Symposium QT10—Emerging Phenomena in Moiré Materials

When two layers of two-dimensional materials are assembled with a relative twist, moiré patterns can arise, giving rise to a tremendous wealth of exotic phenomena. In this work, we consider twisting two triangular lattices hosting Dirac quantum spin liquids. The single-layer parent state is stable, since the proliferation of monopole operators, inducing conventional long-range magnetic or valence bond solid order, is symmetry forbidden. Conversely in the bilayer system, interlayer monopole tunneling is a symmetry-allowed relevant perturbation and can lead to ordered bilayer states. Adding a relative twist between the two layers reduces the stability of the ordered phases compared with the untwisted case and results in tunable moiré modulations of antiferromagnetic and valence bond solid order parameters.

In this talk I will discuss the latest research from my group on moiré quantum systems, including superconductivity, ferroelectricity and other correlated phenomena.

Van der Waals heterostructures (vdW-HS) offer a unique material platform with rich electronic and optical properties highly tunable via a wide selection of layer composition, strain, electric gating, and doping. The twist angle between two monolayers has emerged as a unique control knob to engineer the moiré superlattice formed by periodic variations of inter-layer atomic alignment. A type-II band alignment is typically found in a transition metal dichalcogenide hetero-bilayer, leading to the formation of interlayer excitons. [1] Using a low-energy continuum model, we theoretically separate two leading mechanisms that influence interlayer exciton radiative lifetimes. [2] The shift to indirect transitions in momentum space with an increasing twist angle and the energy modulation from the moiré potential both have a significant impact on interlayer exciton lifetimes. We further predict a distinct temperature dependence of interlayer exciton lifetimes in twisted bilayers (TBLs) with different twist angles. The results are compared to time-resolved PL measurements on a series of stacked MoSe₂ / WSe₂ TBLs with accurately controlled twist angles. Our studies are a first step toward using the twist angle of a vdW-HS to control exciton dynamics.

[1] P. Rivera, J. R. Schaibley, A. M. Jones, J. S. Ross, S. Wu, G. Aivazian, P. Klement, K. Seyler, G. Clark, N. J. Ghimire, J. Yan, D. G. Mandrus, W. Yao, and X. Xu, "Observation of long-lived interlayer excitons in monolayer MoSe₂–WSe₂ heterostructures", Nature Communications 6, 6242 (2015)
[2] J. Choi, M. Florian, A. Steinhoff, D. Erben, K. Tran, L. Sun, J. Quan, R. Claassen, S. Majumder, J. A. Hollingsworth, T. Taniguchi, K. Watanabe, K. Ueno, A. Singh, G. Moody, F. Jahnke, and X. Li, "Twist Angle-Dependent Interlayer Exciton Lifetimes in van der Waals Heterostructures", Phys. Rev. Lett. 126, 047401 (2021)

