arXiv subjects

Mitchell Luskin

Publications and source records attributed to Mitchell Luskin.

At least 37 records · Page 2Linked to original sources

arXiv2022

Seeing moiré: convolutional network learning applied to twistronics

Moiré patterns made of two-dimensional (2D) materials represent highly tunable electronic Hamiltonians, allowing a wide range of quantum phases to emerge in a single material. Current modeling techniques for moiré electrons requires significant technical work specific to each material, impeding large-scale searches for useful moiré materials. In order to address this difficulty, we have developed a material-agnostic machine learning approach and test it here on prototypical one-dimensional (1D) moiré tight-binding models. We utilize the stacking dependence of the local density of states (SD-LDOS) to convert information about electronic bandstructure into physically relevant images. We then train a neural network that successfully predicts moiré electronic structure from the easily computed SD-LDOS of aligned bilayers. This network can satisfactorily predict moiré electronic structures, even for materials that are not included in its training data.

cond-mat.mes-hall↗

arXiv2022

Non-Abelian topological defects and strain mapping in 2D moiré materials

We present a general method to analyze the topological nature of the domain boundary connectivity that appeared in relaxed moiré superlattice patterns at the interface of 2-dimensional (2D) van der Waals (vdW) materials. At large enough moiré lengths, all moiré systems relax into commensurated 2D domains separated by networks of dislocation lines. The nodes of the 2D dislocation line network can be considered as vortex-like topological defects. We find that a simple analogy to common topological systems with an $S^1$ order parameter, such as a superconductor or planar ferromagnet, cannot correctly capture the topological nature of these defects. For example, in twisted bilayer graphene, the order parameter space for the relaxed moiré system is homotopy equivalent to a punctured torus. Here, the nodes of the 2D dislocation network can be characterized as elements of the fundamental group of the punctured torus, the free group on two generators, endowing these network nodes with non-Abelian properties. Extending this analysis to consider moiré patterns generated from any relative strain, we find that antivortices occur in the presence of anisotropic heterostrain, such as shear or anisotropic expansion, while arrays of vortices appear under twist or isotropic expansion between vdW materials. Experimentally, utilizing the dark field imaging capability of transmission electron microscopy (TEM), we demonstrate the existence of vortex and antivortex pair formation in a moiré system, caused by competition between different types of heterostrains in the vdW interfaces. We also present a methodology for mapping the underlying heterostrain of a moiré structure from experimental TEM data, which provides a quantitative relation between the various components of heterostrain and vortex-antivortex density in moiré systems.

cond-mat.mes-hall↗

arXiv2021

Existence of the first magic angle for the chiral model of bilayer graphene

We consider the chiral model of twisted bilayer graphene introduced by Tarnopolsky-Kruchkov-Vishwanath (TKV). TKV have proved that for inverse twist angles $α$ such that the effective Fermi velocity at the moiré $K$ point vanishes, the chiral model has a perfectly flat band at zero energy over the whole Brillouin zone. By a formal expansion, TKV found that the Fermi velocity vanishes at $α\approx .586$. In this work, we give a proof that the Fermi velocity vanishes for at least one $α$ between $.57$ and $.61$ by rigorously justifying TKV's formal expansion of the Fermi velocity over a sufficiently large interval of $α$ values. The idea of the proof is to project the TKV Hamiltonian onto a finite dimensional subspace, and then expand the Fermi velocity in terms of explicitly computable linear combinations of modes in the subspace, while controlling the error. The proof relies on two propositions whose proofs are computer-assisted, i.e., numerical computation together with worst-case estimates on the accumulation of round-off error which show that round-off error cannot possibly change the conclusion of the computation. The propositions give a bound below on the spectral gap of the projected Hamiltonian, an Hermitian $80 \times 80$ matrix whose spectrum is symmetric about $0,$ and verify that two real 18th order polynomials, which approximate the numerator of the Fermi velocity, take values with definite sign when evaluated at specific values of $α$. Together with TKV's work our result proves existence of at least one perfectly flat band of the chiral model.

