Theory of charge-to-spin conversion under quantum confinement

The interplay between spin and charge degrees of freedom in low-dimensional systems is a cornerstone of modern spintronics, where achieving all-electrical control of spin currents is a major goal. Spin-orbit interactions provide a promising mechanism for such control, yet understanding how spin and charge transport emerge from microscopic principles remains a fundamental challenge. Here, we develop a spin-dependent scattering matrix approach to describe spin and charge transport in a multiterminal system in the presence of Rashba spin-orbit interaction. Our framework generalizes the Büttiker formalism by offering expressions for spin and charge current densities as a function of the lead position, along with the corresponding linear response function. It simultaneously captures the effects of quantum confinement, the response to external magnetic fields, and the intrinsic properties of the electronic bands, offering a comprehensive description of the spin-charge interconversion mechanisms at play in a Hall bar, in agreement with experiments.