We argue that edge electrons in monolayer WTe$_2$ can possess a "cooperative" orbital moment (COM) that critically impacts its edge magnetoresistance behavior. Arising from the cooperative action of both Rashba and Ising spin orbit coupling, COM quickly achieves large magnitudes (of order few Bohr magnetons) even for relatively small spin-orbit coupling strengths. As we explain, such large COM magnitudes arise from an unconventional cooperative spin canting of edge spins when Rashba and Ising spin orbit coupling act together. Strikingly, COM can compete with spin moments to produce an unusual anisotropic edge magnetoresistance oriented at an oblique angle. In particular, this competition produces a direction along which $\mathbf{B}$ is ineffective at gapping out the edge spectrum leaving it nearly gapless. As a result, large contrasts in gap sizes manifest as $\mathbf{B}$ is rotated granting giant anisotropic magnetoresistance of 0.1-10 million % at 10 T and low temperature.
A two-dimensional topological insulator (2DTI) is guaranteed to have a helical 1D edge mode in which spin is locked to momentum, producing the quantum spin Hall effect and prohibiting elastic backscattering at zero magnetic field. No monolayer material has yet been shown to be a 2DTI, but recently the Weyl semimetal WTe2 was predicted to become a 2DTI in monolayer form if a bulk gap opens. Here, we report that at temperatures below about 100 K monolayer WTe2 does become insulating in its interior, while the edges still conduct. The edge conduction is strongly suppressed by in-plane magnetic fi…
The electronic structure of WTe2 bulk and layers are investigated by using the first principles calculations. The perfect electron-hole (n-p) charge compensation and high carrier mobilities are found in WTe2 bulk, which may result in the large and non-saturating magnetoresistance (MR) observed very recently in the experiment [Ali et al., Nature 514, 205 (2014)]. The monolayer and bilayer of WTe2 preserve the semimetallic property, with the equal hole and electron carrier concentrations. Moreover, the very high carrier mobilities are also found in WTe2 monolayer, indicating that the WTe2 monola…
We develop a symmetry-based low-energy theory for monolayer $\mathrm{WTe}_2$ in its 1T$^{\prime}$ phase, which includes eight bands (four orbitals, two spins). This model reduces to the conventional four-band spin-degenerate Dirac model near the Dirac points of the material. We show that measurements of the spin susceptibility, and of the magnitude and time dependence of the anomalous Hall conductivity induced by injected or equilibrium spin polarization can be used to determine the magnitude and form of the spin-orbit coupling Hamiltonian, as well as the dimensionless tilt of the Dirac bands.
A monolayer of WTe$_2$ has been shown to display quantum spin Hall (QSH) edge modes persisting up to 100~K in transport experiments. Based on density-functional theory calculations and symmetry-based model building including the role of correlations and substrate support, we develop an effective electronic model for WTe$_2$ which fundamentally differs from other prototypical QSH settings: we find that the extraordinary robustness of quantum spin Hall edge modes in WTe$_2$ roots in a glide symmetry due to which the topological gap opens away from high-symmetry points in momentum space. While th…
The spin orientation of electronic wavefunctions in crystals is an internal degree of freedom, typically insensitive to electrical knobs. We argue from a general symmetry analysis and a $\vec k \cdot \vec p$ perspective, that monolayer 1T'-WTe$_2$ possesses a gate-activated canted spin texture that produces an electrically tunable bulk band quantum geometry. In particular, we find that due to its out-of-plane asymmetry, an applied out-of-plane electric field breaks inversion symmetry to induce both in-plane and out-of-plane electric dipoles. These in-turn generate spin-orbit coupling to lift t…
We study current-induced torques in WTe2/permalloy bilayers as a function of WTe2 thickness. We measure the torques using both second-harmonic Hall and spin-torque ferromagnetic resonance measurements for samples with WTe2 thicknesses that span from 16 nm down to a single monolayer. We confirm the existence of an out-of-plane antidamping torque, and show directly that the sign of this torque component is reversed across a monolayer step in the WTe2. The magnitude of the out-of-plane antidamping torque depends only weakly on WTe2 thickness, such that even a single-monolayer WTe2 device provides…
