Valley Polarization and Inversion in Strained Graphene via Pseudo-Landau Levels, Valley Splitting of Real Landau Levels, and Confined States

Manipulating 2D Materials through Strain Engineering

This review explores the growing interest in 2D layered materials, such as graphene, h-BN, transition metal dichalcogenides (TMDs), and black phosphorus (BP), with a specific focus on recent advances in strain engineering. Both experimental and theoretical results are delved into, highlighting the potential of strain to modulate physical properties, thereby enhancing device performance. Various strain engineering methods are summarized, and the impact of strain on the electrical, optical, magnetic, thermal, and valleytronic properties of 2D materials is thoroughly examined. Finally, the review concludes by addressing potential applications and challenges in utilizing strain engineering for functional devices, offering valuable insights for further research and applications in optoelectronics, thermionics, and spintronics.

Acoustomagnonic Spin Hall Effect in Honeycomb Antiferromagnets

The recently discovered Van der Waals antiferromagnets have suffered from the lack of a comprehensive method to study their magnetic properties. Here, we propose an ac intrinsic magnon spin Hall current driven by surface acoustic waves as a novel probe for such antiferromagnets. Our results pave the way towards mechanical detection and manipulation of the magnetic order in two-dimensional antiferromagnets. Furthermore, they will overcome the difficulties with weak magnetic responses inherent in the use of antiferromagnets and hence provide a building block for future antiferromagnetic spintronics.

Mechanical Control of Quantum Transport in Graphene

2D materials (2DMs) are fundamentally electro-mechanical systems. Their environment unavoidably strains them and modifies their quantum transport properties. For instance, a simple uniaxial strain can completely turn off the conductance of ballistic graphene or switch on/off the superconducting phase of magic-angle bilayer graphene. This article reports measurements of quantum transport in strained graphene transistors which agree quantitatively with models based on mechanically-induced gauge potentials. A scalar potential is mechanically induced in situ to modify graphene's work function by up to 25 meV. Mechanically generated vector potentials suppress the ballistic conductance of graphene by up to 30% and control its quantum interferences. The data are measured with a custom experimental platform able to precisely tune both the mechanics and electrostatics of suspended graphene transistors at low-temperature over a broad range of strain (up to 2.6%). This work opens many opportunities to harness quantitative strain effects in 2DM quantum transport and technologies.

Tunneling Valley Hall Effect Driven by Tilted Dirac Fermions

Valleytronics is a research field utilizing a valley degree of freedom of electrons for information processing and storage. A strong valley polarization is critical for realistic valleytronic applications. Here, we predict a tunneling valley Hall effect (TVHE) driven by tilted Dirac fermions in all-in-one tunnel junctions based on a two-dimensional (2D) valley material. Different doping of the electrode and spacer regions in these tunnel junctions results in momentum filtering of the tunneling Dirac fermions, generating a strong transverse valley Hall current dependent on the Dirac-cone tilting. Using the parameters of an existing 2D valley material, we demonstrate that such a strong TVHE can host a giant valley Hall angle even in the absence of the Berry curvature. Finally, we predict that resonant tunneling can occur in a tunnel junction with properly engineered device parameters such as the spacer width and transport direction, providing significant enhancement of the valley Hall angle. Our work opens a new approach to generate valley polarization in realistic valleytronic systems.

