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A real Chern insulator (RCI) featuring a real Chern number and a second-order boundary mode appears in a two-dimensional (2D) system with the space-time inversion symmetry (PT ). Here, we propose a kind of RCI: mirror real Chern insulator (MRCI) which emerges from the system having additional horizontal mirror symmetry Mz. The MRCI generally is characterized by two independent real Chern numbers, respectively defined in the two mirror subsystems of the system. Hence, the MRCI may host the second-order boundary modes different from the conventional RCI. We show that for spinless systems, the definition of the MRCI is straightforward, as PT keeps each mirror subsystem invariant. For the spinful systems with both PT and Mz, the real Chern number for the total system remain well defined, as MzPT = C2zT , and (C2zT )2= 1. However, since C2zT exchanges the two mirror subsystems, the definition of the MRCI in spinful systems requires the help of projective symmetry algebra. We also discuss the MRCIs in 3D systems, where the MRCI is defined on certain mirror-invariant 2D planes. Compared with its 2D counterpart, the 3D MRCI can exhibit more abundant physics when the systems have additional nonsymmorphic operators. Several concrete MRCI models including 2D and 3D, spinless and spinful models are constructed to further demonstrate our ideas.

A two-dimensional (2D) quantum electron system is characterized by the quantized energy levels, or subbands, in the out-of-plane direction. Populating higher subbands and controlling the inter-subband transitions have wide technological applications such as optical modulators and quantum cascade lasers. In conventional materials, however, the tunability of intersubband spacing is limited. Here we demonstrate electrostatic population and characterization of the second subband in few-layer InSe quantum wells, with giant tunability of its energy, population, and spin-orbit coupling strength, via the control of not only layer thickness but also out-of-plane displacement field. A modulation of as much as 350% or over 250 meV is achievable, underscoring the promise of InSe for tunable infrared and THz sources, detectors and modulators.

The study of topological semimetals has been extended to more general topological nodal systems such as metamaterials and artificial periodic structures. Among various nodal structures, triply degenerate nodal line (TNL) is rare and hence lack of attention. In this work, we have proposed a simple tight-binding model which hosts a topological non-trivial TNL. This TNL not only has the drumhead surface states as usual nodal line systems, but also has surface states which form a contracted-drumhead shape. And the shape and area of this contracted-drumhead can be tuned by the hopping parameters of the model. This provides an effective way to modulate surface states as well as their density of states, which can be important in future applications of topological nodal systems., Comment: 6 pages, 5 figures

Recently, Heusler ferromagnets have been found to exhibit unconventional anomalous electric, thermal, and thermoelectric transport properties. In this study, we employed first-principles density functional theory calculations to systematically investigate both intrinsic and extrinsic contributions to the anomalous Hall effect (AHE), anomalous Nernst effect (ANE), and anomalous thermal Hall effect (ATHE) in two Heusler ferromagnets: Fe$_2$CoAl and Fe$_2$NiAl. Our analysis reveals that the extrinsic mechanism originating from disorder dominates the AHE and ATHE in Fe$_2$CoAl , primarily due to the steep band dispersions across the Fermi energy and corresponding high longitudinal electronic conductivity. Conversely, the intrinsic Berry phase mechanism, physically linked to nearly flat bands around the Fermi energy and gapped by spin-orbit interaction band crossings, governs the AHE and ATHE in Fe$_2$NiAl. With respect to ANE, both intrinsic and extrinsic mechanisms are competing in Fe$_2$CoAl as well as in Fe$_2$NiAl. Furthermore, Fe$_2$CoAl and Fe$_2$NiAl exhibit tunable and remarkably pronounced anomalous transport properties. For instance, the anomalous Nernst and anomalous thermal Hall conductivities in Fe$_2$NiAl attain giant values of 8.29 A/Km and 1.19 W/Km, respectively, at room temperature. To provide a useful comparison, we also thoroughly investigated the anomalous transport properties of Co$_2$MnGa. Our findings suggest that Heusler ferromagnets Fe$_2$CoAl and Fe$_2$NiAl are promising candidates for spintronics and spin-caloritronics applications., Comment: 9 pages, 5 figures

The recent discovery of long-range magnetic order in atomically thin films has triggered particular interest in two-dimensional (2D) van der Waals (vdW) magnetic materials. In this paper, we perform a systematic theoretical study of the magneto-optical Sch\"{a}fer-Hubert effect (MOSHE) in 2D vdW antiferromagnetic \textit{M}PS$_3$ (\textit{M} = Mn, Fe, Ni) with multifold intralayer and interlayer magnetic orders. The formula for evaluating the MOSHE in 2D magnets is derived by considering the influence of a non-magnetic substrate. The MOSHE of monolayer and bilayer \textit{M}PS$_3$ are considerably large ($>2^{\circ}$), originating from the strong anisotropy of in-plane optical conductivity. The Sch\"{a}fer-Hubert rotation angles are surprisingly insensitive to the orientations of the N\'{e}el vector, while the Sch\"{a}fer-Hubert ellipticities are identified to be a good criterion to distinguish different interlayer magnetic orders. Our work establishes a theoretical framework for exploring novel 2D vdW magnets and facilitates the promising applications of the 2D \textit{M}PS$_3$ family in antiferromagnetic nanophotonic devices.

