6 pages, 6 figures, paper is accompanied by supplementary material available in Ancillary files or inside the directory anc/ included in the source of this manuscript
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 scales, yet rougher at short scales. The angular fluctuation distribution of the normals can be quantitatively described by coexistence of two Gaussians, one classical strongly T-dependent and one quantum about $2^{\circ}$ wide, of zero-point character. The quantum evolution of ripple-induced height and angular spread should be observable in electron diffraction in graphene and other two-dimensional materials like MoS$_2$, bilayer graphene, boron nitride, etc.
Intrinsic ripples with various configurations and sizes were reported to affect the physical and chemical properties of 2D materials. By performing molecular dynamics simulations and theoretical analysis, we use two geometric models of the ripple shape to explore numerically the distribution of ripples in graphene membrane. We focus on the ratio of ripple height to its diameter (t/D) which was recently shown to be the most relevant for chemical activity of graphene membranes. Our result demonstrates that the ripple density decreases as the coefficient t/D increases, in a qualitative agreement …
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 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…
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…
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…
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 …
We study the mechanism of wrinkling of suspended graphene, by means of atomistic simulations. We argue that the structural instability under edge compression is the essential physical reason for the formation of periodic ripples in graphene. The ripple wavelength and out-of-plane amplitude are found to obey 1/4-power scaling laws with respect to edge compression. Our results also show that parallel displacement of the clamped boundaries can induce periodic ripples, with oscillation amplitude roughly proportional to the 1/4 power of edge displacement. The results are fundamental to graphene's a…
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.
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 …
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 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…
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…
The stability of two-dimensional (2D) layers and membranes is subject of a long standing theoretical debate. According to the so called Mermin-Wagner theorem, long wavelength fluctuations destroy the long-range order for 2D crystals. Similarly, 2D membranes embedded in a 3D space have a tendency to be crumpled. These dangerous fluctuations can, however, be suppressed by anharmonic coupling between bending and stretching modes making that a two-dimensional membrane can exist but should present strong height fluctuations. The discovery of graphene, the first truly 2D crystal and the recent exper…
We report here on a method to fabricate and characterize highly perfect, periodically rippled graphene monolayers and islands, epitaxially grown on single crystal metallic substrates under controlled UHV conditions. The periodicity of the ripples is dictated by the difference in lattice parameters of graphene and substrate, and, thus, it is adjustable. We characterize its perfection at the atomic scale by means of STM and determine its electronic structure in the real space by local tunnelling spectroscopy. There are periodic variations in the geometric and electronic structure of the graphene…
Theoretical research has predicted that ripples of graphene generates effective gauge field on its low energy electronic structure and could lead to zero-energy flat bands, which are the analog of Landau levels in real magnetic fields. Here we demonstrate, using a combination of scanning tunneling microscopy and tight-binding approximation, that the zero-energy Landau levels with vanishing Fermi velocities will form when the effective pseudomagnetic flux per ripple is larger than the flux quantum. Our analysis indicates that the effective gauge field of the ripples results in zero-energy flat …
Graphene outstanding properties directly come from its pecular electronic structure and thus from the honeycomb lattice symmetry. The way interaction with the substrate impact this lattice is of primary importance. This is peculiarly true for epitaxial graphene because of the SiC substrate. The advantage of this system that produces macroscopic samples in registry with a substrate could turn to a major drawback if the graphene lattice reveals to be strongly distorted. Extensive ab initio calculations supported by Scanning Tunneling Microscopy experiments, demonstrate here that the substrate in…
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…
Emergence of a Helical Metal in Rippled Ultrathin Topological Insulator Sb\textsubscript{2}Te\textsubscript{3} on Graphene
Universidad de Chile（智利大学）；CEDENNA；Universidad Adolfo Ibáñez（阿道夫·伊巴涅斯大学）；Stanford University（斯坦福大学）；Universidad Técnica Federico Santa María（弗雷德里科·圣塔玛利亚技术大学）
Ever since the discovery of graphene and subsequent explosion of interest in single atom thick materials, studying their mechanical properties has been an active area of research. New length scales often necessitate a rethinking of physical laws, making such studies crucial for understanding and ultimately utilizing novel material properties. Here we report on the investigation of nanoscale periodic ripples in suspended, single layer graphene sheets by scanning tunneling microscopy and atomistic scale simulations. Unlike the sinusoidal ripples found in classical fabrics, we find that graphene …
We study the changes induced by the effective gauge field due to ripples on the low energy electronic structure of graphene. We show that zero energy Landau levels will form, associated to the smooth deformation of the graphene layer, when the height corrugation, $h$, and the length of the ripple, $l$, are such that $h^2 / (l a) \gtrsim 1$, where $a$ is the lattice constant. The existence of localized levels gives rise to a large compressibility at zero energy, and to the enhancement of instabilities arising from electron-electron interactions including electronic phase separation. The combine…
The nature of its intrinsic ripples is the key factor for understanding the stability of suspended graphene, and for unraveling the long-standing theoretical debate of the existence of low-dimensional crystalline state. The rippling morphology of graphene, discovered also in other 2D materials, has a profound impact on its electronic, mechanical and chemical properties. Actually, before the discovery of graphene, rippling phenomena are widely observed: for example, the roughing transition of crystalline interface, the rippled phase in biomembrane, and crumpling of flexible sheet polymers model…
