Researchers have developed a theoretical framework to control spin- and valley-resolved quantum transport in monolayer tungsten diselenide (WSe$_2$). This two-dimensional transition metal dichalcogenide (TMD) is described by an effective massive Dirac Hamiltonian. The study focuses on a barrier region where both Fermi velocity and scalar potential are simultaneously modulated, allowing for precise control over the transport properties of quasiparticles.

The research explores how the velocity ratio between the barrier and the external region, $\xi = v_2/v_1$, inspired by Snell-Descartes law, influences the refraction of charge carriers. It is observed that this simple ratio is only recovered in the massless, symmetric limit. The interplay of intrinsic spin-orbit coupling in the conduction ($\lambda_c$) and valence ($\lambda_v$) bands, along with spin- (M_s) and valley-dependent (M_v) Zeeman fields, significantly alters the quasiparticle dispersion, modifying the transport characteristics.

By solving the Dirac equation and enforcing current-conserving matching conditions at the interfaces, the scientists computed the spin- and valley-dependent transmission probability and conductance. The results demonstrate that the barrier velocity, scalar potential, incidence angle, incident energy, and barrier width act as effective control parameters. This leads to strong anisotropy and resonant tunneling features. Furthermore, it is shown that both the magnitude and orientation of spin- and valley-polarized currents can be continuously tuned via velocity and potential modulation.

These findings establish combined velocity and potential engineering as a powerful theoretical tool for manipulating spin-valley physics in two-dimensional TMDs. This control opens new avenues for the development of advanced spintronic and valleytronic devices, leveraging the unique quantum properties of these materials for future applications in computing and sensing.