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Spin-orbit coupling (SOC) and electron-electron interaction can mutually influence each other and give rise to a plethora of intriguing phenomena in condensed matter systems. In pristine bilayer graphene, which has weak SOC, intrinsic Lifshitz transitions and concomitant van-Hove singularities lead to the emergence of many-body correlated phases. Layer-selective SOC can be proximity induced by adding a layer of tungsten diselenide (WSe2) on its one side. By applying an electric displacement field, the system can be tuned across a spectrum wherein electronic correlation, SOC, or a combination of both dominates. Our investigations reveal an intricate phase diagram of proximity-induced SOC-selective bilayer graphene. Not only does this phase diagram include those correlated phases reminiscent of SOC-free doped bilayer graphene, but it also hosts unique SOC-induced states allowing a compelling measurement of valley g-factor and a seemingly impossible correlated insulator at charge neutrality, thereby showcasing the remarkable tunability of the interplay between interaction and SOC in WSe2 enriched bilayer graphene.

Rashba spin-orbit coupling is a quintessential spin interaction appearing in virtually any electronic heterostructure. Its paradigmatic spin texture in the momentum space forms a tangential vector field. Using first-principles investigations, we demonstrate that in twisted homobilayers and hetero-multilayers, the Rashba coupling can be predominantly radial, parallel to the momentum. Specifically, we study four experimentally relevant structures: twisted bilayer graphene (Gr), twisted bilayer WSe$_2$, and twisted multilayers WSe$_2$/Gr/WSe$_2$ and WSe$_2$/Gr/Gr/WSe$_2$. We show, that the Rashba spin-orbit field texture in such structures can be controlled by an electric field, allowing to tune it from radial to tangential. Such spin-orbit engineering should be useful for designing novel spin-charge conversion and spin-orbit torque schemes, as well as for controlling correlated phases and superconductivity in van der Waals materials.

A promising approach to attain long-distance coherent spin propagation is accessing topological spin-polarized edge states in graphene. Achieving this without external magnetic fields necessitates engineering graphene band structure, obtainable through proximity effects in van der Waals heterostructures. In particular, proximity-induced staggered potentials and spin-orbit coupling are expected to form a topological bulk gap in graphene with gapless helical edge states that are robust against disorder. In this work, we detect the spin-polarized helical edge transport in graphene at zero external magnetic field, allowed by the proximity of an interlayer antiferromagnet, CrPS$_4$. We show the coexistence of the quantum spin Hall (QSH) states and magnetism in graphene, where the induced spin-orbit and exchange couplings also give rise to a large anomalous Hall (AH) effect. The detection of the QSH states at zero external magnetic field, together with the AH signal that persists up to room temperature, opens the route for practical applications of magnetic graphene in quantum spintronic circuitries.

We present a DFT-based investigation of the twist-angle dependent proximity spin-orbit coupling (SOC) in graphene/TMDC structures. We find that for Mo-based TMDCs the proximity valley-Zeeman SOC exhibits a maximum at around 15--20{\deg}, and vanishes at 30{\deg}, while for W-based TMDCs we find an almost linear decrease of proximity valley-Zeeman SOC when twisting from 0{\deg} to 30{\deg}. The induced Rashba SOC is rather insensitive to twisting, while acquiring a nonzero Rashba phase angle, $\varphi \in [-20;40]${\deg}, for twist angles different from 0{\deg} and 30{\deg}. This finding contradicts earlier tight-binding predictions that the Rashba angle can be 90{\deg} in the studied systems. In addition, we study the influence of several tunability knobs on the proximity SOC for selected twist angles. By applying a transverse electric field in the limits of $\pm 2$ V/nm, mainly the Rashba SOC can be tuned by about 50\%. The interlayer distance provides a giant tunability, since the proximity SOC can be increased by a factor of 2--3, when reducing the distance by about 10\%. Encapsulating graphene between two TMDCs, both twist angles are important to control the interference of the individual proximity SOCs, allowing to precisely tailor the valley-Zeeman SOC in graphene, while the Rashba SOC becomes suppressed. Finally, based on our effective Hamiltonians with fitted parameters, we calculate experimentally measurable quantities such as spin lifetime anisotropy and charge-to-spin conversion efficiencies. The spin lifetime anisotropy can become giant, up to $10^4$, in encapsulated structures. The charge-to-spin conversion, which is due to spin-Hall and Rashba-Edelstein effects, can lead to twist-tunable non-equilibrium spin-density polarizations that are perpendicular and parallel to the applied charge current., Comment: 23 pages, 14 figures, 4 tables + supplemental material

