Two-dimensional (2D) atomic crystals can radically change their properties in response to external influences such as substrate orientation or strain, resulting in essentially new materials in terms of the electronic structure 1-5 . A striking example is the creation of flat-bands in bilayer-graphene for certain "magic" twist-angles between the orientations of the two layers 6 . The quenched kineticenergy in these flat-bands promotes electron-electron interactions and facilitates the emergence of strongly-correlated phases such as superconductivity and correlated-insulators. However, the exquisite fine-tuning required for finding the magic-angle where flat-bands appear in twisted-bilayer graphene, poses challenges to fabrication and scalability. Here we present an alternative route to creating flat-bands that does not involve fine tuning. Using scanning tunneling microscopy and spectroscopy, together with numerical simulations, we demonstrate that graphene monolayers placed on an atomically-flat substrate can be forced to undergo a buckling-transition 7-9 , resulting in a periodically modulated pseudo-magnetic field 10-14 , which in turn creates a post-graphene material with flat electronic bands. Bringing the Fermi-level into these flat-bands by electrostatic doping, we observe a pseudogap-like depletion in the density-of-states (DOS), which signals the emergence of a correlated-state 15-17 . The described approach of 2D crystal buckling offers a strategy for creating other superlattice systems and, in particular, for exploring interaction phenomena characteristic of flat-bands.

The time evolution of a wavepacket in strained graphene is studied within the tight-binding model and continuum model. The effect of an external magnetic field, as well as a strain-induced pseudomagnetic field, on the wave packet trajectories and zitterbewegung are analyzed. Combining the effects of strain with those of an external magnetic field produces an effective magnetic field which is large in one of the Dirac cones, but can be practically zero in the other. We construct an efficient valley filter, where for a propagating incoming wave packet consisting of momenta around the K and K ′ Dirac points, the outgoing wave packet exhibits momenta in only one of these Dirac points, while the components of the packet that belong to the other Dirac point are reflected due to the Lorentz force. We also found that the zitterbewegung is permanent in time in the presence of either external or strain-induced magnetic fields, but when both the external and strain-induced magnetic fields are present, the zitterbewegung is transient in one of the Dirac cones, whereas in the other cone the wave packet exhibits permanent spatial oscillations.

We describe an efficient numerical approach to calculate the longitudinal and transverse Kubo conductivities of large systems using Bastin's formulation [1]. We expand the Green's functions in terms of Chebyshev polynomials and compute the conductivity tensor for any temperature and chemical potential in a single step. To illustrate the power and generality of the approach, we calculate the conductivity tensor for the quantum Hall effect in disordered graphene and analyze the effect of the disorder in a Chern insulator in Haldane's model on a honeycomb lattice.PACS numbers: 71.23. An,72.15.Rn,71.30.+h One of the most important experimental probes in condensed matter physics is the electrical response to an external electrical field. In addition to the longitudinal conductivity, in specific circumstances, a system can present a transverse conductivity under an electrical perturbation. The Hall effect [2] and the anomalous Hall effect in magnetic materials [3] are two examples of this type of response. Paramagnetic materials with spin-orbit interaction can also present transverse spin currents [4]. There are also the quantized versions of the three phenomena: while the quantum Hall effect (QHE) was observed more than 30 years ago [5], the quantum spin Hall effect (QSHE) and the quantum anomalous Hall effect (QAHE) could only be observed [6, 7] with the recent discovery of topological insulators, a new class of quantum matter [8].In the linear response regime, the conductivity tensor can be calculated using the Kubo formalism [9]. The Hall conductivity can be easily obtained in momentum space in terms of the Berry curvature associated with the bands [10]. The downside of working in momentum space, however, is that the robustness of a topological state in the presence of disorder can only be calculated perturbatively [11]. Real-space implementations of the Kubo formalism for the Hall conductivity, on the other hand, allow the incorporation of different types of disorder in varying degrees, while providing flexibility to treat different geometries. Real-space techniques, however, normally require a large computational effort. This has generally restricted their use to either small systems at any temperature [12,13], or large systems at zero temperature [14].In this Letter, we propose a new efficient numerical approach to calculate the conductivity tensor in solids. We use a real space implementation of the Kubo formalism where both diagonal and off-diagonal conductivities are treated in the same footing. We adopt a formulation of the Kubo theory that is known as Bastin formula [1] and expand the Green's functions involved in terms of Chebyshev polynomials using the kernel polynomial method [16]. There are few numerical methods that use Chebyshev expansions to calculate the longitudinal DC conductivity [17][18][19][20] and transverse conductivity [14,21] at zero temperature. An advantage of our approach is the possibility of obtaining both conductivities for large systems in a single calculation step, independe...