math-ph↗

arXiv2021

Gate-tunable Veselago Interference in a Bipolar Graphene Microcavity

The relativistic charge carriers in monolayer graphene can be manipulated in manners akin to conventional optics (electron-optics): angle-dependent Klein tunneling collimates an electron beam (analogous to a laser), while a Veselago refraction process focuses it (analogous to an optical lens). Both processes have been previously investigated, but the collimation and focusing efficiency have been reported to be relatively low even in state-of-the-art ballistic pn-junction devices. These limitations prevented the realization of more advanced quantum devices based on electron-optical interference, while understanding of the underlying physics remains elusive. Here, we present a novel device architecture of a graphene microcavity defined by carefully-engineered local strain and electrostatic fields. We create a controlled electron-optic interference process at zero magnetic field as a consequence of consecutive Veselago refractions in the microcavity and provide direct experimental evidence through low-temperature electrical transport measurements. The experimentally observed first-, second-, and third-order interference peaks agree quantitatively with the Veselago physics in a microcavity. In addition, we demonstrate decoherence of the interference by an external magnetic field, as the cyclotron radius becomes comparable to the interference length scale. For its application in electron-optics, we utilize Veselago interference to localize uncollimated electrons and characterize its contribution in further improving collimation efficiency. Our work sheds new light on relativistic single-particle physics and provides important technical improvements toward next-generation quantum devices based on the coherent manipulation of electron momentum and trajectory.

cond-mat.mes-hall↗

arXiv2020

Correlated Insulating States and Transport Signature of Superconductivity in Twisted Trilayer Graphene Moiré of Moiré Superlattices

Layers of two-dimensional materials stacked with a small twist-angle give rise to beating periodic patterns on a scale much larger than the original lattice, referred to as a moiré superlattice. When the stacking involves more than two layers with independent twist angles between adjacent layers, it generates moiré of moiré superlattices, with multiple length scales that control the system's behavior. Here we demonstrate these effects of a high-order moiré superlattice in twisted trilayer graphene with two consecutive small twist angles. We report correlated insulating states near the half filling of the moiré of moiré superlattice at an extremely low carrier density (~1010 cm-2), near which we also report a zero-resistance transport behavior typically expected in a 2D superconductor. Moreover, the temperature dependence of the measured resistances at full-occupancy (v = -4 and v = 4) states are semi-metallic, distinct from the insulating behavior of twisted bilayer systems, providing the first demonstration of emergent correlated transport behaviors from continuous, non-isolated higher-order moiré flat bands. Our findings shed new insights into the microscopic mechanisms of moiré correlated states and provide the impetus for future studies on this material platform, such as the demonstration of phase coherence and Meissner-like effect.

cond-mat.mes-hall↗

arXiv2020

Twisted Trilayer Graphene: a Precisely Tunable Platform for Correlated Electrons

We introduce twisted trilayer graphene (tTLG) with two independent twist angles as an ideal system for the precise tuning of the electronic interlayer coupling to maximize the effect of correlated behaviors. As established by experiment and theory in the related twisted bilayer graphene system, van Hove singularities (VHS) in the density of states can be used as a proxy of the tendency for correlated behaviors. To explore the evolution of VHS in the twist-angle phase space of tTLG, we present a general low-energy electronic structure model for any pair of twist angles. We show that the basis of the model has infinite dimensions even at a finite energy cutoff and that no Brillouin zone exists even in the continuum limit. Using this model, we demonstrate that the tTLG system exhibits a wide range of magic angles at which VHS merge and the density of states has a sharp peak at the charge-neutrality point through two distinct mechanisms: the incommensurate perturbation of twisted bilayer graphene's flat bands or the equal hybridization between two bilayer moiré superlattices.

