Ising Spin-orbit Coupling Research Articles

Spin-orbit proximity in MoS2/bilayer graphene heterostructures

Van der Waals heterostructures provide a versatile platform for tailoring electronic properties through the integration of two-dimensional materials. Among these combinations, the interaction between bilayer graphene and transition metal dichalcogenides (TMDs) stands out due to its potential for inducing spin–orbit coupling (SOC) in graphene. Future devices concepts require the understanding of the precise nature of SOC in TMD/bilayer graphene heterostructures and its influence on electronic transport phenomena. Here, we experimentally confirm the presence of two distinct types of SOC – Ising (ΔI = 1.55 meV) and Rashba (ΔR = 2.5 meV) – in bilayer graphene when interfaced with molybdenum disulfide. Furthermore, we reveal a non-monotonic trend in conductivity with respect to the electric displacement field at charge neutrality. This phenomenon is ascribed to the existence of single-particle gaps induced by the Ising SOC, which can be closed by a critical displacement field. Our findings also unveil sharp peaks in the magnetoconductivity around the critical displacement field, challenging existing theoretical models.

Unconventional pairing in Ising superconductors: application to monolayer NbSe2

Abstract The presence of a non-centrosymmetric crystal structure and in-plane mirror symmetry allows an Ising spin–orbit coupling to form in some two-dimensional materials. Examples include transition metal dichalcogenide superconductors like monolayer NbSe2, MoS2, TaS2, and PbTe2, where a nontrivial nature of the superconducting state is currently being explored. In this study, we develop a microscopic formalism for Ising superconductors that captures the superconducting instability arising from a momentum-dependent spin- and charge-fluctuation-mediated pairing interaction. We apply our pairing model to the electronic structure of monolayer NbSe2, where first-principles calculations reveal the presence of strong paramagnetic fluctuations. Our calculations provide a quantitative measure of the mixing between the even- and odd-parity superconducting states and its variation with Coulomb interaction. Further, numerical analysis in the presence of an external Zeeman field reveals the role of Ising spin–orbit coupling and mixing of odd-parity superconducting state in influencing the low-temperature enhancement of the critical magnetic field.

Tunable superconductivity in electron- and hole-doped Bernal bilayer graphene.

Graphene-based, high-quality, two-dimensional electronic systems have emerged as a highly tunable platform for studying superconductivity1-21. Specifically, superconductivity has been observed in both electron- and hole-doped twisted graphene moiré systems1-17, whereas in crystalline graphene systems, superconductivity has so far been observed onlyin hole-doped rhombohedral trilayer graphene(RTG)18 and hole-doped Bernal bilayer graphene (BBG)19-21. Recently, enhanced superconductivity has been demonstrated20,21 in BBG because of the proximity to a monolayer WSe2. Here we report the observation of superconductivity and a series of flavour-symmetry-breaking phases in electron- and hole-doped BBG/WSe2 devices by electrostatic doping. The strength of the observed superconductivity is tunable by applied vertical electric fields. The maximum Berezinskii-Kosterlitz-Thouless transition temperature for the electron- and hole-doped superconductivity is about 210 mK and 400 mK, respectively. Superconductivities emerge only when the applied electric fields drive the BBG electron or hole wavefunctions towards the WSe2 layer, underscoring the importance of the WSe2 layer in the observed superconductivity. The hole-doped superconductivity violates the Pauli paramagnetic limit, consistent with an Ising-like superconductor. By contrast, the electron-doped superconductivity obeys the Pauli limit, although the proximity-induced Ising spin-orbit coupling is also notable in the conduction band. Our findings highlight the rich physics associated with the conduction band in BBG, paving the way for further studies into the superconducting mechanisms of crystalline graphene and the development of superconductor devices based on BBG.

Large quantum anomalous Hall effect in spin-orbit proximitized rhombohedral graphene

The quantum anomalous Hall effect (QAHE) is a robust topological phenomenon that features quantized Hall resistance at zero magnetic field. We report the QAHE in a rhombohedral pentalayer graphene-monolayer tungsten disulfide (WS 2 ) heterostructure. Distinct from other experimentally confirmed QAHE systems, this system has neither magnetic element nor moiré superlattice effect. The QAH states emerge at charge neutrality and feature Chern numbers C = ±5 at temperatures of up to about 1.5 kelvin. This large QAHE arises from the synergy of the electron correlation in intrinsic flat bands of pentalayer graphene, the gate-tuning effect, and the proximity-induced Ising spin-orbit coupling. Our experiment demonstrates the potential of crystalline two-dimensional materials for intertwined electron correlation and band topology physics and may enable a route for engineering chiral Majorana edge states.