We propose a highly efficient numerical method to describe inhomogeneous superconductivity by using the kernel polynomial method in order to calculate the Green's functions of a superconductor. Broken translational invariance of any type (impurities, surfaces, or magnetic fields) can be easily incorporated. We show that limitations due to system size can be easily circumvented and therefore this method opens the way for the study of scenarios and/or geometries that were unaccessible before. The proposed method is highly efficient and amenable to large scale parallel computation. Although we only use it in the context of superconductivity, it is applicable to other inhomogeneous mean-field theories.

When two-dimensional atomic crystals are brought into close proximity to form a van der Waals heterostructure, neighbouring crystals can start influencing each other's electronic properties. Of particular interest is the situation when the periodicity of the two crystals closely match and a moiré pattern forms, which results in specific electron scattering, reconstruction of electronic and excitonic spectra, crystal reconstruction, and many other effects. Thus, formation of moiré patterns is a viable tool of controlling the electronic properties of 2D materials. At the same time, the difference in the interatomic distances for the two crystals combined, determines the range in which the electronic spectrum is reconstructed, and thus is a barrier to the low energy regime. Here we present a way which allows spectrum reconstruction at all energies. By using graphene which is aligned simultaneously to two hexagonal boron nitride layers, one can make electrons scatter in the differential moiré pattern, which can have arbitrarily small wavevector and, thus results in spectrum reconstruction at arbitrarily low energies. We demonstrate that the strength of such a potential relies crucially on the atomic reconstruction of graphene within the differential moiré super-cell. Such structures offer further opportunity in tuning the electronic spectra of two-dimensional materials.Introduction: Van der Waals heterostructures allow combining different two-dimensional (2D) materials into functional stacks(1, 2), which has already produced a range of interesting electronic(3, 4) and optoelectronic(5-8) devices and resulted in observation of exciting physical phenomena. The large variety of the heterostructures is mainly due to the large selection of 2D materials. However, the assembly of van der Waals heterostructures allow one extra degree of freedom: apart from the selection of the sequence of the 2D crystalsthe individual crystals can be differently oriented with respect to each other. Previously such control over the rotational alignment between crystals resulted in the observation of the resonant tunnelling(9-11), renormalisation of exciton binding energy(12) insulating(13) and superconducting(4) states. 01129a). J.Y. and A.M. acknowledge the support of EPSRC Early Career Fellowship EP/N007131/1.