Heterobilayers made by stacking two monolayers of transition-metal dichalcogenide semiconductors feature a moiré pattern that spatially modulates the electronic band-structure, and with it the energies of interlayer excitons (ILXs) – electron-hole pairs bound across the interlayer gap. It is thus expected to confine ILXs in the moiré potential wells. This confinement is anticipated to support topological exciton transport [1], quantum emitter arrays [2], and to be significant in the exploration of many-body exciton-exciton interactions and condensed phases. However, so far this promise remains unfulfilled. On one hand, moiré patterns with large periods, that are predicted to confine ILXs, suffer from spatial inhomogeneities due to reconstruction and strain fields [3]. On the other hand, when the moiré period is smaller, it approaches the size of the ILX, making it unclear whether it confines the ILX or not. On top of that, there is a continuous debate about the valley configuration of the ILX-bound electron and hole (see [4,5] for examples).
In this talk, I will address these issues by looking at the momentum-space signatures of ILXs in the MoS₂/WSe₂ heterobilayer, using time- and angle-resolved photoemission spectroscopy. In an unprecedented measurement, the experiment reveals the momentum-space distributions of both the electron and hole components of the ILX. First, this clearly illustrates the momentum-direct character of this exciton. Second, accompanied by a theoretical analysis, these images yield a quantitative analysis of the real-space structure of the ILXs along all its defining coordinates: the relative electron-hole separation and the center-of-mass. Thus, we extract both the size and the surprisingly strong center-of-mass localization of the ILX by a moiré potential whose period is comparable to the ILX size. It thus shows that moiré patterns with short periods, which are periodic and homogeneous over large domains, are suitable to serve as platforms for ILX-based quantum studies. More generally, looking at the ILX momentum-space signatures would allow to reveal the ILX structure in other moiré systems, as well as to directly visualize the impact of many-body interactions on it.
References:
[1] F. Wu et al. PRL 118, 147401 (2017)
[2] H. Yu et al. Sci. Adv. 3, e1701696 (2017)
[3] L. J. McGilly et al. Nature Nanotech. 15, 580-584 (2020)
[4] B. Miller et al Nano Lett. 17, 5229-5237 (2017)
[5] A. T. Hanbicki et al. ACS Nano 12, 4719-4726 (2018)

Moiré superlattices can be created by stacking two-dimensional crystals with a small lattice mismatch or twist-angle between the layers. The associated moiré potential can give rise to new excitations in the system, such as moiré phonons: phonons from the individual layers downfolded to zone center through the periodicity of the moiré potential. Hydrostatic pressure can be a powerful tool to study and further tune those excitations. Here, we investigate the evolution of moiré phonons in a rotationally-aligned MoS₂/WSe₂ moiré heterostructure at high-pressures via Raman spectroscopy. Moiré phonons are revealed as satellite Raman peaks that appear exclusively in the heterostructure region,which are found to blueshift and increase in intensity under the application of pressure. By combining density functional theory and an effective model of the pressure-dependent moiré potential with zone folding analysis, we are able to assign the satellite peaks to downfolded MoS₂ optical phonons and accurately capture their evolution with pressure. Our theoretical analysis further illustrates the microscopic process that generates moiré phonons, which can be associated to an electron-phonon scattering process within the mini-Brillouin zone. Crucially, this connection to the electronic system establishes moiré phonons as a sensitive probe of the mini-band electronic structure in moiré systems. Our work may shed light on the interplay between moiré potentials, phonons and electron-phonon scattering in novel moiré systems.

In heterostructures of two-dimensional (2D) van der Waals semiconductors, the moiré superlattice reconstruction modulates the electronic and excitonic properties. The signature of moiré excitons in rotationally-aligned WS₂/WSe₂ is in the form of emergent intralayer excitonic resonances in the optical absorption [1,2]. Theoretically, the excitons corresponding to these new resonances are predicted to have a distinct character compared to the A exciton. The moiré excitons can exhibit spatially modulated Wannier character and an intralayer charge-transfer character. While the new resonances have been probed using optical techniques, variations within the moiré unit cell cannot be determined this way due to the optical diffraction limit . Low-loss electron energy loss spectroscopy (EELS) is a unique experimental technique that can measure the imaginary part of the dielectric constant and provide information similar to optical absorption, but with nanoscale spatial resolution. Here we use rotationally aligned WS₂/WSe₂ heterostructures as a model system to probe excitonic spectra variations within a moiré unit cell at 77K using low-loss EELS. We determine the variation of excitons on a length scale of about ~1-2 nm and correlate it with the different atomic stacking configurations within the moiré supercell using high-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM).
[1] C. Jin et al. Nature 567, 76-80 (2019)
[2] Y. Tang et al. Nature 579, 353–358 (2020)

