Band-offset-induced lateral shift of valley electrons in ferromagnetic MoS$_2$/WS$_2$ planar heterojunctions

Low-energy coherent transport and Goos-H\"{a}nchen (GH) lateral shift of valley electrons in planar heterojunctions composed of normal MoS$_2$ and ferromagnetic WS$_2$ monolayers are theoretically investigated. Two types of heterojunctions in the forms of WS$_2$/MoS$_2$/WS$_2$ (type-A) and MoS$_2$/WS$_2$/MoS$_2$ (type-B) with incident electrons in MoS$_2$ region are considered in which the lateral shift of electrons is induced by band alignments of the two constituent semiconductors. It is shown that the type-A heterojunction can act as an electron waveguide due to electron confinement between the two WS$_2$/MoS$_2$ interfaces which cause the incident electrons with an appropriate incidence angle to propagate along the interfaces. In this case the spin- and valley-dependent GH shifts of totally reflected electrons from the interface lead to separated electrons with distinct spin-valley indexes after traveling a sufficiently long distance. In type-B heterojunction, however, transmission resonances occur for incident electron beams passing through the structure, and large spin- and valley-dependent lateral shift values in propagating states can be achieved. Consequently, the transmitted electrons are spatially well-separated into electrons with distinct spin-valley indexes. Our findings reveal that the planar heterojunctions of transition metal dichalcogenides can be utilized as spin-valley beam filter and/or splitter without external gating.