Motivated by easy access to complete corepresentation (corep) data of all the 1651 magnetic space groups (MSGs) in three-dimensional space, we have developed a Mathematica package MSGCorep to provide an offline database of coreps and various functions to manipulate them, based on our previous package SpaceGroupIrep. One can use the package MSGCorep to obtain the elements of any MSG and magnetic little group, to calculate the multiplication of group elements, to obtain the small coreps at any k-point and full coreps of any magnetic k-star for any MSG and show them in a user-friendly table form, to calculate and show the decomposition of direct products of full coreps between any two specified magnetic k-stars, and to determine the small coreps of energy bands. Both single-valued and double-valued coreps are supported. In addition, the 122 magnetic point groups (MPGs) and their coreps are also supported by this package. To the best of our knowledge, MSGCorep is the first package that is able to calculate the direct product of full coreps for any MSG and able to determine small coreps of energy bands for general purpose. In a word, the MSGCorep package is an offline database and tool set for MSGs, MPGs, and their coreps, and it is very useful to study the symmetries in magnetic and nonmagnetic materials., Comment: 23 pages, 7 figures, 6 tables, 1 supplementary material

We present a systematic classification of emergent particles in all 528 magnetic layer groups and 394 magnetic rod groups, which describe two-dimensional and one-dimensional crystals respectively. Our approach is via constructing a correspondence between a given magnetic layer/rod group and one of the magnetic space group, such that all irreducible representations of the layer/rod group can be derived from those of the corresponding space group. Based on these group representations, we explicitly construct the effective models for possible band degeneracies and identify all emergent particles, including both spinless and spinful cases. We find that there are six kinds of particles protected by magnetic layer groups and three kinds by magnetic rod groups. Our work provides a useful reference for the search and design of emergent particles in lower dimensional crystals., Comment: 7 pages, 2 figures, 4 tables

We propose an efficient algorithm to construct $\boldsymbol{k}\cdot \boldsymbol{p}$ effective Hamiltonians, which is much faster than the previously proposed algorithms. This algorithm is implemented in MagneticKP package. The package applies to both single-valued (spinless) and double-valued (spinful) cases, and it works for both magnetic and nonmagnetic systems. By interfacing with SpaceGroupIrep or MSGCorep packages, it can directly output the $\boldsymbol{k}\cdot \boldsymbol{p}$ Hamiltonian around arbitrary momentum and expanded to arbitrary order in $k$., Comment: 10 pages, 2 figures, 1 table

The irreducible representation of band structure is important for physical properties. Based on phonopy and recently developed SpaceGroupIrep package, we developed a package PhononIrep, which can get the band irreducible representation for arbitrary $\boldsymbol{k}$ point at first-principles level. As an application, we for the first time predict the cubic crossing Dirac point can exist in conventional crystal phonon systems X$_3$Y (X=Ti, Nb, Ta, Y=Au, Sb). We hope this package will facilitate phonon research in the future., Comment: 6 pages, 3 figures

The research on emergent particles in condensed matters has been attracting tremendous interest, and recently it is extended to magnetic systems. Here, we study the emergent particles stabilized by the symmetries of type-IV magnetic space groups (MSGs). Type-IV MSGs feature a special time reversal symmetry $\{\mathcal{T}|\boldsymbol{t}_0\}$, namely, the time reversal operation followed by a half lattice translation, which significantly alters the symmetry conditions for stabilizing the band degeneracies. In this work, based on symmetry analysis and modeling, we present a complete classification of emergent particles in type-IV MSGs by studying all possible (spinless and spinful, essential and accidental) particles in each of the 517 type-IV MSGs. Particularly, the detailed correspondence between the emergent particles and the type-IV MSGs that can host them are given in easily accessed interactive tables, where the basic information of the emergent particles, including the symmetry conditions, the effective Hamiltonian, the band dispersion and the topological characters can be found. According to the established encyclopedia, we find that several emergent particles that are previously believed to exist only in spinless systems will occur in spinful systems here, and vice versa, due to the $\{{\cal T}|\boldsymbol{t}_{0}\}$ symmetry. Our work not only deepens the understanding of the symmetry conditions for realizing emergent particles but also provides specific guidance for searching and designing materials with target particles., Comment: 7pages, 1 figures, 3 tables

