Rhombohedral multilayer graphene is a physical realization of the chiral two-dimensional electron gas that can host zero-line modes (ZLMs), also known as kink states, when the local ap opened by inversion symmetry breaking potential changes sign in real space. Here we study how the variations in the local stacking coordination of multilayer graphene affects the formation of the ZLMs. Our analysis indicates that the valley Hall effect develops whenever an interlayer potential difference is able to open up a band gap in stacking faulted multilayer graphene, and that ZLMs can appear at the domain walls separating two distinct regions with imperfect rhombohedral stacking configurations. Based on a tight-binding formulation with distant hopping terms between carbon atoms, we first show that topologically distinct domains characterized by the valley Chern number are separated by a metallic region connecting AA and AA$'$ stacking line in the layer translation vector space. We find that gapless states appear at the interface between the two stacking faulted domains with different layer translation or with opposite perpendicular electric field if their valley Chern numbers are different.
Interlayer coupling in rotationally faulted graphene multilayers breaks the local sublattice-symmetry of the individual layers. We present a theory of this mechanism, which reduces to an effective Dirac model with space-dependent mass in an important limit. It thus makes a wealth of existing knowledge available for the study of rotationally faulted graphene multilayers. We demonstrate quantitative agreement between our theory and a recent experiment.
Interlayer coupling in rotationally faulted graphene multilayers breaks the local sublattice-symmetry of the individual layers. Earlier we have presented a theory of this mechanism, which reduces to an effective Dirac model with space-dependent mass in an important limit. It thus makes a wealth of existing knowledge available for the study of rotationally faulted graphene multilayers. Agreement of this theory with a recent experiment in a strong magnetic field was demonstrated. Here we explore some of the predictions of this theory for the system in zero magnetic field at large interlayer bias…
We show that valley degeneracy in rotationally faulted multilayer graphene may be broken in the presence of a magnetic field and interlayer commensurations. This happens due to a simultaneous breaking of both time-reversal and inversion symmetries leading to a splitting of Landau levels linear in the field. Our theoretical work is motivated by an experiment [Y. J. Song et al., Nature 467, 185 (2010)] on epitaxially grown multilayer graphene where such linear splitting of Landau levels was observed at moderate fields. We consider both bilayer and trilayer configurations and, although a linear s…
This article reviews progress in the theoretical modelling of the electronic structure of rotationally faulted multilayer graphenes. In these systems the crystallographic axes of neighboring layers are misaligned so that the layer stacking does not occur in the Bernal structure observed in three dimensional graphite and frequently found in exfoliated bilayer graphene. Notably, rotationally faulted graphenes are commonly found in other forms of multilayer graphene including epitaxial graphenes thermally grown on ${\rm SiC \, (000 \bar 1)}$, graphenes grown by chemical vapor deposition, folded m…
The electronic spectra of rotationally faulted graphene bilayers are calculated using a continuum formulation for small fault angles that identifies two distinct electronic states of the coupled system. The low energy spectra of one state features a Fermi velocity reduction which ultimately leads to pairwise annihilation and regeneration of its low energy Dirac nodes. The physics in the complementary state is controlled by pseudospin selection rules that prevent a Fermi velocity renormalization and produce second generation symmetry-protected Dirac singularities in the spectrum. These results …
We identify qualitative trends in the stacking sequence dependence of carrier-carrier interaction phenomena in multilayer graphene. Our theory is based on a new approach which explicitly exhibits the important role in interaction phenomena of the momentum-direction dependent intersite phases determined by the stacking sequence. Using this method, we calculate and compare the self-energies, density--density response functions, collective modes, and ground-state energies of several different few layer graphene systems. The influence of electron--electron interactions on important electronic prop…
The crystal structure of solid-state matter greatly affects its electronic properties. For example in multilayer graphene, precise knowledge of the lateral layer arrangement is crucial, since the most stable configurations, Bernal and rhombohedral stacking, exhibit very different electronic properties. Nevertheless, both stacking orders can coexist within one flake, separated by a strain soliton that can host topologically protected states. Clearly, accessing the transport properties of the two stackings and the soliton is of high interest. However, the stacking orders can transform into one a…
We study the effect of perpendicular electric fields on the band structures of ABA and ABC graphene multilayers, and find that the electronic screening effect is significantly different between them. In ABA multilayers, the field produces a band overlap and gives a linear screening, while in ABC multilayers, in contrast, it opens an energy gap in the surface-state band at low energy, leading to a strong screening effect essentially non-linear to the field amplitude. The energy gap of a large ABC stack sharply rises when the external field exceeds a certain critical value.
