F. Guinea, B. Horovitz, P. Le Doussal
We analyze elastic deformations of graphene sheets which lead to effective gauge fields acting on the charge carriers. Corrugations in the substrate induce stresses, which, in turn, can give rise to mechanical instabilities and the formation of wrinkles. Similar effects may take place in suspended graphene samples under tension.
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We study the discrete energy spectrum of curved graphene sheets in the presence of a magnetic field. The shifting of the Landau levels is determined for complex and realistic geometries of curved graphene sheets. The energy levels follow a similar square root dependence on the energy quantum number as for rippled and flat graphene sheets. The Landau levels are shifted towards lower energies proportionally to the average deformation and the effect is larger compared to a simple uni-axially rippled geometry. Furthermore, the resistivity of wrinkled graphene sheets is calculated for different ave…
A continuum theory to describe periodic ripple formation in planar graphene/boron nitride superlattices is formulated. Due to the lattice mismatch between the two materials, it is shown that flat superlattices are unstable with respect to ripple formation of appropriate wavelengths. A competition between bending energy and transverse stretching energy gives rise to an optimal ripple wavelength that depends on the superlattice pitch. The optimal wavelengths predicted by the continuum theory are in good agreement with atomic scale total energy calculations previously reported in Nandwana and Ert…
We study the effects of quenched height fluctuations (ripples) in graphene on the density of states (DOS). We show that at strong ripple disorder a divergence in the DOS can lead to an ordered ground state. We also discuss the formation of dislocations in corrugated systems, buckling effects in suspended samples, and the changes in the Landau levels due to the interplay between a real magnetic field and the gauge potential induced by ripples.
We study the formation of wrinkles in graphene upon wet transfer onto a target substrate, whereby draining of water appears to play an important role. We are able to control the orientation of the wrinkles by tuning the surface morphology. Wrinkles are absent in flakes transferred to strongly hydrophobic substrates, a further indication of the role of the interaction of water with the substrate in wrinkle formation. The electrical and structural integrity of the graphene is not affected by the wrinkles, as inferred from Raman measurements and electrical conductivity measurements.
Formation of wrinkles is a common phenomenon in the large area growth of two dimensional layered materials on metallic substrates. Wrinkles can significantly affect the working of 2D materials based large scale electronic devices and therefore, it is of utmost importance to investigate local electrical properties of such wrinkled/folded structures on 2D materials. Here, we report local conductivity measurements by conducting atomic force microscopy (CAFM) and surface potential mapping by Kelvin Probe Force microscopy (KPFM) on large area wrinkled turbostratic graphene films grown on nickel foi…
The large-scale production of graphene monolayer greatly relies on epitaxial samples which often display stress-relaxation features in the form of wrinkles. Wrinkles of graphene on Ir(111) are found to exhibit a fairly well ordered interconnecting network which is characterized by low-energy electron microscopy (LEEM). The high degree of quasi-hexagonal network arrangement for the graphene aligned to the underlying substrate can be well described as a (non-Poissonian) Voronoi partition of a plane. The results obtained strongly suggest that the wrinkle network is frustrated at low temperatures,…
Wrinkles in graphene suspended on flat substrates: structure and collapse under hydrostatic pressure
本研究通过分子动力学与分子力学模拟，揭示了平坦衬底上石墨烯在单轴、双轴压缩下的多种稳定褶皱态，并发现静水压存在临界值，可使褶皱展平或塌缩为双层折叠。
Wrinkle formation during uniaxial compression of a graphene sheet lying on a soft (polymer) substrate
本文通过建模石墨烯/聚合物体系的单轴压缩，揭示软基底渗入与穿透对褶皱形成的影响，并提出利用基底熔化—冷却消除石墨烯褶皱的方法。
Graphene is only one atom thick and becomes the ultimate thin film to explore membrane physics and mechanics. Here we study hierarchy of graphene wrinkles induced by thermal strain engineering and demonstrate that the wrinkling hierarchy can be accounted for by the wrinklon theory. We derive an equation λ = (ky)0.5 explaining evolution of wrinkling wavelength λ with the distance to the edge y observed in our experiment by considering both bending energy and stretching energy of the graphene flakes. The prefactor k in the equation is determined to be about 55 nm, which is independent of the siz…
The deformation of monolayer graphene, produced by chemical vapor deposition (CVD), on a polyester film substrate has been investigated through the use of Raman spectroscopy. It has been found that the microstructure of the CVD graphene consists of a hexagonal array of islands of flat monolayer graphene separated by wrinkled material. During deformation, it was found that the rate of shift of the Raman 2D band wavenumber per unit strain was less than 25% of that of flat flakes of mechanically-exfoliated graphene, whereas the rate of band broadening per unit strain was about 75% of that of the …