The electromagnetic radiation of electrons in the corrugated graphene in the presence of the transport electric current in the ballistic regime is studied. Radiation of the similar nature can be observed in undulator and wiggler. We considered here an impact of the ripples in the monolayer graphene on its electromagnetic properties. The electromagnetic radiation was actually calculated with a use of the standard electromagnetic theory. Two cases, of regular and random structures are analyzed. Nonlinear relation between the random height function h(x,y) and the gauge field is shown to create th…
We study the effect of the curved ripples observed in the free standing graphene samples on the electronic structure of the system. We model the ripples as smooth curved bumps and compute the Green's function of the Dirac fermions in the curved surface. Curved regions modify the Fermi velocity that becomes a function of the point on the graphene surface and induce energy dependent oscillations in the local density of states around the position of the bump. The corrections are estimated to be of a few percent of the flat density at the typical energies explored in local probes such as scanning …
Meyer et al (2007) found that free-standing graphene sheets, just one atom thick, display spontaneous ripples. The ripples are of order 2-20 Å high and 20-200 Å wide. The sheets in which they appear are only one atom thick, and extend for around 5000 Å through vacuum between metal struts that support them. Other groups have since created free-standing graphene as well (Garcia-Sanchez 2008 ; Bolotin et al. 2008), and similar ripples have been found for graphene on a glass substrate (Geringer et al. 2008). Here we show that these ripples can be explained as a consequence of adsorbed molecules si…
The surface structure of Few-Layer Graphene (FLG) epitaxially grown on the C-face of SiC has been investigated by TM-AFM in ambient air and upon interaction with diluted aqueous solutions of bio-organic molecules (dimethyl sulfoxide, DMSO, and L-Methionine). On pristine FLG we observe nicely ordered, three-fold oriented rippled domains, with a 4.7+/-0.2 nm periodicity (small periodicity, SP) and a peak-to-valley distance in the range 0.1-0.2 nm. Upon mild interaction of the FLG surface with the molecular solution, the ripple periodicity relaxes to 6.2+/-0.2 nm (large periodicity, LP), while th…
3D topological insulators, similar to the Dirac material graphene, host linearly dispersing states with unique properties and a strong potential for applications. A key, missing element in realizing some of the more exotic states in topological insulators is the ability to manipulate local electronic properties. Analogy with graphene suggests a possible avenue via a topographic route by the formation of superlattice structures such as a moiré patterns or ripples, which can induce controlled potential variations. However, while the charge and lattice degrees of freedom are intimately coupled in…
We have carried out scanning tunneling spectroscopy measurements on exfoliated monolayer graphene on SiO$_2$ to probe the correlation between its electronic and structural properties. Maps of the local density of states are characterized by electron and hole puddles that arise due to long range intravalley scattering from intrinsic ripples in graphene and random charged impurities. At low energy, we observe short range intervalley scattering which we attribute to lattice defects. Our results demonstrate that the electronic properties of graphene are influenced by intrinsic ripples, defects and…
Only one atom thick and not inclined to lattice defects, graphene represents the ultimate crystalline membrane. However, its structure reveals unique features not found in other crystalline membranes, in particular the existence of ripples with wavelength of 100-300 Angstroms. Here, I trace the origin of this difference to the free electrons in the membrane. The deformation energy of the lattice creates a coupling between charge fluctuations and the structure, resulting in ripples on the membrane, correlated with charge inhomogeneities. In graphene this mechanism reproduces the experimental re…
Simulation of chemical activity of corrugated graphene within density functional theory predicts an enhancement of its chemical activity if the ratio of height of the corrugation (ripple) to its radius is larger than 0.07. Further growth of the curvature of the ripples results in appearance of midgap states which leads to an additional strong increase of chemisororption energy. These results open a way for tunable functionalization of graphene, namely, depending of curvature of the ripples one can provide both homogeneous (for small curvatures) and spot-like (for large curvatures) functionaliz…
We have studied temperature dependences of electron transport in graphene and its bilayer and found extremely low electron-phonon scattering rates that set the fundamental limit on possible charge carrier mobilities at room temperature. Our measurements have shown that mobilities significantly higher than 200,000 cm2/Vs are achievable, if extrinsic disorder is eliminated. A sharp (threshold-like) increase in resistivity observed above approximately 200K is unexpected but can qualitatively be understood within a model of a rippled graphene sheet in which scattering occurs on intra-ripple flexur…
We delineated the role of thermally excited ripples on thermal expansion properties of 2D honeycomb materials (free-standing graphene, 2D h-BN, and ML-MoS2), by explicitly carrying out three-dimensional (3D) and two-dimensional (2D) molecular dynamics simulations. In 3D simulations, the in-plane lattice parameter (a-lattice) of graphene and 2D h-BN shows thermal contraction over a wide range of temperatures and exhibits a strong system size dependence. The 2D simulations of the very same system show a reverse trend, where the a-lattice is expanding in the whole computed temperature range. Cont…
Structural and electronic properties of the all-Si analogue of graphene, silicene have elucidated through DFT calculations. Silicene differs considerably from graphene in being `chair-type' puckered in each 6-membered ring which leads to ordered ripples across the surface. Binding energies suggest stability for such rippled silicenes and are predicted to behave as a finite gap semi-conductor with electron-hole symmetry quenched. Inter-layer coupling between the silicenes is suggested as the mechanism for the formation of the bulk-Si in its only known diamond form.