Josephson junction of nodal superconductors with a Rashba and Ising spin-orbit coupling

Josephson junction of nodal superconductors with a Rashba and Ising spin-orbit coupling

Prediction of superconductivity in a series of tetragonal transition metal dichalcogenides.

Transition metal dichalcogenides (TMDCs) represent a well-known material family with diverse structural phases and rich electronic properties; they are thus an ideal platform for studying the emergence and exotic phenomenon of superconductivity (SC). Herein, we propose the existence of tetragonal TMDCs with a distorted Lieb (dLieb) lattice structure and the stabilized transition metal disulfides (MS2), including dLieb-ZrS2, dLieb-NbS2, dLieb-MnS2, dLieb-FeS2, dLieb-ReS2, and dLieb-OsS2. Except for semiconducting dLieb-ZrS2 and magnetic dLieb-MnS2, the rest of metallic dLieb-MS2 was found to exhibit intrinsic SC with the transition temperature (TC) ranging from ∼5.4 to ∼13.0 K. The TC of dLieb-ReS2 and dLieb-OsS2 exceeded 10 K and was higher than that of the intrinsic SC in the known metallic TMDCs, which is attributed to the significant phonon-softening enhanced electron-phonon coupling strength. Different from the Ising spin-orbit coupling (SOC) effect in existing non-centrosymmetric TMDCs, the non-magnetic dLieb-MS2 monolayers exhibit the Dresselhaus SOC effect, which is featured by in-plane spin orientations and will give rise to the topological SC under proper conditions. In addition to enriching the structural phases of TMDCs, our work predicts a series of SC candidates with high intrinsic TC and topological non-triviality used for fault-tolerant quantum computation.

Enhancement and switch effect of equal-spin Andreev reflection in ferromagnet/insulator/Ising superconductor junctions

We study the property of equal-spin Andreev reflection (ESAR) in the ferromagnet/insulator/Ising superconductor junction where Ising spin–orbit coupling is taken into account in the insulator. It is found that the ESAR exhibits a regular oscillation with the insulating barrier, the amplitude and period of which can be effectively controlled by the chemical potentials. Compared to that in the ferromagnet/Ising superconductor junction, the ESAR is greatly increased due to the resonant mode, suggesting an enhanced spin-triplet pairing. As an application, the proposed junction may work as a switch to turn on and off the ESAR. Furthermore, the insulating barrier does not change the magnetoanisotropic period of ESAR because of the invariant symmetry of the system, however, the magnetoanisotropy is strengthened.

Photoinduced valley-dependent equal-spin Andreev reflection in Ising superconductor junction

The Ising spin–orbit coupling could give rise to the spin-triplet Cooper pairs and equal-spin Andreev reflection (AR) in Ising superconductors. Here we theoretically study the valley-dependent equal-spin AR in a ferromagnet/Ising superconductor junction with a circularly polarized light applied to the ferromagnet. Because of the spin-triplet Cooper pairs and the optical irradiation, eight kinds of AR processes appear in the junction, including equal-spin AR and normal AR, the strengths and properties of which strongly depend on the valley degree of freedom. The AR probabilities for the incident electron from the two valleys exhibit certain symmetry with respect to the magnetization angle and the effective energy of light. The equal-spin AR and normal AR present different features and resonant behaviors near the superconducting gap edges. Due to equal-spin-triplet Cooper pairs, not only charge supercurrent but also spin supercurrent can transport in the Ising superconductors. The differential spin conductance for electron injecting from the two valleys can be controlled by the circularly polarized light.

Orbital Fulde-Ferrell Pairing State in Moiré Ising Superconductors.

In this Letter, we study superconducting moiré homobilayer transition metal dichalcogenides where the Ising spin-orbit coupling (SOC) is much larger than the moiré bandwidth. We call such noncentrosymmetric superconductors, moiré Ising superconductors. Because of the large Ising SOC, the depairing effect caused by the Zeeman field is negligible and the in-plane upper critical field (B_{c2}) is determined by the orbital effects. This allows us to study the effect of large orbital fields. Interestingly, when the applied in-plane field is larger than the conventional orbital B_{c2}, a finite-momentum pairing phase would appear which we call the orbital Fulde-Ferrell (FF) state. In this state, the Cooper pairs acquire a net momentum of 2q_{B}, where 2q_{B}=eBd is the momentum shift caused by the magnetic field B and d denotes the layer separation. This orbital field-driven FF state is different from the conventional FF state driven by Zeeman effects in Rashba superconductors. Remarkably, we predict that the FF pairing would result in a giant superconducting diode effect under electric gating when layer asymmetry is induced. An upturn of the B_{c2} as the temperature is lowered, coupled with the giant superconducting diode effect, would allow the detection of the orbital FF state.