Twisted and/or lattice mismatched van der Waals heterostructures have emerged as a kind of universal platform, capable of manifesting a wide range of electronic and optical properties spanning metallic, insulating, semiconducting, semimetal, magnetic, superconducting, and topological behaviours. Perhaps most remarkable is the opportunity to transform between these states within a single material platform - an exciting new degree of freedom that has been termed, twistronics. So far the vast majority of studies have reported static devices, where the moire pattern in a heterostructure is established by the lattice alignment during the fabrication process such that each device represents a single twist angle. However the weak interlayer bonding between van der Waals materials means that it is possible to realize devices where the twist angle can be dynamically varied post-fabrication. Here I will discuss our ongoing efforts to develop improved in-situ control with the goal towards realizing fully in-situ tunable devices structures. I will discuss recent progress in terms of mechanical rotation of individual monolayers, improved understanding of interfacial dissipation mechanisms, implementation of in-situ feedback of twist angle and new approaches towards developing non-mechanical control of the heterostructure geometry at the nanoscale. Prospects for realizing functional twistronic devices with real-time in-situ control will be discussed.

Stacking angle control in van der Waals layer assembly has enabled engineering moire superlattices with the tunable length scale. More importantly, interplay between intralayer and interlayer interaction induces periodic atomic modulation at the interface, creating commensurate domains and domain boundaries. We note that the domain boundaries can host emergent functionalities which is associated with the atomic displacement that occurs during the interface reconstruction process. For instance, recent theoretical investigation suggests that piezoelectricity can be induced along the domain boundaries in the twisted bilayer transition metal dichalcogenides. This attracts attention to the twisted bilayer transition metal dichalcogenides structure because of its potential application to low dimensional devices. However, magnitude as well as spatial distribution of strain in the domain wall depending on rotation angle, and atomic scale understanding of charge distribution in the domain boundary still needs progress.
In this study, we performed atomic scale structural investigation on the twisted bilayer transition metal dichalcogenides using aberration corrected scanning transmission electron microscopy (STEM). We measured the magnitudes of induced strain near the domain boundaries of reconstructed bilayers with varying stacking angle. We further analyzed internal electric field, and charge distribution in the reconstructed domains and domain walls with STEM. Our work elucidates the role of stacking angle on structural deformation and emergent functionalities induced by local atomic rearrangement that sensitively depends on the stacking angle.

ABC-stacked trilayer graphene/hexagonal boron nitride moiré superlattice (TLG/hBN) has emerged as a playground for correlated electron physics. A spectroscopy study of this system is, however, challenging due to the device configuration. In this talk, I will introduce our recent efforts on FTIR photocurrent spectroscopy measurements of dual-gated TLG/hBN. We observed strong gate-tunable optical transitions that originated from the moire flat band. At half-filling of the valence flat band, a broad absorption peak emerges at ~18 meV, indicating direct optical excitation across an emerging Mott gap. Furthermore, I will talk about a unique moire-enabled interlayer electron-phonon coupling in this system and its implications.

The realization of magic angle twisted bilayer graphene (TBG) reshaped our understanding of how complex many body physics can be generated in relatively simple materials with straightforward single particle physics. In particular, the superconducting states in TBG hold the possibility of being entirely interaction driven, although many works also point to a phonon-based origin. This difficulty in understanding the nature of the superconductivity extends to newly discovered superconducting graphene systems such as multilayer twisted graphene, normal Bernal-bilayer and rhombohedral-trilayer graphene. These systems all show some features that are suggestive of an unconventional nature, but the interpretations, thus far, are not conclusive. Getting a full picture of one system may help bring us closer to a unified understanding of superconductivity in all graphene systems. Motivated by this, I will discuss our recent efforts in developing a Josephson interferometry technique for moiré superlattices. Fabricated Josephson junctions using a s-wave superconductor have been crucial in identifying the d-wave pairing in cuprate superconductors because they are phase sensitive and can be used to controllably interfere the superconducting order parameter along different spatial directions. Here we report for the first time the realization of Josephson junctions between the TTG superconductors and a bulk 3D s-wave superconductor niobium nitride (NbN). We fabricate NbN contacts that are highly transparent as evidenced by conduction enhancement when TTG is tuned to the normal state, indicating Andreev reflection. When the TTG is tuned to be superconducting, we observe a supercurrent and a Fraunhofer pattern that arises from the junction with a magnetic field period that is distinct from that of the intrinsic Josephson junctions in TTG. More interestingly, the Fraunhofer pattern is significantly modulated when the TTG superconducting phase is tuned. With this successful realization, we are equipped to create corner junctions or SQUIDs that can interfere tunneling in different spatial directions and reveal the pairing symmetry of the TTG superconducting order parameter.