The effect of different stacking order of graphene multilayers on the electric field induced band gap is investigated. We considered a positively charged top and a negatively charged back gate in order to independently tune the band gap and the Fermi energy of three and four layer graphene systems. A tight-binding approach within a self-consistent Hartree approximation is used to calculate the induced charges on the different graphene layers. We found that the gap for trilayer graphene with the ABC stacking is much larger than the corresponding gap for the ABA trilayer. Also we predict that fo…
We study the electronic structure of multilayer graphenes with a mixture of Bernal and rhombohedral stacking and propose a general scheme to understand the electronic band structure of an arbitrary configuration. The system can be viewed as a series of finite Bernal graphite sections connected by stacking faults. We find that the low-energy eigenstates are mostly localized in each Bernal section, and, thus, the whole spectrum is well approximated by a collection of the spectra of independent sections. The energy spectrum is categorized into linear, quadratic and cubic bands corresponding to sp…
Graphene layers are known to stack in two stable configurations, namely ABA or ABC stacking, with drastically distinct electronic properties. Unlike the ABA stacking, little has been done to experimentally investigate the electronic properties of ABC graphene multilayers. Here, we report the first magneto optical study of a large ABC domain in a graphene multilayers flake, with ABC sequences exceeding 17 graphene sheets. The ABC-stacked multilayers can be fingerprinted with a characteristic electronic Raman scattering response, which persists even at room temperatures. Tracing the magnetic fie…
We study interlayer transport of multilayer graphenes in magnetic field with various stacking structures (AB, ABC, and AA types) by calculating the Hall and longitudinal conductivities as functions of Fermi energy. Their behavior depends strongly on the stacking structures and selection of the layers. The Hall conductivity between different layers is no longer quantized. Moreover, for AB stacking, the interlayer conductivity vanishes around zero energy with increasing layer separation, and shows negative values in particular cases. The fact that longitudinal interlayer conductivity suppressed …
Peeling, shearing, and sliding are important mechanical phenomena in van der Waals solids. However, theoretically they have been studied mostly using minimal periodic cells and in the context of accurate quantum simulations. Here, we investigate the peeling of large-scale multilayer graphene stacks with varying thicknesses, stackings, and peeling directions by using classical molecular dynamics simulations with a registry-dependent interlayer potential. Simulations show that, while at large scale the peeling proceeds smoothly, at small scale the registry shifts and sliding patterns of the laye…
We study the effect of vertical electric field (E-field) on the electronic properties of multilayer graphene. We show that the effective mass, electron velocity and density-of-state of a bilayer graphene are modified under the E-field. We also study the transformation of the band structure of multilayer graphenes. E-field induces finite (zero) bandgap in the even (odd)-layer ABA-stacking graphene. On the other hand, finite bandgap is induced in all ABC-stacking graphene. We also identify the optimum E-field to obtain the maximum bandgap in the multilayer graphenes. Finally we compare our resul…
We present an analytical model to study the electronic properties, including full band structure, low energy dispersions around the Dirac point and density of states of the ABC-stacking $N$-layer graphene (ABCNLG). An ABCNLG can be simulated by a linear atomic chain with $2N$ atoms. With only nearest-neighbor inter- and intra-layer hopping integrals taken into account, the Hamiltonian representation is a complex $2N \times 2N$ tridiagonal matrix $H_0$. Through a unitary transformation, we can reduce the $2N \times 2N$ Hamiltonian matrix into two real $N \times N$ tridiagonal matrices $\mathbb{…
We use a tight-binding model and the random-phase approximation to study the Coulomb excitations in simple-hexagonal-stacking multilayer graphene and discuss the field effects. The calculation results include the energy bands, the response functions, and the plasmon dispersions. A perpendicular electric field is predicted to induce significant charge transfer and thus capable of manipulating the energy, intensity, and the number of plasmon modes. This could be further validated by inelastic light scattering or electron-energy-loss spectroscopy.
