Figure - available from: Nature
This content is subject to copyright. Terms and conditions apply.
![Structure of the Landau levels and derivation of the twist angle along a line scan
a, Bzac(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${B}_{z}^{{\rm{ac}}}(x)$$\end{document} versus Vbg for device A acquired along the dashed line in Fig. 3a. The top axis denotes ne/(ns(x)/4) for x = 0 and the separation between the yellow dashed lines describes the evolution of ns(x). The dispersive-band regions are marked in yellow. The signal in the flat bands is amplified seven times and multiplied by −1 for p doping such that incompressible strips are bright. b–d, The derived position-dependent ns(x) (b), θ(x) (c) and the charge disorder δnd(x) (d).](publication/341194869/figure/fig12/AS:888422310891528@1588827793874/Structure-of-the-Landau-levels-and-derivation-of-the-twist-angle-along-a-line-scan-a.png)
Structure of the Landau levels and derivation of the twist angle along a line scan a, Bzac(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${B}_{z}^{{\rm{ac}}}(x)$$\end{document} versus Vbg for device A acquired along the dashed line in Fig. 3a. The top axis denotes ne/(ns(x)/4) for x = 0 and the separation between the yellow dashed lines describes the evolution of ns(x). The dispersive-band regions are marked in yellow. The signal in the flat bands is amplified seven times and multiplied by −1 for p doping such that incompressible strips are bright. b–d, The derived position-dependent ns(x) (b), θ(x) (c) and the charge disorder δnd(x) (d).
Source publication
The recently discovered flat electronic bands and strongly correlated and superconducting phases in magic-angle twisted bilayer graphene (MATBG)1,2 crucially depend on the interlayer twist angle, θ. Although control of the global θ with a precision of about 0.1 degrees has been demonstrated1,2,3,4,5,6,7, little information is available on the distr...
Similar publications
Germanene, a two-dimensional honeycomb germanium crystal, is grown at graphene/Ag(111) and hexagonal boron nitride (h-BN)/Ag(111) interfaces by segregating germanium atoms. A simple annealing process in N2 or H2/Ar at ambient pressure leads to the formation of germanene, indicating that an ultrahigh-vacuum condition is not necessary. The grown germ...
Citations
... As such, we speculate that, when the twist angle is equal or close to 30 • , small fluctuations in the proximity-induced spin-orbit fields will dominate the SHE due to λ vZ approaching zero. These fluctuations can arise from ripples in the graphene flake [70] and nonuniform twisting across the sample [71]. Both of these will yield spatially varying spin-orbit couplings that can engender anomalous spin Hall responses [52,53]. ...
Atomically thin materials based on transition-metal dichalcogenides and graphene offer a promising avenue for unlocking the mechanisms underlying the spin Hall effect (SHE) in heterointerfaces. Here we develop a microscopic theory of the SHE for twisted van der Waals heterostructures that fully incorporates twisting and disorder effects and illustrate the critical role of symmetry breaking in the generation of spin Hall currents. We find that an accurate treatment of vertex corrections leads to a qualitatively and quantitatively different SHE than that obtained from the popular i η and ladder approximations. A pronounced oscillatory behavior of skew-scattering processes with twist angle θ is predicted, reflecting a nontrivial interplay of Rashba and valley-Zeeman effects and yields a vanishing SHE for θ = 30 ∘ and, for graphene- WSe 2 heterostructures, an optimal SHE for θ ≈ 17 ∘ . Our findings reveal disorder and broken symmetries as important knobs to optimize interfacial SHEs.
Published by the American Physical Society 2024
... We target twist angle uncertainty of 0.01°and heterostrain uncertainty of 0.1%, motivated by present limits on how well twist angle can be determined from transport measurements, how uniform the "best" samples appear to be over micron-scale lengths [17][18][19][20], and how small a change in these parameters influences electronic properties seen in transport. ...
