Group delay time of fermions in graphene through tilted potential barrier

The group delay time of Dirac fermions subjected to a tilting barrier potential along the $x$-axis is investigated in graphene. We start by finding the eigenspinor solution of the Dirac equation and then relating it to incident, reflected, and transmitted beam waves. This relationship allows us to compute the group delay time in transmission and reflection by obtaining the corresponding phase shifts. We discovered that the barrier width, incident energy, and incident angle can all be used to modify the group delay time, and that the particles travel through the barrier at the Fermi velocity $v_F$. Our findings also show that the transmission group delay might be controlled, and that gate voltage control could be useful in graphene-based tilting barriers.