In three-dimensional (3D) crystals, emergent particles arise when two or multiple bands contact and form degeneracy (band crossing) in the Brillouin zone. Recently a complete classification of emergent particles in 3D nonmagnetic crystals, which described by the type-II magnetic space groups (MSGs), has been established. However, a systematic investigation of emergent particles in magnetic crystals has not yet been performed, due to the complexity of the symmetries of magnetically ordered structures. Here, we address this challenging task by exploring the possibilities of the emergent particles in the 674 type-III MSGs. Based on effective k.p Hamiltonian and our classification of emergent particles [Yu et al., Sci. Bull. 67, 375 (2022) DOI:10.1016/j.scib.2021.10.023], we identify all possible emergent particles, including spinful and spinless, essential and accidental particles in the type-III MSGs. We find that all emergent particles in type-III MSGs also exist in type-II MSGs, with only one exception, i.e. the combined quadratic nodal line and nodal surface (QNL/NS). Moreover, tabulations of the emergent particles in each of the 674 type-III MSGs, together with the symmetry operations, the small corepresentations, the effective k.p Hamiltonians, and the topological character of these particles, are explicitly presented. Remarkably, combining this work and our homemade SpaceGroupIrep and MSGCorep packages will provide an effcient way to search topological magnetic materials with novel quasiparticles., Comment: 8 pages, 1 figure, 3 tables

Based on first-principles calculations and symmetry analysis, we predict atomically thin ($1-N$ layers) 2H group-VIB TMDs $MX_2$ ($M$ = Mo, W; $X$ = S, Se, Te) are large-gap higher-order topological crystalline insulators protected by $C_3$ rotation symmetry. We explicitly demonstrate the nontrivial topological indices and existence of the hallmark corner states with quantized fractional charge for these familiar TMDs with large bulk optical band gaps ($1.64-1.95$ eV for the monolayers), which would facilitate the experimental detection by STM. We find that the well-defined corner states exist in the triangular finite-size flakes with armchair edges of the atomically thin ($1-N$ layers) 2H group-VIB TMDs, and the corresponding quantized fractional charge is the number of layers $N$ divided by 3 modulo integers, which will simply double including spin degree of freedom.

We present a Mathematica program package MagneticTB, which can generate the tight-binding model for arbitrary magnetic space group. The only input parameters in MagneticTB are the (magnetic) space group number and the orbital information in each Wyckoff positions. Some useful functions including getting the matrix expression for symmetry operators, manipulating the energy band structure by parameters and interfacing with other software are also developed. MagneticTB can help to investigate the physical properties in both magnetic and non-magnetic system, especially for topological properties., Comment: 16 pages, 3 figures

The past decade has witnessed a surge of interest in exploring emergent particles in condensed matter systems. Novel particles, emerged as excitations around exotic band degeneracy points, continue to be reported in real materials and artificially engineered systems, but so far, we do not have a complete picture on all possible types of particles that can be achieved. Here, via systematic symmetry analysis and modeling, we accomplish a complete list of all possible particles in time reversal-invariant systems. This includes both spinful particles such as electron quasiparticles in solids, and spinless particles such as phonons or even excitations in electric-circuit and mechanical networks. We establish detailed correspondence between the particle, the symmetry condition, the effective model, and the topological character. This obtained encyclopedia concludes the search for novel emergent particles and provides concrete guidance to achieve them in physical systems., Comment: 5 pages for main text and 1228 pages for SM; comments are welcome, please contact Z. M. Yu, Y. Yao, or S. A. Yang

We have developed a Mathematica program package SpaceGroupIrep which is a database and tool set for irreducible representations (IRs) of space group in BC convention, i.e. the convention used in the famous book "The mathematical theory of symmetry in solids" by C. J. Bradley & A. P. Cracknell. Using this package, elements of any space group, little group, Herring little group, or central extension of little co-group can be easily obtained. This package can give not only little-group (LG) IRs for any k-point but also space-group (SG) IRs for any k-stars in intuitive table form, and both single-valued and double-valued IRs are supported. This package can calculate the decomposition of the direct product of SG IRs for any two k-stars. This package can determine the LG IRs of Bloch states in energy bands in BC convention and this works for any input primitive cell thanks to its ability to convert any input cell to a cell in BC convention. This package can also provide the correspondence of k-points and LG IR labels between BCS (Bilbao Crystallographic Server) and BC conventions. In a word, the package SpaceGroupIrep is very useful for both study and research, e.g. for analyzing band topology or determining selection rules., Comment: 28 pages (main text) + 629 pages (supplementary material), 11 figures