We present a theoretical study on the electron transmission through the AB-BA stacking boundary in multilayer graphene. Using the tight-binding model and the transfer matrix method, we calculate the electron transmission probability through the boundary as a function of electron Fermi energy in multilayers from bilayer to five-layer. We find that the transmission is strongly suppressed particularly near the band touching point, suggesting that the electronic conductivity in general multilayer graphenes is significantly interfered by stacking fault. The conductivity suppression by stacking faul…
Resonant Structure of Second Harmonic Generation in Multilayer Graphene Polytypes
National Graphene Institute, University of Manchester（国家石墨烯研究所，曼彻斯特大学）；Department of Physics & Astronomy, University of Manchester（曼彻斯特大学物理与天文系）；Department of Physics, University of Osaka（大阪大学物理系）
Chromium triiodide, CrI$_3$, is emerging as a promising magnetic two-dimensional semiconductor where spins are ferromagnetically aligned within a single layer. Potential applications in spintronics arise from an antiferromagnetic ordering between adjacent layers that gives rise to spin filtering and a large magnetoresistance in tunnelling devices. This key feature appears only in thin multilayers and it is not inherited from bulk crystals, where instead neighbouring layers share the same ferromagnetic spin orientation. This discrepancy between bulk and thin samples is unexpected, as magnetic o…
An efficient method with no numerical diagonalization of a huge Hamiltonian matrix and calculation of a tedious Green's function is proposed to acquire the exact energy spectrum and dynamical conductivity in a gated AA-stacking $N$-layer Graphene (AANLG) with the intrinsic spin-orbital coupling (SOC). $2N \times 2N$ tight-binding Hamiltonian matrix, velocity operator and Green's function representation of an AANLG are simultaneously reduced to $N$ $2\times 2$ diagonal block matrices through a proper transformation matrix. A gated AANLG with intrinsic SOC is reduced to $N$ graphene-like layers.…
Stackings in graphene have a pivotal role in properties to be discussed in the future, as seen in the recently found superconductivity of twisted bilayer graphene. Beyond bilayer graphene, the stacking order of multilayer graphene can be rhombohedral, which shows flat bands near the Fermi level that are associated with interesting phenomena, such as tunable conducting surface states expected to exhibit spontaneous quantum Hall effect, surface superconductivity, and even topological order. However, the difficulty in exploring rhombohedral graphenes is that in experiments, the alternative, hexag…
Using a numerical tight-binding approach based on the Chebyshev-Bogoliubov-de Gennes method we describe Josephson junctions made of multilayer graphene contacted by top superconducting gates. Both Bernal (ABA) and rhombohedral (ABC) stacking are considered and we find that the type of stacking has a strong effect on the proximity effect and the supercurrent flow. For both cases the pair amplitude shows a polarization between dimer and non-dimer atoms, being more pronounced for rhombohedral stacking. Even though the proximity effect in non-dimer sites is enhanced when compared to single layer g…
Spatial modulations, such as multilayers, to realize topological materials have recently been studied in theoretical and experimental works. In this paper, we investigate properties of the multilayers of the nodal-line semimetal (NLS) and the normal insulator. We consider two types of multilayers, with the stacking direction being perpendicular or parallel to the plane where the nodal line lies. In particular, we show that when the stacking direction is parallel to the plane, the nodal lines remain but they change their shapes because of the folding of the Brillouin zone. We also study the mul…
We investigate multilayer systems of a normal insulator and a Weyl semimetal. We calculate the bulk band structure and determine phase diagrams by changing the thickness of the normal insulator and that of the Weyl semimetal layer using two models; one is from the effective model for a Weyl semimetal, and the other is the lattice model. We compare the results between the two models, and found that they agree well. As a result, we find that the multilayer system shows a qualitatively different behavior depending on the stacking direction relative to the displacement vector connecting between tw…
The orbital structure of the quantum Hall ferromagnetic states in the zero-energy Landau level in chiral multilayer graphene (AB, ABC, ABCA, etc.\ stackings) is determined by the exchange interaction with all levels, including deep-lying states in the Dirac sea. This exchange field favors orbitally coherent states with a U(1) orbital symmetry if the filling factor $ν$ is not a multiple of the number of layers. If electrons fill the orbital sector of a fixed spin/valley component to one-half, e.g., at $ν=\pm3,\pm1$ in the bilayer and at $ν=\pm2,\pm6$ in the ABCA four-layer, there is a transitio…