Van der Waals (vdW) heterostructures composed of two-dimensional (2D) transition metal dichalcogenides (TMD) and vdW magnetic materials offer an intriguing platform to functionalize valley and excitonic properties in non-magnetic TMDs. Here, we report magneto-photoluminescence (PL) investigations of monolayer (ML) MoSe$_2$ on the layered A-type antiferromagnetic (AFM) semiconductor CrSBr under different magnetic field orientations. Our results reveal a clear influence of the CrSBr magnetic order on the optical properties of MoSe$_2$, such as an anomalous linear-polarization dependence, changes of the exciton/trion energies, a magnetic-field dependence of the PL intensities, and a valley $g$-factor with signatures of an asymmetric magnetic proximity interaction. Furthermore, first principles calculations suggest that MoSe$_2$/CrSBr forms a broken-gap (type-III) band alignment, facilitating charge transfer processes. The work establishes that antiferromagnetic-nonmagnetic interfaces can be used to control the valley and excitonic properties of TMDs, relevant for the development of opto-spintronics devices., Comment: 14 pages, 4 figures

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 tables

The $GW$ method is widely used for calculating the electronic band structure of materials. The high computational cost of $GW$ algorithms prohibits their application to many systems of interest. We present a periodic, low-scaling and highly efficient $GW$ algorithm that benefits from the locality of the Gaussian basis and the polarizability. The algorithm enables $G_0W_0$ calculations on a MoSe$_2$/WS$_2$ bilayer with 984 atoms per unit cell, in 42 hours using 1536 cores. This is four orders of magnitude faster than a plane-wave $G_0W_0$ algorithm, allowing for unprecedented computational studies of electronic excitations at the nanoscale.

The impact of the core mass on the compact/neutron-star mass-radius relation is studied. Besides the mass, the core is parameterized by its radius and surface pressure, which supports the outside one-component Standard Model (SM) matter. The core may accommodate SM matter with unspecified (or poorly known) equation-of-state or several components, e.g. consisting of admixtures of Dark Matter and/or Mirror World matter etc. beyond the SM. Thus, the admissible range of masses and radii of compact stars can be considerably extended., Comment: 20 pages, 9 figures, 1 table. Contribution to a Special Issue of PARTICLES: "Strong Interactions in the Standard Model: Massless Bosons to Compact Stars". Version accepted for publication. arXiv admin note: text overlap with arXiv:2203.17228

We theoretically investigate 3D layered crystals of alternating graphene and hBN layers with different symmetries. Depending on the hopping parameters between the graphene layers, we find that these synthetic 3D materials can feature semimetallic, gapped, or Weyl semimetal phases. Our results demonstrate that 3D crystals stacked from individual 2D materials represent a synthetic materials class with emergent properties different from their constituents.

We investigate the twist-angle and gate dependence of the proximity-induced exchange coupling in the monolayer transition-metal dichalcogenides (TMDCs) MoSe$_2$ and WSe$_2$ due to the vdW coupling to the ferromagnetic semiconductor CrI$_3$, from first-principles calculations. A model Hamiltonian, that captures the relevant band edges at the $K/K^{\prime}$ valleys of the proximitized TMDCs, is employed to quantify the proximity-induced exchange. Upon twisting from 0{\deg} to 30{\deg}, we find a transition of the TMDC valence band (VB) edge exchange splitting from about $-2$ to $2$ meV, while the conduction band (CB) edge exchange splitting remains nearly unchanged at around $-3$ meV. For the VB of WSe$_2$ (MoSe$_2$) on CrI$_3$, the exchange coupling changes sign at around 8{\deg} (16{\deg}). We find that even at the angles with almost zero spin splittings of the VB, the real-space spin polarization profile of holes at the band edge is highly non-uniform, with alternating spin up and spin down orbitals. Furthermore, a giant tunability of the proximity-induced exchange coupling is provided by a transverse electric field of a few V/nm. We complement our \textit{ab initio} results by calculating the excitonic valley splitting to provide experimentally verifiable optical signatures of the proximity exchange. Specifically, we predict that the valley splitting increases almost linearly as a function of the twist angle. Furthermore, the proximity exchange is highly tunable by gating, allowing to tailor the valley splitting in the range of 0 to 12 meV in WSe$_2$/CrI$_3$, which is equivalent to external magnetic fields of up to about 60 Tesla. Our results highlight the important impact of the twist angle and gating when employing magnetic vdW heterostructures in experimental geometries., Comment: 16 pages, 7 figures, 2 tables + Supplemental Material