We discuss various scattering mechanisms for Dirac fermions in single-layer graphene. It is shown that scattering on a short-range potential (due to, for example, neutral impurities) is mostly irrelevant for electronic quality of graphene, which is likely to be controlled by charged impurities and ripples (microscopic corrugations of a graphene sheet). The latter are an inherent feature of graphene due to its two-dimensional nature and can also be an important factor in defining the electron mean free path. We show that certain types of ripples create a long-range scattering potential, similar…
Graphene is locally two-dimensional but not flat. Nanoscale ripples appear in suspended samples and rolling-up often occurs when boundaries are not fixed. We address this variety of graphene geometries by classifying all ground-state deformations of the hexagonal lattice with respect to configurational energies including two- and three-body terms. As a consequence, we prove that all ground-state deformations are either periodic in one direction, as in the case of ripples, or rolled up, as in the case of nanotubes.
With use of simulated friction force microscopy, we study the effect of varying temperature on the frictional properties of suspended graphene. In contrast with the theory of thermally activated friction on the surfaces of three-dimensional materials, kinetic friction is demonstrated to both locally increase and decrease with increasing temperature, depending on sample size, scanning tip diameter, scanning rate, and the externally applied normal load. We attribute the observed effects to the thermally excited flexural ripples intrinsically present in graphene, demonstrating a unique case of te…
Graphene is the stiffest material known so far but, due to its one-atom thickness, it is also very bendable. Consequently, free-standing graphene exhibit ripples that has major effects on its elastic properties. Here we will summarize three experiments where the influence of rippling is essential to address the results. Firstly, we observed that atomic vacancies lessen the negative thermal expansion coefficient of free-standing graphene. We also observed an increase of the Young's modulus with global applied strain and with the introduction of small density defects that we attributed to the de…
In this report we present the temperature evolution of magnetic coercivity of graphene oxide (GO) and reduced graphene oxide (RGO). We report an anamolous decrease in coercivity of GO and RGO with decreasing temperature. We could explain this anamolous behavior invoking the inherent presence of ripple in graphene. We observe antiferromagnetic and ferromagnetic behavior at room temperature for GO and RGO respectively, but at low temperatures both shows paramagnetic behavior.
The melting of fluorographene is very unusual and depends strongly on the degree of fluorination. For temperatures below 1000 K, fully fluorinated graphene (FFG) is thermo-mechanically more stable than graphene but at T$_m\approx$2800 K FFG transits to random coils which is almost twice lower than the melting temperature of graphene, i.e. 5300 K. For fluorinated graphene (PFG) up to 30 % ripples causes detachment of individual F-atoms around 2000 K while for 40-60 % fluorination, large defects are formed beyond 1500 K and beyond 60% of fluorination F-atoms remain bonded to graphene until melti…
One of the salient features of graphene is the very high carrier mobility that implies tremendous potential for use in electronic devices. Unfortunately, transport measurements find the expected high mobility only in freely suspended graphen. When supported on a surface, graphene shows a strongly reduced mobility, and an especially severe reduction for temperatures above 200 K. A temperature-dependent mobility reduction could be explained by scattering of carriers with phonons, but this is expected to be weak for pristine, weakly-doped graphene. The mobility reduction has therefore been ascrib…
Thermal fluctuations of single layer hydrogenated graphene (graphane) are investigated using large scale atomistic simulations. By analyzing the mean square value of the height fluctuations $<h^2>$ and the height-height correlation function $H(q)$ for different system sizes and temperatures we show that hydrogenated graphene is an un-rippled system in contrast to graphene. The height fluctuations are bounded, which is confirmed by a $ H(q) $ tending to a constant in the long wavelength limit instead of showing the characteristic scaling law $ q^{4-η} (η\simeq 0.85)$ predicted by membrane theor…
