Bulk valley transport and Berry curvature spreading at the edge of flat bands

Nonlinear Valley Hall Effect

The valley Hall effect arises from valley-contrasting Berry curvature and requires inversion symmetry breaking. Here, we propose a nonlinear mechanism to generate a valley Hall current in systems with both inversion and time-reversal symmetry, where the linear and second-order charge Hall currents vanish along with the linear valley Hall current. We show that a second-order valley Hall signal emerges from the electric field correction to the Berry curvature, provided a valley-contrasting anisotropic dispersion is engineered. We demonstrate the nonlinear valley Hall effect in tilted massless Dirac fermions in strained graphene and organic semiconductors. Our Letter opens up the possibility of controlling the valley degree of freedom in inversion symmetric systems via nonlinear valleytronics.

A quantum ruler for orbital magnetism in moiré quantum matter

For almost a century, magnetic oscillations have been a powerful "quantum ruler" for measuring Fermi surface topology. In this study, we used Landau-level spectroscopy to unravel the energy-resolved valley-contrasting orbital magnetism and large orbital magnetic susceptibility that contribute to the energies of Landau levels of twisted double-bilayer graphene. These orbital magnetism effects led to substantial deviations from the standard Onsager relation, which manifested as a breakdown in scaling of Landau-level orbits. These substantial magnetic responses emerged from the nontrivial quantum geometry of the electronic structure and the large length scale of the moiré lattice potential. Going beyond traditional measurements, Landau-level spectroscopy performed with a scanning tunneling microscope offers a complete quantum ruler that resolves the full energy dependence of orbital magnetic properties in moiré quantum matter.

Perpendicular electric field drives Chern transitions and layer polarization changes in Hofstadter bands

Moiré superlattices engineer band properties and enable observation of fractal energy spectra of Hofstadter butterfly. Recently, correlated-electron physics hosted by flat bands in small-angle moiré systems has been at the foreground. However, the implications of moiré band topology within the single-particle framework are little explored experimentally. An outstanding problem is understanding the effect of band topology on Hofstadter physics, which does not require electron correlations. Our work experimentally studies Chern state switching in the Hofstadter regime using twisted double bilayer graphene (TDBG), which offers electric field tunable topological bands, unlike twisted bilayer graphene. Here we show that the nontrivial topology reflects in the Hofstadter spectra, in particular, by displaying a cascade of Hofstadter gaps that switch their Chern numbers sequentially while varying the perpendicular electric field. Our experiments together with theoretical calculations suggest a crucial role of charge polarization changing concomitantly with topological transitions in this system. Layer polarization is likely to play an important role in the topological states in few-layer twisted systems. Moreover, our work establishes TDBG as a novel Hofstadter platform with nontrivial magnetoelectric coupling.

Spontaneous time-reversal symmetry breaking in twisted double bilayer graphene

Twisted double bilayer graphene (tDBG) comprises two Bernal-stacked bilayer graphene sheets with a twist between them. Gate voltages applied to top and back gates of a tDBG device tune both the flatness and topology of the electronic bands, enabling an unusual level of experimental control. Metallic states with broken spin and valley symmetries have been observed in tDBG devices with twist angles in the range 1.2–1.3°, but the topologies and order parameters of these states have remained unclear. We report the observation of an anomalous Hall effect in the correlated metal state of tDBG, with hysteresis loops spanning hundreds of mT in out-of-plane magnetic field (B_⊥) that demonstrate spontaneously broken time-reversal symmetry. The B_⊥ hysteresis persists for in-plane fields up to several Tesla, suggesting valley (orbital) ferromagnetism. At the same time, the resistivity is strongly affected by even mT-scale values of in-plane magnetic field, pointing to spin-valley coupling or to a direct orbital coupling between in-plane field and the valley degree of freedom.

Twisted double bilayer graphene (tDBG) comprises two Bernal-stacked bilayer graphene sheets with a twist between them. Here, the authors report a strong anomalous Hall effect in the correlated-metal regime of tDBG, indicating time reversal symmetry breaking from orbital ferromagnetism, likely associated with valley polarization.

Tunable and giant valley-selective Hall effect in gapped bilayer graphene

Berry curvature is analogous to magnetic field but in momentum space and is commonly present in materials with nontrivial quantum geometry. It endows Bloch electrons with transverse anomalous velocities to produce Hall-like currents even in the absence of a magnetic field. We report the direct observation of in situ tunable valley-selective Hall effect (VSHE), where inversion symmetry, and thus the geometric phase of electrons, is controllable by an out-of-plane electric field. We use high-quality bilayer graphene with an intrinsic and tunable bandgap, illuminated by circularly polarized midinfrared light, and confirm that the observed Hall voltage arises from an optically induced valley population. Compared with molybdenum disulfide (MoS₂), we find orders of magnitude larger VSHE, attributed to the inverse scaling of the Berry curvature with bandgap. By monitoring the valley-selective Hall conductivity, we study the Berry curvature's evolution with bandgap. This in situ manipulation of VSHE paves the way for topological and quantum geometric optoelectronic devices, such as more robust switches and detectors.

