Low-energy coherent transport and Goos-Hänchen (GH) lateral shift of valley electrons in planar heterojunctions composed of normal MoS2 and ferromagnetic WS2 monolayers are theoretically investigated. Two types of heterojunctions in the forms of WS2/MoS2/WS2 (type-A) and MoS2/WS2/MoS2 (type-B) with incident electrons in MoS2 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 WS2/MoS2 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.

Based on a Dirac-like Hamiltonian and coherent scattering formalism, we study the spin-valley transport and Goos-Hänchen-like (GHL) effect of transmitted and reflected electrons in a gated monolayer WS. Our results show that the lateral shift of spin-polarized electrons is strongly dependent on the width of the gated region and can be positive or negative in both Klein tunneling and classical motion regimes. The absolute values of the lateral displacements at resonance positions can be considerably enhanced when the incident angle of electrons is close to the critical angle. In contrast to the time reversal symmetry for the transmitted electrons, the GHL shift of the reflected beams is not invariant under simultaneous interchange of spins and valleys, indicating the lack of spin-valley symmetry induced by the tunable potential barrier on the WS monolayer. Our findings provide evidence for electrical control of valley filtering and valley beam splitting by tuning the incident angle of electrons in nanoelectronic devices based on monolayer transition metal dichalcogenides.

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In this paper we use the fuzzy Caputo derivatives under generalized Hukuhara difference to introduce fuzzy fractional Volterra-Fredholm integro-differential equations and prove the existence and uniqueness of the solutions for this class of fractional equations.

The electron scattering from periodic line defects on the surface of topological insulators with hexagonal warping effect is investigated theoretically by means of a transfer matrix method. The influence of surface line defects, acting as structural ripples on propagation of electrons are studied in two perpendicular directions due to the asymmetry of warped energy contour under momentum exchange. The transmission profiles and the details of resonant peaks which vary with the number of defects and the strength of their potentials are strongly dependent on the direction in which the line defects extend. At low energies, the quantum interference between the incident and reflected propagating electrons has the dominant contribution in transmission resonances, while at high energies the multiple scattering processes on the constant-energy contour also appear because of the strong warping effect. By increasing the spatial separation between the line defects, the minimum value of electrical conductance remains significantly high at low incident energies, while the minimum value may approach zero at high energies as the number of defects is increased. Our findings suggest that the potential ripples on the surface of topological insulators can be utilized to control the local electronic properties of these materials.