Symmetry breaking in two-dimensional layered materials plays a significant role in their macroscopic electrical, optical, magnetic and topological properties, including but not limited to spin-polarization effects, valley-contrasting physics, nonlinear Hall effects, nematic order, ferroelectricity, Bose-Einstein condensation and unconventional superconductivity. Engineering symmetry breaking of two-dimensional layered materials not only offers extraordinary opportunities to tune their physical properties, but also provides unprecedented possibilities to introduce completely new physics and technological innovations in electronics, photonics and optoelectronics. Indeed, over the past 15 years, a wide variety of physical, structural and chemical approaches have been developed to engineer symmetry breaking of two-dimensional layered materials. In this Review, we focus on the recent progresses on engineering the breaking of inversion, rotational, time reversal and spontaneous gauge symmetries in two-dimensional layered materials, and illustrate our perspectives on how these may lead to potential new physics and applications., Comment: 35 pages, 4 figures

Crystal-symmetry-protected photonic topological edge states (PTESs) based on air rods in conventional dielectric materials are designed as photonic topological waveguides (PTWs) coupled with side optical cavities. We demonstrate that the cavity coupled with the PTW can change the reflection-free transport of the PTESs, where the cavities with single mode and twofold degenerate modes are taken as examples. The single-mode cavities are able to perfectly reflect the PTESs at their resonant frequencies, forming a dip in the transmission spectra. The dip full width at half depth depends on the coupling strength between the cavity and PTW and thus on the cavity geometry and distance relative to the PTW. While the cavities with twofold degenerate modes lead to a more complex PTES transport whose transmission spectra can be in the Fano form. These effects well agree with the one-dimensional PTW-cavity transport theory we build, in which the coupling of the PTW with cavity is taken as $\delta$ or non-$\delta$ type. Such PTWs coupled with side cavities, combining topological properties and convenient tunability, have wide diversities for topological photonic devices., Comment: 9 pages, 9 figures

Electrons hopping in two-dimensional honeycomb lattices possess a valley degree of freedom in addition to charge and spin. In the absence of inversion symmetry, these systems were predicted to exhibit opposite Hall effects for electrons from different valleys. Such valley Hall effects have been achieved only by extrinsic means, such as substrate coupling, dual gating, and light illuminating. Here, we report the first observation of intrinsic valley Hall transport without any extrinsic symmetry breaking in the non-centrosymmetric monolayer and trilayer MoS2, evidenced by considerable nonlocal resistance that scales cubically with local resistance. Such a hallmark survives even at room temperature with a valley diffusion length at micron scale. By contrast, no valley Hall signal is observed in the centrosymmetric bilayer MoS2. Our work elucidates the topological quantum origin of valley Hall effects and marks a significant step towards the purely electrical control of valley degree of freedom in topological valleytronics., Comment: 22 pages, 4 figures

We investigate the optical properties of spin-triplet interlayer excitons in heterobilayer transition metal dichalcogenides in comparison with the spin-singlet ones. Surprisingly, the optical transition dipole of the spin-triplet exciton is found to be in the same order of magnitude to that of the spin-singlet exciton, in sharp contrast to the monolayer excitons where the spin triplet species is considered as dark compared to the singlet. Unlike the monolayer excitons whose spin-conserved (spin-flip) transition dipole can only couple to light of in-plane (out-of-plane) polarization, such restriction is removed for the interlayer excitons due to the breaking of the out-of-plane mirror symmetry. We find that as the interlayer atomic registry changes, the optical transition dipole of interlayer exciton crosses between in-plane ones of opposite circular polarization and the out-of-plane one for both the spin-triplet and spin-singlet species. As a result, excitons of both species have non-negligible coupling into photon modes of both in-plane and out-of-plane propagations, another sharp difference from the monolayers where the exciton couples predominantly into the out-of-plane propagation channel. At given atomic registry, the spin-triplet and spin-singlet excitons have distinct valley polarization selection rules, allowing the selective optical addressing of both the valley configuration and the spin singlet/triplet configuration of interlayer excitons.

In first-principles calculations, hybrid functional is often used to improve accuracy from local exchange correlation functionals. A drawback is that evaluating the hybrid functional needs significantly more computing effort. When spin-orbit coupling (SOC) is taken into account, the non-collinear spin structure increases computing effort by at least eight times. As a result, hybrid functional calculations with SOC are intractable in most cases. In this paper, we present an approximate solution to this problem by developing an efficient method based on a mixed linear combination of atomic orbital (LCAO) scheme. We demonstrate the power of this method using several examples and we show that the results compare very well with those of direct hybrid functional calculations with SOC, yet the method only requires a computing effort similar to that without SOC. The presented technique provides a good balance between computing efficiency and accuracy, and it can be extended to magnetic materials., Comment: 10 pages, 3 figures