We investigate the electronic band structure of graphene on a series of two-dimensional magnetic transition-metal phosphorus trichalcogenide monolayers, MPX$_3$ with M={Mn,Fe,Ni,Co} and X={S,Se}, with first-principles calculations. A symmetry-based model Hamiltonian is employed to extract orbital parameters and sublattice resolved proximity-induced exchange couplings ($\lambda_{\textrm{ex}}^\textrm{A}$ and $\lambda_{\textrm{ex}}^\textrm{B}$) from the low-energy Dirac bands of the proximitized graphene. Depending on the magnetic phase of the MPX$_3$ layer (ferromagnetic and three antiferromagnetic ones), completely different Dirac dispersions can be realized with exchange splittings ranging from 0 to 10~meV. Surprisingly, not only the magnitude of the exchange couplings depends on the magnetic phase, but also the global sign and the type. Important, one can realize uniform ($\lambda_{\textrm{ex}}^\textrm{A} \approx \lambda_{\textrm{ex}}^\textrm{B}$) and staggered ($\lambda_{\textrm{ex}}^\textrm{A} \approx -\lambda_{\textrm{ex}}^\textrm{B}$) exchange couplings in graphene. From selected cases, we find that the interlayer distance, as well as a transverse electric field are efficient tuning knobs for the exchange splittings of the Dirac bands. More specifically, decreasing the interlayer distance by only about 10\%, a giant 5-fold enhancement of proximity exchange is found, while applying few V/nm of electric field, provides tunability of proximity exchange by tens of percent. We have also studied the dependence on the Hubbard $U$ parameter and find it to be weak. Moreover, we find that the effect of SOC on the proximitized Dirac dispersion is negligible compared to the exchange coupling., Comment: 15 pages, 10 figures, 2 tables + Supplemental Material

An effective model of the hexagonal (NiAs-structure) manganese telluride valence band in the vicinity of the A-point of the Brillouin zone is derived. It is shown that while for the usual antiferromagnetic order (magnetic moments in the basal plane) band splitting at A is small, their out-of-plane rotation enhances the splitting dramatically (to about 0.5 eV). We propose extensions of recent experiments (Moseley et al., Phys. Rev. Materials 6, 014404) where such inversion of magnetocrystalline anisotropy has been observed in Li-doped MnTe, to confirm this unusual sensitivity of a semiconductor band structure to magnetic order., Comment: 5+epsilon pages, 4 figures; to appear in PRB

We investigate the proximity spin-orbit and exchange couplings in ABA and ABC trilayer graphene encapsulated within monolayers of semiconducting transition-metal dichalcogenides and the ferromagnetic semiconductor Cr$_2$Ge$_2$Te$_6$. Employing first-principles calculations we obtain the electronic structures of the multilayer stacks and extract the relevant proximity-induced orbital and spin interaction parameters by fitting the low-energy bands to model Hamiltonians. We also demonstrate the tunability of the proximity effects by a transverse electric field. Using the model Hamiltonians we also study mixed spin-orbit/exchange coupling encapsulation, which allows to tailor the spin interactions very efficiently by the applied field. We also summarize the spin-orbit physics of bare ABA, ABC, and ABB trilayers, and provide, along with the first-principles results of the electronic band structures, density of states, spin splittings, and electric-field tunabilities of the bands, qualitative understanding of the observed behavior and realistic model parameters as a resource for model simulations of transport and correlation physics in trilayer graphene., Comment: 27 pages, 22 figures, 5 tables

