Vibronic States and Their Effect on the Temperature and Strain Dependence of Silicon-Vacancy Qubits in $4H$-$\mathrm{Si}\mathrm{C}$

Silicon-vacancy qubits in silicon carbide ($\mathrm{Si}\mathrm{C}$) are emerging tools in quantum-technology applications due to their excellent optical and spin properties. In this paper, we explore the effect of temperature and strain on these properties by focusing on the two silicon-vacancy qubits, V1 and V2, in $4H$-$\mathrm{Si}\mathrm{C}$. We apply density-functional theory beyond the Born-Oppenheimer approximation to describe the temperature-dependent mixing of electronic excited states assisted by phonons. We obtain a polaronic gap of around 5 and 22 meV for the V1 and V2 centers, respectively, which results in a significant difference in the temperature-dependent dephasing and zero-field splitting of the excited states, which explains recent experimental findings. We also compute how crystal deformations affect the zero-phonon line of these emitters. Our predictions are important ingredients in any quantum applications of these qubits sensitive to these effects.

Figure 1

Geometric and electronic structures of the negatively charged silicon-vacancy (${\mathrm{V}}_{\mathrm{Si}}$) defect in $4H$-$\mathrm{Si}\mathrm{C}$. (a) Atomic model of the defect at the $h$ site showing $C_{3v}$ symmetry. (b) Electronic structure of the spin-polarized ${}^4{A}_2$ ground state and the one-particle excitation schemes corresponding to the ${}^4{A}_2$ and ${}^{4}E$ excited states. (c) Fine structure of the excited states regarding spin-orbit (SO) and dipolar spin-spin (SS) interactions (see Ref. [28]). CB, conduction band; VB, valence band.

Figure 2

Temperature dependence of the excited-state zero-field splitting in V1 and V2 centers (a) at cryogenic temperatures and (b) in a wider temperature range. Solid curves are calculated from the Boltzmann statistics of the polaronic states with specific zero-field-splitting values from the vibronic mixing. The calculated ZFS curves are shifted by $211\mathrm{MHz}$ (V1 DFT) and $378\mathrm{MHz}$ (V2 DFT) for comparison with the trends of our observed PL data in (a) and previous experimental data in (b) from Ref. [13] derived from optically-detected-magnetic-resonance (ODMR; solid circles) and level-anticrossing (LAC; hollow circles) measurements.

Figure 3

Temperature-dependent linewidth of the observed PL signal fit by a single-phonon-absorption model [see Eq. (9)] to the averaged $m_S=\pm 1/2$ (hollow points) and $m_S=\pm 3/2$ (solid points) transitions of V1 and V2 centers.

Figure 4

Experimental and calculated photoluminescence sideband of (a) V1 and (b) V2 centers. Calculated Huang-Rhys factors ($S$) are shown. All experimental data are corrected for spectrometer efficiency.

Figure 5

Size scaling of parallel spin-orbit coupling in the quasi-$T_d$ excited-state configuration of V1 and V2 centers.

Figure 6

Strain dependence of the Kohn-Sham levels in the ground state of the V1 center. $u$, $v$, and $e$ are the defect levels depicted in Fig. 1. VBM, valence-band maximum.