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Every noncompact surface is shown to have a (3,6)-tight triangulation and a characterisation of such triangulations is given in terms of length constraints on the boundary walks of superfaces of a given genus. Applications are given to the generic rigidity of associated countable bar-joint frameworks in R^3. In particular, every noncompact surface has a (3,6)-tight triangulation that is minimally 3-rigid. A proof of Ker\'ekj\'art\'o's classification of noncompact surfaces is also given., Comment: Improved introduction, added references and minor changes. 17 pages and 7 figures

Barnette and Edelson have shown that there are finitely many minimal triangulations of a connected compact 2-manifold M. A similar finiteness result is obtained for cellular partial triangulations that satisfy the Maxwell count 3v-e=6 and girth inequality constraints for certain embedded cycles. Also a characterisation of cellular M-embedded (3,6)-tight graphs is given in terms of the satisfaction of higher genus girth inequalities., Comment: Additional references and figures, and improved proofs. 22 pages, 10 figures

The domain walls between AB- and BA-stacked gapped bilayer graphene have garnered intense interest as they host topologically-protected, valley-polarised transport channels. The introduction of a twist angle between the bilayers and the associated formation of a Moire pattern has been the dominant method used to study these topological channels, but heterostrain can also give rise to similar stacking domains and interfaces. Here, we theoretically study the electronic structure of a uniaxially heterostrained bilayer graphene. We discuss the formation and evolution of interface-localized channels in the one-dimensional Moire pattern that emerges due to the different stacking registries between the two layers. We find that a uniform heterostrain is not sufficient to create one-dimensional topological channels in biased bilayer graphene. Instead, using a simple model to account for the in-plane atomic reconstruction driven by the changing stacking registry, we show that the resulting expanded Bernal-stacked domains and sharper interfaces are required for robust topological interfaces to emerge. These states are highly localised in the AA- or SP-stacked interface regions and exhibit differences in their layer and sublattice distribution depending on the interface stacking. We conclude that heterostrain can be used as a mechanism to tune the presence and distribution of topological channels in gapped bilayer graphene systems, complementary to the field of twistronics., Comment: 10 pages, 7 figures

The relative strength of different proximity spin-orbit couplings in graphene on transition metal dichalcogenides (TMDC) can be tuned via the metal composition in the TMDC layer. While Gr/MoSe$_2$, has a normal gap, proximity to WSe$_2$ instead leads to valley-Zeeman-driven inverted bands. Although the $\mathbb{Z}_2$ index vanishes, these systems enable a concentration-dependent topological crossover with band gap closure when graphene is stacked on a composite or alloyed TMDC layer. This is due to a nonzero Berry curvature at the individual valleys and a change of the valley Chern index at a critical composition ratio. Therefore, inherently, we also expect that stacked heterostructures of graphene on composite TMDC layers should host localised boundary modes due to the presence of Gr/WSe$_2$- and Gr/MoSe$_2$-like domains with opposite valley Chern indices. In this study, we show that a Gr/(Mo-W)Se$_2$ heterostructure with a lateral interface in the TMDC layer can indeed host topologically protected in-gap propagating modes, similar to those at the border of commensurate AB and BA domains in biased minimally-twisted bilayer graphene. However, the stability of these modes depends crucially on the system size. We demonstrate that the electronic behaviour of Gr/(Mo-W)Se$_2$ heterostructures evolves from a homogeneous effective medium to a superposition of domain-localised bands and zero-energy branch crossings as the domain size in the alloyed TMDC layer is increased., Comment: 10 pages and 4 figures

Asymptotic equilibrium stresses are defined for countably infinite tensegrities and generalisations of the Roth-Whiteley characterisation of first-order rigidity are obtained. Generalisations of prestress stability and second order rigidity are given for countably infinite bar-joint frameworks and are shown to give sufficient conditions for continuous rigidity relative to certain prescribed motions. The proofs are based on a new short proof for finite frameworks that prestress stability ensures continuous rigidity., Comment: Journal accepted version, to appear in the Journal of Mathematical Analysis and Applications. 19 pages, 8 figures. More detail and examples given. The implication "BPS implies BSR (boundedly smoothly rigid)" has been strengthened to "BPS implies DCR (directedly continuously rigid)"