Stabilizing Fluctuating Spin-Triplet Superconductivity in Graphene via Induced Spin-Orbit Coupling.

A recent experiment showed that a proximity-induced Ising spin-orbit coupling enhances the spin-triplet superconductivity in Bernal bilayer graphene. Here, we show that, due to the nearly perfect spin rotation symmetry of graphene, the fluctuations of the spin orientation of the triplet order parameter suppress the superconducting transition to nearly zero temperature. Our analysis shows that both an Ising spin-orbit coupling and an in-plane magnetic field can eliminate these low-lying fluctuations and can greatly enhance the transition temperature, consistent with the recent experiment. Our model also suggests the possible existence of a phase at small anisotropy and magnetic field which exhibits quasilong-range ordered spin-singlet charge 4e superconductivity, even while the triplet 2e superconducting order only exhibits short-ranged correlations. Finally, we discuss relevant experimental signatures.

Superconductivity from electronic interactions and spin-orbit enhancement in bilayer and trilayer graphene

What drives superconductivity in non-moir\'e graphene? Here, an electron-electron mechanism is applied to the continuum framework of Bernal bilayer and rhombohedral trilayer graphene. Within a Kohn-Luttinger-like approach, the internal screening of the long-range Coulomb potential leads to an effective attraction between carriers, enabling the formation of Cooper pairs. The calculations provide a filling-dependent critical temperature that is remarkably enhanced due to Ising spin-orbit coupling, and determine the spin-triplet valley-singlet symmetry of the superconducting order parameter, in line with recent experimental advances.

Quasisymmetry protected Ising spin-orbit coupling

Quasisymmetry is an approximated symmetry in the sense that it is not the crystalline symmetry but commutes with the Hamiltonian at a lower order. Unlike exact symmetry, quasisymmetry emerges in the context of low-energy effective theory. One of the most desirable properties added by quasisymmetry is the large Berry curvatures due to the perturbatively small band gaps, which are robust against chemical and physical perturbations. Ising spin-orbit coupling (SOC) is a special type of SOC with electron spins polarizing to out-of-plane direction and can protect the spin coherence against the relaxation. For Ising-type superconductors, Ising SOC significantly affects the superconducting properties such as Pauli limit. It is commonly believed that the out-of-plane mirror symmetry ${M}_{z}$ is necessary for Ising SOC, which puts a constraint on the allowed materials. Here we demonstrate that the strong Ising SOC exists beyond the system with ${M}_{z}$ symmetry and is protected by the quasisymmetry. Based on the group-theoretical analysis and density-functional theory calculations, we show that the lower-order SOC Hamiltonian for certain representation is Ising type, which is protected by the quasisymmetry. Several promising systems with strong Ising SOC of several hundreds of meV near the Fermi energy are identified. In addition, we propose the general design principles to search for the realistic materials. Our results advance the fundamental understanding of Ising spin-orbit physics and greatly broaden the range of the materials hosting the Ising SOC.

Kondo screening of single magnetic impurity doped type-II Ising superconductors

We theoretically study the Kondo effect in type-II Ising superconductors with a single magnetic impurity. Type-II Ising superconductivity was found in two-dimensional centrosymmetric materials with multiple degenerate orbitals, such as in few-layer stanene [Falson et al., Science 367, 1454 (2020)] and ultrathin ${\mathrm{PdTe}}_{2}$ films [Liu et al., Nano Lett. 20, 5728 (2020)]. The type-II Ising spin-orbit coupling (SOC) in these materials generates out-of-plane effective Zeeman fields orienting opposite directions for opposing orbitals, which strongly protects the interorbital superconducting pairing states against in-plane magnetic fields. We show that the SOC-induced band splitting and the chemical potential significantly influence the formation of a localized magnetic moment, with particle-hole asymmetry in the phase boundary between magnetic and nonmagnetic states. The behaviors of spin-induced Yu-Shiba-Rusinov bound states and low-temperature magnetic susceptibility demonstrate that the SOC suppresses the Kondo screening of the magnetic moment, while the interorbital mixing weakens the SOC at finite momentum, then enhances the Kondo effect. The quantum phase transition between magnetic doublet and Kondo singlet ground states can be tuned through chemical potential in experiments.