Recent experimental evidence for the quantum spin Hall (QSH) state in monolayer WTe$_2$ has bridged two of the most active fields of condensed matter physics, 2D materials and topological physics. This 2D topological crystal also displays unconventional spin-torque and gate-tunable superconductivity. While the realization of QSH has demonstrated the nontrivial topology of the electron wavefunctions of monolayer WTe$_2$, the geometrical properties of the wavefunction, such as the Berry curvature, remain unstudied. On the other hand, it has been increasingly recognized that the Berry curvature p…
Recently, quantum oscillation of the resistance in insulating monolayer WTe$_2$ was reported. An explanation in terms of gap modulation in the hybridized Landau levels of an excitonic insulator was also proposed by one of us. However, the previous picture of gap modulation in the Landau levels spectrum was built on a pair of well nested electron and hole Fermi surfaces, while the monolayer WTe$_2$ has one hole and two electron Fermi pockets with relative anisotropy. Here we demonstrate that for system like monolayer WTe$_2$, the excitonic insulating state arising from the coupled one hole and …
The quantum spin Hall (QSH) state was recently demonstrated in monolayers of the transition metal dichalcogenide 1T'-WTe$_2$ and is characterized by a band gap in the two-dimensional (2D) interior and helical one-dimensional (1D) edge states. Inducing superconductivity in the helical edge states would result in a 1D topological superconductor, a highly sought-after state of matter. In the present study, we use a novel dry-transfer flip technique to place atomically-thin layers of WTe$_2$ on a van der Waals superconductor, NbSe$_2$. Using scanning tunneling microscopy and spectroscopy (STM/STS)…
We study the influence of sample termination on the electronic properties of the novel quantum spin Hall insulator monolayer $1T'$-WTe$_2$. For this purpose, we construct an accurate, minimal 4-orbital tight-binding model with spin-orbit coupling by employing a combination of density-functional theory calculations, symmetry considerations, and fitting to experimental data. Based on this model, we compute energy bands and 2-terminal conductance spectra for various ribbon geometries with different terminations, with and without magnetic field. Because of the strong electron-hole asymmetry we fin…
Monolayer WTe2 is predicted to be a quantum spin Hall insulator (QSHI) and electron transport along its edges has been experimentally observed. However, the 'smoking gun' of QSHI, spin momentum locking of the edge electrons, has not been experimentally demonstrated. We propose a model to establish the relationship between the anisotropic magnetoresistance (AMR) and spin orientation of the helical electrons in WTe2. Based on the predictions of the model, angular dependent magnetoresistance measurements were carried out. The experimental results fully supported the model and the spin orientation…
We develop a theory of the excitonic phase recently proposed as the zero-field insulating state observed near charge neutrality in monolayer WTe$_2$. Using a Hartree-Fock approximation, we numerically identify two distinct gapped excitonic phases: a spin density wave state for weak non-zero interaction strength and spin spiral order at stronger interactions, separated by a narrow window of non-excitonic quantum spin Hall insulator. We introduce a simplified model capturing key features of the WTe$_2$ band structure, in which these phases appear as distinct valley ferromagnetic orders. We link …
We evidence the possibility for coherent electrical manipulation of the spin orientation of topologically protected edge states in a low-symmetry quantum spin Hall insulator. By using a combination of ab-initio simulations, symmetry-based modeling, and large-scale calculations of the spin Hall conductivity, it is shown that small electric fields can efficiently vary the spin textures of edge currents in monolayer 1T'-WTe2 by up to a 90-degree spin rotation, without jeopardizing their topological character. These findings suggest a new kind of gate-controllable spin-based device, topologically …
Tungsten ditelluride (WTe$_2$) exhibits extremely large and unsaturated magnetoresistance (MR). Due to the large spatially extensions of Te-5p and W-5d orbitals, the electronic properties of WTe$_2$ are sensitive to the lattice structures, which can probably affect the strongly temperature dependent MR found in experiment. Based on first-principle calculations, we investigate the temperature effect on the lattice and electronic structures of WTe$_2$. Our numerical results show that the thermal expansion coefficients of WTe$_2$ are highly anisotropic and considerably large. However, the tempera…
