Theory of proximity-induced exchange coupling in graphene on hBN/(Co, Ni)

We perform systematic first-principles calculations of the proximity exchange coupling, induced by cobalt (Co) and nickel (Ni) in graphene, via a few (up to three) layers of hexagonal boron nitride (hBN). We find that the induced spin splitting of the graphene bands is of the order of 10 meV for a monolayer of hBN, decreasing in magnitude but alternating in sign by adding each new insulating layer. We find that the proximity exchange can be giant if there is a resonant $d$ level of the transition metal close to the Dirac point. Our calculations suggest that this effect could be present in Co heterostructures, in which a $d$ level strongly hybridizes with the valence-band orbitals of graphene. Since this hybridization is spin dependent, the proximity spin splitting is unusually large, about 10 meV even for two layers of hBN. An external electric field can change the offset of the graphene and transition-metal orbitals and can lead to a reversal of the sign of the exchange parameter. This we predict to happen for the case of two monolayers of hBN, enabling electrical control of proximity spin polarization (but also spin injection) in graphene/hBN/Co structures. Nickel-based heterostructures show weaker proximity effects than cobalt heterostructures. We introduce two phenomenological models to describe the first-principles data. The minimal model comprises the graphene (effective) $p_z$ orbitals and can be used to study transport in graphene with proximity exchange, while the $p_z$-$d$ model also includes hybridization with $d$ orbitals, which is important to capture the giant proximity exchange. Crucial to both models is the pseudospin-dependent exchange coupling, needed to describe the different spin splittings of the valence and conduction bands.Comment: 14 pages, 17 figures, 2 table

Density functional calculations on graphene structures with WIEN2k

This work deals with the problem how to calculate some electronic properties of graphene, within the framework of density functional theory on the basis of the WIEN2k code

Spin-orbit coupling in methyl functionalized graphene

We present first-principles calculations of the electronic band structure and spin-orbit effects in graphene functionalized with methyl molecules in dense and dilute limits. The dense limit is represented by a 2$\times$2 graphene supercell functionalized with one methyl admolecule. The calculated spin-orbit splittings are up to $0.6$ meV. The dilute limit is deduced by investigating a large, 7$\times$7, supercell with one methyl admolecule. The electronic band structure of this supercell is fitted to a symmetry-derived effective Hamiltonian, allowing us to extract specific hopping parameters including intrinsic, Rashba, and PIA (pseudospin inversion asymmetry) spin-orbit terms. These proximity-induced spin-orbit parameters have magnitudes of about 1 meV, giant compared to pristine graphene whose intrinsic spin-orbit coupling is about 10 $\mu$eV. We find that the origin of this giant local enhancement is the $sp^3$ corrugation and the breaking of local pseudospin inversion symmetry, as in the case of hydrogen adatoms. Also similar to hydrogen, methyl acts as a resonant scatterer, with a narrow resonance peak near the charge neutrality point. We also calculate STM-like images showing the local charge densities at different energies around methyl on graphene.Comment: 9 pages, 10 figure