cond-mat.mes-hall↗

arXiv2020

Modeling and Computation of Kubo Conductivity for 2D Incommensurate Bilayers

This paper presents a unified approach to the modeling and computation of the Kubo conductivity of incommensurate bilayer heterostructures at finite temperature. Firstly, we derive an expression for the large-body limit of Kubo-Greenwood conductivity in terms of an integral of the conductivity function with respect to a current-current correlation measure. We then observe that the incommensurate structure can be exploited to decompose the current-current correlation measure into local contribution and deduce an approximation scheme which is exponentially convergent in terms of domain size. Secondly, we analyze the cost of computing local conductivities via Chebyshev approximation. Our main finding is that if the inverse temperature $β$ is sufficiently small compared to the inverse relaxation time $η$, namely $β\lesssim η^{-1/2}$, then the dominant computational cost is $\mathcal{O}\bigl(η^{-3/2}\bigr)$ inner products for a suitably truncated Chebyshev series, which significantly improves on the $\mathcal{O}\bigl(η^{-2}\bigr)$ inner products required by a naive Chebyshev approximation. Thirdly, we propose a rational approximation scheme for the low temperature regime $η^{-1/2} \lesssim β$, where the cost of the polynomial method increases up to $\mathcal{O}\bigl(β^2\bigr),$ but the rational scheme scales much more mildly with respect to $β$.

math.NA↗

arXiv2020

Nonlinear eigenvalue problems for coupled Helmholtz equations modeling gradient-index graphene waveguides

We discuss a quartic eigenvalue problem arising in the context of an optical waveguiding problem involving atomically thick 2D materials. The waveguide configuration we consider consists of a gradient-index (spatially dependent) dielectric equipped with conducting interior interfaces. This leads to a quartic eigenvalue problem with mixed transverse electric and transverse magnetic modes, and strongly coupled electric and magnetic fields. We derive a weak formulation of the quartic eigenvalue problem and introduce a numerical solver based on a quadratification approach in which the quartic eigenvalue problem is transformed to a spectrally equivalent companion problem. We verify our numerical framework against analytical solutions for prototypical geometries. As a practical example, we demonstrate how an improved quality factor (defined by the ratio of the real and the imaginary part of the computed eigenvalues) can be obtained for a family of gradient-index host materials with internal conducting interfaces. We outline how this result lays the groundwork for solving related shape optimization problems.

physics.comp-ph↗

arXiv2020

Finite-size effects in wave transmission through plasmonic crystals: A tale of two scales

We study optical coefficients that characterize wave propagation through layered structures called plasmonic crystals. These consist of a finite number of stacked metallic sheets embedded in dielectric hosts with a subwavelength spacing. By adjustment of the frequency, spacing, number as well as geometry of the layers, these structures may exhibit appealing transmission properties in a range of frequencies from the terahertz to the mid-infrared regime. Our approach uses a blend of analytical and numerical methods for the distinct geometries with infinite, translation invariant, flat sheets and nanoribbons. We describe the transmission of plane waves through a plasmonic crystal in comparison to an effective dielectric slab of equal total thickness that emerges from homogenization, in the limit of zero interlayer spacing. We demonstrate numerically that the replacement of the discrete plasmonic crystal by its homogenized counterpart can accurately capture a transmission coefficient akin to the extinction spectrum, even for a relatively small number of layers. We point out the role of a geometry-dependent corrector field, which expresses the effect of subwavelength surface plasmons. In particular, by use of the corrector we describe lateral resonances inherent to the nanoribbon geometry.

cond-mat.mes-hall↗

arXiv2020

Dipole excitation of collective modes in viscous two-dimensional electron systems

We describe the structure of the time-harmonic electromagnetic field of a vertical Hertzian electric dipole source radiating over an infinite, translation invariant two-dimensional electron system. Our model for the electron flow takes into account the effects of shear and Hall viscosities as well as an external static magnetic field perpendicular to the sheet. We identify two wave modes, namely, a surface plasmon and a diffusive mode. In the presence of an external static magnetic field, the diffusive mode combines the features of both the conventional and Hall diffusion and may exhibit a negative group velocity. In our analysis, we solve exactly a boundary value problem for the time-harmonic Maxwell equations coupled with linearized hydrodynamic equations for the flat, two-dimensional material. By numerically evaluating the integrals for the electromagnetic field on the sheet, we find that the plasmon contribution dominates in the intermediate-field region of the dipole source. In contrast, the amplitude of the diffusive mode reaches its maximum value in the near-field region, and quickly decays with the distance from the source. We demonstrate that the diffusive mode can be distinguished from the plasmon in the presence of the static magnetic field, when the highly oscillatory plasmon is gapped and tends to disappear.