Highly uniform and ordered nanodot arrays are crucial for high performance quantum optoelectronics including new semiconductor lasers and single photon emitters, and for synthesizing artificial lattices of interacting quasiparticles towards quantum information processing and simulation of many-body physics. Van der Waals heterostructures of 2D semiconductors are naturally endowed with an ordered nanoscale landscape, i.e. the moir\'e pattern that laterally modulates electronic and topographic structures. Here we find these moir\'e effects realize superstructures of nanodot confinements for long-lived interlayer excitons, which can be either electrically or strain tuned from perfect arrays of quantum emitters to excitonic superlattices with giant spin-orbit coupling (SOC). Besides the wide range tuning of emission wavelength, the electric field can also invert the spin optical selection rule of the emitter arrays. This unprecedented control arises from the gauge structure imprinted on exciton wavefunctions by the moir\'e, which underlies the SOC when hopping couples nanodots into superlattices. We show that the moir\'e hosts complex-hopping honeycomb superlattices, where exciton bands feature a Dirac node and two Weyl nodes, connected by spin-momentum locked topological edge modes., Comment: To appear in Science Advances

The interlayer couplings in commensurate and incommensurate bilayer structures of transition metal dichalcogenides are investigated with perturbative treatment. The interlayer coupling in \pm K valleys can be decomposed into a series of hopping terms with distinct phase factors. In H-type and R-type commensurate bilayers, the interference between the three main terms leads to a sensitive dependence of the interlayer coupling strength on the translation. The magnitudes of the main hopping terms are found to some dozen of meV by fitting to the ab initio results. The interlayer couplings in \Gamma valley of valence band and Q valley of conduction band are also studied. The obtained strong coupling strengths of several hundred meV can play important roles in mediating the ultrafast interlayer charge transfer in heterobilayers of transition metal dichalcogenides., Comment: 12 pages, 8 figures

Effective gravity and gauge fields are emergent properties intrinsic for low-energy quasiparticles in topological semimetals. Here, taking two Dirac semimetals as examples, we demonstrate that applied lattice strain can generate warped spacetime, with fascinating analogues in astrophysics. Particularly, we study the possibility of simulating black-hole/white-hole event horizons and gravitational lensing effect. Furthermore, we discover strain-induced topological phase transitions, both in the bulk materials and in their thin films. Especially in thin films, the transition between the quantum spin Hall and the trivial insulating phases can be achieved by a small strain, naturally leading to the proposition of a novel piezo-topological transistor device. Possible experimental realizations and analogue of Hawking radiation effect are discussed. Our result bridges multiple disciplines, revealing topological semimetals as a unique table-top platform for exploring interesting phenomena in astrophysics and general relativity; it also suggests realistic materials and methods to achieve controlled topological phase transitions with great potential for device applications., Comment: 29 pages,6 figures

We investigate the optical control possibilities of spin-valley qubit carried by single electrons localized in nanostructures of monolayer TMDs, including small quantum dots formed by lateral heterojunction and charged impurities. The quantum controls are discussed when the confinement induces valley hybridization and when the valley hybridization is absent. We show that the bulk valley and spin optical selection rules can be inherited in different forms in the two scenarios, both of which allow the definition of spin-valley qubit with desired optical controllability. We also investigate nuclear spin induced decoherence and quantum control of electron-nuclear spin entanglement via intervalley terms of the hyperfine interaction. Optically controlled two-qubit operations in a single quantum dot are discussed., Comment: 17pages, 10 figures

Atomically thin group-VIB transition metal dichalcogenides (TMDs) have recently emerged as a new class of two-dimensional (2D) semiconductors with extraordinary properties including the direct band gap in the visible frequency range, the pronounced spin-orbit coupling, the ultra-strong Coulomb interaction, and the rich physics associated with the valley degree of freedom. These 2D TMDs exhibit great potentials for device applications and have attracted vast interest for the exploration of new physics. 2D TMDs have complex electronic structures which underlie their physical properties. Here we review the bulk electronic structures in these new 2D materials as well as the theoretical models developed at different levels, along which we sort out the understandings on the origins of a variety of properties observed or predicted., Comment: 25 pages, 6 figures. Review article, published online on Chem Soc Rev in Dec 2014. Posted with permission after the 12 months' embargo period

In few-layer (FL) transition metal dichalcogenides (TMDC), the conduction bands along the Gamma-K directions shift downward energetically in the presence of interlayer interactions, forming six Q valleys related by three-fold rotational symmetry and time reversal symmetry. In even-layers the extra inversion symmetry requires all states to be Kramers degenerate, whereas in odd-layers the intrinsic inversion asymmetry dictates the Q valleys to be spin-valley coupled. In this Letter, we report the transport characterization of prominent Shubnikov-de Hass (SdH) oscillations for the Q valley electrons in FL transition metal disulfide (TMDs), as well as the first quantum Hall effect (QHE) in TMDCs. Our devices exhibit ultrahigh field-effect mobilities (~16,000 cm2V-1s-1 for FL WS2 and ~10,500 cm2V-1s-1 for FL MoS2) at cryogenic temperatures. Universally in the SdH oscillations, we observe a valley Zeeman effect in all odd-layer TMD devices and a spin Zeeman effect in all even-layer TMD devices., Comment: 20 pages, 4 figures

