Spin relaxation in graphite due to spin-orbital-phonon interaction from first-principles density-matrix approach

We predict "intrinsic" spin relaxation times ($T_{1}$) of graphite due to spin-orbit-phonon interaction, i.e., the combination of spin-orbit coupling and electron-phonon interaction, using our developed first-principles density-matrix approach. We obtain ultralong $T_{1}$, e.g., $\sim$600 ns at 300 K, which leads to ultralong in-plane spin diffusion length $\sim$110 $\mu$m within the drift-diffusion model. Our prediction sets the upper bound of $T_{1}$ of graphite at each given temperature and Fermi level. The anisotropy ratios of $T_{1}$ or values of $T_{1z}/T_{1x}$ are found small and around 0.6. We examine the applicability of the well-known Elliot-Yafet (EY) relation, which declares that spin relaxation rate $T_{1\alpha}^{-1}$ ($\alpha=x,y,z$) is proportional to the product of the ensemble average of spin mixing parameter $\left\langle b_{\alpha}^{2}\right\rangle$ and carrier relaxation rate $\tau_{p}^{-1}$. Our numerical tests suggest that the EY relation works qualitatively if the degeneracy threshold $t^{\mathrm{deg}}$ for evaluating $b_{\alpha}^{2}$ is elatively large (not much smaller than or comparable to $k_{B}T$), e.g., $10^{-3}$ eV or larger, but fails if $t^{\mathrm{deg}}$ is too tiny (much smaller than $k_{B}T$), e.g., $10^{-6}$ eV or smaller.