We study the electronic structures and topological properties of $(M+N)$-layer twisted graphene systems. We consider the generic situation that $N$-layer graphene is placed on top of the other $M$-layer graphene, and is twisted with respect to each other by an angle $θ$. In such twisted multilayer graphene (TMG) systems, we find that there exists two low-energy flat bands for each valley emerging from the interface between the $M$ layers and the $N$ layers. These two low-energy bands in the TMG system possess valley Chern numbers that are dependent on both the number of layers and the stacking…
There has been a lot of excitement around the observation of superconductivity in twisted bilayer graphene, associated to flat bands close to the Fermi level. Such correlated electronic states also occur in multilayer rhombohedral stacked graphene (RG), which has been receiving increasing attention in the last years. In both natural and artificial samples however, multilayer stacked Bernal graphene (BG) occurs more frequently, making it desirable to determine what is their relative stability and under which conditions RG might be favored. Here, we study the energetics of BG and RG in bulk and …
We calculate the conductivity of arbitrarily stacked multilayer graphene sheets within a relaxation time approximation, considering both short-range and long-range impurities. We theoretically investigate the feasibility of identifying the stacking order of these multilayers using transport measurements. For relatively clean samples, the conductivities of the various stacking configurations depend on the carrier density as a power-law for over two decades. This dependence arises from a low density decomposition of the multilayer band structure into a sum of chiral Hamiltonians. For dirty sampl…
The thermodynamic properties of the electron gas in multilayer graphene depend strongly on the number of layers and the type of stacking. Here we analyse how those properties change when we vary the number of layers for rhombohedral stacked multilayer graphene and compare our results with those from a conventional two dimensional electron gas. We show that the highly degenerate zero energy Landau level which is partly filled with electrons and partly with holes has a strong influence on the value of the different thermodynamic quantities.
We use first-principle density functional theory (DFT) to study the transport properties of single and double barrier heterostructures realized by stacking multilayer h-BN or BC$_{2}$N, and graphene films between graphite leads. The heterostructures are lattice matched. The considered single barrier systems consist of layers of up to five h-BN or BC$_{2}$N monoatomic layers (Bernal stacking) between graphite electrodes. The transmission probability of an h-BN barrier exhibits two unusual behaviors: it is very low also in a classically allowed energy region, due to a crystal momentum mismatch b…
We report on a theoretical study of one-dimensional (1D) states localized at few-layer graphene system ribbon edges, and at interfaces between few-layer graphene systems with different valley Hall conductivities. These 1D states are topologically protected when valley mixing is neglected. We address the influence on their properties of stacking arrangement, interface structure, and external electric field perpendicular to the layers. We find that 1D states are generally absent at multilayer ribbon armchair direction edges, but present irrespective of crystallographic orientation at any interna…
Successful synthesis of the nitrogenated holey two-dimensional structures C2N (Nat. Commun. 2015, 6, 1-7) using simply wet-chemical reaction offer a cost-effective way to generate other 2D materials with novel optical and electronic properties. Using the few-layer C2N as models, we have performed an ab initio study of electronic properties of layered C2N. Band gaps of this system exhibit monotone decreasing as the number of layers increase. And a direct-gap to indirect-gap transition at the bulk C2N. Besides, when we apply an out-of-plane electric field on few-layer C2N, the band gap of multil…
Multilayer graphene with rhombohedral and Bernal stacking are supposed to be metallic, as predicted by density functional theory calculations using semi-local functionals. However recent angular resolved photoemission and transport data have questioned this point of view. In particular, rhombohedral flakes are suggested to be magnetic insulators. Bernal flakes composed of an even number of layers are insulating, while those composed of an odd number of layers are pseudogapped. Here, by systematically benchmarking with plane waves codes, we develop very accurate all-electron Gaussian basis sets…
A transfer matrix approach to study ballistic charge transport in few-layer graphene with chiral-symmetric stacking configurations is developed. We demonstrate that the chiral symmetry justifies a non-Abelian gauge transformation at the spectral degeneracy point (zero energy). This transformation proves the equivalence of zero-energy transport properties of the multilayer to those of the system of uncoupled monolayers. Similar transformation can be applied in order to gauge away an arbitrary magnetic field, weak strain, and hopping disorder in the bulk of the sample. Finally, we calculate the …
Twisted bilayers of nodal superconductors were recently proposed as a promising platform to host superconducting phases that spontaneously break time-reversal symmetry. Here we extend this analysis to twisted multilayers, focusing on two high-symmetry stackings with alternating ($\pm θ$) and constant ($θ$) twist angles. In analogy to alternating-twist multilayer graphene, the former can be mapped to twisted bilayers with renormalized interlayer couplings, along with a remnant gapless monolayer when the number of layers $L$ is odd. In contrast, the latter exhibits physics beyond twisted bilayer…