Nano-rippled graphene, a structurally modified graphene, presents a novel material with a large range of possible applications including sensors, electrodes, coatings, optoelectronics, spintronics and straintronics. In this work we have synthesized macroscopic single layer graphene with well-defined uniaxial periodic modulation on a stepped Ir(332) substrate and transferred it to a dielectric support. The applied fast transfer process does not damage the Ir crystal which can be repeatedly used for graphene synthesis. Upon transfer, a millimeter sized graphene flake with a uniform periodic nano…
Wrinkling is a ubiquitous phenomenon in two-dimensional membranes. In particular, in the large-scale growth of graphene on metallic substrates, high densities of wrinkles are commonly observed. Despite their prevalence and potential impact on large-scale graphene electronics, relatively little is known about their structural morphology and electronic properties. Surveying the graphene landscape using atomic force microscopy, we found that wrinkles reach a certain maximum height before folding over. Calculations of the energetics explain the morphological transition, and indicate that the tall …
Experimentally produced graphene sheets exhibit a wide range of mobility values. Both extrinsic charged impurities and intrinsic ripples (corrugations) have been suggested to induce long-range disorder in graphene and could be a candidate for the dominant source of disorder. Here, using large-scale molecular dynamics and quantum transport simulations, we find that the hopping disorder and the gauge and scalar potentials induced by the ripples are short-ranged, in strong contrast with predictions by continuous models, and the transport fingerprints of the ripple disorder are very different from…
Wrinkles in monolayer graphene (GN) affect the GN electronic and transport properties. Defined network of wrinkles can be reached by placing the GN on the substrate decorated with the nanoparticles (NPs). In order to explain mechanism behind the topographically induced changes of the GN electronic structure and to correlate it with the wrinkling, correct description of the GN morphology is of high demand. We demonstrate here how to determine from the atomic force microscopy (AFM) images whether the GN is in the contact with the substrate or it is delaminated, how is the wrinkling of the layer …
We report measurements of the effects of a random vector potential generated by applying an in-plane magnetic field to a graphene flake. Magnetic flux through the ripples cause orbital effects: phase-coherent weak localization is suppressed, while quasi-random Lorentz forces lead to anisotropic magnetoresistance. Distinct signatures of these two effects enable an independent estimation of the ripple amplitude and correlation length.
We analyze spin scattering in ballistic transport of electrons through a ripple at a normal incidence of an electron flow. The model of a ripple consists of a curved graphene surface in the form of an arc of a circle connected from the left-hand and right-hand sides to two flat graphene sheets. At certain conditions the curvature induced spin-orbit coupling creates a transparent window for incoming electrons with one spin polarization simultaneously with a backscattering of those with opposite polarization. This window is equally likely transparent for electrons with spin up and spin down that…
As the thinnest atomic membrane, graphene presents an opportunity to combine geometry, elasticity and electronics at the limits of their validity. The availability of reliable atomistic potentials for graphene allows unprecedented precise simulations of the mechanical response of atomic membranes. Here we describe the transport and electronic structure in the neighbourhood of conical singularities, the elementary excitations of the ubiquitous wrinkled and crumpled graphene that occur in non-epitaxial suspended samples where shear stresses are unavoidable. We use a combination of atomistic mech…
We report a quantum study of the carrier scattering induced by graphene ripples. Crucial differences between the scattering induced by the ripple and ordinary scattering were found. In contrast to the latter, in which the Born approximation is valid for high-energy process, the former is valid for the low-energy process with a quite broad energy range. Furthermore, in polar symmetry ripples, the scattering amplitude exhibits a pseudo-spin structure, an additional factor $\cosθ/2$, which leads to an absence of backward scattering. We also elucidate that the scattering cross sections are proport…
It is well known that an eddy current will be induced in a conductor subject to a varying magnetic field. Here we propose another mechanism of generating nano-scale eddy current in rippled graphene nanoribbons(GNRs), which is only driven by an electric potential. In particular, it is found that under appropriate gate voltages, a local deformation may induce some unexpected global eddy currents, which form vortices in both rippled and entire flat areas of the GNR. We will explain that these vortices in flat areas is a manifestation of the nonlocality of quantum interference.
