Giant Second-Order Nonlinear Hall Effect in Twisted Bilayer Graphene

Metallic Electro-optic Effect in Gapped Bilayer Graphene

Electro-optic (EO) modulation is a critical device action in photonics. Recently, the non-Drude dynamics induced by the Berry curvature dipole (BCD) in metals have attracted attention as a potential candidate for terahertz EO modulation. However, such BCD-induced EO effects can be challenging to realize, often requiring flat bands and complex materials such as a strained magic-angle twisted bilayer graphene on hexagonal boron nitride. Here, we argue that metallic EO can be achieved with a much simpler material, gapped bilayer graphene, with EO coefficients comparable to that in flat-band materials in the terahertz range. In particular, we find metallic EO can be realized without a Berry curvature dipole; we identify skew-scattering and a "snap" (third-order derivative of velocity) can readily produce pronounced Pockels and Kerr EO effects in clean metals. These yield nonreciprocal and field-activated birefringence and field-induced modulations to transmission and reflection, essential components for terahertz EO modulators.

Colossal room-temperature non-reciprocal Hall effect

Non-reciprocal charge transport has gained significant attention due to its potential in exploring quantum symmetry and its promising applications. Traditionally, non-reciprocal transport has been observed in the longitudinal direction, with non-reciprocal resistance being a small fraction of the ohmic resistance. Here we report a transverse non-reciprocal transport phenomenon featuring a quadratic current-voltage characteristic and divergent non-reciprocity, termed the non-reciprocal Hall effect. This effect is observed in microscale Hall devices fabricated from platinum (Pt) deposited by a focused ion beam on silicon substrates. The transverse non-reciprocal Hall effect arises from the geometrically asymmetric scattering of textured Pt nanoparticles within the focused-ion-beam-deposited Pt structures. Notably, the non-reciprocal Hall effect generated in focused-ion-beam-deposited Pt electrodes can propagate to adjacent conductors such as Au and NbP through Hall current injection. Additionally, this pronounced non-reciprocal Hall effect facilitates broadband frequency mixing. These findings not only validate the non-reciprocal Hall effect concept but also open avenues for its application in terahertz communication, imaging and energy harvesting.

Nonlinear edge transport in a quantum Hall system

Nonlinear transport phenomena in condensed matter reflect the geometric nature, quantum coherence, and many-body correlation of electronic states. Electric currents in solids are classified into (i) ohmic current, (ii) supercurrent, and (iii) geometric or topological current. While the nonlinear current-voltage (I-V) characteristics of the former two categories have been extensive research topics recently, those of the last category remains unexplored. Among them, the quantum Hall current is a representative example. Realized in two-dimensional electronic systems under a strong magnetic field, the topological protection quantizes the Hall conductance in the unit of e2/h (e, elementary charge; and h, Planck constant), of which the edge transport picture gives a good account. Here, we theoretically study the nonlinear I-VH characteristic of the edge transport up to third order in VH. We find that nonlinearity arises in the Hall response from electron-electron interaction between the counterpropagating edge channels with the nonlinear energy dispersions. We also discuss possible experimental observations.

Colossal anomalous Hall effect in the layered antiferromagnetic EuAl2Si2 compound

The anomalous Hall effect (AHE), significantly enhanced by the extrinsic mechanism, has attracted attention for its almost unlimited Hall response, which exceeds the upper limit of the Berry curvature mechanism. However, due to the high conductivity in the clean regime and weak skew scattering, it is a great challenge to obtain large anomalous Hall conductivities and large anomalous Hall angles at the same time. Here, we unveil a new magnetic metal system, EuAl2Si2, which hosts both colossal anomalous Hall conductivity (σAxy ≥ 104 Ω-1 cm-1) and large anomalous Hall angle (AHA >10%). The scaling relation suggests that the skew scattering mechanism is dominant in the colossal anomalous Hall response and gives rise to a large skew scattering constant. The large effective SOC and large magnetic moment may account for this anomaly. Our results indicate that EuAl2Si2 is a good platform to study the extrinsic AHE mechanism.