cond-mat.mes-hall↗

arXiv2020

Duality between atomic configurations and Bloch states in twistronic materials

The relative orientation (twist) of successive layers of stacked two-dimensional (2D) materials creates variations in the interlayer atomic registry. The variations often form a super lattice, called a moiré pattern, which can alter electronic properties. In this work we introduce a classification of the single-particle electronic structures that can occur in twisted stacks of 2D layers by characterizing them as "moiré molecules" or "moiré crystals". The molecules generate localized electronic states and moiré flat bands, while the crystals are sometimes unconventional and produce electronic banding in the configuration basis. The underpinning of this classification is the duality between interlayer configuration and monolayer Bloch momentum in moiré Hamiltonians. We apply this understanding to diagrams of local electron density in untwisted geometries to produce intuitive and quantitative predictions of twistronic properties. We provide a conceptual introduction to this framework through a one-dimensional model, and then apply it to 2D twisted bilayers of the semi-metal graphene, and of MoS$_2$, a representative material of the transition metal dichalcogenide (TMDC) family of semiconductors. This level of thorough understanding of twistronic phenomena is vital in the search for new material platforms for localized moiré electrons.

cond-mat.mes-hall↗

arXiv2020

Homogenization of hydrodynamic transport in Dirac fluids

Large-scale electrical and thermal currents in ordinary metals are well approximated by effective medium theory: global transport properties are governed by the solution to homogenized coupled diffusion equations. In some metals, including the Dirac fluid of nearly charge neutral graphene, microscopic transport is not governed by diffusion, but by a more complicated set of linearized hydrodynamic equations, which form a system of degenerate elliptic equations coupled with the Stokes equation for fluid velocity. In sufficiently inhomogeneous media, these hydrodynamic equations reduce to homogenized diffusion equations. We re-cast the hydrodynamic transport equations as the infimum of a functional over conserved currents, and present a functional framework to model and compute the homogenized diffusion tensor relating electrical and thermal currents to charge and temperature gradients. We generalize to this system two well-known results in homogenization theory: Tartar's proof of local convergence to the homogenized theory in periodic and highly oscillatory media, and sub-additivity of the above functional in random media with highly oscillatory, stationary and ergodic coefficients.

math.AP↗

arXiv2020

Moiré of Moiré: Modeling Mechanical Relaxation in Incommensurate Trilayer van der Waals Heterostructures

The incommensurate stacking of multi-layered two-dimensional materials is a challenging problem from a theoretical perspective and an intriguing avenue for manipulating their physical properties. Here we present a multi-scale model to obtain the mechanical relaxation pattern of twisted trilayer van der Waals (vdW) heterostructures with two independent twist angles, a generally incommensurate system without a supercell description. We adopt the configuration space as a natural description of such incommensurate layered materials, based on the local environment of atomic positions, bypassing the need for commensurate approximations. To obtain the relaxation pattern, we perform energy minimization with respect to the relaxation displacement vectors. We use a continuum model in combination with the Generalized Stacking Fault energy to describe the interlayer coupling, obtained from first-principles calculations based on Density Functional Theory. We show that the relaxation patterns of twisted trilayer graphene and $\mathrm{WSe_2}$ are "moiré of moiré", as a result of the incommensurate coupling two bilayer moiré patterns. We also show that, in contrast to the symmetry-preserving in-plane relaxation in twisted bilayers, trilayer relaxation can break the two-fold rotational symmetry about the xy-plane when the two twist angles are equal.

cond-mat.mes-hall↗

arXiv2019

Efficient Computation of Kubo Conductivity for Incommensurate 2D Heterostructures

Here we introduce a numerical method for computing conductivity via the Kubo Formula for incommensurate 2D bilayer heterostructures using a tight-binding framework. We begin with deriving the momentum space formulation and Kubo Formula from the real space tight-binding model using the appropriate Bloch transformation operator. We further discuss the resulting algorithm along with its convergence rate and computation cost in terms of parameters such as relaxation time and temperature. In particular, we show that for low frequencies, low temperature, and long relaxation times conductivity can be computed very efficiently using momentum space for a wide class of materials. We then demonstrate our method by computing conductivity for twisted bilayer graphene (tBLG) for small twist angles.

