The ground state of the two-dimensional electron systems in Bernal bilayer and ABC-stacked multilayer graphenes in the presence of a strong magnetic field is investigated with the Hartree-Fock approximation. Phase diagrams of the systems are obtained, focusing on charge density wave states including states with vortices of valley pseudospins (called a Skyrme crystal). The single-electron states in these stacked graphenes are given by two-component wave functions. That of the first excited Landau level has the same component as the lowest Landau level of the ordinary two-dimensional electrons. Because of this localized wave functions, the Skyrme crystal has low energy in this first excited level up to four layers of graphene, when the inter-layer distance is assumed to be infinitesimal. At the same time, bubble crystals are suppressed, so the phase diagram is different from that of a single-layer graphene.
Additional component in multi-layer epitaxial graphene grown on the C-terminated surface of SiC, which exhibits the characteristic electronic properties of a AB-stacked graphene bilayer, is identified in magneto-optical response of this material. We show that these inclusions represent a well-defined platform for accurate magneto-spectroscopy of unperturbed graphene bilayers.
We study the effects of site dilution disorder on the electronic properties in graphene multilayers, in particular the bilayer and the infinite stack. The simplicity of the model allows for an easy implementation of the coherent potential approximation and some analytical results. Within the model we compute the self-energies, the density of states and the spectral functions. Moreover, we obtain the frequency and temperature dependence of the conductivity as well as the DC conductivity. The c-axis response is unconventional in the sense that impurities increase the response for low enough dopi…
We discuss the quantum Hall effect of bilayer graphene with finite gate voltage where the Fermi energy exceeds the interlayer hopping energy. We calculated magnetic susceptibility, diagonal and off-diagonal conductivities in finite-magnetic-field formalism, and observed crossover of integer quantum Hall effect from two independent monolayer type system to strongly coupled bilayer systems by changing the ratio of interlayer hopping energy and the gate voltage. We also discuss the case of multilayer systems with Bernal stacking
We study the effect of vertical electric-field (E-field) on the electronic properties of the multilayer armchair graphene nanoribbon (aGNR). Under E-field, the band structure of a bilayer aGNR undergoes interesting transformations such as change in the electron velocity, sign of the electron effective mass, band gap, the position of the band gap in the momentum space. Depending on the width of the aGNR and the applied E-field, the band gap of the aGNR may either be increased or decreased. When the applied E-field is above a critical value, the band gap of the bilayer aGNR is identical to that …
We present a theoretical study on the orbital magnetism in multilayer graphenes within the effective mass approximation. The Hamiltonian and thus susceptibility can be decomposed into contributions from sub-systems equivalent to monolayer or bilayer graphene. The monolayer-type subband exists only in odd layers and exhibits a delta-function susceptibility at $E_F=0$. The bilayer-type subband appearing in every layer number gives a singular structure in the vicinity of $E_F=0$ due to the trigonal warping as well as a logarithmic tail away from $E_F=0$. The integral of the susceptibility over en…
We uncover a new type of magic-angle phenomena when an AA-stacked graphene bilayer is twisted relative to another graphene system with band touching. In the simplest case this constitutes a trilayer system formed by an AA-stacked bilayer twisted relative to a single layer of graphene. We find multiple anisotropic Dirac cones coexisting in such twisted multilayer structures at certain angles, which we call "Dirac magic." We trace the origin of Dirac magic angles to the geometric structure of the twisted AA-bilayer Dirac cones relative to the other band-touching spectrum in the moiré reciprocal …
CrN/MgO(111) multilayers modeled via \textit{ab initio} calculations give rise to nanoscale, scalable, spatially separated two-dimensional electron and hole gases (2DEG+2DHG), each confined to its own CrN interface. Due to the Cr $3d^3$ configuration, both electron and hole gases are based on correlated transition metal layers involving bands of $3d$ character. Transport calculations predict each subsystem will have a large thermopower, on the order of 250 $μ$$V /K$ at room temperature. These heterostructures combine a large thermoelectric efficiency with scalable nanoscale conducting sheets; …
Results of a study on the electrochemical properties of exfoliated single and multilayer graphene flakes are presented. Graphene flakes were deposited on silicon/silicon oxide wafers to enable fast and accurate characterization by optical microscopy and Raman spectroscopy. Conductive silver paint and silver wires were used to fabricate contacts; epoxy resin was employed as masking coating in order to expose a stable, well defined area of graphene. Both multilayer and monolayer graphene microelectrodes showed quasi-reversible behavior during voltammetric measurements in potassium ferricyanide. …