Scanning probe techniques are popular, non-destructive ways to visualize the real space structure of Van der Waals moir\'es. The high lateral spatial resolution provided by these techniques enables extracting the moir\'e lattice vectors from a scanning probe image. We have found that the extracted values, while precise, are not necessarily accurate. Scan-to-scan variations in the behavior of the piezos which drive the scanning probe, and thermally-driven slow relative drift between probe and sample, produce systematic errors in the extraction of lattice vectors. In this Letter, we identify the errors and provide a protocol to correct for them. Applying this protocol to an ensemble of ten successive scans of near-magic-angle twisted bilayer graphene, we are able to reduce our errors in extracting lattice vectors to less than 1%. This translates to extracting twist angles with a statistical uncertainty less than 0.001{\deg} and uniaxial heterostrain with uncertainty on the order of 0.002%.
... An external magnetic field B, which preserves C 2z but breaks T, has been argued to be an alternative way to reveal the band topology [11,19,22], as evidenced by the experimental observations of correlated Chern insulating states (CCIs) with a finite Chern number t and extrapolating to a band filling s at B ¼ 0 [6][7][8]11,[13][14][15][16]18,19,[21][22][23]25,26]. Specifically, the most prominent sequence of CCIs has ðs; tÞ ¼ ð0; AE4Þ; AEð1; 3Þ; AEð2; 2Þ; AEð3; 1Þ, consistent with selective population of the aforementioned B ¼ 0 Chern AE 1 bands [11,21,22]. ...
Magic-angle twisted bilayer graphene is the best-studied physical platform featuring moiré potential-induced narrow bands with nontrivial topology and strong electronic correlations. Despite their significance, the Chern insulating states observed at a finite magnetic field—and extrapolating to a band filling s at zero field—remain poorly understood. Unraveling their nature is among the most important open problems in the province of moiré materials. Here, we present the first comprehensive study of interacting electrons in finite magnetic field while varying the electron density, twist angle, and heterostrain. Within a panoply of correlated Chern phases emerging at a range of twist angles, we uncover a unified description for the ubiquitous sequence of states with the Chern number t for ( s , t ) = ± ( 0 , 4 ) , ± ( 1 , 3 ) , ± ( 2 , 2 ) , and ± ( 3 , 1 ) . We also find correlated Chern insulators at unconventional sequences with s + t ≠ ± 4 , as well as with fractional s , and elucidate their nature.
Published by the American Physical Society 2024
... Such a kind of energy band yields distinguished low-energy excitations, which are in sharp contrast to those of conventional Dirac materials [1,2]. Besides, the RTG is notable for being free from twist-related defects [45][46][47][48][49][50][51][52][53] and possessing a much simpler band structure compared to other materials [43,54,55]. In consequence, the RTG has garnered significant in recent years [55][56][57][58][59][60][61]. ...
The effects of short-range fermion-fermion interactions on the low-energy properties of the rhombohedral trilayer graphene are comprehensively investigated by virtue of the momentum-shell renormalization group method. We take into account all one-loop corrections and establish the energy-dependent coupled evolutions of independent fermionic couplings that carry the physical information stemming from the interplay of various fermion-fermion interactions. With the help of the detailed numerical analysis, we notice that the ferocious competition among all fermion-fermion interactions can drive fermionic couplings to four distinct kinds of fixed points, dubbed $\textrm{FP}_{1}$, $\textrm{FP}_{2}$, $\textrm{FP}_{3}$ and $\textrm{FP}_{4}$, in the interaction-parameter space. Such fixed points principally dictate the fate of the system in the low-energy regime, which are always associated with some instabilities with specific symmetry breakings and thus accompanied by certain phase transitions. In order to judge the favorable states arising from the potential phase transitions, we bring out a number of fermion-bilinear source terms to characterize the underlying candidate states. By comparing their related susceptibilities, it is determined that the dominant states correspond to a spin-singlet superconductivity, a spin-triplet pair-density-wave, and a spin-triplet superconductivity for approaching the fixed points $\textrm{FP}_{1,3}$, $\textrm{FP}_{2}$, and $\textrm{FP}_{4}$, respectively. These results would be helpful to further reveal the low-energy properties of the rhombohedral trilayer graphene and analogous materials.