The negligible intrinsic spin-orbit coupling (SOC) in graphene can be enhanced by proximity effects in stacked heterostructures of graphene and transition metal dichalcogenides (TMDCs). The composition of the TMDC layer plays a key role in determining the nature and strength of the resultant SOC induced in the graphene layer. Here, we study the evolution of the proximity-induced SOC as the TMDC layer is deliberately defected. Alloyed ${\rm G/W_{\chi}Mo_{1-\chi}Se_2}$ heterostructures with diverse compositions ($\chi$) and defect distributions are simulated using density functional theory. Comparison with continuum and tight-binding models allows both local and global signatures of the metal-atom alloying to be clarified. Our findings show that, despite some dramatic perturbation of local parameters for individual defects, the low-energy spin and electronic behaviour follow a simple effective medium model which depends only on the composition ratio of the metallic species in the TMDC layer. Furthermore, we demonstrate that the topological state of such alloyed systems can be feasibly tuned by controlling this ratio., Comment: 11 pages, 5 figures

Strain, both naturally occurring and deliberately engineered, can have a considerable effect on the structural and electronic properties of 2D and layered materials. Uniaxial or biaxial heterostrain modifies the stacking arrangement of bilayer graphene (BLG) which subsequently influences the electronic structure of the bilayer. Here, we use Density Functional Theory (DFT) calculations to investigate the interplay between an external applied heterostrain and the resulting stacking in BLG. We determine how a strain applied to one layer is transferred to a second, 'free' layer and at what critical strain the ground-state AB-stacking is disrupted. To overcome limitations introduced by periodic boundary conditions, we consider an approximate system consisting of an infinite graphene sheet and an armchair graphene nanoribbon (AGNR). We find that above a critical strain of ~1%, it is energetically favourable for the free layer to be unstrained, indicating a transition between uniform AB-stacking and non-uniform mixed stacking. This is in agreement with a simple model estimate based on the individual energy contributions of strain and stacking effects. Our findings suggest that small levels of strain provide a platform to reversibly engineer stacking order and Moir\'e features in bilayers, providing a viable alternative to twistronics to engineer topological and exotic physical phenomena in such systems., Comment: 9 pages, 5 figures

Magnetic moments near zigzag edges in graphene allow complex nanostructures with customised spin properties to be realised. However, computational costs restrict theoretical investigations to small or perfectly periodic structures. Here we demonstrate that a machine-learning approach, using only geometric input, can accurately estimate magnetic moment profiles, allowing arbitrarily large and disordered systems to be quickly simulated. Excellent agreement is found with mean-field Hubbard calculations, and important electronic, magnetic and transport properties are reproduced using the estimated profiles. This approach allows the magnetic moments of experimental-scale systems to be quickly and accurately predicted, and will speed-up the identification of promising geometries for spintronic applications. While machine-learning approaches to many-body interactions have largely been limited to exact solutions of complex models at very small scales, this work establishes that they can be successfully applied at very large scales at mean-field levels of accuracy., Comment: submitted

We provide a critical discussion concerning the claim of topological valley currents, driven by a global Berry curvature and valley Hall effect proposed in recent literature. After pointing out a major inconsistency of the theoretical scenario proposed to interpret giant nonlocal resistance, we discuss various possible alternative explanations and open directions of research to solve the mystery of nonlocal transport in graphene superlattices., Comment: 3 pages, 1 figure

A converse is given to the well-known fact that a hyperplane localised zero mode of a crystallographic bar-joint framework gives rise to a line or lines in the zero mode (RUM) spectrum. These connections motivate definitions of linear zero mode spectra, for an aperiodic bar-joint framework $G$, that are based on relatively dense sets of linearly localised flexes. For a Delone framework in the plane the limit spectrum ${\bf L}_{lim}(G, a)$ is defined in this way, as a subset of the reciprocal space for a reference basis $a$ of the ambient space. A smaller spectrum, the slippage spectrum ${\bf L}_{slip}(G, a)$, is also defined. In the case of the quasicrystal parallelogram frameworks associated with regular multi-grids, in the sense of de Bruijn and Beenker, these spectra coincide and are determined in terms of the geometry of $G$., Comment: 21 pages, 4 figures. Some minor corrections and a notation adjustment have been made