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…
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 present an experimental and theoretical investigation of all-optical switching by single femtosecond laser pulses. Our experimental results demonstrate that, unlike rare earth-transition metal ferrimagnetic alloys, Pt/Co/[Ni/Co]$_N$/Gd can be switched in the absence of a magnetization compensation temperature, indicative for strikingly different switching conditions. In order to understand the underlying mechanism, we model the laser-induced magnetization dynamics in Co/Gd bilayers and GdCo alloys on an equal footing, using an extension of the microscopic three-temperature model to multiple…
Ferromagnetic/antiferromagnetic bilayers are interfaced with normal metal/ferromagnetic bilayers to form F*/AF/N/F valves. The N-spacer thickness is chosen such that it mediates strong indirect exchange (RKKY) between the outer magnetic layers, which varies in strength/direction depending on the N thickness and in direction on switching F. The system exhibits a strong modulation of the F*/AF exchange bias, oscillating in strength syncronously with the oscillation in the interlayer RKKY exchange across the normal metal spacer. The effect is explained as due to a superposition taking place withi…
The absence of an energy gap separating valence and conduction bands makes the low-energy electronic properties of graphene and its multi-layers sensitive to electron-electron interactions. In bilayers, for instance, interactions are predicted to open a gap at charge neutrality, turning the system into an insulator, as observed experimentally. In mono and (Bernal-stacked) trilayers, interactions, although still important, do not have an equally drastic effect, and these systems remain conducting at low temperature. It may be expected that interaction effects become weaker for thicker multilaye…
We present a simple analytical approach to study anharmonic effects in single layer, bilayer, and multilayer graphene. The coupling between in plane and out of plane modes leads to negative Grüneisen coefficients and negative thermal expansion. The value of the thermal expansion coefficient depends on the coupling to the substrate. The bending rigidity in bilayer graphene shows a crossover between a long wavelength regime where its value is determined by the in plane elastic properties and a short wavelength regime where its value approaches twice that of a single layer.
The magneto-optical absorption properties of graphene multilayers are theoretically studied. It is shown that the spectrum can be decomposed into sub-components effectively identical to the monolayer or bilayer graphene, allowing us to understand the spectrum systematically as a function of the layer number. Odd-layered graphenes always exhibit absorption peaks which shifts in proportion to sqrt(B), with B being the magnetic field, due to the existence of an effective monolayer-like subband. We propose a possibility of observing the monolayer-like spectrum even in a mixture of multilayer graph…
Bilayer graphene has been a subject of intense study in recent years. We extend a structural phase field crystal method to include an external potential from adjacent layer(s), which is generated by the corresponding phase field and changes over time. Moreover, multiple layers can be added into the structure. Using the thickness of the boundaries between different stacking variants of the bilayer structure as the key parameter, we quantify the strength of the adjacent layer potential by comparing with atomistic simulation results. We then test the multiple graphene structures, including bilaye…
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 study spatial symmetry in general ABA-stacked multilayer graphene to illustrate how electronic spectra at the two valleys are related in a magnetic field. We show that the lattice of multilayers with an even number of layers, as well as that of monolayer graphene, satisfy spatial inversion symmetry, which rigorously guarantees valley degeneracy in the absence of time-reversal symmetry. A multilayer with an odd number of layers (three or more) lacks inversion symmetry, but there is another transformation imposing an approximate valley degeneracy, which arises because the low-energy Hamiltoni…
Crystal fields occur due to a potential difference between chemically different atomic species. In Van-der-Waals heterostructures such fields are naturally present perpendicular to the planes. It has been realized recently that twisted graphene multilayers provide powerful playgrounds to engineer electronic properties by the number of layers, the twist angle, applied electric biases, electronic interactions and elastic relaxations, but crystal fields have not received the attention they deserve. Here we show that the bandstructure of large-angle twisted double bilayer graphene is strongly modi…
