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