We propose an effective lattice Hamiltonian for monolayer MoS2 in order to describe the lowenergy band structure and investigate the effect of perpendicular electric and magnetic fields on its electronic structure. We derive a tight-binding model based on the hybridization of the d orbitals of molybdenum and p orbitals of sulfur atoms and then introduce a modified two-band continuum model of monolayer MoS2 by exploiting the quasi-degenerate partitioning method. Our theory proves that the low-energy excitations of the system are no longer massive Dirac fermions. It reveals a difference between electron and hole masses and provides trigonal warping effects. Furthermore, we predict a valley degeneracy breaking effect in the Landau levels. Besides, we also show that applying a gate voltage perpendicular to the monolayer modifies the electronic structure including the band gap and effective masses.

We theoretically demonstrate the capability of a ferromagnetic-normal interface in graphene to focus an electron wave with a certain spin direction. The essential feature is the negative refraction Klein tunneling, which is spin resolved when the exchange energy of ferromagnetic graphene exceeds its Fermi energy. Exploiting this property, we propose a graphene normal-ferromagnetic-normal electronic spin lens through which an unpolarized electronic beam can be collimated with a finite spin polarization. Our study reveals that magnetic graphene has the potential to be the electronic counterpart of the recently discovered photonic chiral metamaterials that exhibit a negative refractive index for only one direction of the circular polarization of the photon wave.

Conventional wisdom expects that making semiconductors ferromagnetic requires doping with magnetic ions, and that superconductivity cannot coexist with magnetism. However, recent concerted efforts exploring new classes of materials have established that intrinsic ferromagnetic semiconductors exist and that certain types of strongly correlated metals can be ferromagnetic and superconducting at the same time. Here we show that the trifecta of semiconducting behavior, ferromagnetism and superconductivity can be achieved in a single material. Samarium nitride (SmN) is a well-characterised intrinsic ferromagnetic semiconductor, hosting strongly spin-ordered 4f electrons below a Curie temperature of 27 K. We have now observed that it also hosts a superconducting phase below 4 K when doped to electron concentrations above 10 21 cm −3 . The large exchange splitting of the conduction band in SmN favors equal-spin triplet pairing with p-wave symmetry. An analysis of the robustness of such a superconducting phase against disorder leads to the conclusion that the 4f bands are crucial for superconductivity, making SmN a heavy-fermion-type superconductor.

We show that Andreev reflection in a junction between ferromagnetic (F) and superconducting (S) graphene regions is fundamentally different from the common FS junctions. For a weakly doped F graphene with an exchange field h larger than its Fermi energy EF, Andreev reflection of massless Dirac fermions is associated with a Klein tunneling through an exchange field p-n barrier between two spin-split conduction and valence subbands. We find that this Andreev-Klein process results in an enhancement of the subgap conductance of a graphene FS junction by h up to the point at which the conductance at low voltages eV ≪ ∆ is greater than its value for the corresponding nonferromagnetic junction. We also demonstrate that the Andreev reflection can be of retro or specular types in both convergent and divergent ways with the reflection direction aligned, respectively, closer to and farther from the normal to the junction as compared to the incidence direction.PACS numbers: 74.45.+c, 74.78.Na Transmission of low energy electrons through a normalmetal -superconductor (NS) junction is realized via a peculiar scattering process, known as Andreev reflection (AR) [1]. In AR an electron excitation with energy ε and spin direction σ upon hitting the NS-interface is converted into a hole excitation with the same energy but opposite spin directionσ = −σ. Under AR the momentum change is of order ε/v F which is negligibly small for a degenerate N metal with large Fermi energy E F ≫ ∆. Thus the hole velocity is almost opposite to the velocity of the incident electron (since a hole moves opposite to its momentum), which implies that Andreev process is retro reflection. Andreev reflection results in a finite conductance of a NS junction at the voltages below the superconducting gap ∆ [2].The fact that the Andreev reflected electron-hole belong to different spin-subbands has an important consequence for Andreev conductance when N metal is a ferromagnet (F). The exchange splitting energy h of F-metal induces an extra momentum change 2h/v F of the reflected hole which diminishes the amplitude of AR. As the result the subgap Andreev condutance of ferromagnetsuperconductor (FS) junctions decreases with increasing h and vanishes for a half-metal F with h = E F [3]. Suppression of AR at FS interface is a manifestation of the common fact that ferromagnetism and spin singlet superconductivity are opposing phenomena. In this letter, however, we show that the situation differs significantly if the FS junction is realized in graphene, the recently discovered two-dimensional (2D) carbon atoms arranged in hexagonal lattice [4,5,6]. We find that in a graphene FS junction, the exchange interaction can enhance the subgap Andreev conductance, depending on the doping of F graphene. In particular we show that at low voltage eV ≪ ∆ the conductance of a graphene FS junction with a strong exchange field h ≫ E F is larger than its value for the corresponding NS structure. We explain this effect in terms of Andreev-Klein reflection in which the superconductin...