Decoupling the effects of ripples from tensile strain on the thermal conductivity of graphene and understanding the role of curvature on the thermal conductivity of graphene with grain boundaries
本文通过平衡分子动力学与 Green-Kubo 方法解耦了波纹和拉伸应变对石墨烯热导率的影响，并揭示了曲率和晶界倾斜角对含晶界石墨烯热导率的调控规律。
Recently graphene was introduced with tunable ripple texturing, a nanofabric enabled by graphene's remarkable elastic properties. However, one can further envision sandwiching the ripples, thus constructing composite nanomaterial, graphene cardboard. Here the basic mechanical properties of such structures are investigated computationally. It turns out that graphene cardboard is highly tunable material, for its elastic figures of merit vary orders of magnitude, with Poisson ratio tunable from 10 to -0.5 as one example. These trends set a foundation to guide the design and usage of metamaterials…
Graphene on a dielectric substrate exhibits spatial doping inhomogeneities, forming electron-hole puddles. Understanding and controlling the latter is of crucial importance for unraveling many of graphene's fundamental properties at the Dirac point. Here we show the coexistence and correlation of charge puddles and topographic ripples in graphene decoupled from the metallic substrate it was grown on. The analysis of interferences of Dirac fermion-like electrons yields a linear dispersion relation, indicating that graphene on a metal can recover its intrinsic electronic properties.
Graphene, dubbed as a two-dimensional material represents the topological concept of "surface" embedded in a three-dimensional space. This regard enables to employ existing theories/tools in topology to understand different properties/observations in graphene. Under the light of the long-established "Gauss's Theorema Egregium" we study wrinkled graphene, observing a peculiar correlation between morphology and strain distribution. Compressing graphene on water serves as an effectual platform to realize wrinkles; we explain the evolution of the wrinkles and the global distribution of the strain …
The electronic and optical properties of graphene can be precisely tuned by generating deterministic arrangements of strain features. In this paper, we report the formation of widespread and controlled buckling delamination of monolayer graphene deposited on hexagonal boron-nitride promoted by a significant squeezing of the graphene flake and induced by polymeric micro-actuators. The flexibility of this method offers a promising technique to create arbitrary buckling geometries and arrays of wrinkles which could also be subjected to iterative folding-unfolding cycles. Further development of th…
Graphene ripples possess peculiar essential properties owing to the strong chemical bonds, as an investigation using first principle calculations clearly revealed. Various charge distributions, bond lengths, energy bands, and densities of states strongly depend on the corrugation structures, ripple curvatures and periods. Armchair ripples belonging to a zero-gap semiconductor display split middle-energy states, while the zigzag ripples exhibit highly anisotropic energy bands, semi-metallic behavior implicated by the destruction of the Dirac cone, and the newly created critical points. Their de…
We study the effect of disordered ripples on the conductivity of monolayer graphene flakes. We calculate the relaxation times and the Boltzmann conductivities associated with two mechanisms. First, we study the conductivity correction due to an external in-plane magnetic field $B_\parallel$. Due to the irregular local curvature found at graphene sheets deposited over a substrate, $B_\parallel$ can be mapped into an effective random magnetic field perpendicular to the graphene surface. Second, we study the electron momentum relaxation due to intrinsic pseudo magnetic fields originated from defo…
We propose a theory of ripples in suspended graphene sheets based on two-dimensional elasticity equations that are made discrete on the honeycomb lattice and then periodized. At each point carbon atoms are coupled to Ising spins whose values indicate the atoms local trend to move vertically off-plane. The Ising spins are in contact with a thermal bath and evolve according to Glauber dynamics. In the limit of slow spin flip compared to membrane vibrations, ripples with no preferred orientation appear as long-lived metastable states for any temperature. Numerical solutions confirm this picture.