Imaging quantum oscillations and millitesla pseudomagnetic fields in graphene

The exceptional control of the electronic energy bands in atomically thin quantum materials has led to the discovery of several emergent phenomena1. However, at present there is no versatile method for mapping the local band structure in advanced two-dimensional materials devices in which the active layer is commonly embedded in the insulating layers and metallic gates. Using a scanning superconducting quantum interference device, here we image the de Haas-van Alphen quantum oscillations in a model system, the Bernal-stacked trilayer graphene with dual gates, which shows several highly tunable bands2-4. By resolving thermodynamic quantum oscillations spanning more than 100 Landau levels in low magnetic fields, we reconstruct the band structure and its evolution with the displacement field with excellent precision and nanoscale spatial resolution. Moreover, by developing Landau-level interferometry, we show shear-strain-induced pseudomagnetic fields and map their spatial dependence. In contrast to artificially induced large strain, which leads to pseudomagnetic fields of hundreds of tesla5-7, we detect naturally occurring pseudomagnetic fields as low as 1 mT corresponding to graphene twisting by 1 millidegree, two orders of magnitude lower than the typical angle disorder in twisted bilayer graphene8-11. This ability to resolve the local band structure and strain at the nanoscale level enables the characterization and use of tunable band engineering in practical van der Waals devices.

Mechanical, electronic, optical, piezoelectric and ferroic properties of strained graphene and other strained monolayers and multilayers: an update

This is an update of a previous review (Naumiset al2017Rep. Prog. Phys.80096501). Experimental and theoretical advances for straining graphene and other metallic, insulating, ferroelectric, ferroelastic, ferromagnetic and multiferroic 2D materials were considered. We surveyed (i) methods to induce valley and sublattice polarisation (P) in graphene, (ii) time-dependent strain and its impact on graphene's electronic properties, (iii) the role of local and global strain on superconductivity and other highly correlated and/or topological phases of graphene, (iv) inducing polarisationPon hexagonal boron nitride monolayers via strain, (v) modifying the optoelectronic properties of transition metal dichalcogenide monolayers through strain, (vi) ferroic 2D materials with intrinsic elastic (σ), electric (P) and magnetic (M) polarisation under strain, as well as incipient 2D multiferroics and (vii) moiré bilayers exhibiting flat electronic bands and exotic quantum phase diagrams, and other bilayer or few-layer systems exhibiting ferroic orders tunable by rotations and shear strain. The update features the experimental realisations of a tunable two-dimensional Quantum Spin Hall effect in germanene, of elemental 2D ferroelectric bismuth, and 2D multiferroic NiI2. The document was structured for a discussion of effects taking place in monolayers first, followed by discussions concerning bilayers and few-layers, and it represents an up-to-date overview of exciting and newest developments on the fast-paced field of 2D materials.

Untwisting Moiré Physics: Almost Ideal Bands and Fractional Chern Insulators in Periodically Strained Monolayer Graphene

Moiré systems have emerged in recent years as a rich platform to study strong correlations. Here, we will propose a simple, experimentally feasible setup based on periodically strained graphene that reproduces several key aspects of twisted moiré heterostructures-but without introducing a twist. We consider a monolayer graphene sheet subject to a C_{2}-breaking periodic strain-induced pseudomagnetic field with period L_{M}≫a, along with a scalar potential of the same period. This system has almost ideal flat bands with valley-resolved Chern number ±1, where the deviation from ideal band geometry is analytically controlled and exponentially small in the dimensionless ratio (L_{M}/l_{B})^{2}, where l_{B} is the magnetic length corresponding to the maximum value of the pseudomagnetic field. Moreover, the scalar potential can tune the bandwidth far below the Coulomb scale, making this a very promising platform for strongly interacting topological phases. Using a combination of strong-coupling theory and self-consistent Hartree-Fock, we find quantum anomalous Hall states at integer fillings. At fractional filling, exact diagonaliztion reveals a fractional Chern insulator at parameters in the experimentally feasible range. Overall, we find that this system has larger interaction-induced gaps, smaller quasiparticle dispersion, and enhanced tunability compared to twisted graphene systems, even in their ideal limit.

Topological Flat Bands in Strained Graphene: Substrate Engineering and Optical Control

The discovery of correlated phases in twisted moiré superlattices accelerated the search for low-dimensional materials with exotic properties. A promising approach uses engineered substrates to strain the material. However, designing substrates for tailored properties is hindered by the incomplete understanding of the relationship between the substrate's shapes and the electronic properties of the deposited materials. By analyzing effective models of graphene under periodic deformations with generic crystalline profiles, we identify strong C2z symmetry breaking as the critical substrate geometric feature for emerging energy gaps and quasi-flat bands. We find continuous strain profiles producing connected pseudomagnetic field landscapes are important for band topology. We show that the resultant electronic and topological properties from a substrate can be controlled with circularly polarized light, which also offers unique signatures for identifying the band topology imprinted by strain. Our results can guide experiments on strain engineering for exploring interesting transport and topological phenomena.

