Influence of the velocity barrier on the massive Dirac electron transport in a monolayer MoS$_{2}$ quantum structure

Using the transfer matrix method, spin- and valley-dependent electron transport properties modulated by the velocity barrier were studied in the normal/ferromagnetic/normal monolayer MoS$_{2}$ quantum structure. Based on Snell's Law in optics, we define the velocity barrier as $\xi=v_{2}/v_{1}$ by changing the Fermi velocity of the intermediate ferromagnetic region to obtain a deflection condition during the electron transport process in the structure. The results show that both the magnitude and the direction of spin- and valley-dependent electron polarization can be regulated by the velocity barrier. $-100\%$ polarization of spin- and valley-dependent electron can be achieved for $\xi>1$, while $100\%$ polarization can be obtained for $\xi<1$. Furthermore, it is determined that perfect spin and valley transport always occur at a large incident angle. In addition, the spin- and valley-dependent electron transport considerably depends on the length $k_{F}L$ and the gate voltage $U(x)$ of the intermediate ferromagnetic region. These findings provide an effective method for designing novel spin and valley electronic devices.