Quantum geometry quadrupole-induced third-order nonlinear transport in antiferromagnetic topological insulator MnBi2Te4

The study of quantum geometry effects in materials has been one of the most important research directions in recent decades. The quantum geometry of a material is characterized by the quantum geometric tensor of the Bloch states. The imaginary part of the quantum geometry tensor gives rise to the Berry curvature while the real part gives rise to the quantum metric. While Berry curvature has been well studied in the past decades, the experimental investigation on the quantum metric effects is only at its infancy stage. In this work, we measure the nonlinear transport of bulk MnBi2Te4, which is a topological anti-ferromagnet. We found that the second order nonlinear responses are negligible as required by inversion symmetry, the third-order nonlinear responses are finite. The measured third-harmonic longitudinal (V x x 3 ω ) and transverse (V x y 3 ω ) voltages with frequency 3 ω , driven by an a.c. current with frequency ω , show an intimate connection with magnetic transitions of MnBi2Te4 flakes. Their magnitudes change abruptly as MnBi2Te4 flakes go through magnetic transitions from an antiferromagnetic state to a canted antiferromagnetic state and to a ferromagnetic state. In addition, the measuredV x x 3 ω is an even function of the applied magnetic field B whileV x y 3 ω is odd in B. Amazingly, the field dependence of the third-order responses as a function of the magnetic field suggests thatV x x 3 ω is induced by the quantum metric quadrupole andV x y 3 ω is induced by the Berry curvature quadrupole. Therefore, the quadrupoles of both the real and the imaginary part of the quantum geometry tensor of bulk MnBi2Te4 are revealed through the third order nonlinear transport measurements. This work greatly advanced our understanding on the connections between the higher order moments of quantum geometry and nonlinear transport.

Nonlinear physics of moiré superlattices

Nonlinear physics is one of the most important research fields in modern physics and materials science. It offers an unprecedented paradigm for exploring many fascinating physical phenomena and realizing diverse cutting-edge applications inconceivable in the framework of linear processes. Here we review the recent theoretical and experimental progress concerning the nonlinear physics of synthetic quantum moiré superlattices. We focus on the emerging nonlinear electronic, optical and optoelectronic properties of moiré superlattices, including but not limited to the nonlinear anomalous Hall effect, dynamically twistable harmonic generation, nonlinear optical chirality, ultralow-power-threshold optical solitons and spontaneous photogalvanic effect. We also present our perspectives on the future opportunities and challenges in this rapidly progressing field, and highlight the implications for advances in both fundamental physics and technological innovations.

Emergent phases in graphene flat bands

Electronic correlations in two-dimensional materials play a crucial role in stabilising emergent phases of matter. The realisation of correlation-driven phenomena in graphene has remained a longstanding goal, primarily due to the absence of strong electron-electron interactions within its low-energy bands. In this context, magic-angle twisted bilayer graphene has recently emerged as a novel platform featuring correlated phases favoured by the low-energy flat bands of the underlying moiré superlattice. Notably, the observation of correlated insulators and superconductivity, and the interplay between these phases have garnered significant attention. A wealth of correlated phases with unprecedented tunability was discovered subsequently, including orbital ferromagnetism, Chern insulators, strange metallicity, density waves, and nematicity. However, a comprehensive understanding of these closely competing phases remains elusive. The ability to controllably twist and stack multiple graphene layers has enabled the creation of a whole new family of moiré superlattices with myriad properties. Here, we review the progress and development achieved so far, encompassing the rich phase diagrams offered by these graphene-based moiré systems. Additionally, we discuss multiple phases recently observed in non-moiré multilayer graphene systems. Finally, we outline future opportunities and challenges for the exploration of hidden phases in this new generation of moiré materials.