... Several single-gated local probe techniques have been devel- * mitali.banerjee@epfl.ch oped to enrich the understanding of band structure in two-dimensional materials, like scanning tunneling microscopy/spectroscopy (STM/STS) [4,8,[23][24][25][26][27], single electron transistor (SET) [28], planar tunneling junction [29] and nano-SQUID [30]. In addition to local probes, global measurements, including magneto-transport [5][6][7][8][9][10][11][12][13][14][15][19][20][21][22][23], nano Angle-Resolved Photoemission Spectroscopy (Nano-ARPES) [31,32], electronic compressibility [33], nano-infrared imaging [34] and Fourier transform infrared (FTIR) spectroscopy [35,36] are widely adopted. ...
... Although electronic compressibility measurements, nano-infrared imaging, Nano-SQUID-on-tip microscopy, and FTIR spectroscopy can investigate dual gate devices, these measurements each have their own drawbacks [30][31][32][34][35][36]. For example, the large beam spot (∼ 1 mm) used in FTIR spectroscopy, compared with the size of typical devices, makes the experiments substantially challenging. ...
Moir\'e systems featuring flat electronic bands exhibit a vast landscape of emergent exotic quantum states, making them one of the resourceful platforms in condensed matter physics in recent times. Tuning these systems via twist angle and the electric field greatly enhances our comprehension of their strongly correlated ground states. Here, we report a technique to investigate the nuanced intricacies of band structures in dual-gated multilayer graphene systems. We utilize the Landau levels of a decoupled monolayer graphene to extract the electric field-dependent bilayer graphene charge neutrality point gap. Then, we extend this method to analyze the evolution of the band gap and the flat bandwidth in twisted mono-bilayer graphene. The band gap maximizes at the same displacement field where the flat bandwidth minimizes, indicating the strongest electron-electron correlation, which is supported by directly observing the emergence of a strongly correlated phase. Moreover, we extract integer and fractional gaps to further demonstrate the strength of this method. Our technique gives a new perspective and paves the way for improving the understanding of electronic band structure in versatile flat band systems.
... The complexity of these rich and diverse phase spaces is influenced by numerous external factors. These factors include the twist-angle 20 , the twist-angle disorder 21 , the dielectric environment 22,23 , the relative alignment to the encapsulating layer used, in particular hexagonal boron nitride (hBN) 6 , and strain 21,24,25 . Some of these parameters, such as the selection of specific dielectric thickness ranges or controlling the alignment of graphene with the hBN, are integrated into the fabrication process. ...
... It also aids to avoid unexpected rapid movements or "jumps" in the stamp during the stacking, which can give rise to sudden stress release and so enhance bubble formation 33 (an example is shown in Suppl. Video 1), which we try minimize as much as possible since these significantly contribute to angle inhomogeneity 21 . However, if the flakes are too thin (i.e. ...
... In order to achieve the cleanest 2D interfaces and highest twist-angle homogeneity possible, we aim to avoid bubbles from forming during the stacking process as much as possible, as has been explained above. Bubbles significantly contribute to angle inhomogeneity and can even lead to the absence of the magic-angle condition in an area up to ca. 0.5 μm around the bubble 21 , and can induce quite strong strain field in the device. Typically, bubbles form during the stacking process mainly because of accumulation of dirt on the surfaces of the different 2D materials 32,45 or due to fast wave-front approaches, which can trap air along the interface 46 . ...