We report an exact map into one dimensional effective chains, of the tight-binding Hamiltonian for electrons in armchair and zigzag graphene nanoribbons with any uniaxial ripple. This mapping is used for studying the effect of uniaxial periodic ripples, taking into account the relative orientation changes between $π$ orbitals. Such effects are important for short wavelength ripples, while for long-wave ones, the system behaves nearly as strained graphene. The spectrum has a complex nature, akin to the Hofstadter butterfly with a rich localization behavior. Gaps at the Fermi level and dispersio…
Graphene is the nature's thinnest elastic membrane, with exceptional mechanical and electrical properties. We report the direct observation and creation of one-dimensional (1D) and 2D periodic ripples in suspended graphene sheets, using spontaneously and thermally induced longitudinal strains on patterned substrates, with control over their orientations and wavelengths. We also provide the first measurement of graphene's thermal expansion coefficient, which is anomalously large and negative, ~ -7x10^-6 K^-1 at 300K. Our work enables novel strain-based engineering of graphene devices.
Ripples in pristine freestanding graphene naturally orient themselves in an array that is alternately curved-up and curved-down; maintaining an average height of zero. Using scanning tunneling microscopy (STM) to apply a local force, the graphene sheet will reversibly rise and fall in height until the height reaches 60-70 percent of its maximum at which point a sudden, permanent jump occurs. We successfully model the ripples as a spin-half Ising magnetic system, where the height of the graphene is the spin. The permanent jump in height, controlled by the tunneling current, is found to be equiv…
In nanofabrication, just as in any other craft, the scale of spatial details is limited by the dimensions of the tool at hand. For example, the smallest details for direct laser writing with far-field light are set by the diffraction limit, which is approximately half of the used wavelength. In this work, we overcome this universal assertion by optically forging graphene ripples that show features with dimensions unlimited by diffraction. Thin sheet elasticity simulations suggest that the scaled-down ripples originate from the interplay between substrate adhesion, in-plane strain, and circular…
We propose a model of ripples in suspended graphene sheets based on plate equations that are made discrete with the periodicity of the honeycomb lattice and then periodized. In addition, the equation for the displacements with respect to the planar configuration contains a double-well site potential, a nonlinear friction and a multiplicative white noise term satisfying the fluctuation-dissipation theorem. The nonlinear friction terms agree with those proposed by Eichler et al [Nature Nanotech. {\bf 6}, 339 (2011)] to explain their experiments with a graphene resonator. The site double-well pot…
Using electrical transport experiments and shot noise thermometry, we find strong evidence that "supercollision" scattering processes by flexural modes are the dominant electron-phonon energy transfer mechanism in high-quality, suspended graphene around room temperature. The power law dependence of the electron-phonon coupling changes from cubic to quintic with temperature. The change of the temperature exponent by two is reflected in the quadratic dependence on chemical potential, which is an inherent feature of two-phonon quantum processes.
In previous studies, it proved difficult to realize periodic graphene ripples with wavelengths of few nanometers. Here we show that one-dimensional periodic graphene ripples with wavelengths from 2 nm to tens of nanometers can be implemented in the intrinsic areas of a continuous mosaic, locally N-doped, graphene monolayer by simultaneously using both the thermal strain engineering and the anisotropic surface stress of Cu substrate. Our result indicates that the constraint imposed at the boundaries between the intrinsic and the N-doped regions play a vital role in creating these 1D ripples. We…
Using the example of graphene, we have extended the classic ripple tank experiment to illustrate the behavior of waves in periodic lattices. A loudspeaker driving air through a periodically perforated plate onto the tank's water surface creates wave patterns that are in agreement with numerical simulations and are explained in terms of solid state theory. From an educational point of view, the experiment provides an illustrative example of the concepts of reciprocal space and symmetry.
