell structure is shown in an orthographic view, and the lattice is represented by a wire structure. In the core of the defects, Si and C atoms are represented by orange and cyan balls, respectively, while dashed balls stand for the ${\mathrm{V}}_{\text{Si}}$. We indicate the $c$-axis for all views.

Figure 8

Distortion of the tetrahedron formed by the $4\times \mathrm{C}$ neighboring the ${\mathrm{V}}_{\text{Si}}$ for (a) isolated ${\mathrm{V}}_{\text{Si}}^{-}-h$, (b) ${\mathrm{V}}_{\text{Si}}^{-}-{\mathrm{C}}_{\text{Si}}-h{k}^{\text{(a)}}$, and (c) ${\mathrm{V}}_{\text{Si}}^{-}-{\mathrm{C}}_{\text{Si}}-h{k}^{\text{(b)}}$. The $4\times \mathrm{C}$ nuclei are represented by cyan balls, while the rest of the supercell is shown as a wire structure. The bond lengths are given in $\AA$ units.

Figure 9

Angular dependencies of the two representative ${\mathrm{V}}_{\text{Si}}$-related centers in $4H$ SiC with $C_{\text{1h}}$ symmetry (a) ${\mathrm{V}}_{\text{Si}}-{\mathrm{C}}_{\text{Si}}-h{k}^{\text{(a)}}$ and (b) ${\mathrm{V}}_{\text{Si}}-{\mathrm{C}}_{\text{Si}}-h{k}^{\text{(b)}}$ with the magnetic field rotating in the ($11\overline{2}0$) plane and the microwave frequency of 9.415 GHz. In the simulations, the $g$-value of the negative Si vacancy ($g=2.0029$) is assumed for these $S=3/2$ centers. The fine-structure parameters for the centers are $D^{\text{(a)}}=-226.26$ MHz, $E^{\text{(a)}}=-20.66$ MHz, and $D^{\text{(b)}}=-295.64$ MHz, $E^{\text{(b)}}=-59.36$ MHz are from the calculations of the ZFS tensor where the direction of the axial component has an angle of about $0.8^{\circ }$ and $70.6^{\circ }$, respectively, with the $c$-axis of $4H$ SiC.

Figure 10

Formation energies of isolated ${\mathrm{V}}_{\text{Si}}, {\mathrm{C}}_{\text{Si}}$, and ${\mathrm{Si}}_{\text{C}}$ defects under stoichiometric conditions calculated by the PBE functional in the Fermi-level region between the $(0/-)$ and $(-/2-)$ charge transition levels of ${\mathrm{V}}_{\text{Si}}-h/k$.