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In this dissertation we are mainly interested in presenting a mathematical formalism in order to derive the Spin Boltzmann Transport equation (SBTE). Our preeminent research interest is spintronics, specifically, spin relaxation phenomena. The aforesaid, is a key phenomena in the field of spintronics and could be direct linked to future technology development. The SBTE provides a spin drift-diffusion equation that carries the information about the spin decay. Aiming to derive the Boltzmann Transport equation (BTE), one has to resort to mathematical tools, viz., the Keldysh formalism and Non-equilibrium Green’s functions (NEGF). A rigorously analyses and a deep understanding of the formalism, is crucial in order to generalize the BTE to include spin, higher order spin-orbit interactions, scattering from impurities, electrons, phonons, etc. As it turns out, we have been able to master the formalism and deriving the spin drift-diffusion equation, which is a formidable task. We present steps necessary to not only interpret such equation but also to include other types of interactions one might be interested in. We also show how to apply the spin-drift equation to a regime known as the Persistent Spin Helix (PSH). By the end of our research we have also managed to formally derive a novel spin drift-diffusion equation for heterostructures with two occupied subbands. This is a preliminary result, however, new in the literature. In addition, we have applied it to a regime known as crossed Persistent Spin helix (cPSH), for which the subbands are set into orthogonal PSH regimes. In this case we find that the cPSH dynamics depends on the relative intensity between the spin-orbit energy splitting and the impurity induced broadening of the states. Therefore, mastering the formalism paid off by allowing us to formally generalize the PSH dynamics to the two subband problem, whilst further extensions towards novel systems and additional interactions (collision integrals) are now at reach.

From the Keldysh formalism to the Boltzmann equation for spin drift and diffusion