We study various mechanisms of electron transmission across the corrugations in the graphene sheet. The spin dependence of the electron transmission probability in the rippled graphene is found. The electrons mean free path and transmission probabilities are analysed for different distributions of ripples in the graphene sheet as well. We demonstrate that the periodically repeated rippled graphene structure (the superlattice) leads to the suppression of the transmission of the ballistic electrons with one spin orientation in contrast to the other, depending on the direction of the incoming ele…
A new mechanism that induces charge density variations in corrugated graphene is proposed. Here it is shown how the interplay between lattice deformations and exchange interactions can induce charge separation, i.e., puddles of electrons and holes, for realistic deformation values of the graphene sheet. The induced charge density lies in the range of $10^{11}-10^{12}$ cm$^{-2}$, which is compatible with recent measurements.
We discuss the possibility to create hybrid quantum systems that combine ultracold atoms with graphene membranes. We investigate a setup in which a cold atom cloud is placed close to a free-standing sheet of graphene at distances on the order of a few hundred nanometers. The atoms then couple to the graphene membrane via Casimir-Polder forces. Temporal changes in the atomic state of the atomic cloud changes the Casimir-Polder interaction, thereby leading to the creation of a backaction force in the graphene sheet. This setup provides a controllable way to engineer ripples in a graphene sheet.
We study the electronic properties of rippled freestanding graphene membranes under central load from a sharp tip. To that end, we develop a gauge field theory on a honeycomb lattice valid beyond the continuum theory. Based on the proper phase conjugation of the tight-binding pseudospin Hamiltonian, we develop a method to determine conditions under which continuum elasticity can be used to extract gauge fields from strain. Along the way, we resolve a recent controversy on the theory of strain engineering in graphene: There are no K-point dependent gauge fields. We combine this lattice gauge fi…
We investigate how the criticality of the quantum Hall plateau transition in disordered graphene differs from those in the ordinary quantum Hall systems, based on the honeycomb lattice with ripples modeled as random hoppings. The criticality of the graphene-specific n=0 Landau level is found to change dramatically to an anomalous, almost exact fixed point as soon as we make the random hopping spatially correlated over a few bond lengths. We attribute this to the preserved chiral symmetry and suppressed scattering between K and K' points in the Brillouin zone. The results suggest that a fixed p…
With use of carefully designed molecular dynamics simulations, we demonstrate tuning of dynamic ripples in free-standing graphene by applying a local out-of-plane sinusoidal excitation. Depending on the boundary conditions and external modulation, we show control of the local dynamic morphology, including flattening and stable rippling patterns. In addition to studying the dynamic response of atomically thin layers to external time-varying excitation, our results open intriguing possibilities for modulating their properties via local dynamic morphology control.
