Electronic structure and anisotropic Rashba spin-orbit coupling in monolayer black phosphorus

Abstract

We investigate the electronic structure of the monolayer black phosphorus (BP) using density functional methods both with and without an applied electric field. We find that a simple one-band tight-binding Hamiltonian based on the pz orbitals and nearest-neighbor hopping is sufficient to describe the band structure in the gap region rather well and justification for this is given from symmetry arguments. The anisotropic nature of the band structure leads in turn to an anisotropic Rashba effect, where the magnitude of the spin splitting caused by an applied electric field is not only momentum dependent, but also depends on the direction of k. The Rashba Hamiltonian is generalized for the anisotropic case, which reads: HR = αR( σ × k ′ ) ·ẑ, where the scaled momentum k ′ contains the anisotropy effect. The Rashba effect is studied quantitatively for BP from ab initio density-functional calculations in the presence of an applied electric field. A by-product of this work is the demonstration that the strength of the spin-orbit coupling for the outermost electrons in the atoms, which are relevant for the solids, increases only as the Landau-Lifshitz Z 2 scaling with the atomic number Z, rather than the higher power Z 4 scaling, as sometimes thought.

Electronic structure and thermoelectric transport of black phosphorus

“…2 confirm that the active electronic states in this system involve the 3p carriers, where all three p bands have appreciable weight below and above the Fermi energy, E F (=ω = 0). Also consistent with previous calculations [13], our results reveal an anisotropic one-particle band structure.…”

“…The semiconducting behavior of bulk BP is, in our approach, that of a correlated Kondo insulator. As discussed earlier [13], within the puckered lattice structure of BP a narrow semiconducting band gap opens up because of the orbital-dependent p-band hybridization. Additionally, as shown here, correlation effects in conjunction with band structure details play a crucial role.…”

“…1. Due to its intrinsic semiconducting band gap [12][13][14] and peculiar electronic structure [15] it displays anisotropic particle-hole excitations [16], magnetoresistance [10], and thermal transport properties [4]. Moreover, the band gap of bulk BP is tunable by strain [3] and moderated pressures [17].…”

Topological spin-singlet superconductors with underlying sublattice structure

“…Two of the four orbitals experience an electric potential V E , while the two others experience −V E . For an electric potential V E 1 eV, the two different Rashba spin-orbit hoppings have been estimated as λ 0.004 eV and λ = αλ [62]. As shown in Appendix E, this description preserves inversion symmetry within the phosphorene sheet.…”

“…[62]. Crucially, the spinless description of phosphorene characterized by diagonal blocks of the type K σσ (k) is still invariant under inversion symmetry since…”