Scattering of relativistic electrons and analogies with optical phenomena: A study of longitudinal and transverse shifts at step potentials

We investigate the behavior of relativistic electrons encountering a potential step through analogies with optical phenomena. By accounting for the conservation of the Dirac current, we elucidate that the Goos-Hänchen shift can be understood as a combination of two components: one arising from the current entering the transmission region and the other originating from the interference between the incident and reflected beams. This result has been proven to be consistent with findings obtained utilizing the stationary phase method. Moreover, we explore the transverse Imbert-Fedorov shift by applying both current conservation and total angular momentum conservation, revealing intriguing parallel to the spin Hall effect. Beyond enriching our comprehension of fundamental quantum phenomena, our findings have potential applications for designing and characterizing devices using Dirac and topological materials.

Figure 1

(a) Schematic diagram of relativistic electrons encountering a step potential, depicting spin-flip reflection and transmission. Projection in the (b) $yoz$ plane and in the (c) $xoz$ plane, showcasing the GH ($S_y$) and IF ($S_x$) shifts, respectively, along with the corresponding currents.

Figure 2

Separation of the various energy zones by fixing, respectively, the incident angle $\theta$ and the potential $V_0$, where $V_0=5m$. Two blue solid lines are the critical angles of the existence of tunneling and orange dotted line presents $E=2.5m$, corresponding to ${\theta }_c=\pi /2$, solved from Eq. (8).

Figure 3

The longitudinal GH shift $S_y=S_y^{\text{e}}+S_y^{\text{ir}}$ (in units of the reduced Compton wavelength $\overline{\lambda }=\hslash /mc$) is depicted as a function of the incidence angle $\theta$. The shifts contributed from $S_y^{\text{e}}$ (dashed, red) and $S_y^{\text{ir}}$ (dot-dashed, black) are compared to the one obtained from the stationary phase method $S_y^p=S_y$ (solid, blue). The orange dotted line denotes the GH shift around the critical angle ${\theta }_c$ for visual reference. The parameters are set to $E_0=8.5m$, $V_0=5m$, and $m=1$, $\hslash =1$ and $c=1$ are used for simplicity.

Figure 4

The dependence of the transverse IF shift, denoted by $S_x$ (in units of the reduced Compton wavelength $\overline{\lambda }=\hslash /mc$) on the incidence angle $\theta$ is illustrated. Blue solid and red dashed lines represent spin-up and spin-down polarizations of incidence. For comparison, the shifts $\delta S_x$ (in units of $\overline{\lambda }=\hslash /mc$) calculated from Eq. (23) are also depicted by purple dotted and black dotted-dash lines, corresponding to two different polarizations, respectively. All parameters remain consistent with those in Fig. 3.