Engineering the Strain and Interlayer Excitons of 2D Materials via Lithographically Engraved Hexagonal Boron Nitride

Strain engineering has quickly emerged as a viable option to modify the electronic, optical, and magnetic properties of 2D materials. However, it remains challenging to arbitrarily control the strain. Here we show that, by creating atomically flat surface nanostructures in hexagonal boron nitride, we achieve an arbitrary on-chip control of both the strain distribution and magnitude on high-quality molybdenum disulfide. The phonon and exciton emissions are shown to vary in accordance with our strain field designs, enabling us to write and draw any photoluminescence color image in a single chip. Moreover, our strain engineering offers a powerful means to significantly and controllably alter the strengths and energies of interlayer excitons at room temperature. This method can be easily extended to other material systems and offers promise for functional excitonic devices.

Giant Periodic Pseudomagnetic Fields in Strained Kagome Magnet FeSn Epitaxial Films on SrTiO3(111) Substrate

Quantum materials, particularly Dirac materials with linearly dispersing bands, can be effectively tuned by strain-induced lattice distortions leading to a pseudomagnetic field that strongly modulates their electronic properties. Here, we grow kagome magnet FeSn films, consisting of alternatingly stacked Sn₂ honeycomb (stanene) and Fe₃Sn kagome layers, on SrTiO₃(111) substrates by molecular beam epitaxy. Using scanning tunneling microscopy/spectroscopy, we show that the Sn honeycomb layer can be periodically deformed by epitaxial strain for a film thickness below 10 nm, resulting in differential conductance peaks consistent with Landau levels generated by a pseudomagnetic field greater than 1000 T. Our findings demonstrate the feasibility of strain engineering the electronic properties of topological magnets at the nanoscale.

Theory of quantized photonic spin Hall effect in strained graphene under a sub-Tesla external magnetic field

The quantized photonic spin Hall effect (PSHE) in the strained graphene-substrate system is predicted under a sub-Tesla external magnetic field, which is two orders of magnitude smaller than required to produce the quantized effect in the conventional graphene-substrate system. It is found that in-plane and transverse spin-dependent splittings in the PSHE, exhibit different quantized behaviors and are closely related to the reflection coefficients. Unlike the quantized PSHE in the conventional graphene-substrate system formed by the splitting of real Landau levels, the quantized PSHE in the strained graphene-substrate system is attributed to the splitting of pseudo-Landau levels caused by the pseudo-magnetic field and the lifting of valley degeneracy of the n ≠ 0 pseudo-Landau levels induced by the sub-Tesla external magnetic field. At the same time, the pseudo-Brewster angles of the system are also quantized with the change of Fermi energy. The sub-Tesla external magnetic field and the PSHE appear as quantized peak values near these angles. The giant quantized PSHE is expected to be used for direct optical measurements of the quantized conductivities and pseudo-Landau levels in the monolayer strained graphene.

Chiral excitonic order from twofold van Hove singularities in kagome metals

Recent experiments on kagome metals AV₃Sb₅ (A=K,Rb,Cs) identify twofold van Hove singularities (TvHS) with opposite concavity near the Fermi energy, generating two approximately hexagonal Fermi surfaces - one electron-like and the other hole-like. Here we propose that a TvHS generates a novel time-reversal symmetry breaking excitonic order - arising due to bound pairs of electrons and holes located at opposite concavity van Hove singularities. We introduce a minimal model for the TvHS and investigate interaction induced many-body instabilities via the perturbative renormalisation group technique and a free energy analysis. Specialising to parameters appropriate for the kagome metals AV₃Sb₅, we construct a phase diagram comprising chiral excitons, charge density wave and a region of coexistence. We propose this as an explanation of a diverse range of experimental observations in AV₃Sb₅. Notably, the chiral excitonic state gives rise to a quantum anomalous Hall conductance, providing an appealing interpretation of the observed anomalous Hall effect in kagome metals. Possible alternative realisations of the TvHS mechanism in bilayer materials are also discussed. We suggest that TvHS open up interesting possibilities for correlated phases, enriching the set of competing ground states to include excitonic order.