Monolayer (ML) transition metal dichalcogenides (TMDs) have been attracting great research attentions lately for their extraordinary properties, in particular the exotic spin-valley coupled electronic structures that promise future spintronic and valleytronic applications1-3. The energy bands of ML TMDs have well separated valleys that constitute effectively an extra internal degree of freedom for low energy carriers3-12. The large spin-orbit coupling in the TMDs makes the spin index locked to the valley index, which has some interesting consequences such as the magnetoelectric effects in 2H bilayers13. A direct experimental characterization of the spin-valley coupled electronic structure can be of great interests for both fundamental physics and device applications. In this work, we report the first experimental observation of the quasi-particle interference (QPI) patterns in ML WSe2 using low-temperature (LT) scanning tunneling microscopy/spectroscopy (STM/S). We observe intervalley quantum interference involving the Q-valleys in the conduction band due to spin-conserved scattering processes, while spin-flip intervalley scattering is absent. This experiment establishes unequivocally the presence of spin-valley coupling and affirms the large spin-splitting at the Q valleys. Importantly, the inefficient spin-flip intervalley scattering implies long valley and spin lifetime in ML WSe2, which represents a key figure of merit for valley-spintronic applications., Comment: 15 pages, 4 figures

Monolayer transition metal dichalcogenides (TMDs) offer new opportunities for realizing quantum dots (QDs) in the ultimate two-dimensional (2D) limit. Given the rich control possibilities of electron valley pseudospin discovered in the monolayers, this quantum degree of freedom can be a promising carrier of information for potential quantum spintronics exploiting single electrons in TMD QDs. An outstanding issue is to identify the degree of valley hybridization, due to the QD confinement, which may significantly change the valley physics in QDs from its form in the 2D bulk. Here we perform a systematic study of the intervalley coupling by QD confinement potentials on extended TMD monolayers. We find that the intervalley coupling in such geometry is generically weak due to the vanishing amplitude of the electron wavefunction at the QD boundary, and hence valley hybridization shall be well quenched by the much stronger spin-valley coupling in monolayer TMDs and the QDs can well inherit the valley physics of the 2D bulk. We also discover sensitive dependence of intervalley coupling strength on the central position and the lateral length scales of the confinement potentials, which may possibly allow tuning of intervalley coupling by external controls, Comment: 17 pages, 14 figures

Topological edge states at the boundary of quantum spin Hall (QSH) insulators hold great promise for dissipationless electron transport. The device application of topological edge states has several critical requirements for the QSH insulator materials, e.g., large band gap, appropriate insulating substrates, and multiple conducting channels. In this paper, based on first-principle calculations, we show that Bi$_{4}$Br$_{4}$ is a suitable candidate. Single-layer Bi$_{4}$Br$_{4}$ was demonstrated to be QSH insulator with sizable gap recently. Here, we find that, in multilayer systems, both the band gaps and low-energy electronic structures are only slightly affected by the interlayer coupling. On the intrinsic insulating substrate of Bi$_{4}$Br$_{4}$, the single-layer Bi$_{4}$Br$_{4}$ well preserves its topological edge states. Moreover, at the boundary of multilayer Bi$_{4}$Br$_{4}$, the topological edge states stemming from different single-layers are weakly coupled, and can be fully decoupled via constructing a stair-stepped edge. The decoupled topological edge states are well suitable for multi-channel dissipationless transport., Comment: 4 pages, 4 figures

Controlled flow of spin and valley pseudospin is key to future electronics exploiting these internal degrees of freedom of carriers. Here we discover a universal possibility for generating spin and valley currents by electric bias or temperature gradient only, which arises from the anisotropy of Fermi pockets in crystalline solids. We find spin and valley currents to the second order in the electric field, as well as their thermoelectric counterparts, i.e. the nonlinear spin and valley Seebeck effects. These second-order nonlinear responses allow two unprecedented possibilities to generate pure spin and valley flows without net charge current: (i) by an AC bias; or (ii) by an arbitrary inhomogeneous temperature distribution. As examples, we predict appreciable nonlinear spin and valley currents in two-dimensional (2D) crystals including graphene, monolayer and trilayer transition metal dichalcogenides, and monolayer gallium selenide. Our finding points to a new route towards electrical and thermal generations of spin and valley currents for spintronic and valleytronic applications based on 2D quantum materials.