Nonlinear Electrical Transport Unveils Fermi Surface Malleability in a Moiré Heterostructure

Van Hove singularities enhance many-body interactions and induce collective states of matter ranging from superconductivity to magnetism. In magic-angle twisted bilayer graphene, van Hove singularities appear at low energies and are malleable with density, leading to a sequence of Lifshitz transitions and resets observable in Hall measurements. However, without a magnetic field, linear transport measurements have limited sensitivity to the band's topology. Here, we utilize nonlinear longitudinal and transverse transport measurements to probe these unique features in twisted bilayer graphene at zero magnetic field. We demonstrate that the nonlinear responses, induced by the Berry curvature dipole and extrinsic scattering processes, intricately map the Fermi surface reconstructions at various fillings. Importantly, our experiments highlight the intrinsic connection of these features with the moiré bands. Beyond corroborating the insights from linear Hall measurements, our findings establish nonlinear transport as a pivotal tool for probing band topology and correlated phenomena.

Giant nonlinear Hall and wireless rectification effects at room temperature in the elemental semiconductor tellurium

The second-order nonlinear Hall effect (NLHE) in non-centrosymmetric materials has recently drawn intense interest, since its inherent rectification could enable various device applications such as energy harvesting and wireless charging. However, previously reported NLHE systems normally suffer from relatively small Hall voltage outputs and/or low working temperatures. In this study, we report the observation of a pronounced NLHE in tellurium (Te) thin flakes at room temperature. Benefiting from the semiconductor nature of Te, the obtained nonlinear response can be readily enhanced through electrostatic gating, leading to a second-harmonic output at 300 K up to 2.8 mV. By utilizing such a giant NLHE, we further demonstrate the potential of Te as a wireless Hall rectifier within the radiofrequency range, which is manifested by the remarkable and tunable rectification effect also at room temperature. Extrinsic scattering is then revealed to be the dominant mechanism for the NLHE in Te, with symmetry breaking on the surface playing a key role. As a simple elemental semiconductor, Te provides an appealing platform to advance our understanding of nonlinear transport in solids and to develop NLHE-based electronic devices.

Effective Manipulation of a Colossal Second-Order Transverse Response in an Electric-Field-Tunable Graphene Moiré System

The second-order nonlinear transport illuminates a frequency-doubling response emerging in quantum materials with a broken inversion symmetry. The two principal driving mechanisms, the Berry curvature dipole and the skew scattering, reflect various information including ground-state symmetries, band dispersions, and topology of electronic wave functions. However, effective manipulation of them in a single system has been lacking, hindering the pursuit of strong responses. Here, we report on the effective manipulation of the two mechanisms in a single graphene moiré superlattice, AB-BA stacked twisted double bilayer graphene. Most saliently, by virtue of the high tunability of moiré band structures and scattering rates, a record-high second-order transverse conductivity ∼ 510 μm S V-1 is observed, which is orders of magnitude higher than any reported values in the literature. Our findings establish the potential of electrically tunable graphene moiré systems for nonlinear transport manipulations and applications.

Creating chirality in the nearly two dimensions

Structural chirality, defined as the lack of mirror symmetry in materials' atomic structure, is only meaningful in three-dimensional space. Yet two-dimensional (2D) materials, despite their small thickness, can show chirality that enables prominent asymmetric optical, electrical and magnetic properties. In this Perspective, we first discuss the possible definition and mathematical description of '2D chiral materials', and the intriguing physics enabled by structural chirality in van der Waals 2D homobilayers and heterostructures, such as circular dichroism, chiral plasmons and the nonlinear Hall effect. We then summarize the recent experimental progress and approaches to induce and control structural chirality in 2D materials from monolayers to superlattices. Finally, we postulate a few unique opportunities offered by 2D chiral materials, the synthesis and new properties of which can potentially lead to chiral optoelectronic devices and possibly materials for enantioselective photochemistry.