Surface acoustic wave induced phenomena in two-dimensional materials

Surface acoustic wave (SAW)-matter interaction provides a fascinating key for inducing and manipulating novel phenomena and functionalities in two-dimensional (2D) materials. The dynamic strain field and piezo-electric field associated with propagating SAWs determine the coherent manipulation and transduction between 2D excitons and phonons. Over the past decade, many intriguing acoustic-induced effects, including the acousto-electric effect, acousto-galvanic effect, acoustic Stark effect, acoustic Hall effect and acoustic exciton transport, have been reported experimentally. However, many more phenomena, such as the valley acousto-electric effect, valley acousto-electric Hall effect and acoustic spin Hall effect, were only theoretically proposed, the experimental verification of which are yet to be achieved. In this minireview, we attempt to overview the recent breakthrough of SAW-induced phenomena covering acoustic charge transport, acoustic exciton transport and modulation, and coherent acoustic phonons. Perspectives on the opportunities of the proposed SAW-induced phenomena, as well as open experimental challenges, are also discussed, attempting to offer some guidelines for experimentalists and theorists to explore the desired exotic properties and boost practical applications of 2D materials.

Magnetic-Field-Tunable Valley-Contrasting Pseudomagnetic Confinement in Graphene

Introducing quantum confinement has uncovered a rich set of interesting quantum phenomena and allows one to directly probe the physics of confined (quasi-)particles. In most experiments, however, an electrostatic potential is the only available method to generate quantum dots in a continuous system to confine (quasi-)particles. Here we demonstrate experimentally that inhomogeneous pseudomagnetic fields in strained graphene can introduce exotic quantum confinement of massless Dirac fermions. The pseudomagnetic fields have opposite directions in the two distinct valleys of graphene. By adding and tuning real magnetic fields, the total effective magnetic fields in the two valleys are imbalanced. By that we realized valley-contrasting spatial confinement, which lifts the valley degeneracy and results in field-tunable valley-polarized confined states in graphene. Our results provide a new avenue to manipulate the valley degree of freedom.

Realizing One-Dimensional Electronic States in Graphene via Coupled Zeroth Pseudo-Landau Levels

Strain-induced pseudomagnetic fields can mimic real magnetic fields to generate a zero-magnetic-field analog of the Landau levels (LLs), i.e., the pseudo-Landau levels (PLLs), in graphene. The distinct nature of the PLLs enables one to realize novel electronic states beyond what is feasible with real LLs. Here, we show that it is possible to realize exotic electronic states through the coupling of zeroth PLLs in strained graphene. In our experiment, nanoscale strained structures embedded with PLLs are generated along a one-dimensional (1D) channel of suspended graphene monolayer. Our results demonstrate that the zeroth PLLs of the strained structures are coupled together, exhibiting a serpentine pattern that snakes back and forth along the 1D suspended graphene monolayer. These results are verified theoretically by large-scale tight-binding calculations of the strained samples. Our result provides a new approach to realizing novel quantum states and to engineering the electronic properties of graphene by using localized PLLs as building blocks.