Electronic and Spin-Orbit Properties of hBN Encapsulated Bilayer Graphene

Van der Waals (vdW) heterostructures consisting of Bernal bilayer graphene (BLG) and hexagonal boron nitride (hBN) are investigated. By performing first-principles calculations we capture the essential BLG band structure features for several stacking and encapsulation scenarios. A low-energy model Hamiltonian, comprising orbital and spin-orbit coupling (SOC) terms, is employed to reproduce the hBN-modified BLG dispersion, spin splittings, and spin expectation values. Most important, the hBN layers open an orbital gap in the BLG spectrum, which can range from zero to tens of meV, depending on the precise stacking arrangement of the individual atoms. Therefore, large local band gap variations may arise in experimentally relevant moir\'{e} structures. Moreover, the SOC parameters are small (few to tens of $\mu$eV), just as in bare BLG, but are markedly proximity modified by the hBN layers. Especially when BLG is encapsulated by monolayers of hBN, such that inversion symmetry is restored, the orbital gap and spin splittings of the bands vanish. In addition, we show that a transverse electric field mainly modifies the potential difference between the graphene layers, which perfectly correlates with the orbital gap for fields up to about 1~V/nm. Moreover, the layer-resolved Rashba couplings are tunable by $\sim 5~\mu$eV per V/nm. Finally, by investigating twisted BLG/hBN structures, with twist angles between 6$^{\circ}$ -- 20$^{\circ}$, we find that the global band gap increases linearly with the twist angle. The extrapolated $0^{\circ}$ band gap is about 23~meV and results roughly from the average of the stacking-dependent local band gaps. Our investigations give new insights into proximity spin physics of hBN/BLG heterostructures, which should be useful for interpreting experiments on extended as well as confined (quantum dot) systems.Comment: 16 pages, 9 figures, 7 table

Proximity induced exchange coupling in graphene

This thesis deals with the proximity induced exchange coupling, induced from the ferromagnets cobalt and nickel through different insulators, in graphene

Twist-angle dependent proximity induced spin-orbit coupling in graphene/transition-metal dichalcogenide heterostructures

We investigate the proximity-induced spin-orbit coupling in heterostructures of twisted graphene and monolayers of transition-metal dichalcogenides (TMDCs) MoS$_2$, WS$_2$, MoSe$_2$, and WSe$_2$ from first principles. We identify strain, which is necessary to define commensurate supercells, as the key factor affecting the band offsets and thus magnitudes of the proximity couplings. We establish that for biaxially strained graphene the band offsets between the Dirac point and conduction (valence) TMDC bands vary linearly with strain, regardless of the twist angle. This relation allows to identify the apparent zero-strain band offsets and find a compensating transverse electric field correcting for the strain. The resulting corrected band structure is then fitted around the Dirac point to an established spin-orbit Hamiltonian. This procedure yields the dominant, valley-Zeeman and Rashba spin-orbit couplings. The magnitudes of these couplings do not vary much with the twist angle, although the valley-Zeeman coupling vanishes for 30$^{\circ}$ and Mo-based heterostructures exhibit a maximum of the coupling at around 20$^{\circ}$. The maximum for W-based stacks is at 0$^{\circ}$. The Rashba coupling is in general weaker than the valley-Zeeman coupling, except at angles close to 30$^{\circ}$. We also identify the Rashba phase angle which measures the deviation of the in-plane spin texture from tangential, and find that this angle is very sensitive to the applied transverse electric field. We further discuss the reliability of the supercell approach with respect to atomic relaxation (rippling of graphene), relative lateral shifts of the atomic layers, and transverse electric field.Comment: 14 pages, 9 figures, 7 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

Heterostructures of Graphene and Topological Insulators Bi2Se3, Bi2Te3, and Sb2Te3

Prototypical 3D topological insulators of the Bi2Se3 family provide a beautiful example of the appearance of the surface states inside the bulk bandgap caused by spin-orbit coupling-induced topology. The surface states are protected against backscattering by time reversal symmetry, and exhibit spin-momentum locking whereby the electron spin is polarized perpendicular to the momentum, typically in the plane of the surface. In contrast, graphene is a prototypical 2D material, with negligible spin-orbit coupling. When graphene is placed on the surface of a topological insulator, giant spin-orbit coupling is induced by the proximity effect, enabling interesting novel electronic properties of its Dirac electrons. A detailed theoretical study of the proximity effects of monolayer graphene and topological insulators Bi2Se3, Bi2Te3, and Sb2Te3 is presented, and the appearance of the qualitatively new spin-orbit splittings, well described by a phenomenological Hamiltonian, is elucidated by analyzing the orbital decomposition of the involved band structures. This should be useful for building microscopic models of the proximity effects between the surfaces of the topological insulators and graphene