Optical properties and plasmons in moiré structures

The discoveries of numerous exciting phenomena in twisted bilayer graphene (TBG) are stimulating significant investigations on moiré structures that possess a tunable moiré potential. Optical response can provide insights into the electronic structures and transport phenomena of non-twisted and twisted moiré structures. In this article, we review both experimental and theoretical studies of optical properties such as optical conductivity, dielectric function, non-linear optical response, and plasmons in moiré structures composed of graphene, hexagonal boron nitride (hBN), and/or transition metal dichalcogenides. Firstly, a comprehensive introduction to the widely employed methodology on optical properties is presented. After, moiré potential induced optical conductivity and plasmons in non-twisted structures are reviewed, such as single layer graphene-hBN, bilayer graphene-hBN and graphene-metal moiré heterostructures. Next, recent investigations of twist-angle dependent optical response and plasmons are addressed in twisted moiré structures. Additionally, we discuss how optical properties and plasmons could contribute to the understanding of the many-body effects and superconductivity observed in moiré structures.

Odd Nonlinear Conductivity under Spatial Inversion in Chiral Tellurium

Electrical transport in noncentrosymmetric materials departs from the well-established phenomenological Ohm's law. Instead of a linear relation between current and electric field, a nonlinear conductivity emerges along specific crystallographic directions. This nonlinear transport is fundamentally related to the lack of spatial inversion symmetry. However, the experimental implications of an inversion symmetry operation on the nonlinear conductivity remain to be explored. Here, we report on a large, nonlinear conductivity in chiral tellurium. By measuring samples with opposite handedness, we demonstrate that the nonlinear transport is odd under spatial inversion. Furthermore, by applying an electrostatic gate, we modulate the nonlinear output by a factor of 300, reaching the highest reported value excluding engineered heterostructures. Our results establish chiral tellurium as an ideal compound not just to study the fundamental interplay between crystal structure, symmetry operations and nonlinear transport; but also to develop wireless rectifiers and energy-harvesting chiral devices.

Berry Curvature Spectroscopy from Bloch Oscillations

Artificial crystals such as moiré superlattices can have a real-space periodicity much larger than the underlying atomic scale. This facilitates the presence of Bloch oscillations in the presence of a static electric field. We demonstrate that the optical response of such a system, when dressed with a static field, becomes resonant at the frequencies of Bloch oscillations, which are in the terahertz regime when the lattice constant is of the order of 10 nm. In particular, we show within a semiclassical band-projected theory that resonances in the dressed Hall conductivity are proportional to the lattice Fourier components of the Berry curvature. We illustrate our results with a low-energy model on an effective honeycomb lattice.

Gate-Tunable Berry Curvature Dipole Polarizability in Dirac Semimetal Cd_{3}As_{2}

We reveal the gate-tunable Berry curvature dipole polarizability in Dirac semimetal Cd_{3}As_{2} nanoplates through measurements of the third-order nonlinear Hall effect. Under an applied electric field, the Berry curvature exhibits an asymmetric distribution, forming a field-induced Berry curvature dipole, resulting in a measurable third-order Hall voltage with a cubic relationship to the longitudinal electric field. Notably, the magnitude and polarity of this third-order nonlinear Hall effect can be effectively modulated by gate voltages. Furthermore, our scaling relation analysis demonstrates that the sign of the Berry curvature dipole polarizability changes when tuning the Fermi level across the Dirac point, in agreement with theoretical calculations. The results highlight the gate control of nonlinear quantum transport in Dirac semimetals, paving the way for promising advancements in topological electronics.

Roses in the nonperturbative current response of artificial crystals

In two-dimensional artificial crystals with large real-space periodicity, the nonlinear current response to a large applied electric field can feature a strong angular dependence, which encodes information about the band dispersion and Berry curvature of isolated electronic Bloch minibands. Within the relaxation-time approximation, we obtain analytic expressions up to infinite order in the driving field for the current in a band-projected theory with time-reversal and trigonal symmetry. For a fixed field strength, the dependence of the current on the direction of the applied field is given by rose curves whose petal structure is symmetry constrained and is obtained from an expansion in real-space translation vectors. We illustrate our theory with calculations on periodically buckled graphene and twisted double bilayer graphene, wherein the discussed physics can be accessed at experimentally relevant field strengths.