Magnetic interaction with the gapless surface states in topological insulator (TI) has been predicted to give rise to a few exotic quantum phenomena. However, the effective magnetic doping of TI is still challenging in experiment. Using first-principles calculations, the magnetic doping properties (V, Cr, Mn and Fe) in three strong TIs (Bi$_{2}$Se$_{3}$, Bi$_{2}$Te$_{3}$ and Sb$_{2}$Te$_{3}$) are investigated. We find that for all three TIs the cation-site substitutional doping is most energetically favorable with anion-rich environment as the optimal growth condition. Further our results show that under the nominal doping concentration of 4%, Cr and Fe doped Bi$_{2}$Se$_{3}$, Bi$_{2}$Te$_{3}$, and Cr doped Sb$_{2}$Te$_{3}$ remain as insulator, while all TIs doped with V, Mn and Fe doped Sb$_{2}$Te$_{3}$ become metal. We also show that the magnetic interaction of Cr doped Bi$_{2}$Se$_{3}$ tends to be ferromagnetic, while Fe doped Bi$_{2}$Se$_{3}$ is likely to be antiferromagnetic. Finally, we estimate the magnetic coupling and the Curie temperature for the promising ferromagnetic insulator (Cr doped Bi$_{2}$Se$_{3}$) by Monte Carlo simulation. These findings may provide important guidance for the magnetism incorporation in TIs experimentally., Comment: 10 pages, 9 figures

Majorana fermions, quantum particles with non-Abelian exchange statistics, are not only of fundamental importance, but also building blocks for fault-tolerant quantum computation. Although certain experimental breakthroughs for observing Majorana fermions have been made recently, their conclusive dection is still challenging due to the lack of proper material properties of the underlined experimental systems. Here we propose a new platform for Majorana fermions based on edge states of certain non-topological two-dimensional semiconductors with strong spin-orbit coupling, such as monolayer group-VI transition metal dichalcogenides (TMD). Using first-principles calculations and tight-binding modeling, we show that zigzag edges of monolayer TMD can host well isolated single edge band with strong spin-orbit coupling energy. Combining with proximity induced s-wave superconductivity and in-plane magnetic fields, the zigzag edge supports robust topological Majorana bound states at the edge ends, although the two-dimensional bulk itself is non-topological. Our findings points to a controllable and integrable platform for searching and manipulating Majorana fermions., Comment: 12 pages, 7 figures

We present a three-band tight-binding (TB) model for describing the low-energy physics in monolayers of group-VIB transition metal dichalcogenides $MX_2$ ($M$=Mo, W; $X$=S, Se, Te). As the conduction and valence band edges are predominantly contributed by the $d_{z^{2}}$, $d_{xy}$, and $d_{x^{2}-y^{2}}$ orbitals of $M$ atoms, the TB model is constructed using these three orbitals based on the symmetries of the monolayers. Parameters of the TB model are fitted from the first-principles energy bands for all $MX_2$ monolayers. The TB model involving only the nearest-neighbor $M$-$M$ hoppings is sufficient to capture the band-edge properties in the $\pm K$ valleys, including the energy dispersions as well as the Berry curvatures. The TB model involving up to the third-nearest-neighbor $M$-$M$ hoppings can well reproduce the energy bands in the entire Brillouin zone. Spin-orbit coupling in valence bands is well accounted for by including the on-site spin-orbit interactions of $M$ atoms. The conduction band also exhibits a small valley-dependent spin splitting which has an overall sign difference between Mo$X_{2}$ and W$X_{2}$. We discuss the origins of these corrections to the three-band model. The three-band TB model developed here is efficient to account for low-energy physics in $MX_2$ monolayers, and its simplicity can be particularly useful in the study of many-body physics and physics of edge states., Comment: 11 pages, 10 figures, 4 tables

In monolayer group-VI transition metal dichalcogenides (TMDC), charge carriers have spin and valley degrees of freedom, both associated with magnetic moments. On the other hand, the layer degree of freedom in multilayers is associated with electrical polarization. Here, we show that TMDC bilayers offer an unprecedented platform to realize a strong coupling between the spin, layer pseudospin, and valley degrees of freedom of holes. Such coupling not only gives rise to the spin Hall effect and spin circular dichroism in inversion symmetric bilayer, but also leads to a variety of magnetoelectric effects permitting quantum manipulation of these electronic degrees of freedom. Oscillating electric and magnetic fields can both drive the hole spin resonance where the two fields have valley-dependent interference, making possible a prototype interplay between the spin and valley as information carriers for potential valley-spintronic applications. We show how to realize quantum gates on the spin qubit controlled by the valley bit., Comment: 15 pages, 1 table, 4 figures