Two-bands Ising superconductivity from Coulomb interactions in monolayer

The nature of superconductivity in monolayer transition metal dichalcogenides is still under debate. It has already been argued that repulsive Coulomb interactions, combined with the disjoint Fermi surfaces around the K, Kʹ valleys and at the Γ point, can lead to superconducting instabilities in monolayer . Here, we demonstrate the two-bands nature of superconductivity in . It arises from the competition of repulsive long range intravalley and short range intervalley interactions together with Ising spin–orbit coupling. The two distinct superconducting gaps, one for each spin-orbit split band, consist of a mixture of s-wave and f-wave components. Their different amplitudes are due to the difference between the normal densities of states of the two bands at the Fermi level. Using a microscopic multiband BCS approach, we derive and self-consistently solve the gap equation, demonstrating the stability of nontrivial solutions in a realistic parameter range. We find a universal behavior of the temperature dependence of the gaps and of the critical in-plane field which is consistent with various sets of existing experimental data.

Enhanced superconductivity through virtual tunneling in Bernal bilayer graphene coupled to WSe2

Motivated by a recent experiment [arXiv:2205.05087], we investigate a possible mechanism that enhances superconductivity in hole-doped Bernal bilayer graphene due to a proximate WSe$_2$ monolayer. We show that the virtual tunneling between WSe$_2$ and Bernal bilayer graphene, which is known to induce Ising spin-orbit coupling, can generate an additional attraction between two holes, providing a potential explanation for enhancing superconductivity in Bernal bilayer graphene. Using microscopic interlayer tunneling, we derive the Ising spin-orbit coupling and the effective attraction as functions of the twist angle between Bernal bilayer graphene and the WSe$_2$ monolayer. Our theory provides an intuitive and physical explanation for the intertwined relation between Ising spin-orbit coupling and superconductivity enhancement, which should motivate future studies.

Equal-spin and oblique-spin crossed Andreev reflections in ferromagnet/Ising superconductor/ferromagnet junction

We study the subgap transport through a ferromagnet/Ising superconductor/ferromagnet (F/ISC/F) junction by solving the Bogoliubov--de Gennes equations. It is found that the crossed Andreev reflection (CAR) and local Andreev reflection (LAR) depend strongly on the spin-polarized F, the magnetization direction, and the Ising superconducting phase. For the same magnetization directions of the two F leads, the equal-spin CAR could take place due to the spin-flip mechanism induced by the Ising spin-orbit coupling and equal-spin-triplet pairing. Both equal-spin CAR and equal-spin LAR exhibit a remarkable magnetoanisotropy with period $\ensuremath{\pi}$ and show oscillatory behavior with a chemical potential. The equal-spin CAR is more prominent for half-metal F and double-band ISC while the normal CAR is completely suppressed. When the magnetization directions of the two F leads are different, the oblique-spin CAR occurs and its magnetoanisotropic period generally becomes $2\ensuremath{\pi}$ instead of $\ensuremath{\pi}$. In the oblique-spin CAR process, the spins of the electron and hole are neither parallel nor antiparallel. Furthermore, the property of oblique-spin CAR is very sensitive to the spin and valley degrees of freedom. The spin- and valley-polarized CAR can be achieved and controlled by the chemical potentials and the magnetization directions.

Spin-valley locked instabilities in moiré transition metal dichalcogenides with conventional and higher-order Van Hove singularities

Recent experiments have observed correlated insulating and possible superconducting phases in twisted homobilayer transition metal dichalcogenides (TMDs). Besides the spin-valley locked moir\'e bands due to the intrinsic Ising spin-orbit coupling, homobilayer moir\'e TMDs also possess either logarithmic or power-law divergent Van Hove singularities (VHSs) near the Fermi surface, controllable by an external displacement field. The former and the latter are dubbed conventional and higher-order VHSs, respectively. Here, we perform a perturbative renormalization group (RG) analysis to unbiasedly study the dominant instabilities in homobilayer TMDs for both the conventional and higher-order VHS cases. We find that the spin-valley locking largely alters the RG flows and leads to instabilities unexpected in the corresponding extensively studied graphene-based moir\'e systems, such as spin- and valley-polarized ferromagnetism and topological superconductivity with mixed parity. In particular, for the case with two higher-order VHSs, we find a spin-valley-locking-driven metallic state with no symmetry breaking in the TMDs despite the diverging bare susceptibility. Our results show how the spin-valley locking significantly affects the RG analysis and demonstrate that moir\'e TMDs are suitable platforms to realize various interaction-induced spin-valley locked phases, highlighting physics fundamentally different from the well-studied graphene-based moir\'e systems.