Extensive efforts have been devoted to the study of spin-orbit torque in ferromagnetic metal/heavy metal bilayers and exploitation of it for magnetization switching using an in-plane current. As the spin-orbit torque is inversely proportional to the thickness of the ferromagnetic layer, sizable effect has only been realized in bilayers with an ultrathin ferromagnetic layer. Here we demonstrate that, by stacking ultrathin Pt and FeMn alternately, both ferromagnetic properties and current induced spin-orbit torque can be achieved in FeMn/Pt multilayers without any constraint on its total thickne…
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…
We discuss the chiral decomposition of non-symmetric stacking structures. It is shown that the low-energy electronic structure of Bernal stacked graphene multilayer deposited on h-BN consists of chiral pseudospin doublets. N-layer graphene stocks on h-BN layer have N/2 effective bilayer graphene systems and one effective h-BN layer if N is even or (N-1)/2 effective graphene bilayers plus one graphene monolayer modified by h-BN layer if N is odd. We present the decomposition procedure and we derive the recurrence relations for the effective parameters characterizing the chiral subsystems. The e…
Whereas massless Dirac fermions in monolayer graphene exhibit Klein tunneling when passing through a potential barrier upon normal incidence, such a barrier totally reflects massive Dirac fermions in bilayer graphene due to difference in chirality. We show that, in the presence of magnetic barriers, such massive Dirac fermions can have transmission through even at normal incidence. The general consequence of this behaviour for multilayer graphene consisting of massless and massive modes are mentioned. We also briefly discuss the effect of a bias voltage on such magnetotransport.
Zero-modes, their topological degeneracy and relation to index theorems have attracted attention in the study of single- and bilayer graphene. For negligible scalar potentials, index theorems explain why the degeneracy of the zero-energy Landau level of a Dirac hamiltonian is not lifted by gauge field disorder, for example due to ripples, whereas other Landau levels become broadened by the inhomogenous effective magnetic field. That also the bilayer hamiltonian supports such protected bulk zero-modes was proved formally by Katsnelson and Prokhorova to hold on a compact manifold by using the At…
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 …
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…
The morphology and electronic structure of pentacene (Pn) deposited on Cu(111) was studied using scanning tunneling microscopy (STM) and spectroscopy (STS). Deposition of a multilayer followed by annealing to reduce coverage to a monolayer results in the formation of either of two unique phases: a 2D herringbone structure previously unobserved for any linear acene, or a 'random- tiling' structure. Coverage greater than a monolayer promotes the formation of a bilayer phase similar to that observed for Pn/Ag(111). STS shows that the electronic structure of the first layer is strongly modified du…
Suspended Bernal-stacked graphene multilayers up to an unexpectedly large thickness exhibit a broken-symmetry ground state, whose origin remains to be understood. Here we show that a finite-temperature second order phase transition occurs in multilayers whose critical temperature Tc increases from 12 K in bilayers to 100 K in heptalayers. A comparison of the data to a phenomenological model inspired by a mean field approach suggests that the transition is associated with the appearance of a self-consistent valley- and spin-dependent staggered potential changing sign from one layer to the next,…
We propose a generalized Peierls substitution method in conjunction with the tight-binding model to explore the magnetic quantization and quantum Hall effect in twisted multilayer graphene under a magnetic field. The Bloch-basis tight-binding Hamiltonian is constructed for large twist angle while a simplified tight-binding model is employed for the magic angle. We investigate extensively the band structures, Landau levels (LLs), and quantum Hall conductivity (QHC) of twisted bilayer graphene and twisted double bilayer graphene, as well as their dependence on the twist angle. Comparison between…
The experimental control over the twist angle in twisted bilayer graphene has not been reported and its realistic structure is most likely incommensurate. In this paper, we develop a tight-binding virtual crystal approximation theory to study the electronic properties in incommensurate twisted bilayer graphene. The theory yields the electronic band structure and the local density of states for any incommensurate twist angle θ between the graphene sheets. Angle dependent Van Hove singularities are observed in the numerically calculated local density of states. In accord with observations in sca…
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…
