Signatures of electric field and layer separation effects on the spin-valley physics of MoSe2_2/WSe2_2 heterobilayers: from energy bands to dipolar excitons

We investigate the spin-valley physics (SVP) in MoSe$_2$/WSe$_2$ heterobilayers under external electric field (EF) and changes of the interlayer distance (ID). We analyze the spin ($S_z$) and orbital ($L_z$) degrees of freedom, and the symmetry properties of relevant band edges (at K, Q, and $\Gamma$ points) in high-symmetry stackings at 0 (R-type) and 60 (H-type) degree angles, the important building blocks of moir\'e or atomically reconstructed structures. We reveal distinct hybridization signatures of $S_z$ and $L_z$ in low-energy bands due to the wave function mixing between the layers, which are stacking-dependent and can be further modified by EF and ID. The H-type stackings favor large changes in the g-factors under EF, e. g. from $-5$ to $3$ in the valence bands of the H$^h_h$ stacking, due to the opposite orientation of $S_z$ and $L_z$ in the individual monolayers. For the low-energy dipolar excitons (DEs), direct and indirect in $k$-space, we quantify the electric dipole moments and polarizabilities, reflecting the layer delocalization of the constituent bands. We found that direct DEs carry a robust valley Zeeman effect nearly independent of the EF but tunable by the ID, which can be experimentally accessible via applied external pressure. For the momentum-indirect DEs, our symmetry analysis indicates that phonon-mediated optical processes can easily take place. For the indirect DEs with conduction bands at the Q point for H-type stackings, we found marked variations of the valley Zeeman ($\sim 4$) as a function of the EF that notably stand out from the other DE species. Stronger signatures of the coupled SVP are favored in H-type stackings, which can be experimentally investigated in $\sim 60^\text{o}$ samples. Our study provides fundamental insights into the SVP of van der Waals heterostructures, relevant to understand the valley Zeeman of DEs and intralayer excitons.Comment: 19 pages, 15 figures, 8 table

Giant proximity exchange and valley splitting in TMDC/hBN/(Co, Ni) heterostructures

We investigate the proximity-induced exchange coupling in transition-metal dichalcogenides (TMDCs), originating from spin injector geometries composed of hexagonal boron-nitride (hBN) and ferromagnetic (FM) cobalt (Co) or nickel (Ni), from first-principles. We employ a minimal tight-binding Hamiltonian that captures the low energy bands of the TMDCs around K and K' valleys, to extract orbital, spin-orbit, and exchange parameters. The TMDC/hBN/FM heterostructure calculations show that due to the hBN buffer layer, the band structure of the TMDC is preserved, with an additional proximity-induced exchange splitting in the bands. We extract proximity exchange parameters in the 1-10 meV range, depending on the FM. The combination of proximity-induced exchange and intrinsic spin-orbit coupling (SOC) of the TMDCs, leads to a valley polarization, translating into magnetic exchange fields of tens of Tesla. The extracted parameters are useful for subsequent exciton calculations of TMDCs in the presence of a hBN/FM spin injector. Our calculated absorption spectra show a large splitting of the first exciton peak; in the case of MoS$_2$/hBN/Co we find a value of about 8 meV, corresponding to about 50 Tesla external magnetic field in bare TMDCs. The reason lies in the band structure, where a hybridization with Co $d$ orbitals causes a giant valence band exchange splitting of more than 10 meV. Structures with Ni do not show any $d$ level hybridization features, but still sizeable proximity exchange and exciton peak splittings of around 2 meV are present in the TMDCs

Signatures of Electric Field and Layer Separation Effects on the Spin-Valley Physics of MoSe2/WSe2 Heterobilayers: From Energy Bands to Dipolar Excitons

Multilayered van der Waals heterostructures based on transition metal dichalcogenides are suitable platforms on which to study interlayer (dipolar) excitons, in which electrons and holes are localized in different layers. Interestingly, these excitonic complexes exhibit pronounced valley Zeeman signatures, but how their spin-valley physics can be further altered due to external parameters—such as electric field and interlayer separation—remains largely unexplored. Here, we perform a systematic analysis of the spin-valley physics in MoSe2/WSe2 heterobilayers under the influence of an external electric field and changes of the interlayer separation. In particular, we analyze the spin (Sz) and orbital (Lz) degrees of freedom, and the symmetry properties of the relevant band edges (at K, Q, and Γ points) of high-symmetry stackings at 0° (R-type) and 60° (H-type) angles—the important building blocks present in moiré or atomically reconstructed structures. We reveal distinct hybridization signatures on the spin and the orbital degrees of freedom of low-energy bands, due to the wave function mixing between the layers, which are stacking-dependent, and can be further modified by electric field and interlayer distance variation. We find that H-type stackings favor large changes in the g-factors as a function of the electric field, e.g., from −5 to 3 in the valence bands of the Hhh stacking, because of the opposite orientation of Sz and Lz of the individual monolayers. For the low-energy dipolar excitons (direct and indirect in k-space), we quantify the electric dipole moments and polarizabilities, reflecting the layer delocalization of the constituent bands. Furthermore, our results show that direct dipolar excitons carry a robust valley Zeeman effect nearly independent of the electric field, but tunable by the interlayer distance, which can be rendered experimentally accessible via applied external pressure. For the momentum-indirect dipolar excitons, our symmetry analysis indicates that phonon-mediated optical processes can easily take place. In particular, for the indirect excitons with conduction bands at the Q point for H-type stackings, we find marked variations of the valley Zeeman (∼4) as a function of the electric field, which notably stands out from the other dipolar exciton species. Our analysis suggests that stronger signatures of the coupled spin-valley physics are favored in H-type stackings, which can be experimentally investigated in samples with twist angle close to 60°. In summary, our study provides fundamental microscopic insights into the spin-valley physics of van der Waals heterostructures, which are relevant to understanding the valley Zeeman splitting of dipolar excitonic complexes, and also intralayer excitons