In 1964, Zak's discovery of the magnetic translation group demonstrated the possibility of reentrant electronic phases when the flux through a single unit cell is 2π. For the first time, the large unit cell of twisted bilayer graphene (TBG) has made it possible to test Zak's prediction in a real material. We use a newly developed gauge-invariant formalism to determine the exact single-particle band structure, topology, and correlated insulator states of magic angle TBG at 25T. We find that the characteristic flat bands reemerge at 2π flux, but, due to the magnetic field breaking C₂T, they split and acquire nonzero Chern number. We then show that reentrant correlated insulators reappear at 2π flux driven by the Coulomb interaction, and we predict the characteristic Landau fans from their excitation spectrum. Initial experiments are consistent with these predictions. Finally, we conjecture that superconductivity can also be re-entrant at 2π flux due to an emergent Hofstadter symmetry.

In a flat band superconductor, the charge carriers’ group velocity vF is extremely slow, quenching their kinetic energy. The emergence of superconductivity thus appears paradoxical, as conventional BCS theory implies a vanishing coherence length, superfluid stiffness, and critical current. Here, using twisted bilayer graphene (tBLG), we explore the profound effect of vanishingly small vF in a Dirac superconducting flat band system. Using Schwinger-limited non-linear transport studies, we demonstrate an extremely slow vF ~ 1000 m/s for filling fraction n between -1/2 and -3/4 of the moiré superlattice. In the superconducting state, the same velocity limit constitutes a new limiting mechanism for the critical current, analogous to a relativistic superfluid. Importantly, our measurement of superfluid stiffness, which controls the superconductor’s electrodynamic response, shows that it is not dominated by the kinetic energy, but instead by the interaction-driven superconducting gap, consistent with recent theories on a quantum geometric contribution. We find evidence for small pairs, characteristic of the BCS to Bose-Einstein condensation (BEC) crossover, with an unprecedented ratio of the superconducting transition temperature to the Fermi temperature exceeding unity. Our results places tBLG in the regime of very strong coupling superconductivity, and underscores the important role played by quantum geometry in flat-band systems.

The properties of low energy charge fluctuations in twisted bilayer graphene, and their role in superconducting pairing are analyzed. These damped collective modes are modified by Coulomb interactions and by longitudinal phonons, and they define basic excitations of twisted bilayer graphene and related materials. Charge fluctuations, modified by phonons and by moiré Umklapp processes, lead to a net attraction for small momentum transfers, and to superconductivity. The role of these excitations in other graphene arrangements is also studied.

Charge carriers in magic angle graphene come in eight flavors described by a combination of their spin, valley, and sublattice polarizations. When the inversion and time reversal symmetries are broken by the substrate or by strong interactions, eight low-energy electronic flat bands are formed, each accommodating a single flavor, which can be filled sequentially. Due to their non-trivial band topology and Berry curvature, each of the bands is classified by a topological Chern number, leading to the quantum anomalous Hall and Chern insulator states at integer fillings ν of the bands. It has been recently predicted, however, that depending on the local atomic-scale arrangements of the graphene and the encapsulating hBN lattices, rather than being a global topological invariant, the Chern number C may become position dependent, altering transport and magnetic properties of the itinerant electrons. Using scanning superconducting quantum interference device on a tip (SQUID-on-tip), we directly image the nanoscale Berry-curvature-induced equilibrium orbital magnetism, the polarity of which is governed by the local Chern number, and detect its two constituent components associated with the drift and the self-rotation of the electronic wave packets. At ν=1, we observe local zero-field valley-polarized Chern insulators forming a mosaic of microscopic patches of C=-1, 0, or 1, governed by the local sublattice polarization, consistent with predictions. Upon further filling, we find abrupt recondensation of electrons from valley K to K', which leads to irreversible flips of the local Chern number and magnetization and to the formation of valley domain walls giving rise to hysteretic global anomalous Hall resistance. The findings shed new light on the structure and dynamics of topological phases and call for exploration of the controllable formation of flavor domain walls and their utilization in twistronic devices.