Intrinsic and extrinsic valley Hall effects are predicted to emerge in graphene systems with uniform or spatially-varying mass terms. Extrinsic mechanisms, mediated by the valley-dependent scattering of electrons at the Fermi surface, can be directly linked to quantum transport simulations. This is a promising route towards more complete experimental investigation of valleytronic phenomena in graphene, but a major obstacle is the difficulty in applying the sublattice-dependent potentials required. Here we show that strongly valley-dependent scattering also emerges from bilayer graphene quantum dots, where the gap size can be easily modulated using the interlayer potentials in dual-gated devices. Robust valley-dependent scattering and concomitant valley currents are observed for a range of systems, and we investigate the role of dot size, mass strength and additional potential terms. Finally, we note that a strong valley splitting of electronic current also emerges when a biased bilayer dot is embedded in a single layer of graphene, but that the effect is less robust than for a bilayer host. Our findings suggest that bilayer graphene devices with custom mass profiles provide an excellent platform for future valleytronic exploration of two-dimensional materials., Comment: 14 pages, 8 figures, submitted

Necessary and sufficient conditions are obtained for the infinitesimal rigidity of braced grids in the plane with respect to non-Euclidean norms. Component rectangles of the grid may carry 0, 1 or 2 diagonal braces, and the combinatorial part of the conditions is given in terms of a matroid for the bicoloured bipartite multigraph defined by the braces., Comment: 12 pages, 4 figures. v2 has an erroneous corollary removed and a remark corrected

The parabolic algebra A_p is the weakly closed algebra on L^2(R) generated by the unitary semigroup of right translations and the unitary semigroup of multiplication by the analytic exponential functions e^{i\lambda x}, \lambda \geq 0. This algebra is reflexive with an invariant subspace lattice, Lat A_p, which is naturally homeomorphic to the unit disc (Katavolos and Power, 1997). This identification is used here to classify strongly irreducible isometric representations of the partial Weyl commutation relations. The notion of a synthetic subspace lattice is extended from commutative to noncommutative lattices and it is shown that Lat A_p is nonsynthetic relative to the maximal abelian multiplication subalgebra of A_p. Also, operator algebras derived from isometric representations of A_p and from compact perturbations are defined and determined., Comment: 20 pages, 2 figures

It is shown that a simple graph which is embeddable in the real projective plane is minimally 3-rigid if and only if it is (3,6)-tight. Moreover the topologically uncontractible embedded graphs of this type are constructible from one of 8 embedded graphs by a sequence of vertex splitting moves. In particular the characterisation of minimal 3-rigidity holds for a triangulated Mobius strip., Comment: 34 pages (including a 9 page appendix). This is a substantial revision with fuller details and improved terminology. The main results are unchanged but there are new proofs and diagrams

Entangled embedded periodic nets and crystal frameworks are defined, along with their dimension type, homogeneity type, adjacency depth and periodic isotopy type. We obtain periodic isotopy classifications for various families of embedded nets with small quotient graphs. We enumerate the 25 periodic isotopy classes of depth 1 embedded nets with a single vertex quotient graph. Additionally, we classify embeddings of n-fold copies of pcu with all connected components in a parallel orientation and n vertices in a repeat unit, and determine their maximal symmetry periodic isotopes. We also introduce the methodology of linear graph knots on the flat 3-torus [0, 1)^3. These graph knots, with linear edges, are spatial embeddings of the labelled quotient graphs of an embedded net which are associated with its periodicity bases., Comment: 51 pages, 19 Figures and 3 tables

