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

We predict intrinsic spin relaxation times $({T}_{1})$ of graphite due to spin-orbit--phonon interaction, i.e., spin-phonon interaction caused by spin-orbit coupling and electron-phonon interaction, using our developed first-principles density-matrix approach. We obtain ultralong ${T}_{1}$, e.g., $\ensuremath{\approx}600\phantom{\rule{0.16em}{0ex}}\mathrm{ns}$ at 300 K, which leads to ultralong in-plane spin diffusion length $\ensuremath{\approx}110\phantom{\rule{4pt}{0ex}}\textmu{}\mathrm{m}$ 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 ${T}_{1\ensuremath{\alpha}}^{\ensuremath{-}1}$ $(\ensuremath{\alpha}=x,y,z)$ is proportional to the product of the ensemble average of spin mixing parameter $\ensuremath{\langle}{b}_{\ensuremath{\alpha}}^{2}\ensuremath{\rangle}$ and carrier relaxation rate ${\ensuremath{\tau}}_{p}^{\ensuremath{-}1}$. Our numerical tests suggest that the EY relation works qualitatively if the degeneracy threshold ${t}^{\mathrm{deg}}$ for evaluating ${b}_{\ensuremath{\alpha}}^{2}$ is relatively large (not much smaller than or comparable to ${k}_{B}T$), e.g., ${10}^{\ensuremath{-}3}\phantom{\rule{0.16em}{0ex}}\mathrm{eV}$ or larger, but fails if ${t}^{\mathrm{deg}}$ is too tiny (much smaller than ${k}_{B}T$), e.g., ${10}^{\ensuremath{-}6}\phantom{\rule{0.16em}{0ex}}\mathrm{eV}$ or smaller.