We show that the superconducivity in twisted graphene multilayers originates from a common feature, which is the strong valley symmetry breaking characteristic of these moiré systems at the magic angle. This leads to a breakdown of the rotational symmetry of the flat moiré bands down to $C_3$, and to ground states in which the time-reversal symmetry is broken for a given spin projection. However, this symmetry can be recovered upon exchange of spin-up and spin-down electrons, as we illustrate by means of a self-consistent microscopic Hartree-Fock resolution where the states for the two spin pr…
Using the parton construction, we build a three-dimensional (3D) multilayer fractional quantum Hall state with average filling ν= 1/3 per layer that is qualitatively distinct from a stacking of weakly coupled Laughlin states. The state supports gapped charge e/3 fermionic quasiparticles that can propagate both within and between the layers, in contrast to the quasiparticles in a multilayer Laughlin state which are confined within each layer. Moreover, the state has gapless neutral collective modes, a manifestation of an emergent "photon", which is minimally coupled to the fermionic quasipartic…
We investigate the competition between acoustic-phonon-mediated superconductivity and the long-range Coulomb interaction in moiréless graphene multilayers, specifically, Bernal bilayer graphene, rhombohedral trilayer graphene, and ABCA-stacked tetralayer graphene. In these graphene multilayers, the acoustic phonons can realize, through electron-phonon coupling, both spin-singlet and spin-triplet pairings, and the intra-sublattice pairings ($s$-wave spin-singlet and $f$-wave spin-triplet) are the dominant channels. Our theory naturally explains the distinct recent experimental findings in Berna…
Spin currents can be generated on an ultrafast timescale by excitation of a ferromagnetic (FM) thin film with a femtosecond laser-pulse. Recently, it has been demonstrated that these ultrafast spin currents can transport angular momentum to neighbouring FM layers, being able to change both the magnitude and orientation of the magnetization in the adjacent layer. In this work, both the generation and absorption of these optically excited spin currents are investigated. This is done using non-collinear magnetic bilayers, i.e. two FM layers separated by a conductive spacer. Spin currents are gene…
When two monolayers of graphene are stacked with a small relative twist angle, the resulting band structure exhibits a remarkably flat pair of bands at a sequence of 'magic angles' where correlation effects can induce a host of exotic phases. Here, we study a class of related models of $n$-layered graphene with alternating relative twist angle $\pm θ$ which exhibit magic angle flat bands coexisting with several Dirac dispersing bands at the Moiré K point. Remarkably, we find that the Hamiltonian for the multilayer system can be mapped exactly to a set of decoupled bilayers at {\it different} a…
We apply the semi-classical quantum Boltzmann formalism for the computation of transport properties to multilayer graphene. We compute the electrical conductivity as well as the thermal conductivity and thermopower for Bernal-stacked multilayers with an even number of layers. We show that the window for hydrodynamic transport in multilayer graphene is similar to the case of bilayer graphene. We introduce a simple hydrodynamic model which we dub the multi-fluid model and which can be used to reproduce the results for the electrical conductivity and thermopower from the quantum Boltzmann equatio…
We study the electronic properties of commensurate faulted bilayer graphene by diagonalizing the one-particle Hamiltonian of the bilayer system in a complete basis of Bloch states of the individual graphene layers. Our novel approach is very general and can be easily extended to any commensurate graphene-based heterostructure. Here, we consider three cases: i) twisted bilayer graphene, ii) bilayer graphene where triaxial stress is applied to one layer, and iii) bilayer graphene where uniaxial stress is applied to one layer. We show that the resulting superstructures can be divided into distinc…
We develop a theory for the optical conductivity of doped multilayer graphene including the effects of electron-electron interactions. Applying the quantum kinetic formalism, we formulate a set of pseudospin Bloch equations that governs the dynamics of the nonequilibrium density matrix driven by an external \emph{a.c.} electric field under the influence of Coulomb interactions. These equations reveal a dynamical mechanism that couples the Drude and interband responses arising from the chirality of pseudospin textures in multilayer graphene systems. We demonstrate that this results in an intera…
We investigate the electronic band structure and the proximity exchange effect in bilayer graphene on a family of ferromagnetic multilayers Cr$_2$X$_2$Te$_6$, X=Ge, Si, and Sn, with first principles methods. In each case the intrinsic electric field of the heterostructure induces an orbital gap on the order of 10 meV in the graphene bilayer. The proximity exchange is strongly band dependent. For example, in the case of Cr$_2$Ge$_2$Te$_6$, the low-energy valence band of bilayer graphene has exchange splitting of 8 meV, while the low energy conduction band's splitting is 30 times less (0.3 meV).…