Spin-orbit coupling and interactions in quantum Hall states of graphene/ WSe2 heterobilayers

We use magnetotransport measurements to probe quantum Hall ground states in graphene/${\mathrm{WSe}}_{2}$ heterobilayers. Compared to pristine graphene, inter-Landau level (LL) gaps at half-filled quartets away from filling factor $\ensuremath{\nu}=0$ show significantly weaker dependence on the magnetic field $B$, while odd $\ensuremath{\nu}$ fillings show a stronger dependence. We interpret this behavior using a model incorporating Ising and Rashba spin-orbit coupling (SOC) along with Coulomb interactions within the self-consistent Hartree-Fock framework. A model fit yields Ising SOC in range $\ensuremath{\sim}1--2 \mathrm{meV}$, Rashba $\ensuremath{\sim}10 \mathrm{meV}$, and the in-plane dielectric constant $\ensuremath{\sim}12$, in agreement to previously found values. In the zeroth LL quartet, the $\ensuremath{\nu}=0$ gap as a function of magnetic field exhibits a plateau near 5 T, compared to $\ensuremath{\sim}20--25 \mathrm{T}$ for pristine graphene. This behavior is in agreement with a model in which the SOC causes a phase transition from a canted antiferromagnetic state to a ferromagnetic state to occur at a much lower field. Our studies demonstrate how the interplay of SOC and electronic interactions affect graphene's electronic structure.

Role of Two-Dimensional Ising Superconductivity in the Nonequilibrium Quasiparticle Spin-to-Charge Conversion Efficiency

Nonequilibrium studies of two-dimensional (2D) superconductors (SCs) with Ising spin–orbit coupling are prerequisite for their successful application to equilibrium spin-triplet Cooper pairs and, potentially, Majorana Fermions. By taking advantage of the recent discoveries of 2D SCs and their compatibility with any other materials, we fabricate here nonlocal magnon devices to examine how such 2D Ising superconductivity affects the conversion efficiency of magnon spin to quasiparticle charge in superconducting flakes of 2H-NbSe2 transferred onto ferrimagnetic insulating Y3Fe5O12. Comparison with a reference device based on a conventionally paired superconductor shows that the Y3Fe5O12-induced in-plane (IP) exchange spin-splitting in the NbSe2 flake is hindered by its inherent out-of-plane (OOP) spin–orbit field, which, in turn, limits the transition-state enhancement of the spin-to-charge conversion efficiency. Our out-of-equilibrium study highlights the significance of symmetry matching between underlying Cooper pairs and exchange-induced spin-splitting for the giant transition-state spin-to-charge conversion and may have implications toward proximity-engineered spin-polarized triplet pairing via tuning the relative strength of IP exchange and OOP spin–orbit fields in ferromagnetic insulator/2D Ising SC bilayers.

Evidence for the Fulde\u2013Ferrell\u2013Larkin\u2013Ovchinnikov state in bulk NbS2

We present measurements of the magnetic torque, specific heat and thermal expansion of the bulk transition metal dichalcogenide (TMD) superconductor NbS2 in high magnetic fields, with its layer structure aligned strictly parallel to the field using a piezo rotary positioner. The upper critical field of superconducting TMDs in the 2D form is known to be dramatically enhanced by a special form of Ising spin orbit coupling. This Ising superconductivity is very robust to the Pauli paramagnetic effect and can therefore exist beyond the Pauli limit for superconductivity. We find that superconductivity beyond the Pauli limit still exists in bulk single crystals of NbS2 for a precisely parallel field alignment. However, the comparison of our upper critical field transition line with numerical simulations rather points to the development of a Fulde-Ferrell-Larkin-Ovchinnikov state above the Pauli limit as a cause. This is also consistent with the observation of a magnetic field driven phase transition in the thermodynamic quantities within the superconducting state near the Pauli limit.