Improved fabrication techniques have enabled the possibility of ballistic transport and unprecedented spin manipulation in ultraclean graphene devices. Spin transport in graphene is typically probed in a nonlocal spin valve and is analyzed using spin diffusion theory, but this theory is not necessarily applicable when charge transport becomes ballistic or when the spin diffusion length is exceptionally long. Here, we study these regimes by performing quantum simulations of graphene nonlocal spin valves. We find that conventional spin diffusion theory fails to capture the crossover to the ballistic regime as well as the limit of long spin diffusion length. We show that the latter can be described by an extension of the current theoretical framework. Finally, by covering the whole range of spin dynamics, our study opens a new perspective to predict and scrutinize spin transport in graphene and other two-dimensional material-based ultraclean devices., Comment: 6 pages, 4 figures. Physical Review Letters

We report on the emergence of bulk, valley-polarized currents in graphene-based devices, driven by spatially varying regions of broken sublattice symmetry, and revealed by non-local resistance ($R_\mathrm{NL}$) fingerprints. By using a combination of quantum transport formalisms, giving access to bulk properties as well as multi-terminal device responses, the presence of a non-uniform local bandgap is shown to give rise to valley-dependent scattering and a finite Fermi surface contribution to the valley Hall conductivity, related to characteristics of $R_\mathrm{NL}$. These features are robust against disorder and provide a plausible interpretation of controversial experiments in graphene/hBN superlattices. Our findings suggest both an alternative mechanism for the generation of valley Hall effect in graphene, and a route towards valley-dependent electron optics, by materials and device engineering., Comment: 6 pages, 4 figures

We investigate properties of sparse and tight surface graphs. In particular we derive topological inductive constructions for $(2, 2)$-tight surface graphs in the case of the sphere, the plane, the twice punctured sphere and the torus. In the case of the torus we identify all 116 irreducible base graphs and provide a geometric application involving contact graphs of configurations of circular arcs., Comment: 25 pages, 20 figures

We experimentally investigate the charge induction mechanism across gated, narrow, ballistic graphene devices with different degrees of edge disorder. By using magnetoconductance measurements as the probing technique, we demonstrate that devices with large edge disorder exhibit a nearly homogeneous capacitance profile across the device channel, close to the case of an infinitely large graphene sheet. In contrast, devices with lower edge disorder (< 1 nm roughness) are strongly influenced by the fringing electrostatic field at graphene boundaries, in quantitative agreement with theoretical calculations for pristine systems. Specifically, devices with low edge disorder present a large effective capacitance variation across the device channel with a nontrivial, inhomogeneous profile due not only to classical electrostatics but also to quantum mechanical effects. We show that such quantum capacitance contribution, occurring due to the low density of states (DOS) across the device in the presence of an external magnetic field, is considerably altered as a result of the gate electrostatics in the ballistic graphene device. Our conclusions can be extended to any two dimensional (2D) electronic system confined by a hard-wall potential and are important for understanding the electronic structure and device applications of conducting 2D materials., Comment: 3 figures

Anisotropic electronic transport is a possible route towards nanoscale circuitry design, particularly in two-dimensional materials. Proposals to introduce such a feature in patterned graphene have to date relied on large-scale structural inhomogeneities. Here we theoretically explore how a random, yet homogeneous, distribution of zigzag-edged triangular perforations can generate spatial anisotropies in both charge and spin transport. Anisotropic electronic transport is found to persist under considerable disordering of the perforation edges, suggesting its viability under realistic experimental conditions. Furthermore, controlling the relative orientation of perforations enables spin filtering of the transmitted electrons, resulting in a half-metallic anisotropic transport regime. Our findings point towards a co-integration of charge and spin control in a two-dimensional platform of relevance for nanocircuit design. We further highlight how geometrical effects allow finite samples to display finite transverse resistances, reminiscent of Spin Hall effects, in the absence of any bulk fingerprints of such mechanisms, and explore the underlying symmetries behind this behaviour., Comment: 11 pages, 6 figures