Semiconductor moiré materials provide a highly tunable platform for studies of electron correlation. Correlation-driven phenomena, including the Mott insulator, generalized Wigner crystals and continuous Mott transition, have been demonstrated. In this talk, I will discuss how to introduce band topology by applying an out-plane electric field in AB-stacked MoTe₂/WSe₂ moiré heterobilayers. A quantum anomalous Hall (QAH) effect at half band filling (corresponding to one particle per moiré unit cell) is observed. I will discuss the nature of the QAH state based on transport and spectroscopy studies. Our results establish semiconductor moiré materials as a versatile system for exploring the rich phenomenology arising from the combined influence of strong correlation and topology.

We calculate STM signatures of correlated ground-states at integer filling of the magic angle twisted bilayer graphene narrow bands. First, we compute the fully-interacting TBG spectral function at ±4 electrons/moiré unit cell and show that it can be used to experimentally validate the strong-coupling approach. Although variation exists in the data, we find experimental evidence for the strong-coupling regime. For all other integer fillings of the flat bands, we consider the spatial features of the corresponding spectral functions of many states in the large degenerate ground-state manifold, and assess the possibility of Kekulé distortion (KD) emerging at the graphene lattice scale. Remarkably, we find that coupling the two opposite graphene valleys in the intervalley-coherent (IVC) TBG insulators does not always result in KD. As an example, we show that the K-IVC state and all its nonchiral U(4) rotations do not exhibit any KD, while T-IVC does. We analyze 14 different many-body correlated states and show that their combined STM/Chern number signal can be used to uniquely determine the nature of the many-body ground-state. Their STM signal and features are obtained over a large range of energies and model parameters.

I will describe a strong coupling approach to flavor-ordered insulators and superconductivity in twisted bilayer graphene (TBG), in which the quantum geometry of the electronic bands plays a central role. I will discuss recent numerical simulations on realistic models of twisted bilayer graphene, involving >100 electrons, which confirm detailed aspects of this theory. An important outcome of this theory is the prediction of other platforms - the alternating twist multilayer graphene along with their associated magic angles - that retain the essential ingredients. I will highlight new features that are expected in these multilayer structures and compare them with recent experiments. Finally, I will describe how the unique quantum geometry of TBG makes it an ideal platform to realize fractional Chern insulators, and compare theory with new experimental observations.

Band states in graphene multilayers have a 4-fold spin x valley flavor degeneracy. Recent experiments have demonstrated that many-electron ground states of graphene multilayers that break flavor degeneracy without breaking translational symmetry are common when correlations are strong in the absence of a magnetic field, just as they are in the quantum Hall regime. I will discuss properties of these flavor ferromagnets, commenting on the Fermi surface reconstructions [1] and soft collective modes [2] that they produce, on the influence of dynamic screening of the long-range Coulomb interaction on the pattern of broken symmetries [3], and on the relationship to superconductivity [4].

