Controllable Goos-H\"{a}nchen shifts and spin beam splitter for   ballistic electrons in a parabolic quantum well under a uniform magnetic   field
  The quantum Goos-H\"{a}nchen shift for ballistic electrons is investigated in a parabolic potential well under a uniform vertical magnetic field. It is found that the Goos-H\"{a}nchen shift can be negative as well as positive, and becomes zero at transmission resonances. The beam shift depends not only on the incident energy and incidence angle, but also on the magnetic field and Landau quantum number. Based on these phenomena, we propose an alternative way to realize the spin beam splitter in the proposed spintronic device, which can completely separate spin-up and spin-down electron beams by negative and positive Goos-H\"{a}nchen shifts. 
Xi Chen, Xiao-Jing Lu, Yan Wang, Chun-Fang Li
cond-mat.mes-hall