We use confocal Raman microscopy and modified vector analysis methods to investigate the nanoscale origin of strain and carrier concentration in exfoliated graphene-hexagonal boron nitride (hBN) heterostructures on silicon dioxide (SiO2). Two types of heterostructures are studied: graphene on SiO2 partially coved by hBN, and graphene fully encapsulated between two hBN flakes. We extend the vector analysis methods to produce spatial maps of the strain and doping variation across the heterostructures. This allows us to visualise and directly quantify the much-speculated effect of the environment on carrier concentration as well as strain in graphene. Moreover, we demonstrate that variations in strain and carrier concentration in graphene arise from nanoscale features of the heterostructures such as fractures, folds and bubbles trapped between layers. For bubbles in hBN-encapsulated graphene, hydrostatic strain is shown to be greatest at bubble centres, whereas the maximum of carrier concentration is localised at bubble edges. Raman spectroscopy is shown to be a non-invasive tool for probing strain and doping in graphene, which could prove useful for engineering of two-dimensional devices., Comment: 14 pages, 5 figures

A theory of free spanning sets, free bases and their space group symmetric variants is developed for the first order flex spaces of infinite bar-joint frameworks. Such spanning sets and bases are computed for a range of fundamental crystallographic bar-joint frameworks, including the honeycomb (graphene) framework, the octahedron (perovskite) framework and the 2D and 3D kagome frameworks. It is also shown that the existence of crystal flex bases is closely related to linear structure in the rigid unit mode (RUM) spectrum and a more general geometric flex spectrum., Comment: 21 pages

We report the fabrication of one-dimensional (1D) ferromagnetic edge contacts to two-dimensional (2D) graphene/h-BN heterostructures. While aiming to study spin injection/detection with 1D edge contacts, a spurious magnetoresistance signal was observed, which is found to originate from the local Hall effect in graphene due to fringe fields from ferromagnetic edge contacts and in the presence of charge current spreading in the nonlocal measurement configuration. Such behavior has been confirmed by the absence of a Hanle signal and gate-dependent magnetoresistance measurements that reveal a change in sign of the signal for the electron- and hole-doped regimes, which is in contrast to the expected behavior of the spin signal. Calculations show that the contact-induced fringe fields are typically on the order of hundreds of mT, but can be reduced below 100 mT with careful optimization of the contact geometry. There may be additional contribution from magnetoresistance effects due to tunneling anisotropy in the contacts, which need to be further investigated. These studies are useful for optimization of spin injection and detection in 2D material heterostructures through 1D edge contacts.

A rigidity theory is developed for frameworks in a metric space with two types of distance constraints. Mixed sparsity graph characterisations are obtained for the infinitesimal and continuous rigidity of completely regular bar-joint frameworks in a variety of such contexts. The main results are combinatorial characterisations for (i) frameworks restricted to surfaces with both Euclidean and geodesic distance constraints, (ii) frameworks in the plane with Euclidean and non-Euclidean distance constraints, and (iii) direction-length frameworks in the non-Euclidean plane., Comment: Revised version, 23 pages, 6 figures. The analysis of the projective plane frameworks (whose proofs were incomplete) has been removed for discussion elsewhere. This double-distance context is defined in the final section of the revision together with other settings with underlying (2,1)-sparsity

The observation of large nonlocal resistances near the Dirac point in graphene has been related to a variety of intrinsic Hall effects, where the spin or valley degrees of freedom are controlled by symmetry breaking mechanisms. Engineering strong spin or valley Hall signals on scalable graphene devices could stimulate further practical developments of spin- and valleytronics. Here we report on scale-invariant nonlocal transport in large-scale chemical vapour deposition graphene under an applied external magnetic field. Contrary to previously reported Zeeman spin Hall effect, our results are explained by field-induced spin-filtered edge states whose sensitivity to grain boundaries manifests in the nonlocal resistance. This phenomenon, related to the emergence of the quantum Hall regime, persists up to the millimeter scale, showing that polycrystalline morphology can be imprinted in nonlocal transport. This suggests that topological Hall effects in large-scale graphene materials are highly sensitive to the underlying structural morphology, limiting practical realizations., Comment: Main paper (14 pages, 5 figures) and Supplementary information (8 pages, 8 figures)