Crystal symmetry governs the nature of electronic Bloch states. For example, in the presence of time reversal symmetry, the orbital magnetic moment and Berry curvature of the Bloch states must vanish unless inversion symmetry is broken. In certain 2D electron systems such as bilayer graphene, the intrinsic inversion symmetry can be broken simply by applying a perpendicular electric field. In principle, this offers the remarkable possibility of switching on/off and continuously tuning the magnetic moment and Berry curvature near the Dirac valleys by reversible electrical control. Here we demonstrate this principle for the first time using bilayer MoS2, which has the same symmetry as bilayer graphene but has a bandgap in the visible that allows direct optical probing of these Berry-phase related properties. We show that the optical circular dichroism, which reflects the orbital magnetic moment in the valleys, can be continuously tuned from -15% to 15% as a function of gate voltage in bilayer MoS2 field-effect transistors. In contrast, the dichroism is gate-independent in monolayer MoS2, which is structurally non-centrosymmetric. Our work demonstrates the ability to continuously vary orbital magnetic moments between positive and negative values via symmetry control. This represents a new approach to manipulating Berry-phase effects for applications in quantum electronics associated with 2D electronic materials., Comment: 13 pages main text + 4 pages supplementary materials

Motivated by the triumph and limitation of graphene for electronic applications, atomically thin layers of group VI transition metal dichalcogenides are attracting extensive interest as a class of graphene-like semiconductors with a desired band-gap in the visible frequency range. The monolayers feature a valence band spin splitting with opposite sign in the two valleys located at corners of 1st Brillouin zone. This spin-valley coupling, particularly pronounced in tungsten dichalcogenides, can benefit potential spintronics and valleytronics with the important consequences of spin-valley interplay and the suppression of spin and valley relaxations. Here we report the first optical studies of WS2 and WSe2 monolayers and multilayers. The efficiency of second harmonic generation shows a dramatic even-odd oscillation with the number of layers, consistent with the presence (absence) of inversion symmetry in even-layer (odd-layer). Photoluminescence (PL) measurements show the crossover from an indirect band gap semiconductor at mutilayers to a direct-gap one at monolayers. The PL spectra and first-principle calculations consistently reveal a spin-valley coupling of 0.4 eV which suppresses interlayer hopping and manifests as a thickness independent splitting pattern at valence band edge near K points. This giant spin-valley coupling, together with the valley dependent physical properties, may lead to rich possibilities for manipulating spin and valley degrees of freedom in these atomically thin 2D materials.

We show that inversion symmetry breaking together with spin-orbit coupling leads to coupled spin and valley physics in monolayers of MoS2 and other group-VI dichalcogenides, making possible controls of spin and valley in these 2D materials. The spin-valley coupling at the valence band edges suppresses spin and valley relaxation, as flip of each index alone is forbidden by the valley contrasting spin splitting. Valley Hall and spin Hall effects coexist in both electron-doped and hole-doped systems. Optical interband transitions have frequency-dependent polarization selection rules which allow selective photoexcitation of carriers with various combination of valley and spin indices. Photo-induced spin Hall and valley Hall effects can generate long lived spin and valley accumulations on sample boundaries. The physics discussed here provides a route towards the integration of valleytronics and spintronics in multi-valley materials with strong spin-orbit coupling and inversion symmetry breaking., Comment: published version

In this paper, we combine thermal effects with Landau-Zener (LZ) quantum tunneling effects in a dynamical Monte Carlo (DMC) framework to produce satisfactory magnetization curves of single-molecule magnet (SMM) systems. We use the giant spin approximation for SMM spins and consider regular lattices of SMMs with magnetic dipolar interactions (MDI). We calculate spin reversal probabilities from thermal-activated barrier hurdling, direct LZ tunneling, and thermal-assisted LZ tunnelings in the presence of sweeping magnetic fields. We do systematical DMC simulations for Mn$_{12}$ systems with various temperatures and sweeping rates. Our simulations produce clear step structures in low-temperature magnetization curves, and our results show that the thermally activated barrier hurdling becomes dominating at high temperature near 3K and the thermal-assisted tunnelings play important roles at intermediate temperature. These are consistent with corresponding experimental results on good Mn$_{12}$ samples (with less disorders) in the presence of little misalignments between the easy axis and applied magnetic fields, and therefore our magnetization curves are satisfactory. Furthermore, our DMC results show that the MDI, with the thermal effects, have important effects on the LZ tunneling processes, but both the MDI and the LZ tunneling give place to the thermal-activated barrier hurdling effect in determining the magnetization curves when the temperature is near 3K. This DMC approach can be applicable to other SMM systems, and could be used to study other properties of SMM systems., Comment: Phys Rev B, accepted; 10 pages, 6 figures

We propose a nonequilibrium Monte Carlo (MC) approach to explore nonequilibrium dynamical ferromagnetism of interacting single molecule magnets (SMMs). Both quantum spin tunneling and thermally activated spin reversal are successfully implemented in the same MC simulation framework. Applied to a typical example, this simulation method satisfactorily reproduces experimental magnetization curves with experimental parameters. Our results show that both quantum and classical effects are essential to determine the hysteresis behaviors. This method is effective and reliable to gain deep insights into SMMs., Comment: 3 pages