Relations between T₁ and T₂ for spin systems with electron-phonon scattering

A critical requirement for the realization of spintronic devices is the longevity of a spin-polarized ensemble, characterized by the T₁ timescale. Meanwhile, the coherence of the spin ensemble is characterized by the T₂ timescale, which can be very different from T₁, and is vital for quantum information applications. Using ab initio real-time density-matrix dynamics within a Lindbladian framework that includes quantum scatterings, we compute the T₂ timescales for a range of materials subject to electron-phonon scattering. The materials span both isotropic and anisotropic systems, as well as both Elliot-Yafet and Dyakonov-Perel relaxation mechanisms. We show that T₂, for electron-phonon scattering, can be understood simply as an extension of T₁ with appropriate averaging over the directions that the spins rotate in. We then deduce a number of identities relating T₁ and T₂. Importantly, in the presence of magnetic field inhomogeneities (either in real or k-space), a spin ensemble can dephase, characterized by the timescale T₂*. Dephasing is distinct from decoherence since it can be reversed using a spin echo measurement. Our real-time framework enables a simulation of the spin echo setup, which we use to illustrate the relationship between the spin echo and the dephasing magnetic fluctuations in a simple model system.