Temperature anomaly of the V$_Si$ and V$_C$ vacancy spin coherence time in 4H-SiC

TL;DR

This work addresses the counterintuitive increase of spin coherence time $T_2$ with temperature for $V_{Si}$ and $V^{\pm}_{C}$ defects in $4H$-SiC. It develops a theoretical framework that combines motional Jahn-Teller distortions, polaron hopping via the Holstein two-site model, Redfield relaxation, and vibronic coupling to predict $T_2(T)$ and the conditions under which motional narrowing occurs. Key contributions include analytic expressions for $T^{JT}_{2}$ and the full $T_2^{*}$ incorporating phonon-assisted relaxation, plus atomistic calculations of vibronic couplings $F$, $G$, $K$, and the Jahn-Teller energies $E_{JT}$ and barriers $\delta_{JT}$ for $V_{Si}$ and $V^{\pm}_{C}$; the results reproduce experimental trends with turning-on around tens of kelvin and turning-off at higher temperatures. The findings suggest a potentially universal mechanism for coherence-time anomalies in defect-spin systems and offer a quantitative framework for predicting $T_2$ behavior in related materials and defects.

Abstract

Increasing the spin coherence time (T2) is a major area of interest for spin defect systems such as the silicon (V$_Si$) and carbon (V$^\pm _C$) vacancies in 4H-SiC. Usually as temperature increases, T2 decreases due to the thermal bath. Observations of electron-paramagnetic resonance and direct systematic measurements of T2 has seen an anomaly where T2 increases with increasing temperature. In this work, we investigate the mechanisms that cause the T2 temperature anomaly. We find that due to a spontaneous symmetry lowering from a motional Jahn-Teller distortion, a polaron quasi-particle is generated from the vibronic coupling. Initially, for temperatures from 8 to 20 - 40K, the coherence temperature dependence is dominated by phonon-assisted spin relaxation. At temperatures around 20 - 40K, depending on the vacancy, a thermally activated polaron hopping turns on and motional narrowing dominates and increases T2 with increasing temperature. As temperatures reach 120 - 160K, the energy barrier gets high enough to slow the polaron hopping. At this point the Larmor precession dominates, leading to decoherence. Our calculated temperature-dependent coherence agrees with what has been seen experimentally, giving a full theoretical framework for the mechanisms that cause the T2 temperature anomaly of increasing T2 with increasing temperature. The theoretical framework presented here also gives insight into these mechanisms being a probable universal phenomenon that could occur in many other defect center spin systems.

Figures (7)

• Figure 1: The 8-atom unit cell structure section where a vacancy center can arise for the polytype $4H-$SiC used to calculate the baseline dynamical matrix. The green dots are carbon atoms and the yellow dots are silicon atoms. (This image was made with VMD software support. VMD is developed with NIH support by the Theoretical and Computational Biophysics group at the Beckman Institute, University of Illinois at Urbana-Champaign.)
• Figure 2: Linear vibronic coupling, $F$, as a function of temperature. Blue dashed line is for $n_{C,Si} = 3 \times 10^{14} \mathrm{cm}^{-2}$ and red dotted is for $n_{C,Si} = 3 \times 10^{15} \mathrm{cm}^{-2}$. $F$ increases with increasing temperature, which is expected for semiconductors. At every temperature, a higher concentration of defects entails a decrease in the total number of coupled atoms, resulting in a lower $F$ value.
• Figure 3: The simulation environment made up of 10 atoms for the a.) silicon and b.) carbon vacancy ($V_{Si},V_{C}$) of the polytype 4H-SiC to calculate the dynamical matrix. The green dots are carbon atoms and the yellow dots are silicon atoms. (This image was made with VMD software support. VMD is developed with NIH support by the Theoretical and Computational Biophysics group at the Beckman Institute, University of Illinois at Urbana-Champaign.)
• Figure 4: The near zero Kelvin adiabatic potential energy surface for the a.) silicon vacancy ($V_{Si}$) and b.) carbon vacancy ($V^{\pm}_{C}$). The mexican hat shape can be seen which allows the determination of the Jahn-Teller stabilization energy ($E_{JT}$) and the energy barrier $\delta_{JT}$.
• Figure 5: The coherence time due to the motional Jahn-Teller distortion versus temperature for the positively ($V_{C}^{+}$, orange dotted), negatively charged carbon ($V_{C}^{-}$, green dashed) vacancy, silicon vacancy ($V_{Si}$, solid blue), and Embley et al Fig. 8(a) (black squares) and Fig 8(b) (red diamonds) experimental data ECM17. Here the activation energy of for $V_{Si}$ is $E_{a}=20$ meV, $V_{C}^{+}$ is $14$ meV, and $V_{C}^{-}$$15$ meV). At low temperatures, our model predictions for the $V_{Si}$ shows a rise earlier than the experimental data of Embley et al. Whereas the peak lines up well. The reasoning is due to phonon-assisted relaxation explained in Sec. (\ref{['PhononIncorporated']})