Magic-angle twisted bilayer graphene (MATBG) stands as one of the most versatile materials in condensed-matter physics due to its hosting of a wide variety of exotic phases while also offering convenient tunability. However, the fabrication of MATBG is still manual, and remains to be a challenging and inefficient process, with devices being highly dependent on specific fabrication methods, that often result in inconsistency and variability. In this work, we present an optimized protocol for the fabrication of MATBG samples, for which we use deterministic graphene anchoring to stabilize the twist-angle, and a careful bubble removal techniques to ensure a high twist-angle homogeneity. We use low-temperature transport experiments to extract the average twist-angle between pairs of leads. We find that up to 38 percent of the so fabricated devices show micrometer square sized regions with a twist-angle in the range 1.1 plus/minus 0.1 degrees, and a twist-angle variation of only 0.02 degrees, where in some instances such regions were up to 36 micrometer square large. We are certain that the discussed protocols can be directly transferred to non-graphene materials, and will be useful for the growing field of moire materials.
... By quantitatively reconstructing the magnetization, we determine the local thermodynamic gaps of the most robust FCI state at ν = −2/3, finding −2/3 ∆ as large as 7 meV. Spatial mapping of the charge density-and displacement field-tuned magnetic phase diagram further allows us to characterize sample disorder, which we find to be dominated by both inhomogeneity in the effective unit cell area [6] as well as inhomogeneity in the band edge offset and bound dipole moment. Our results highlight both the challenges posed by structural disorder in the study of twisted homobilayer moiré systems and the opportunities afforded by the remarkably robust nature of the underlying correlated topological states. ...
... This universal contribution occurs atop a non-universal background of spin and orbital magnetic moments arising from the filled electronic states in the sample bulk. Previous experiments have used nanoscale superconducting quantum interference device on tip (nSOT) [32,33] to detect orbital currents associated with the formation of integer quantum Hall states in monolayer graphene [6], quantum anomalous Hall states in both graphene and dichalcogenide moiré systems [34,35], and the non-topological orbital magnetization associated with orbitally polarized metallic states in both moiré and non-moiré graphene systems [36,37]. However, magnetic imaging of fractionalized phases has not been reported. ...
... The effective interlayer twist angle θ corresponding to the trajectory in Fig. 4a is plotted in Fig. 4b, while a map of the effective twist angle throughout the device is shown in Fig. 4c. Qualitatively, the twist angle map is reminiscent of twisted bilayer graphene [6] with regions of approximately uniform twist angle separated by domain walls where the effective twist angle changes suddenly. ...
In the absence of time reversal symmetry, orbital magnetization provides a sensitive probe of topology and interactions, with particularly rich phenomenology in Chern insulators where topological edge states carry large equilibrium currents. Here, we use a nanoscale superconducting sensor to map the magnetic fringe fields in twisted bilayers of MoTe$_2$, where transport and optical sensing experiments have revealed the formation of fractional Chern insulator (FCI) states at zero magnetic field. At a temperature of 1.6K, we observe oscillations in the local magnetic field associated with fillings $\nu=-1,-2/3,-3/5,-4/7$ and $-5/9$ of the first moir\'e hole band, consistent with the formation of FCIs at these fillings. By quantitatively reconstructing the magnetization, we determine the local thermodynamic gaps of the most robust FCI state at $\nu=-2/3$, finding $^{-2/3}\Delta$ as large as 7 meV. Spatial mapping of the charge density- and displacement field-tuned magnetic phase diagram further allows us to characterize sample disorder, which we find to be dominated by both inhomogeneity in the effective unit cell area as well as inhomogeneity in the band edge offset and bound dipole moment. Our results highlight both the challenges posed by structural disorder in the study of twisted homobilayer moir\'e systems and the opportunities afforded by the remarkably robust nature of the underlying correlated topological states.
... Two-dimensional (2D) electron systems subjected to magnetic fields form Landau levels with quantized topological orders, which is regarded as a basic model in modern quantum physics and hatched several scientific branches such as topological insulators [1][2][3][4], anyon collisions [5,6] and twistronics [7,8]. Whilst, recent advances of synthetic quantum materials highlighted sophisticated photonic systems to emulate the properties of electrons [9], such as dielectric photonic crystals [10][11][12], continuum photon fluids [13,14], and optical topology [15][16][17], merging the fermionic and bosonic systems and promoting physical frontier of quantum simulation [18]. ...