Commensurability oscillations in the magnetotransport of periodically patterned systems, emerging from the interplay of cyclotron orbit and the pattern periodicity, are a benchmark of mesoscopic physics in electron gas systems. Exploiting similar effects in 2D materials would allow exceptional control of electron behaviour, but is hindered by the requirement to maintain ballistic transport over large length scales. Recent experiments have overcome this obstacle and observed distinct magnetoresistance commensurability peaks for perforated graphene sheets (antidot lattices). Interpreting the exact mechanisms behind these peaks is of key importance, particularly in graphene where a range of regimes are accessible by varying the electron density. In this work a fully atomistic, device-based simulation of magnetoresistance experiments allows us to analyse both the resistance peaks and the current flow at commensurability conditions. Magnetoresistance spectra are found in excellent agreement with experiment, but we show that a semi-classical analysis, in terms of simple skipping or pinned orbits, is insufficient to fully describe the corresponding electron trajectories. Instead, a generalised mechanism in terms of states bound to individual antidots, or to groups of antidots, is required. Commensurability features are shown to arise when scattering between such states is enhanced. The emergence and suppression of commensurability peaks is explored for different antidot sizes, magnetic field strengths and electron densities. The insights gained from our study will guide the design and optimization of future experiments with nanostructured graphene.

Zigzag edges of the honeycomb structure of graphene exhibit magnetic polarization making them attractive as building blocks for spintronic devices. Here, we show that devices with zigzag edged triangular antidots perform essential spintronic functionalities, such as spatial spin-splitting or spin filtering of unpolarized incoming currents. Near-perfect performance can be obtained with optimized structures. The device performance is robust against substantial disorder. The gate-voltage dependence of transverse resistance is qualitatively different for spin-polarized and spin-unpolarized devices, and can be used as a diagnostic tool. Importantly, the suggested devices are feasible within current technologies., Comment: 6 pages, 5 figures, published

A recent "hidden symmetry" conjecture of B. Gin-ge Chen et al is resolved, concerning the dimension of the mechanical modes of a generic $2$-periodic triangulated surface $O$ in $R^3$ whose structure graph corresponds to a triangular tiling of $R^2$. We introduce an indexing of the terms in an associated sparse determinant $p_O(z_1, z_2)$ by means of oriented $3$-colourings of the underlying sparse graph, and use this and vertex splitting arguments to show that $p_O(z_1,z_2)$ is palindromic or antipalindromic, up to a shift index. This implies the conjectured dimension $1$ phenomenon for these surfaces. As a corollary we obtain the topological stability of the generic modes of a $1$-periodic triangulated nanotube., Comment: There is a serious error 3 lines below "Step (II)". It is not true that "It follows that the set C' of all oriented colourings for ... is equal to the set of extension colourings.. ." (In fact they only account for half of the extension colourings.)

The low energy band structure of graphene has two inequivalent valleys at K and K' points of the Brillouin zone. The possibility to manipulate this valley degree of freedom defines the field of valleytronics, the valley analogue of spintronics. A key requirement for valleytronic devices is the ability to break the valley degeneracy by filtering and spatially splitting valleys to generate valley polarized currents. Here we suggest a way to obtain valley polarization using strain-induced inhomogeneous pseudomagnetic fields (PMF) which act differently on the two valleys. Notably, the suggested method does not involve external magnetic fields, or magnetic materials, as previous proposals. In our proposal the strain is due to experimentally feasible nanobubbles (but any local deformation would do): the associated PMFs lead to different real space trajectories for K and K' electrons, thus allowing the two valleys to be addressed individually. In this way, graphene nanobubbles can be exploited in both valley filtering and valley splitting devices, and our simulations reveal that a number of different functionalities are possible depending on the deformation field.

Magnetic fields are often used for characterizing transport in nanoscale materials. Recent magnetotransport experiments have demonstrated that ballistic transport is possible in graphene antidot lattices (GALs). These experiments have inspired the present theoretical study of GALs in a perpendicular magnetic field. We calculate magnetotransport through graphene antidot barriers (GABs), which are finite rows of antidots arranged periodically in a pristine graphene sheet, using a tight-binding model and the Landauer-B\"uttiker formula. We show that GABs behave as ideal Dirac mass barriers for antidots smaller than the magnetic length, and demonstrate the presence of magnetic edge states, which are localized states on the periphery of the antidots due to successive reflections on the antidot edge in the presence of a magnetic field. We show that these states are robust against variations in lattice configuration and antidot edge chirality. Moreover, we calculate the transmittance of disordered GABs and find that magnetic edge states survive a moderate degree of disorder. Due to the long phase-coherence length in graphene and the robustness of these states, we expect magnetic edge states to be observable in experiments as well.