Recent experiments showed that non-uniform strain can be produced by depositing graphene over pillars. We employed atomistic calculations to study the non-uniform strain and the induced pseudomagnetic field up to 5000 Tesla in graphene on top of nano-pillars. By decreasing the distance between the nano-pillars a complex distribution for the pseudo-magnetic field can be generated. Furthermore, we performed tight-binding calculations of the local density of states (LDOS) by using the relaxed graphene configuration obtained from the atomistic calculations. We find that the quasiparticle LDOS are strongly modified near the pillars, both at low energies showing sub-lattice polarization, and at high energies showing shifts of the van Hove singularity. Our study shows that changing the specific pattern of the nano-pillars allows us to create a desired shape of the pseudo-magnetic field profile while the LDOS maps provide an input for experimental verifications by scanning tunneling microscopy.Graphene is a newly discovered atomic thin twodimensional honeycomb lattice consisting of carbon atoms 1 . It is a zero gap semimetal with a conical band structure where the conduction and valence bands touch each other at the Dirac point 2 . Nano-engineered nonuniform strain distribution in graphene is a promising road to generate a band gap and a pseudo-magnetic field. Scanning tunneling microscopy (STM) measurements have shown strain-induced Landau levels 4 which correspond to a large pseudo-magnetic field. Shear strain is essential and neither uniaxial nor isotropic strain produces a strong uniform pseudo-magnetic field 3 . Graphene's highly responses to external forces resulting in mechanical deformations. Over the last few years there have been many efforts to control graphene's electronic properties by strain 5-7 . Elastic deformations create a pseudo-magnetic field which acts on graphene's massless charge carriers [8][9][10] . The resulting variation of the hopping energies can be viewed as an induced pseudo-magnetic field which enters in the Dirac equation. Engineering of the right topology of the induced pseudo-magnetic field provides symmetrical magnetic confinement which confines electrons in specific regions in space 11 . Recently, it was predicted that non-uniform strain may lead to a considerable energy gap and a large gauge field that effectively acts as a uniform magnetic field 12 . Recently, Tomori et al used pillars made of a dielectric material (electron beam resist) which were placed on top of a substrate which is then overlayed with graphene to generate non-uniform strain on a micro-scale 13 . The graphene sections which are located between the pillars are attached to the substrate and the size and separation of the pillars control the strength and distribution of the strain. The length scale in the experiment was micronmeteres and SiO 2 was used as the substrate 13 . Here we study non-uniform strain at the atomistic scale where the continuum approach is no longer applicable. We also study the local density o...

The electronic properties of a triaxially strained hexagonal graphene flake with either armchair or zig-zag edges are investigated using molecular dynamics simulations and tight-binding calculations. We found that: i) the pseudo-magnetic field in the strained graphene flakes is not uniform neither in the center nor at the edge of zig-zag terminated flakes, ii) the pseudo-magnetic field is almost zero in the center of armchair terminated flakes but increases dramatically near the edges, iii) the pseudo-magnetic field increases linearly with strain, for strains lower than 15% while growing nonlinearly beyond this threshold, iv) the local density of states in the center of the zig-zag hexagon exhibits pseudo-Landau levels with broken sub-lattice symmetry in the zero'th pseudo-Landau level, and in addition there is a shift in the Dirac cone due to strain induced scalar potentials. This study provides a realistic model of the electronic properties of inhomogeneously strained graphene where the relaxation of the atomic positions is correctly included together with strain induced modifications of the hopping terms up to next-nearest neighbors.

Using highly efficient GPU-based simulations of the tight-binding Bogoliubov-de Gennes equations we solve self-consistently for the pair correlation in rhombohedral (ABC) and Bernal (ABA) multilayer graphene by considering a finite intrinsic s-wave pairing potential. We find that the two different stacking configurations have opposite bulk/surface behavior for the order parameter. Surface superconductivity is robust for ABC stacked multilayer graphene even at very low pairing potentials for which the bulk order parameter vanishes, in agreement with a recent analytical approach. In contrast, for Bernal stacked multilayer graphene, we find that the order parameter is always suppressed at the surface and that there exists a critical value for the pairing potential below which no superconducting order is achieved. We considered different doping scenarios and find that homogeneous doping strongly suppresses surface superconductivity while non-homogeneous fieldinduced doping has a much weaker effect on the superconducting order parameter. For multilayer structures with hybrid stacking (ABC and ABA) we find that when the thickness of each region is small (few layers), high-temperature surface superconductivity survives throughout the bulk due to the proximity effect between ABC/ABA interfaces where the order parameter is enhanced.