cond-mat.mes-hall↗

arXiv2019

Nonretarded edge plasmon-polaritons in anisotropic two-dimensional materials

By an integral equation approach to the time-harmonic classical Maxwell equations, we describe the dispersion in the nonretarded frequency regime of the edge plasmon-polariton (EPP) on a semi-infinite flat sheet. The sheet has an arbitrary, physically admissible, tensor valued and spatially homogeneous conductivity, and serves as a model for a family of two-dimensional conducting materials. We formulate a system of integral equations for the electric field tangential to the sheet in a homogeneous and isotropic ambient medium. We show how this system is simplified via a length scale separation. This view entails the quasi-electrostatic approximation, by which the tangential electric field is replaced by the gradient of a scalar potential, $φ$. By the Wiener-Hopf method, we solve an integral equation for $φ$ in some generality. The EPP dispersion relation comes from the elimination of a divergent limiting Fourier integral for $φ$ at the edge. We connect the existence, or lack thereof, of the EPP dispersion relation to the index for Wiener-Hopf integral equations, an integer of topological character. We indicate that the values of this index may express an asymmetry due to the material anisotropy in the number of wave modes propagating on the sheet away from the edge with respect to the EPP direction of propagation. We discuss extensions such as the setting of two semi-infinite, coplanar sheets. Our theory forms a generalization of the treatment by Volkov and Mikhailov (1988 Sov. Phys. JETP 67 1639).

math-ph↗

arXiv2019

Energy minimization of 2D incommensurate heterostructures

We derive and analyze a novel approach for modeling and computing the mechanical relaxation of incommensurate 2D heterostructures. Our approach parametrizes the relaxation pattern by the compact local configuration space rather than real space, thus bypassing the need for the standard supercell approximation and giving a true aperiodic atomistic configuration. Our model extends the computationally accessible regime of weakly coupled bilayers with similar orientations or lattice spacing, for example materials with a small relative twist where the widely studied large-scale moire patterns arise. Our model also makes possible the simulation of multi-layers for which no interlayer empirical atomistic potential exists, such as those composed of MoS2 layers, and more generally makes possible the simulation of the relaxation of multi-layer heterostructures for which a planar moire pattern does not exist.

physics.comp-ph↗

arXiv2019

Switchable and unidirectional plasmonic beacons in hyperbolic 2D materials

In hyperbolic 2D materials, energy is channeled to their deep subwavelength polaritonic modes via four narrow beams. Here we consider the launching of surface polaritons in the hyperbolic 2D materials and demonstrate that efficient uni-directional excitation is possible with an elliptically polarized electric dipole, with the optimal choice of dipole ellipticity depending on the materials optical constants. The selection rules afforded by the choice of dipole polarization allow turning off up to two beams, and even three if the dipole is placed close to an edge. This makes the dipole a directionally switchable beacon for the launching of sub-difractional polaritonic beams, a potential logical gate. We develop an analytical approximation of the excitation process which describes the results of the numerical simulations well and affords a simple physical interpretation.

cond-mat.mes-hall↗

arXiv2019

Homogenization of plasmonic crystals: Seeking the epsilon-near-zero effect

By using an asymptotic analysis and numerical simulations, we derive and investigate a system of homogenized Maxwell's equations for conducting material sheets that are periodically arranged and embedded in a heterogeneous and anisotropic dielectric host. This structure is motivated by the need to design plasmonic crystals that enable the propagation of electromagnetic waves with no phase delay (epsilon-near-zero effect). Our microscopic model incorporates the surface conductivity of the two-dimensional (2D) material of each sheet and a corresponding line charge density through a line conductivity along possible edges of the sheets. Our analysis generalizes averaging principles inherent in previous Bloch-wave approaches. We investigate physical implications of our findings. In particular, we emphasize the role of the vector-valued corrector field, which expresses microscopic modes of surface waves on the 2D material. We demonstrate how our homogenization procedure may set the foundation for computational investigations of: effective optical responses of reasonably general geometries, and complicated design problems in the plasmonics of 2D materials.

math.NA↗