Nonlinear Topological Magnon Spin Hall Effect

When a magnon passes through two-dimensional magnetic textures, it will experience a fictitious magnetic field originating from the 3×3 skew-symmetric gauge fields. To date, only one of the three independent components of the gauge fields has been found to play a role in generating the fictitious magnetic field, while the other two are perfectly hidden. In this Letter, we show that they are concealed in the nonlinear magnon transport in magnetic textures. Without loss of generality, we theoretically study the nonlinear magnon-skyrmion interaction in antiferromagnets. By analyzing the scattering features of three-magnon processes between the circularly polarized incident magnon and breathing skyrmion, we predict a giant Hall angle of both the confluence and splitting modes. Furthermore, we find that the Hall angle reverses its sign when one switches the handedness of the incident magnons. We dub this the nonlinear topological magnon spin Hall effect. Our findings are deeply rooted in the bosonic nature of magnons that the particle number is not conserved, which has no counterpart in low-energy fermionic systems and may open the door for probing gauge fields by nonlinear means.

Intrinsic Nonlinear Hall Effect and Gate-Switchable Berry Curvature Sliding in Twisted Bilayer Graphene

Though the observation of the quantum anomalous Hall effect and nonlocal transport response reveals nontrivial band topology governed by the Berry curvature in twisted bilayer graphene, some recent works reported nonlinear Hall signals in graphene superlattices that are caused by the extrinsic disorder scattering rather than the intrinsic Berry curvature dipole moment. In this Letter, we report a Berry curvature dipole induced intrinsic nonlinear Hall effect in high-quality twisted bilayer graphene devices. We also find that the application of the displacement field substantially changes the direction and amplitude of the nonlinear Hall voltages, as a result of a field-induced sliding of the Berry curvature hotspots. Our Letter not only proves that the Berry curvature dipole could play a dominant role in generating the intrinsic nonlinear Hall signal in graphene superlattices with low disorder densities, but also demonstrates twisted bilayer graphene to be a sensitive and fine-tunable platform for second harmonic generation and rectification.

Intrinsic Nonlinear Hall Detection of the Néel Vector for Two-Dimensional Antiferromagnetic Spintronics

The respective unique merit of antiferromagnets and two-dimensional (2D) materials in spintronic applications inspires us to exploit 2D antiferromagnetic spintronics. However, the detection of the Néel vector in 2D antiferromagnets remains a great challenge because the measured signals usually decrease significantly in the 2D limit. Here we propose that the Néel vector of 2D antiferromagnets can be efficiently detected by the intrinsic nonlinear Hall (INH) effect which exhibits unexpected significant signals. As a specific example, we show that the INH conductivity of the monolayer manganese chalcogenides MnX (X=S, Se, Te) can reach the order of nm·mA/V^{2}, which is orders of magnitude larger than experimental values of paradigmatic antiferromagnetic spintronic materials. The INH effect can be accurately controlled by shifting the chemical potential around the band edge, which is experimentally feasible via electric gating or charge doping. Moreover, we explicitly demonstrate its 2π-periodic dependence on the Néel vector orientation based on an effective k·p model. Our findings enable flexible design schemes and promising material platforms for spintronic memory device applications based on 2D antiferromagnets.

Quantum Geometric Oscillations in Two-Dimensional Flat-Band Solids

Two-dimensional van der Waals heterostructures can be engineered into artificial superlattices that host flat bands with significant Berry curvature and provide a favorable environment for the emergence of novel electron dynamics. In particular, the Berry curvature can induce an oscillating trajectory of an electron wave packet transverse to an applied static electric field. Though analogous to Bloch oscillations, this novel oscillatory behavior is driven entirely by quantum geometry in momentum space instead of band dispersion. While the current from Bloch oscillations can be localized by increasing field strength, the current from the geometric orbits saturates to a nonzero plateau in the strong-field limit. In nonmagnetic materials, the geometric oscillations are even under inversion of the applied field, whereas the Bloch oscillations are odd, a property that can be used to distinguish these two coexisting effects.