We investigate the layer localization control of two-dimensional states in multilayer metal-organic frameworks (MOFs). For finite stackings of (NiC4S4)3 MOFs, the weak van der Waals coupling between adjacent layers leads to a Fermi level dependent distribution of the electronic states in the monolayers. Such distribution is reflected in the topological edge states of multilayer nanoribbons. Moreover, by applying an external electric field, parallel to the stacking direction, the spacial localization of the electronic states can be controlled for a chosen Fermi energy. This localization behavio…
Domain walls, topological defects that define the frontier between regions of different stacking in multilayer graphene, have proved to host exciting physics. The ability of tuning these topological defects in-situ in an electronic transport experiment brings a wealth of possibilities in terms of fundamental understanding of domain walls as well as for electronic applications. Here, we demonstrate through a MEMS (micro-electromechanical system) actuator and magnetoresistance measurements the effect of domain walls in multilayer graphene quantum Hall effect. Reversible and controlled uniaxial s…
Essential properties of multilayer graphenes are diversified by the number of layers and the stacking configurations. For an $N$-layer system, Landau levels are divided into $N$ groups, with each identified by a dominant sublattice associated with the stacking configuration. We focus on the main characteristics of Landau levels, including the degeneracy, wave functions, quantum numbers, onset energies, field-dependent energy spectra, semiconductor-metal transitions, and crossing patterns, which are reflected in the magneto-optical spectroscopy, scanning tunneling spectroscopy, and quantum tran…
We apply the mean-field Hartree Fock theory of gapped electronic states at charge neutrality in bilayer graphene to thin films of rhombohedral graphite with up to thirty layers. For the ground state, the order parameter (the separation of bands at the valley center) saturates to a constant non-zero value as the layer number increases, whereas the band gap decreases with layer number. We consider chiral symmetry breaking disorder in the form of random layer potentials and chiral preserving disorder in the form of random values of the interlayer coupling. The former reduces the magnitude of the …
We show that Floquet engineering with circularly polarized light (CPL) can selectively split the valley degeneracy of a twisted multilayer graphene (TMG), and thus generate a controlled valley-polarized Floquet Chern flat band with tunable large Chern number. It offers a feasible optical way to manipulate the valley degree of freedom in moiré flat bands, and hence opens new opportunities to study the valleytronics of morié flat band systems. We thus expect that many of the valley-related properties of TMG, e.g. orbital ferromagnetism, can be switched by CPL with proper doping. We reveal a Cher…
The so called quantum spin Hall phase is a topologically non trivial insulating phase that is predicted to appear in graphene and graphene-like systems. In this work we address the question of whether this topological property persists in multilayered systems. We consider two situations: purely multilayer graphene and heterostructures where graphene is encapsulated by trivial insulators with a strong spin-orbit coupling. We use a four orbital tight-binding model that includes the full atomic spin-orbit coupling and we calculate the $Z_{2}$ topological invariant of the bulk states as well as th…
We discuss the magnetic and topological properties of bulk crystals and quasi-two-dimensional thin films formed by stacking intrinsic magnetized topological insulator ( for example Mn(Sb$_{x}$Bi$_{1-x}$)$_2$X$_4$ with X = Se,Te, including MnBi$_2$Te$_4$) septuple layers and topological insulator quintuple layers in arbitrary order. Our analysis makes use of a simplified model that retains only Dirac-cone degrees of freedom on both surfaces of each septuple or quintuple layer. We demonstrate the model's applicability and estimate its parameters by comparing with {\it ab initio } density-functio…
Supermoiré domain-resolved effective Hamiltonians and valley topology in helical multilayer graphene
Seoul National University（首尔大学）；Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and BIST（加泰罗尼亚纳米科学和纳米技术研究所（ICN2，西班牙高等科学研究委员会和巴塞罗那独立系统大学））；University of Seoul（首尔大学）
Multilayer van der Waals (vdWs) heterostructures assembled by diverse atomically thin layers have demonstrated a wide range of fascinating phenomena and novel applications. Understanding the interlayer coupling and its correlation effect is paramount for designing novel vdWs heterostructures with desirable physical properties. Using a detailed theoretical study of 2D MoS2-graphene (GR)-based heterostructures based on state-of-the-art hybrid density functional theory, we reveal that for 2D few-layer heterostructures, vdWs forces between neighboring layers depend on the number of layers. Compare…