We study the orbital effect of a strong magnetic field parallel to the layers on the energy spectrum of the Bernal-stacked graphene bilayer and multilayers, including graphite. We consider the minimal model with the electron tunneling between the nearest sites in the plane and out of the plane. Using the semiclassical analytical approximation and exact numerical diagonalization, we find that the energy spectrum consists of two domains. In the low- and high-energy domains, the semiclassical electron orbits are closed and open, so the spectra are discrete and continuous, correspondingly. The dis…
We theoretically study the correlated insulator states, quantum anomalous Hall (QAH) states, and field-induced topological transitions between different correlated states in twisted multilayer graphene systems. Taking twisted bilayer-monolayer graphene and twisted double-bilayer graphene as examples, we show that both systems stay in spin polarized, $C_{3z}$-broken insulator states with zero Chern number at 1/2 filling of the flat bands under finite displacement fields. In some cases these spin polarized, nematic insulator states are in the quantum valley Hall phase by virtue of the nontrivial…
We explore the electronic ground states of Bernal-stacked multilayer graphenes using the Hartree-Fock mean-field approximation and the full-parameter band model. We find that the electron-electron interaction tends to open a band gap in multilayer graphenes from bilayer to 8-layer, while the nature of the insulating ground state sensitively depends on the band parameter $γ_2$, which is responsible for the semimetallic nature of graphite. In 4-layer graphene, particularly, the ground state assumes an odd-spatial-parity staggered phase at $γ_2 = 0$, while an increasing, finite value of $γ_2$ sta…
We analyze the phase diagram of multilayer-graphene sandwiched between identical transition metal dichalcogenides. Recently realized in all van-der-Wall heterostructures, these sandwiches induce sizable (1-15 meV) spin orbit coupling in the graphene, offering a way to engineer topological band-structures in a pristine and gate-tunable platform. We find a rich phase diagram that depends on the number of layers $N$ and the gate-tunable perpendicular electric field. For $N > 1$ and odd, the system is a strong 2D topological insulator with a gap equal to the strength of proximity-induced Ising spi…
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 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…
In this paper we investigate the electron-phonon coupling in bilayer graphene, as a paradigmatic case for multilayer graphenes where interlayer hoppings are relevant. Using a frozen-phonon approach within the context of Density Functional Theory (DFT) and using different optical phonon displacements we are able to evaluate quantitatively the electron-phonon coupling $α_i$ associated with each hopping term $γ_i$. This analysis also reveals a simple scaling law between the hopping terms $γ_i$ and the electron-phonon coupling $α_i$ which goes beyond the specific DFT technique employed.
Being used in optoelectronic devices as ultra-thin conductor-insulator junctions, detailed investigations are needed about how exactly h-BN and graphene hybridize. Here, we present a comprehensive ab initio study of hot carrier dynamics governed by electron-phonon scattering at the h-BN/graphene interface, using graphite (bulk), monolayer and bilayer graphene as benchmark materials. In contrast to monolayer graphene, all multilayer structures possess low-energy optical phonon modes that facilitate carrier thermalization. We find that the h-BN/graphene interface represents an exception with com…
We study superfluid behavior of a gas of spatially indirect magnetoexcitons with reference to a system of two graphene layers embedded in a multilayer dielectric structure. The system is considered as an alternative of a double quantum well in a GaAs haterostructure. We determine a range of parameters (interlayer distance, dielectric constant, magnetic field and gate voltage) where magnetoexciton superfluidity can be achieved. Temperature of superfluid transition is computed. A reduction of critical parameters caused by impurities is evaluated and critical impurity concentration is determined.
Twisted graphene multilayers provide tunable platforms to engineer flat bands and exploit the associated strongly correlated physics. The two-dimensional nature of these systems makes them suitable for encapsulation by materials that break specific symmetries. In this context, recently discovered two-dimensional helimagnets, such as the multiferroic monolayer NiI$_2$, are specially appealing for breaking time-reversal and inversion symmetries due to their nontrivial spin textures. Here we show that this spin texture can be imprinted on the electronic structure of twisted bilayer graphene by pr…