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 tables

We investigate the twist-angle and gate dependence of the proximity exchange coupling in twisted graphene on monolayer Cr$_2$Ge$_2$Te$_6$ from first principles. The proximitized Dirac band dispersions of graphene are fitted to a model Hamiltonian, yielding effective sublattice-resolved proximity-induced exchange parameters ($\lambda_{\textrm{ex}}^\textrm{A}$ and $\lambda_{\textrm{ex}}^\textrm{B}$) for a series of twist angles between 0$^{\circ}$ and 30$^{\circ}$. For aligned layers (0$^{\circ}$ twist angle), the exchange coupling of graphene is the same on both sublattices, $\lambda_{\textrm{ex}}^\textrm{A} \approx \lambda_{\textrm{ex}}^\textrm{B} \approx 4$ meV, while the coupling is reversed at 30$^{\circ}$ (with $\lambda_{\textrm{ex}}^\textrm{A} \approx \lambda_{\textrm{ex}}^\textrm{B} \approx -4$ meV). Remarkably, at 19.1$^{\circ}$ the induced exchange coupling becomes antiferromagnetic: $\lambda_{\textrm{ex}}^\textrm{A} < 0, \lambda_{\textrm{ex}}^\textrm{B} > 0$. Further tuning is provided by a transverse electric field and the interlayer distance. The predicted proximity magnetization reversal and emergence of an antiferromagnetic Dirac dispersion make twisted graphene/Cr$_2$Ge$_2$Te$_6$ bilayers a versatile platform for realizing topological phases and for spintronics applications., Comment: 9 pages, 5 figures + Supplemental Material

We explore the effect of temperatures between 80 and 720 K on the energy and linewidth of zone-center transverse (TO) and longitudinal (LO) optical phonons in bulk gallium phosphide (GaP) using Fourier transform infrared ellipsometry from 0.03 to 0.60 eV. We extract the optical phonon parameters of GaP by fitting the ellipsometric angles with the Lowndes-Gervais model, which applies two different broadening parameters to the TO and LO phonons. In GaP, the two-phonon density of states is larger for the decay of TO phonons than for LO phonons. Therefore, we observed a larger TO phonon broadening (compared to the LO phonon) and an asymmetric reststrahlen line shape. This would lead to a negative imaginary part of the dielectric function just above the LO phonon energy, but the addition of two-phonon absorption avoids this. We find a temperature dependent redshift and broadening of TO and LO phonons with increasing temperature due to thermal expansion and anharmonic phonon-phonon scattering, involving three and four phonon decay processes. We also investigate the temperature-dependence of the high-frequency dielectric constant. Its variation is explained by thermal expansion and the temperature dependence of the Penn gap.

Van der Waals (vdW) heterostructures consisting of bilayer graphene (BLG) encapsulated within monolayers of strong spin-orbit semiconductor WS$_2$ or ferromagnetic semiconductor Cr$_2$Ge$_2$Te$_6$ (CGT), are investigated. By performing realistic first-principles calculations we capture the essential BLG band structure features, including layer- and sublattice-resolved proximity spin-orbit or exchange couplings. For different relative twist angles (0 or 60$^{\circ}$) of the WS$_2$ layers, and the magnetizations (parallel or antiparallel) of the CGT layers, with respect to BLG, the low energy bands are found and characterized by a series of fit parameters of model Hamiltonians. These effective models are then employed to investigate the tunability of the relevant energy dispersions by a gate field. For WS$_2$/BLG/WS$_2$ encapsulation we find that twisting allows to turn off the spin splittings away from the $K$ points, due to opposite proximity-induced valley-Zeeman couplings in the two sheets of BLG. Close to the $K$ points the electron spins are polarized out of the plane. This polarization can be flipped by applying a gate field. As for magnetic CGT/BLG/CGT structures, we realize the recently proposed spin-valve effect, whereby a gap opens for antiparallel magnetizations of the CGT layers. Furthermore, we find that for the antiferromagnetic orientation the electron states away from $K$ have vanishingly weak proximity exchange, while the states close to $K$ remain spin polarized in the presence of an electric field. The induced magnetization can be flipped by changing the gate field. These findings should be useful for spin transport, spin filtering, and spin relaxation anisotropy studies of BLG-based vdW heterostructures., Comment: 11 pages, 6 figures, 1 table + Supplemental Material