Realizing Valley-Polarized Energy Spectra in Bilayer Graphene Quantum Dots via Continuously Tunable Berry Phases

The Berry phase plays an important role in determining many physical properties of quantum systems. However, tuning the energy spectrum of a quantum system via Berry phase is comparatively rare because the Berry phase is usually a fixed constant. Here, we report the realization of an unusual valley-polarized energy spectra via continuously tunable Berry phases in Bernal-stacked bilayer graphene quantum dots. In our experiment, the Berry phase of electron orbital states is continuously tuned from about π to 2π by perpendicular magnetic fields. When the Berry phase equals π or 2π, the electron states in the two inequivalent valleys are energetically degenerate. By altering the Berry phase to noninteger multiples of π, large and continuously tunable valley-polarized energy spectra are realized. Our result reveals the Berry phase's essential role in valleytronics and the observed valley splitting, on the order of 10 meV at a magnetic field of 1 T, is about 100 times larger than Zeeman splitting for spin, shedding light on graphene-based valleytronics.

Terahertz tunable optically induced lattice in the magnetized monolayer graphene

The emergence of monolayer material has opened new avenue for manipulating light beyond the capability of traditional optics. However, controlling the terahertz (THz) wave with magnetized monolayer graphene based on multi-beam interference method is interesting but yet reported. In this article, we report an optically induced lattice with tunability in THz by interfering four plane waves in the magnetized monolayer graphene. We show that the optical properties of the induced optical lattice can be efficiently tuned by varying the optical parameter of the interference beams (i.e., the photon detuning and the Rabi frequency), resulting in both amplitude- and phase-type lattice. Based on Fraunhofer diffraction theory, it is found that the far-field diffraction efficiency is adjustable via varying the probe detuning. Moreover, it is also found that the probe field is diffracted into the high-order direction when the photon detuning is within the triangle-like anti-centrosymmetric region. Such a tunable THz lattice may provide a versatile tool for all-optical switching at the few photons level and paves the way for next generation high-speed wireless communication.

Oscillations of the Spacing between van Hove Singularities Induced by sub-Ångstrom Fluctuations of Interlayer Spacing in Graphene Superlattices

Physical properties of two-dimensional van der Waals (vdWs) structures depend sensitively on both stacking orders and interlayer interactions. Yet, in most cases studied to date, the interlayer interaction is considered to be a "static" property of the vdWs structures. Here we demonstrate that applying a scanning tunneling microscopy (STM) tip pulse on twisted bilayer graphene (TBG) can induce sub-Ångstrom fluctuations of the interlayer separation in the TBG, which are equivalent to dynamic vertical external pressure of about 10 GPa on the TBG. The sub-Ångstrom fluctuations of the interlayer separation result in large oscillations of the energy separations between two van Hove singularities (VHSs) in the TBG. The period of the oscillations of the VHSs spacing is extremely long, about 500-1000 sec, attributing to tip-induced local stress in the atomic-thick TBG. Our result provides an efficient method to tune and measure the physical properties of the vdWs structures dynamically.

Pseudo-magnetic field-induced slow carrier dynamics in periodically strained graphene

The creation of pseudo-magnetic fields in strained graphene has emerged as a promising route to investigate intriguing physical phenomena that would be unattainable with laboratory superconducting magnets. The giant pseudo-magnetic fields observed in highly deformed graphene can substantially alter the optical properties of graphene beyond a level that can be feasible with an external magnetic field, but the experimental signatures of the influence of such pseudo-magnetic fields have yet to be unveiled. Here, using time-resolved infrared pump-probe spectroscopy, we provide unambiguous evidence for slow carrier dynamics enabled by the pseudo-magnetic fields in periodically strained graphene. Strong pseudo-magnetic fields of ~100 T created by non-uniform strain in  graphene on nanopillars are found to significantly decelerate the relaxation processes of hot carriers by more than an order of magnitude. Our findings offer alternative opportunities to harness the properties of graphene enabled by pseudo-magnetic fields for optoelectronics and condensed matter physics.

The effect of strain-induced pseudo-magnetic fields on the optical properties of graphene has not been experimentally explored yet. Here, pseudo-magnetic fields reaching values of 100 T are shown to increase by more than an order of magnitude the relaxation lifetime of hot carriers in periodically strained graphene.