We consider non-selfadjoint operator algebras $\mathfrak{L}(G,\lambda)$ generated by weighted creation operators on the Fock-Hilbert spaces of countable directed graphs $G$. These algebras may be viewed as noncommutative generalizations of weighted Bergman space algebras, or as weighted versions of the free semigroupoid algebras of directed graphs. A complete description of the commutant is obtained together with broad conditions that ensure the double commutant property. It is also shown that the double commutant property may fail for $\mathfrak{L}(G,\lambda)$ in the case of the single vertex graph with two edges and a suitable choice of left weight function $\lambda$., Comment: 18 pages, 5 figures. Accepted for publication in Proceedings of the AMS

We have realized nanometer size constrictions in ballistic graphene nanoribbons grown on sidewalls of SiC mesa structures. The high quality of our devices allows the observation of a number of electronic quantum interference phenomena. The transmissions of Fabry-Perot like resonances were probed by in-situ transport measurements at various temperatures. The energies of the resonances are determined by the size of the constrictions which can be controlled precisely using STM lithography. The temperature and size dependence of the measured conductances are in quantitative agreement with tight-binding calculations. The fact that these interference effects are visible even at room temperature makes the reported devices attractive as building blocks for future carbon based electronics., Comment: to appear in Phys. Rev. Lett

Ideal graphene antidot lattices are predicted to show promising band gap behavior (i.e., $E_G\simeq 500$ meV) under carefully specified conditions. However, for the structures studied so far this behavior is critically dependent on superlattice geometry and is not robust against experimentally realistic disorders. Here we study a rectangular array of triangular antidots with zigzag edge geometries and show that their band gap behavior qualitatively differs from the standard behavior which is exhibited, e.g, by rectangular arrays of armchair-edged triangles. In the spin unpolarized case, zigzag-edged antidots give rise to large band gaps compared to armchair-edged antidots, irrespective of the rules which govern the existence of gaps in armchair-edged antidot lattices. In addition the zigzag-edged antidots appear more robust than armchair-edged antidots in the presence of geometrical disorder. The inclusion of spin polarization within a mean-field Hubbard approach gives rise to a large overall magnetic moment at each antidot due to the sublattice imbalance imposed by the triangular geometry. Half-metallic behavior arises from the formation of spin-split dispersive states near the Fermi energy, reducing the band gaps compared to the unpolarized case. This behavior is also found to be robust in the presence of disorder. Our results highlight the possibilities of using triangular perforations in graphene to open electronic band gaps in systems with experimentally realistic levels of disorder, and furthermore, of exploiting the strong spin dependence of the system for spintronic applications., Comment: 9 pages, 11 figures, published

Strain fields in graphene giving rise to pseudomagnetic fields have received much attention due to the possibility of mimicking real magnetic fields with magnitudes of greater than 100 Tesla. We examine systems with such strains confined to finite regions ("pseudomagnetic dots") and provide a transparent explanation for the characteristic sublattice polarization occurring in the presence of pseudomagnetic field. In particular, we focus on a triaxial strain leading to a constant field in the central region of the dot. This field causes the formation of pseudo Landau levels, where the zeroth order level shows significant differences compared to the corresponding level in a real magnetic field. Analytic arguments based on the Dirac model are employed to predict the sublattice and valley dependencies of the density of states in these systems. Numerical tight binding calculations of single pseudomagnetic dots in extended graphene sheets confirm these predictions, and are also used to study the effect of the rotating the strain direction and varying the size of the pseudomagnetic dot.