It is now possible to produce graphene nanoribbons (GNRs) with atomically defined widths. GNRs offer many opportunities for electronic devices and composites, if it is possible to establish the link between edge structure and functionalisation, and resultant GNR properties. Switching hydrogen edge termination to larger more complex functional groups such as hydroxyls or thiols induces strain at the ribbon edge. However we show that this strain is then relieved via the formation of static out-of-plane ripples. The resultant ribbons have a significantly reduced Young's Modulus which varies as a …
Electronic transport in graphene with out-of-plane disorder
研究石墨烯面外无序缺陷对电子输运的影响，通过紧束缚和密度泛函计算发现晶格共格性是决定背散射抑制的关键，为控制石墨烯输运性质提供指导。
Graphene is a famous realization of elastic crystalline 2D membrane. Thermal fluctuations of a 2D membrane tend to destroy the long-range order in the system. Such fluctuations are stabilized by strong anharmonicity effects, which preserve thermodynamic stability. The anharmonic effects demonstrate critical behaviour on scales larger than the Ginzburg scale. In particular, clean suspended flake of graphene shows a power-law increase of the bending rigidity with the system size, $\varkappa\propto L^η,$ due to anharmonic interaction between in-plane and out-of-plane (flexural) phonon modes. We d…
We argue that the unscreened Coulomb interaction in graphene provides a positive, universal, and logarithmic correction to scaling of zero-temperature conductivity with frequency. The combined effect of the disorder due to wrinkling of the graphene sheet and the long range electron-electron interactions is a finite positive contribution to the dc conductivity. This contribution is disorder strength dependent and thus non-universal. The low-energy behavior of such a system is governed by the line of fixed points at which both the interaction and disorder are finite, and the density of states is…
We investigate spin relaxation in rippled graphene where curvature induces a Zeeman-like spin-orbit coupling with opposite effective magnetic fields along the graphene plane in ${\bf K}$ and ${\bf K}^\prime$ valleys. The joint effect of this Zeeman-like spin-orbit coupling and the intervalley electron-optical phonon scattering opens a spin relaxation channel, which manifests itself in low-mobility samples with the electron mean free path being smaller than the ripple size. Due to this spin relaxation channel, with the increase of temperature, the relaxation time for spins perpendicular to the …
A beam of linearly polarized light transmitted through magnetically biased graphene can have its axis of polarization rotated by several degrees after passing the graphene sheet. This large Faraday effect is due to the action of the magnetic field on graphene's charge carriers. As deformations of the graphene membrane result in pseudomagnetic fields acting on the charge carriers, the effect of random mesoscopic corrugations (ripples) can be described as the exposure of graphene to a random pseudomagnetic field. We aim to clarify the interplay of these typically sample inherent fields with the …
The two-dimensional nature of graphene Josephson junctions offers the possibility of creating effective superconductor-ferromagnet-superconductor junctions with tunable Zeeman splitting caused by an in-plane magnetic field. Such junctions would be able to alternate between a conventional superconducting ground state and a ground state with an intrinsic phase difference, making them controllable $0-π$ Josephson junctions. However, in addition to the Zeeman splitting, an in-plane magnetic field will in general also produce an orbital effect because of height variations in graphene, colloquially …
Single-layer graphene sheets are typically characterized by long-wavelength corrugations (ripples) which can be shown to be at the origin of rather strong potentials with both scalar and vector components. We present an extensive microscopic study, based on a self-consistent Kohn-Sham-Dirac density-functional method, of the carrier density distribution in the presence of these ripple-induced external fields. We find that spatial density fluctuations are essentially controlled by the scalar component, especially in nearly-neutral graphene sheets, and that in-plane atomic displacements are as im…
Electronic states in a bilayer graphene quantum ripple
该研究针对双层石墨烯量子波纹表面的电子态问题，基于无自旋定态薛定谔方程与束缚势诱导的Da Costa势，分析了几何结构和轨道角动量对系统电子态的影响。
Thermal ripples of graphene are well understood at room temperature, but their quantum counterparts at low temperatures are still in need of a realistic quantitative description. Here we present atomistic path-integral Monte Carlo simulations of freestanding graphene, which show upon cooling a striking classical-quantum evolution of height and angular fluctuations. The crossover takes place at ever-decreasing temperatures for ever-increasing wavelengths so that a completely quantum regime is never attained. Zero-temperature quantum graphene is flatter and smoother than classical at large scale…