Evidence for the quantum spin Hall (QSH) effect has been reported in several experimental systems in the form of approximately quantized edge conductance. However, the most fundamental feature of the QSH effect, spin-momentum locking in the edge channels, has never been demonstrated experimentally. Here, we report clear evidence for spin-momentum locking in the edge channels of monolayer WTe2, thought to be a two-dimensional topological insulator (2D TI). We observe that the edge conductance is controlled by the component of an applied magnetic field perpendicular to a particular axis, which w…
A two-dimensional (2D) topological insulator (TI) exhibits the quantum spin Hall (QSH) effect, in which topologically protected spin-polarized conducting channels exist at the sample edges. Experimental signatures of the QSH effect have recently been reported for the first time in an atomically thin material, monolayer WTe2. Electrical transport measurements on exfoliated samples and scanning tunneling spectroscopy on epitaxially grown monolayer islands signal the existence of edge modes with conductance approaching the quantized value. Here, we directly image the local conductivity of monolay…
Monolayer 1T'-WTe$_2$ has been the first two-dimensional crystal where a quantum spin Hall phase was experimentally observed. In addition, recent experiments and theoretical modeling reported the presence of a robust excitonic insulating phase. While first-principles calculations with hybrid functionals and several measurements at low temperatures suggest the presence of a band gap of the order of 50 meV, experiments could confirm the presence of the helical edge states only up to 100 K. Here, we study with first-principle simulations the temperature effects on the electronic structure of mono…
Layered van der Waals semimetallic $T_\mathrm{d}$-WTe$_{2}$, exhibiting intriguing properties which include non-saturating extreme positive magnetoresistance (MR) and tunable chiral anomaly, has emerged as model topological type-II Weyl semimetal system. Here, $\sim$45 nm thick mechanically exfoliated flakes of $T_\mathrm{d}$-WTe$_{2}$ are studied $via$ atomic force microscopy, Raman spectroscopy, low-$T$/high-$μ_{0}H$ magnetotransport measurements and optical reflectivity. The contribution of anisotropy of the Fermi liquid state to the origin of the large positive transverse $\mathrm{MR}_\per…
A quantum spin Hall (QSH) insulator is a novel two-dimensional quantum state of matter that features quantized Hall conductance in the absence of magnetic field, resulting from topologically protected dissipationless edge states that bridge the energy gap opened by band inversion and strong spin-orbit coupling. By investigating electronic structure of epitaxially grown monolayer 1T'-WTe2 using angle-resolved photoemission (ARPES) and first principle calculations, we observe clear signatures of the topological band inversion and the band gap opening, which are the hallmarks of a QSH state. Scan…
Monolayer WTe$_2$, a centrosymmetric transition metal dichacogenide, has recently been established as a quantum spin Hall insulator and found superconducting upon gating. Here we study the pairing symmetry and topological nature of superconducting WTe$_2$ with a microscopic model at mean-field level. Surprisingly, we find that the spin-triplet phases in our phase diagram all host Majorana modes localized on two opposite corners. Even when the conventional pairing is favored, we find that an intermediate in-plane magnetic field exceeding the Pauli limit stabilizes an unconventional equal-spin p…
The integration of diverse electronic phenomena, such as magnetism and nontrivial topology, into a single system is normally studied either by seeking materials that contain both ingredients, or by layered growth of contrasting materials. The ability to simply stack very different two dimensional (2D) van der Waals materials in intimate contact permits a different approach. Here we use this approach to couple the helical edges states in a 2D topological insulator, monolayer WTe2, to a 2D layered antiferromagnet, CrI3. We find that the edge conductance is sensitive to the magnetization state of…
Monolayer 1T'-WTe2 is a quantum spin Hall insulator with a gapped bulk and gapless helical edge states persisting to temperatures around 100 K. Recent studies have revealed a topological-to-trivial phase transition as well the emergence of an unconventional, potentially topological superconducting state upon tuning the carrier concentration with gating. However, despite extensive studies, the effects of gating on the band structure and the helical edge states have not yet been established. In this work we present a combined low-temperature STM and first principles study of back-gated monolayer…
Topological quantum materials (TQMs) possess abundant and attractive spin physics, and a Weyl semimetal is the representative material because of the generation of spin polarization that is available for spin devices due to fictitious Weyl monopoles at the edge of the Weyl node. Meanwhile, a Weyl semimetal allows the other but unexplored spin polarization due to local symmetry breaking. Here, we report all-electric spin device operation using a type-II Weyl semimetal, WTe$_2$, at room temperature. The polarization of spins propagating in the all-electric device is perpendicular to the WTe$_2$ …