Valleytronics in two-dimensional materials with line defect

The concept of valley originates from two degenerate but nonequivalent energy bands at the local minimum in the conduction band or local maximum in the valence band. Manipulating the valley states for information storage and processing develops a brand-new electronics-valleytronics. Broken inversion symmetry is a necessary condition to produce pure valley currents. The polycrystalline two-dimensional materials (graphene, silicene, monolayer group-VI transition metal dichalcogenides, etc) with pristine grains stitched together by disordered grain boundaries (GBs) are the natural inversion-symmetry-broken systems and the candidates in the field of valleytronics. Different from their pristine forms, the Dirac valleys on both sides of GBs are mismatched in the momentum space and induce peculiar valley transport properties across the GBs. In this review, we systematically demonstrate the fundamental properties of valley degree of freedom across mostly studied and experimentally feasible polycrystalline structure-the line defect, and the manipulation strategies with electrical, magnetic and mechanical methods to realize the valley polarization. We also introduce an effective numerical method, the non-equilibrium Green's function technique, to tackle the valley transport issues in the line defect systems. The present challenges and the perspective on the further investigations of the line defect in valleytronics are also summarized.

Spin-Polarized Nematic Order, Quantum Valley Hall States, and Field-Tunable Topological Transitions in Twisted Multilayer Graphene Systems

We theoretically study the correlated insulator states, quantum anomalous Hall (QAH) states, and field-induced topological transitions between different correlated states in twisted multilayer graphene systems. Taking twisted bilayer-monolayer graphene and twisted double-bilayer graphene as examples, we show that both systems stay in spin-polarized, C_{3z}-broken insulator states with zero Chern number at 1/2 filling of the flat bands under finite displacement fields. In some cases these spin-polarized, nematic insulator states are in the quantum valley Hall (QVH) phase by virtue of the nontrivial band topology of the systems. The spin-polarized insulator state is quasidegenerate with the valley polarized state if only the dominant intravalley Coulomb interaction is included. Such quasidegeneracy can be split by atomic on-site interactions such that the spin-polarized, nematic state become the unique ground state. Such a scenario applies to various twisted multilayer graphene systems at 1/2 filling, thus can be considered as a universal mechanism. Moreover, under vertical magnetic fields, the orbital Zeeman splittings and the field-induced change of charge density in twisted multilayer graphene systems would compete with the atomic Hubbard interactions, which can drive transitions from spin-polarized zero-Chern-number states to valley-polarized QAH states with small onset magnetic fields.

Flexible Piezoelectricity of Two-Dimensional Materials Governed by Effective Berry Curvature

Two-dimensional piezoelectric materials have been regarded as ideal candidates for flexible and versatile nanoelectromechanical systems, yet their fundamental piezoelectric mechanisms remain to be fully understood. Employing joint theoretical-statistical analyses, we develop universal models for quantifying the piezoelectricity of three-coordinated honeycomb-like monolayers at the atomistic level. The theoretical model within the framework of modern polarization theory suggests that the distribution of effective Berry curvature is essential for interpreting the relation between microscopic/electronic structures and piezoelectric properties. The statistical model based on DFT high-throughput calculation reveals that 2D piezoelectricity crucially depends on the effective mass, bandgap, and atomic distance along the rotation axis. Implementing stress and doping is demonstrated to be effective for optimizing piezoelectricity. Such findings provide valuable guidelines for designing 2D piezoelectric materials.

Long-range nontopological edge currents in charge-neutral graphene

Van der Waals heterostructures display numerous unique electronic properties. Nonlocal measurements, wherein a voltage is measured at contacts placed far away from the expected classical flow of charge carriers, have been widely used in the search for novel transport mechanisms, including dissipationless spin and valley transport^1-9, topological charge-neutral currents^10-12, hydrodynamic flows¹³ and helical edge modes^14-16. Monolayer^1-5,10,15-19, bilayer^9,11,14,20 and few-layer²¹ graphene, transition-metal dichalcogenides^6,7 and moiré superlattices^8,10,12 have been found to display pronounced nonlocal effects. However, the origin of these effects is hotly debated^{3,11,17,22-24}. Graphene, in particular, exhibits giant nonlocality at charge neutrality^1,15-19, a striking behaviour that has attracted competing explanations. Using a superconducting quantum interference device on a tip (SQUID-on-tip) for nanoscale thermal and scanning gate imaging²⁵, here we demonstrate that the commonly occurring charge accumulation at graphene edges^23,26-31 leads to giant nonlocality, producing narrow conductive channels that support long-range currents. Unexpectedly, although the edge conductance has little effect on the current flow in zero magnetic field, it leads to field-induced decoupling between edge and bulk transport at moderate fields. The resulting giant nonlocality at charge neutrality and away from it produces exotic flow patterns that are sensitive to edge disorder, in which charges can flow against the global electric field. The observed one-dimensional edge transport is generic and nontopological and is expected to support nonlocal transport in many electronic systems, offering insight into the numerous controversies and linking them to long-range guided electronic states at system edges.