Density functional theory (DFT) modeling of the physisorption of four different types of molecules (toluene, bromine dimmer, water and nitrogen dioxide) over and above graphene ripples has been performed. For all types of molecules changes of charge transfer and binding energies in respect to flat graphene is found. The changes in electronic structure of corrugated graphene and turn of π-orbitals of carbon atoms in combination with chemical structure of adsorbed molecules are proposed as the causes of difference with the perfect graphene case and variety of adsorption properties of different t…
We consider a single-layer graphene with high ripples, so that the pseudo-magnetic fields due to these ripples are strong. If the magnetic length corresponding to a typical pseudo-magnetic field is smaller than the ripple size, the resulting Landau levels are local. Then the effective properties of the macroscopic sample can be calculated by averaging the local properties over the distribution of ripples. We find that this averaging does not wash out the Landau quantization completely. Average density of states (DOS) contains a feature (inflection point) at energy corresponding to the first La…
The effects of a propagating sinusoidal out-of-plane flexural deformation in the electronic properties of a tense membrane of graphene are considered within a non-perturbative approach, leading to an electron-ripple coupling. The deformation is taken into account by introducing its corresponding pseudo-vector and pseudo-scalar potentials in the Dirac equation. By using a transformation to the time-cone of the strain wave, the Dirac equation is reduced to an ordinary second-order differential Matthieu equation, i.e., to a parametric pendulum, giving a spectrum of bands and gaps determined by re…
The electronic behavior in graphene under arbitrary uniaxial deformations, such as foldings or flexural fields is studied by including in the Dirac equation pseudoelectromagnetic fields. General foldings are thus studied by showing that uniaxial deformations can be considered pseudomagnetic fields in the Coulomb gauge norm. This allows to give an expression for the Fermi (zero) energy modes wavefunctions. For random deformations, contact is made with previous works on the quantum Hall effect under random magnetic fields, showing that the density of states has a power law behavior and that the …
Suspended graphene samples are observed to be gently rippled rather than being flat. In [M. Friedrich, U. Stefanelli. Graphene ground states, arXiv:1802.05049], we have checked that this nonplanarity can be rigorously described within the classical molecular-mechanical frame of configurational-energy minimization. There, we have identified all ground-state configurations with graphene topology with respect to classes of next-to-nearest neighbor interaction energies and classified their fine nonflat geometries. In this second paper on graphene nonflatness, we refine the analysis further and pro…
Upon cooling, branched line defects develop in epitaxial graphene grown at high temperature on Pt(111) and Ir(111). Using atomically resolved scanning tunneling microscopy we demonstrate that these defects are wrinkles in the graphene layer, i.e. stripes of partially delaminated graphene. With low energy electron microscopy (LEEM) we investigate the wrinkling phenomenon in situ. Upon temperature cycling we observe hysteresis in the appearance and disappearance of the wrinkles. Simultaneously with wrinkle formation a change in bright field imaging intensity of adjacent areas and a shift in the …
We have used Ultrahigh Vacuum (UHV) Scanning tunneling microscopy (STM) to investigate the effect of thermal annealing of graphene grown by chemical vapour deposition (CVD) on a Cu(110) foil. We show that the annealing appears to induce a reconstruction of the Cu surface along the [210] direction, with a period of 1.43 nm. Such reconstructions have been ascribed to the tensile strain induced in the Cu surface by differential thermal expansion of it relative to the graphene overlayer, but we show that it is in fact a Moiré pattern due to interference between the graphene and the underlying atom…
Application of an in-plane magnetic field to rippled graphene will make the system be a plane with randomly distributed vector potentials. Massless Dirac fermions carrying charges on graphene are scattered by the vector potentials and magnetoresistance is induced proportional to the square of amplitude of in-plane magnetic field $B_{\parallel}^2$. Recently, Lundeberg and Folk proposed a formula showing dependence of the magnetoresistance on carrier density, in which the coefficient of $B_{\parallel}^2$ is given by a functional of the height-correlation function $c(r)$ of ripples. In the presen…
The amount of rippling in graphene sheets is related to the interactions with the substrate or with the suspending structure. Here, we report on an irreversibility in the response to forces that act on suspended graphene sheets. This may explain why one always observes a ripple structure on suspended graphene. We show that a compression-relaxation mechanism produces static ripples on graphene sheets and determine a peculiar temperature $T_c$, such that for $T<T_c$ the free-energy of the rippled graphene is smaller than that of roughened graphene. We also show that $T_c$ depends on the structur…
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