Spin-orbit coupling in elemental two-dimensional materials

The fundamental spin-orbit coupling and spin mixing in graphene and rippled honeycomb lattice materials silicene, germanene, stanene, blue phosphorene, arsenene, antimonene, and bismuthene is investigated from first principles. The intrinsic spin-orbit coupling in graphene is revisited using multi-band $k\cdot p$ theory, showing the presence of non-zero spin mixing in graphene despite the mirror symmetry. However, the spin mixing itself does not lead to the the Elliott-Yafet spin relaxation mechanism, unless the mirror symmetry is broken by external factors. For other aforementioned elemental materials we present the spin-orbit splittings at relevant symmetry points, as well as the spin admixture $b^2$ as a function of energy close to the band extrema or Fermi levels. We find that spin-orbit coupling scales as the square of the atomic number Z, as expected for valence electrons in atoms. For isolated bands, it is found that $b^2\sim Z^4$. The spin-mixing parameter also exhibits giant anisotropy which, to a large extent, can be controlled by tuning the Fermi level. Our results for $b^2$ can be directly transferred to spin relaxation time due to the Elliott-Yafet mechanism, and therefore provide an estimate of the upper limit for spin lifetimes in materials with space inversion center.Comment: 10 pages, 8 figure

k.p theory for phosphorene: Effective g-factors, Landau levels, and excitons

Phosphorene, a single layer of black phosphorus, is a direct band gap two-dimensional semiconductor with promising charge and spin transport properties. The electronic band structure of phosphorene is strongly affected by the structural anisotropy of the underlying crystal lattice. We describe the relevant conduction and valence bands close to the Gamma-point by four-and six-band (with spin) k . p models, including the previously overlooked interband spin-orbit coupling which is essential for studying anisotropic crystals. All the k . p parameters are obtained by a robust fit to ab initio data, by taking into account the nominal band structure and the k-dependence of the effective mass close to the Gamma-point. The inclusion of interband spin-orbit coupling allows us to determine dipole transitions along both armchair and zigzag directions. The interband coupling is also key to determine the effective g-factors and Zeeman splittings of the Landau levels. We predict the electron and hole g-factor correction of approximate to 0.03 due to the intrinsic contributions in phosphorene, which lies within the existing range of experimental data. Furthermore, we investigate excitonic effects using the k . p models and find exciton binding energy (0.81 eV) and exciton diameters consistent with experiments and ab initio based calculations. The proposed k . p Hamiltonians should be useful for investigating magnetic, spin, transport, and optical properties and many-body effects in phosphorene

Strain-tunable orbital, spin-orbit, and optical properties of monolayer transition-metal dichalcogenides

When considering transition-metal dichalcogenides (TMDCs) in van der Waals heterostructures for periodic ab initio calculations, usually, lattice mismatch is present, and the TMDC needs to be strained. In this study we provide a systematic assessment of biaxial strain effects on the orbital, spin-orbit, and optical properties of the monolayer TMDCs using ab initio calculations. We complement our analysis with a minimal tight-binding Hamiltonian that captures the low-energy bands of the TMDCs around the K and K' valleys. We find characteristic trends of the orbital and spin-orbit parameters as a function of the biaxial strain. Specifically, the orbital gap decreases linearly, while the valence (conduction) band spin splitting increases (decreases) nonlinearly in magnitude when the lattice constant increases. Furthermore, employing the Bethe-Salpeter equation and the extracted parameters, we show the evolution of several exciton peaks, with biaxial strain, on different dielectric surroundings, which are particularly useful for interpreting experiments studying strain-tunable optical spectra of TMDCs

Proximity exchange effects in MoSe2 and WSe2 heterostructures with CrI3: Twist angle, layer, and gate dependence

Proximity effects in two-dimensional (2D) van der Waals heterostructures offer controllable ways to tailor the electronic band structure of adjacent materials. Proximity exchange in particular is important for making materials magnetic without hosting magnetic ions. Such synthetic magnets could be used for studying magnetotransport in high-mobility 2D materials, or magneto-optics in highly absorptive nominally nonmagnetic semiconductors. Using first-principles calculations, we show that the proximity exchange in monolayer MoSe2 and WSe2 due to ferromagnetic monolayer CrI3 can be tuned (even qualitatively) by twisting and gating. Remarkably, the proximity exchange remains the same when using antiferromagnetic CrI3 bilayer, paving the way for optical and electrical detection of layered antiferromagnets. Interestingly, the proximity exchange is opposite to the exchange of the adjacent antiferromagnetic layer. Finally, we show that the proximity exchange is confined to the layer adjacent to CrI3, and that adding a separating hBN barrier drastically reduces the proximity effect. We complement our ab initio results with tight-binding modeling and solve the Bethe-Salpeter equation to provide experimentally verifiable optical signatures (in the exciton spectra) of the proximity exchange effects