Either in bulk form, or when exfoliated into atomically thin crystals, layered transition metal dichalcogenides are continuously leading to the discovery of new phenomena. The latest example is provided by 1T'-WTe$_2$, a semimetal recently found to exhibit the largest known magnetoresistance in bulk crystals, and predicted to become a two-dimensional topological insulator in strained monolayers. Here, we show that reducing the thickness through facile exfoliation provides an effective experimental knob to tune the electronic properties of WTe$_2$, which allows us to identify the microscopic me…
WTe2, as a type-II Weyl semimetal, has 2D Fermi arcs on the (001) surface in the bulk and 1D helical edge states in its monolayer. These features have recently attracted wide attention in condensed matter physics. However, in the intermediate regime between the bulk and monolayer, the edge states have not been resolved owing to its closed band gap which makes the bulk states dominant. Here, we report the signatures of the edge superconductivity by superconducting quantum interference measurements in multilayer WTe2 Josephson junctions and we directly map the localized supercurrent. In thick WT…
We present magnetotransport measurements at classical magnetic fields for three graphene monolayers with various levels of disorder. A square root magnetoresistance (SRMR) behavior is observed in one sample which has the characteristic sub-linear conductivity signaling on the presence of short-range disorder in this sample. No square root MR was observed in other samples where short-range scattering is inessential as it is evident from the gate voltage dependences of their conductivities. Comparing our experimental data for the sample with theoretical calculations we found a good qualitative a…
We report a proof-of-concept study of extraordinary magnetoresistance (EMR) in devices of monolayer graphene encapsulated in hexagonal boron nitride, having metallic edge contacts and a central metal shunt. Extremely large EMR values, $MR=(R(B) - R_0) / R_0\sim 10^5$, are achieved in part because $R_0$ approaches or crosses zero as a function of the gate voltage, exceeding that achieved in high mobility bulk semiconductor devices. We highlight the sensitivity, $dR/dB$, which in two-terminal measurements is the highest yet reported for EMR devices, and in particular exceeds prior results in gra…
We report electron transport studies in an encapsulated few-layer WTe$_2$ at low temperatures and high magnetic fields. The magnetoconductance reveals a temperature-induced crossover between weak antilocalization (WAL) and weak localization (WL) in quantum diffusive regime. We show that the crossover clearly manifests coexistence and competition among several characteristic lengths, including the dephasing length, the spin-flip length, and the mean free path. In addition, low temperature conductance increases logarithmically with the increase of temperature indicating an interplay of electron-…
Using density functional theory calculations and the Greens's function formalism, we report the existence of magnetic edge states with a non-collinear spin texture present on different edges of the 1T' phase of the three monolayer transition metal dichalcogenides (TMDs): MoS$_2$, MoTe$_2$ and WTe$_2$. The magnetic states are gapless and accompanied by a spontaneous breaking of the time-reversal symmetry. This may have an impact on the prospects of utilizing WTe$_2$ as a quantum spin Hall insulator. It has previously been suggested that the topologically protected edge states of the 1T' TMDs co…
Single-crystal materials with sufficiently low crystal symmetry and strong spin-orbit interactions can be used to generate novel forms of spin-orbit torques on adjacent ferromagnets, such as the out-of-plane antidamping torque previously observed in WTe$_2$/ferromagnet heterostructures. Here, we present measurements of spin-orbit torques produced by the low-symmetry material $β$-MoTe$_2$, which unlike WTe$_2$ retains bulk inversion symmetry. We measure spin-orbit torques on $β$-MoTe$_2$/Permalloy heterostructures using spin-torque ferromagnetic resonance as a function of crystallographic align…
The spin polarization induced by the spin Hall effect (SHE) in thin films typically points out of the plane. This is rooted on the specific symmetries of traditionally studied systems, not in a fundamental constraint. Recently, experiments on few-layer ${\rm MoTe}_2$ and ${\rm WTe}_2$ showed that the reduced symmetry of these strong spin-orbit coupling materials enables a new form of {\it canted} spin Hall effect, characterized by concurrent in-plane and out-of-plane spin polarizations. Here, through quantum transport calculations on realistic device geometries, including disorder, we predict …