Topological orders emerge in both microscopic quantum dynamics and macroscopic materials as a fundamental principle to characterize intricate properties in nature with vital significance, for instance, the Landau levels of electron systems in magnetic field. Whilst, recent advances of synthetic photonic systems enable generalized concepts of Landau levels across fermionic and bosonic systems, extending the modern physical frontier. However, the controls of Landau levels of photons were only confined in complex artificial metamaterials or multifolded cavities. Here, we exploit advanced structured light laser technology and propose the theory of high-dimensional frequency-degeneracy, which enables photonic Landau level control in a linear open laser cavity with simple displacement tuning of intracavity elements. This work not only create novel structured light with new topological effects but also provides broad prospects for Bose-analogue quantum Hall effects and topological physics.
... Recent strides in condensed matter systems has suggested that the disorder can trigger a metal-insulator transition (MIT) and modify the conducting properties of MIT [22,23]. Moreover, in the twisted bilayer graphene, the disorder can change the band structure, magnetism, and superconductivity [24][25][26][27]. Thus, investigating the influence of disorder on the flatbands is much important for moiré physics and related applications. ...
Plenty of exotic phenomena in moiré superlattices arise from the emergence of flatbands, but their significance could be diminished by structural disorders that will significantly alter flatbands. Thus, unveiling the effects of disorder on moiré flatbands is crucial. In this work, we explore the disorder effects on two sets of flatbands in silicon-based mismatched moiré superlattices, where the level of disorder is controlled by varying the magnitude of random perturbations of the locations of silicon strips. The results reveal that, after ensemble averaging, the average spectral positions of the four flatbands exhibit stability despite variations in the degree of disorder. However, the δ-like density of states (DOS) related to flatbands in the perfect superlattice evolves into a finite-width envelope of high DOS. By increasing the level of disorder, the width of the DOS envelope increases accordingly. Particularly, we observe a fascinating contrast: the width of bandgap flatbands saturates after initial growth, while the width of dispersive-band-crossed flatbands exhibits a linear increase versus the disorder. This unveils fundamental differences in how flatbands respond to structural imperfections, offering crucial insights into their perturbation characteristics within moiré superlattices. Our work offers new perspectives on flatbands in partially disordered moiré superlattices.
... Despite such small differences in twist angles between 0.94°and 0.98°and between 1.12°and 1.16°, the FTIR measurements show notable energy shifts in the spectral peaks. Such a large sensitivity raises the question as to whether our spectra are skewed by twist-angle inhomogeneity commonly observed in TBG [43,44]. However, from measurements of the two-probe resistance between different contact pairs, we extract a twist-angle inhomogeneity within ±0.02°(Supplementary VIII A). ...
Twisted bilayer graphene (TBG) represents a highly tunable, strongly correlated electron system owed to its unique flat electronic bands. However, understanding the single-particle band structure alone has been challenging due to complex lattice reconstruction effects and a lack of spectroscopic measurements over a broad energy range. Here, we probe the band structure of TBG around the magic angle using infrared spectroscopy. Our measurements reveal spectral features originating from interband transitions whose energies are uniquely defined by the twist angle. By combining with quantum transport, we connect spectral features over a broad energy range (10-700 meV) spanning several superlattice minibands and track their evolution with twist angle. We compare our data with calculations of the band structures obtained via the continuum model and find good agreement only when considering a variation of interlayer/intralayer tunneling parameters with the twist angle. Our analysis suggests that the magic angle also shifts due to lattice relaxation, and is better defined for a wide angular range of 0.9°-1.1°. Our work provides spectroscopic insights into TBG's band structure and offers an optical fingerprint of the magic angle for screening heterostructures before nanofabrication.