A simple graph is $3$-rigid if its generic bar-joint frameworks in $R^3$ are infinitesimally rigid. Necessary and sufficient conditions are obtained for the minimal $3$-rigidity of a simple graph which is obtained from the $1$-skeleton of a triangulated torus by the deletion of edges interior to a triangulated disc., Comment: 31 pages, 37 figures

A simple graph G=(V,E) is 3-rigid if its generic bar-joint frameworks in R3 are infinitesimally rigid. Block and hole graphs are derived from triangulated spheres by the removal of edges and the addition of minimally rigid subgraphs, known as blocks, in some of the resulting holes. Combinatorial characterisations of minimal $3$-rigidity are obtained for these graphs in the case of a single block and finitely many holes or a single hole and finitely many blocks. These results confirm a conjecture of Whiteley from 1988 and special cases of a stronger conjecture of Finbow-Singh and Whiteley from 2013., Comment: 25 pages, 12 figures

Strain-induced deformations in graphene are predicted to give rise to large pseudomagnetic fields. We examine theoretically the case of gas-inflated bubbles to determine whether signatures of such fields are present in the local density of states. Sharp-edged bubbles are found to induce Friedel-type oscillations which can envelope pseudo-Landau level features in certain regions of the bubble. However, bubbles which minimise interference effects are also unsuitable for pseudo-Landau level formation due to more spatially varying field profiles., Comment: Submitted to Edison19

Recent experimental findings and theoretical predictions suggest that nitrogen-doped CVD-grown graphene may give rise to electronic band gaps due to impurity distributions which favour segregation on a single sublattice. Here we demonstrate theoretically that such distributions give rise to more complex behaviour in the presence of edges, where geometry determines whether electrons in the sample view the impurities as a gap-opening average potential or as scatterers. Zigzag edges give rise to the latter case, and remove the electronic bandgaps predicted in extended graphene samples. We predict that such behaviour will give rise to leakage near grain boundaries with a similar geometry or in zigzag-edged etched devices. Furthermore, we examine the formation of one-dimensional metallic channels at interfaces between different sublattice domains, which should be observable experimentally and offer intriguing waveguiding possibilities., Comment: 6 pages, 6 figures, published in PRB

We present a numerically efficient technique to evaluate the Green's function for extended two dimensional systems without relying on periodic boundary conditions. Different regions of interest, or `patches', are connected using self energy terms which encode the information of the extended parts of the system. The calculation scheme uses a combination of analytic expressions for the Green's function of infinite pristine systems and an adaptive recursive Green's function technique for the patches. The method allows for an efficient calculation of both local electronic and transport properties, as well as the inclusion of multiple probes in arbitrary geometries embedded in extended samples. We apply the Patched Green's function method to evaluate the local densities of states and transmission properties of graphene systems with two kinds of deviations from the pristine structure: bubbles and perforations with characteristic dimensions of the order of 10-25 nm, i.e. including hundreds of thousands of atoms. The strain field induced by a bubble is treated beyond an effective Dirac model, and we demonstrate the existence of both Friedel-type oscillations arising from the edges of the bubble, as well as pseudo-Landau levels related to the pseudomagnetic field induced by the nonuniform strain. Secondly, we compute the transport properties of a large perforation with atomic positions extracted from a TEM image, and show that current vortices may form near the zigzag segments of the perforation.

The weak operator topology closed operator algebra on $L^2(R)$ generated by the one-parameter semigroups for translation, dilation and multiplication by $exp(i\lambda x), \lambda \geq 0$, is shown to be a reflexive operator algebra, in the sense of Halmos, with invariant subspace lattice equal to a binest. This triple semigroup algebra, $A_{ph}$, is antisymmetric in the sense that $A_{ph} \cap A_{ph}^*= CI$, it has a nonzero proper weakly closed ideal generated by the finite-rank operators, and its unitary automorphism group is $R$. Furthermore, the 8 choices of semigroup triples provide 2 unitary equivalence classes of operator algebras, with $A_{ph}$ and $A_{ph}^*$ being chiral representatives., Comment: A proof has been corrected and a clarifying figure added (Figure 4). New results have been added (Corollary 5.4 and Theorem 5.7). Minor corrections and changes of order have been made