[1] Xie M, MacDonald AH. Weak-Field Hall Resistivity and Spin-Valley Flavor Symmetry Breaking in Magic-Angle Twisted Bilayer Graphene. Physical Review Letters. 2021 Nov 2;127(19):196401.
[2] Kumar A, Xie M, MacDonald AH. Lattice collective modes from a continuum model of magic-angle twisted bilayer graphene. Physical Review B. 2021 Jul 8;104(3):035119.
[3] Zhu Jihang, Torre Iacoppo, Polini Marco, MacDonald AH. GW correlations and flavor ferromagnetism in magic-angle twisted bilayer graphene. (in preparation).
[4] Huang C, Wei N, Qin W, MacDonald AH. Pseudospin Paramagnons and the Superconducting Dome in Magic Angle Twisted Bilayer Graphene. arXiv preprint arXiv:2110.13351. 2021 Oct 26.

Two-dimensional (2D) moiré materials, formed for example by inducing a relative twist between layers of a bilayer material, exhibit a wide range of exotic properties that are sensitive to the precise structure of the material, both in-plane within the layers and out-of-plane between the layers. For example, the stacking sequence of trilayer graphene enables the electronic character of the material to be controlled, varying between a semimetal and semiconductor.^[1] More strikingly, bilayer graphene undergoes in-plane structural reconstructions when small twist angles are present between the layers, significantly altering the electronic state of the material,^[2] which even transitions to a superconducting state at a specific “magic” angle.^[3] The strong dependence of the properties of these materials on their structure makes them highly tunable, but also means a detailed understanding of the relationships between tunable parameters and resulting structural configurations is necessary to do so purposefully. In addition, real 2D moiré systems can contain heterogeneities in their structure that modify their properties. The ability to determine the precise structural arrangement of real 2D moiré systems with high spatial resolution is therefore key to accurately controlling their exotic properties.

While several techniques to measure the structural properties of 2D moiré materials exist, such as scanning tunneling microscopy (STM),^[4] transmission electron microscopy (TEM),^[5-6] and scanning transmission electron microscopy (STEM),^[5,7-8] no conventional technique can simultaneously measure both structural distortions with nanometer resolution and interlayer spacings in these materials. To enable the properties of real 2D moiré materials to be accurately tuned, a technique is therefore still needed that can directly and precisely reveal in-plane structural information at high spatial resolution and provide out-of-plane interlayer spacing information.

In this contribution, we will describe a recently developed technique, interferometric four-dimensional scanning transmission electron microscopy (4D-STEM), that utilizes Bragg interference to reveal the relative positions of atoms in separate layers of few-layered 2D moiré materials.^[9] We use experiments as well as paired density functional theory and multislice electron scattering calculations^[10] to demonstrate how this technique enables pm-scale in-plane structural distortions to be mapped with nanometer resolution in twisted bilayer and trilayer graphene. In addition, we show how interlayer spacing information and therefore determination of the stacking sequence of trilayer materials can be determined both directly and indirectly in certain cases. By providing the ability to directly and precisely map local structural distortions in 2D moiré materials with high spatial resolution, as well as provide information about interlayer spacings, the interferometric 4D-STEM technique presented here will enable the exotic properties of these materials to be more accurately tuned.


References:

[1] Z. Gao et al., Nat. Commun. 2020, 11 (1), 546.
[2] H. Yoo et al., Nat. Mater. 2019, 18 (5), 448–453.
[3] Y. Cao et al. Nature 2018, 556 (7699), 43–50.
[4] R. Decker et al., Nano Lett. 2011, 11 (6), 2291–2295.
[5] J.S. Alden et al., Proc. Natl. Acad. Sci. 2013, 110 (28), 11256 LP – 11260.
[6] S.H. Sung et al., Phys. Rev. Mater. 2019, 3 (6), 64003.
[7] R. Ishikawa et al., Sci. Rep. 2016, 6 (1), 21273.
[8] J. Lin et al., Nano Lett. 2013, 13 (7), 3262–3268.
[9] M.J. Zachman et al., Small 2021, 17, 2100388
[10] J. Madsen and T. Susi, Open Res. Eur. 2021, 1:24.