We investigate the electronic structure of graphene on a series of 2D hexagonal nitride insulators hXN, X = B, Al, and Ga, with DFT calculations. A symmetry-based model Hamiltonian is employed to extract orbital parameters and spin-orbit coupling (SOC) from the low-energy Dirac bands of proximitized graphene. While commensurate hBN induces a staggered potential of about 10 meV into the Dirac bands, less lattice-matched hAlN and hGaN disrupt the Dirac point much less, giving a staggered gap below 100 $\mu$eV. Proximitized intrinsic SOC surprisingly does not increase much above the pristine graphene value of 12 $\mu$eV; it stays in the window of (1-16) $\mu$eV, depending strongly on stacking. However, Rashba SOC increases sharply when increasing the atomic number of the boron group, with calculated maximal values of 8, 15, and 65 $\mu$eV for B, Al, and Ga-based nitrides, respectively. The individual Rashba couplings also depend strongly on stacking, vanishing in symmetrically-sandwiched structures, and can be tuned by a transverse electric field. The extracted spin-orbit parameters were used as input for spin transport simulations based on Chebyshev expansion of the time-evolution of the spin expectation values, yielding interesting predictions for the electron spin relaxation. Spin lifetime magnitudes and anisotropies depend strongly on the specific (hXN)/graphene/hXN system, and they can be efficiently tuned by an applied external electric field as well as the carrier density in the graphene layer. A particularly interesting case for experiments is graphene/hGaN, in which the giant Rashba coupling is predicted to induce spin lifetimes of 1-10 ns, short enough to dominate over other mechanisms, and lead to the same spin relaxation anisotropy as observed in conventional semiconductor heterostructures: 50\%, meaning that out-of-plane spins relax twice as fast as in-plane spins., Comment: 18 Pages, 12 Figures, 2 Tables

We studied metastable $\alpha$-FeGe$_2$, a novel layered tetragonal material, embedded as a spacer layer in spin valve structures with ferromagnetic Fe$_3$Si and Co$_2$FeSi electrodes. For both types of electrodes, spin valve operation is demonstrated with a metallic transport behavior of the $\alpha$-FeGe$_2$ spacer layer. The spin valve signals are found to increase both with temperature and spacer thickness, which is discussed in terms of a decreasing magnetic coupling strength between the ferromagnetic bottom and top electrodes. The temperature-dependent resistances of the spin valve structures exhibit characteristic features, which are explained by ferromagnetic phase transitions between 55 and 110~K. The metallic transport characteristics as well as the low-temperature ferromagnetism are found to be consistent with the results of first-principles calculations., Comment: 11 pages, 13 figures

The concept of swapping the two most important spin interactions -- exchange and spin-orbit coupling -- is proposed based on two-dimensional multilayer van der Waals heterostructures. Specifically, we show by performing realistic ab initio simulations, that a single device consisting of a bilayer graphene sandwiched by a 2D ferromagnet Cr$_2$Ge$_2$Te$_6$ (CGT) and a monolayer WS$_2$, is able not only to generate, but also to swap the two interactions. The highly efficient swapping is enabled by the interplay of gate-dependent layer polarization in bilayer graphene and short-range spin-orbit and exchange proximity effects affecting only the layers in contact with the sandwiching materials. We call these structures ex-so-tic, for supplying either exchange (ex) or spin-orbit (so) coupling in a single device, by gating. Such bifunctional devices demonstrate the potential of van der Waals spintronics engineering using 2D crystal multilayers., Comment: 6 pages, 3 figures, 1 table + Supplemental Material

Prototypical three-dimensional topological insulators of the Bi$_2$Se$_3$ family provide a beautiful example of the appearance of the surface states inside the bulk band gap caused by spin-orbit coupling-induced topology. The surface states are protected against back scattering 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. On the other hand, graphene is a prototypical two-dimensional 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. We present a detailed theoretical study of the proximity effects of monolayer graphene and topological insulators Bi$_2$Se$_3$, Bi$_2$Te$_3$, and Sb$_2$Te$_3$, and elucidate the appearance of the qualitatively new spin-orbit splittings well described by a phenomenological Hamiltonian, 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., Comment: 10 pages, 9 figures, 1 table

We investigate an effective model of proximity modified graphene (or symmetrylike materials) with broken time-reversal symmetry. We predict the appearance of quantum anomalous Hall phases by computing bulk band gap and Chern numbers for benchmark combinations of system parameters. Allowing for staggered exchange field enables quantum anomalous Hall effect in flat graphene with Chern number $C=1$. We explicitly show edge states in zigzag and armchair nanoribbons and explore their localization behavior. Remarkably, the combination of staggered intrinsic spin-orbit and uniform exchange coupling gives topologically protected (unlike in time-reversal systems) pseudohelical states, whose spin is opposite in opposite zigzag edges. Rotating the magnetization from out of plane to in plane makes the system trivial, allowing to control topological phase transitions. We also propose, using density functional theory, a material platform---graphene on Ising antiferromagnet MnPSe$_3$---to realize staggered exchange (pseudospin Zeeman) coupling.