Anisotropic and gate-tunable valley filtering based on 8-Pmmn borophene

We propose a valley filter based on 8-Pmmn borophene which hosts two tilted Dirac cones. It is found that the application of a magnetic-electric barrier provided by a single ferromagnetic gate is sufficient to create valley-polarized current in 8-Pmmn borophene. The valley polarization of output current depends on the barrier orientation. Due to an intrinsic symmetry, it vanishes when the barrier orientation is along the tilted direction of Dirac cones. For the barrier orientation perpendicular to the tilted direction, the valley polarization for a realistic magnetic barrier can approach nearly 100% at proper Fermi energy and gate voltage. The remarkable valley contrast of conductance in this case is attributed to a new transmission resonance. The tilting of Dirac cones is essential for the predicted valley filtering. Our findings are helpful for valleytronic applications of two-dimensional materials with tilted Dirac cones.

Strongly Absorbing Nanoscale Infrared Domains within Strained Bubbles at hBN-Graphene Interfaces

Graphene has great potential for use in infrared (IR) nanodevices. At these length scales, nanoscale features, and their interaction with light, can be expected to play a significant role in device performance. Bubbles in van der Waals heterostructures are one such feature, which have recently attracted considerable attention, thanks to their ability to modify the optoelectronic properties of two-dimensional (2D) materials through strain. Here, we use scattering-type scanning near-field optical microscopy (sSNOM) to measure the nanoscale IR response from a network of variously shaped bubbles in hexagonal boron nitride (hBN)-encapsulated graphene. We show that within individual bubbles there are distinct domains with strongly enhanced IR absorption. The IR domain boundaries coincide with ridges in the bubbles, which leads us to attribute them to nanoscale strain domains. We further validate the strain distribution in the graphene by means of confocal Raman microscopy and vector decomposition analysis. This shows intricate and varied strain configurations, in which bubbles of different shape induce more bi- or uniaxial strain configurations. This reveals pathways toward future strain-based graphene IR devices.

Tunable Lattice Reconstruction, Triangular Network of Chiral One-Dimensional States, and Bandwidth of Flat Bands in Magic Angle Twisted Bilayer Graphene

The interplay between interlayer van der Waals interaction and intralayer lattice distortion can lead to structural reconstruction in slightly twisted bilayer graphene (TBG) with the twist angle being smaller than a characteristic angle θ_{c}. Experimentally, the θ_{c} is demonstrated to be very close to the magic angle (θ≈1.08°). Here we address the transition between reconstructed and unreconstructed structures of the TBG across the magic angle by using scanning tunneling microscopy (STM). Our experiment demonstrates that both structures are stable in the TBG around the magic angle. By using a STM tip, we show that the two structures can be changed to each other and a triangular network of chiral one-dimensional states hosted by domain boundaries can be switched on and off. Consequently, the bandwidth of the flat band, which plays a vital role in the emergent strongly correlated states in the magic angle TBG, is tuned. This provides an extra control knob to manipulate the exotic electronic states of the TBG near the magic angle.

Movable Valley Switch Driven by Berry Phase in Bilayer-Graphene Resonators

Berry phase, the geometric phase accumulated over a closed loop in parameter space during an adiabatic cyclic evolution, has been demonstrated to play an important role in many quantum systems since its discovery. In gapped Bernal bilayer graphene, the Berry phase can be continuously tuned from zero to 2π, which offers a unique opportunity to explore the tunable Berry phase on physical phenomena. Here, we report experimental observation of Berry-phase-induced valley splitting and crossing in movable bilayer-graphene p-n junction resonators. In our experiment, the resonators are generated by combining the electric field of a scanning tunneling microscope tip with the gap of bilayer graphene. A perpendicular magnetic field changes the Berry phase of the confined bound states in the resonators from zero to 2π continuously and leads to the Berry phase difference for the two inequivalent valleys in the bilayer graphene. As a consequence, we observe giant valley splitting and unusual valley crossing of the lowest bound states. Our results indicate that the bilayer-graphene resonators can be used to manipulate the valley degree of freedom in valleytronics.