Magnetoresistance (MR) of ion irradiated monolayer graphene samples with variable-range hopping (VRH) mechanism of conductivity was measured at temperatures down to $T = 1.8$ K in magnetic fields up to $B = 8$ T. It was observed that in perpendicular magnetic fields, hopping resistivity $R$ decreases, which corresponds to negative MR (NMR), while parallel magnetic field results in positive MR (PMR) at low temperatures. NMR is explained on the basis of the "orbital" model in which perpendicular magnetic field suppresses the destructive interference of many paths through the intermediate sites i…
Giant Anisotropic Magnetoresistance in Magnetic Monolayers CrPX3 (X = S, Se, Te) due to symmetry breaking between the in-plane and out-of-plane crystallographic axes
We consider magnetotransport on a helical edge of a quantum spin Hall insulator, in the presence of bulk midgap states ``side-coupled" to the edge. In the presence of a magnetic field, the midgap levels are spin-split, and hybridization of these levels with the itinerant edge states leads to backscattering, and the ensuing increase in the resistance. We show that there is a singular cusp-like contribution to the positive magnetoresistance stemming from resonant midgap states weakly coupled to the edge. The singular behavior persists for both coherent and incoherent edge transport regimes. We u…
The two-dimensional atomically thin insulator hexagonal boron nitride (h-BN) constitutes a new paradigm in tunnel based devices. A large band gap along with its atomically flat nature without dangling bonds or interface trap states makes it an ideal candidate for tunnel spin transport in spintronic devices. Here, we demonstrate the tunneling of spin-polarized electrons through large area monolayer h-BN prepared by chemical vapor deposition in magnetic tunnel junctions. In ferromagnet/h-BN/ferromagnet heterostructures fabricated over a chip scale, we show tunnel magneto resistance at room tempe…
The layered semimetal WTe_2 has recently been found to be a two-dimensional topological insulator (2D TI) when thinned down to a single monolayer, with conducting helical edge channels. We report here that intrinsic superconductivity can be induced in this monolayer 2D TI by mild electrostatic doping, at temperatures below 1 K. The 2D TI-superconductor transition can be easily driven by applying a just a small gate voltage. This discovery offers new possibilities for gate-controlled devices combining superconductivity and topology, and could provide a basis for quantum information schemes base…
The spin conductance of two-dimensional topological insulators (2D TIs) is not expected to be quantized in the presence of perturbations that break the spin-rotational symmetry. However, the deviation from the pristine-limit quantization has yet to be studied in detail. In this paper, we define the spin current operator for the helical edge modes of a 2D TI and introduce a four-terminal setup to measure spin conductances. Using the developed formalism, we consider the effects of disorder terms that break spin-rotational symmetry or give rise to edge-to-edge coupling. We identify a key role pla…
Persistent spin texture preserved by local symmetry in graphene/WTe$_2$ heterostructure
Zernike Institute for Advanced Materials, University of Groningen（格罗宁根大学泽尔尼克高级材料研究所）；Faculty of Physics and Applied Informatics, University of Łódź（罗兹大学物理与应用信息学院）
The field of topological insulators (TI) was sparked by the prediction of the quantum spin Hall effect (QSHE) in time reversal invariant systems, such as spin-orbit coupled monolayer graphene. Ever since, a variety of monolayer crystals have been proposed as two-dimensional (2D) TIs exhibiting the QSHE, possibly even at high temperatures. However, conclusive evidence for a monolayer QSHE is still lacking, and systems based on semiconductor heterostructures operate at temperatures close to liquid helium. Here we report the observation of the QSHE in monolayer WTe2 at temperatures up to 100 Kelv…
Crystalline two-dimensional (2D) superconductors with low carrier density are an exciting new class of materials in which superconductivity coexists with strong interactions, the effects of complex topology are not obscured by disorder, and electronic properties can be strongly tuned by electrostatic gating. Very recently, two such materials, 'magic-angle' twisted bilayer graphene and monolayer $T_d$-WTe$_2$, have been reported to show superconductivity at temperatures near 1 K. Here we report superconductivity in semimetallic monolayer $T_d$-MoTe$_2$. The critical temperature $T_\textrm{c}$ r…
