Meyer等人（2007）发现，只有一个原子厚的独立石墨烯片显示出自发的波纹。波纹的数量级为2-20厘米高，20-200厘米宽。它们出现的薄片只有一个原子厚，通过支撑它们的金属支柱之间的真空延伸约5000埃。此后，其他研究小组也创造了独立的石墨烯（Garcia-Sanchez 2008 ; Bolotin et al. 2008），并在玻璃基板上发现了石墨烯的类似波纹（Geringer et al. 2008）。在这里，我们表明，这些波纹可以解释为吸附分子坐在随机网站的后果。被吸附物使碳原子间的键稍微变长。然后，静态吸附是由导致叶片弯曲的机制引起的;吸附物覆盖率约为20%所引起的吸附与实验观察一致。我们解释了为什么这种机制更有可能解释波纹比热波动或Mermin-Wagner定理，这是以前被调用。
英文摘要：
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 sitting on random sites. The adsorbates cause the bonds between carbon atoms to lengthen slightly. Static buckles then result from a mechanism like the one that leads to buckling of leaves; buckles caused by roughly 20% coverage of adsorbates are consistent with experimental observations. We explain why this mechanism is more likely to explain rippled than thermal fluctuations or the Mermin-Wagner theorem, which have previously been invoked.
The allotropes of a new layered material, phosphorus carbide (PC), have been predicted recently and a few of these predicted structures have already been successfully fabricated. Herein, by using first-principles calculations we investigated the effects of rippling a PC monolayer, one of the most stable modifications of layered PC, under large compressive strains. Similar to phosphorene, layered PC was found to have the extraordinary ability to bend and form ripples with large curvatures under a sufficiently large strain applied along its armchair direction. The band gap size, workfunction, an…
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…
We calculate Root Mean Square (RMS) deviations from equilibrium for atoms in a two dimensional crystal with local (e.g. covalent) bonding between close neighbors. Large scale Monte Carlo calculations are in good agreement with analytical results obtained in the harmonic approximation. When motion is restricted to the plane, we find a slow (logarithmic) increase in fluctuations of the atoms about their equilibrium positions as the crystals are made larger and larger. We take into account fluctuations perpendicular to the lattice plane, manifest as undulating ripples, by examining dual layer sys…
We introduce an interatomic potential for hexagonal boron nitride (hBN) based on the Gaussian approximation potential (GAP) machine learning methodology. The potential is based on a training set of configurations collected from density functional theory (DFT) simulations and is capable of treating bulk and multilayer hBN as well as nanotubes of arbitrary chirality. The developed force field faithfully reproduces the potential energy surface predicted by DFT while improving the efficiency by several orders of magnitude. We test our potential by comparing formation energies, geometrical properti…
In mechanical systems, Maxwell-Betti reciprocity means that the displacement at point B in response to a force at point A is the same as the displacement at point A in response to the same force applied at point B. Because the notion of reciprocity is general, fundamental, and is operant for other physical systems like electromagnetics, acoustics, and optics, there is significant interest in understanding systems that are not reciprocal, or exhibit non-reciprocity. However, most studies of non-reciprocity have occurred in bulk-scale structures for dynamic problems involving time reversal symme…
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…
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…
Large-scale atomistic simulations using the reactive empirical bond order force field approach is implemented to investigate thermal and mechanical properties of single-layer (SL) and multi-layer (ML) molybdenum disulfide (MoS$_2$). The amplitude of the intrinsic ripples of SL-MoS$_2$ are found to be smaller than those exhibited by graphene (GE). Furthermore, because of the van der Waals interaction between layers, the out-of-plane thermal fluctuations of ML-MoS$_2$ decreases rapidly with increasing number of layers. This trend is confirmed by the buckling transition due to uniaxial stress whi…
Conventional three-dimensional crystal lattices are terminated by surfaces, which can demonstrate complex rebonding and rehybridisation, localised strain and dislocation formation. Two dimensional crystal lattices, of which graphene is the archetype, are terminated by lines. The additional available dimension at such interfaces opens up a range of new topological interface possibilities. We show that graphene sheet edges can adopt a range of topological distortions depending on their nature. Rehybridisation, local bond reordering, chemical functionalisation with bulky, charged, or multi-functi…
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.
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…
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 …
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…
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…
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…
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 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…
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…
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 …
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…
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…
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…
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…
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 …
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.
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）
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…
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…
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…
Giant Intrinsic Carrier Mobilities in Graphene and Its Bilayer
我们研究了石墨烯及其双层中电子输运的温度依赖性，发现极低的电子-声子散射率，这对室温下可能的载流子迁移率设置了基本限制。我们的测量表明，迁移率显着高于200，000 cm 2/Vs是可以实现的，如果外在的障碍被消除。在大约200 K以上观察到的电阻率的急剧（阈值状）增加是出乎意料的，但是可以在波纹石墨烯片的模型内定性地理解，其中散射发生在波纹内弯曲声子上。
We present a new semi-empirical potential for graphene, which includes also an out-of-plane energy term. This novel potential is developed from density functional theory (DFT) calculations for small numbers of atoms, and can be used for configurations with millions of atoms. Our simulations show that buckling caused by typical defects such as the Stone-Wales (SW) defect extends to hundreds of nanometers. Surprisingly, this long-range relaxation lowers the defect formation energy dramatically - by a factor of $2$ or $3$ - implying that previously published DFT-calculated defect formation energi…
Fabrication of ordered structures at the nanoscale limit poses a cornerstone challenge for modern technologies. In this work we show how naturally occurring moiré patterns in Ir(111)-supported graphene template the formation of 2D ordered arrays of substitutional boron species. A complementary experimental and theoretical approach provides a comprehensive description of the boron species distribution, bonding configurations, interfacial interaction with Ir(111) and the impact on graphene's electronic structure. Atomically-resolved scanning tunnelling microscopy images and density functional th…
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.
Using the recently developed technique of microsoldering, we perform a systematic transport study of the influence of PMMA on graphene flakes revealing a doping effect of up to 3.8x10^12 1/cm^2, but a negligible influence on mobility and gate voltage induced hysteresis. Moreover, we show that the microsoldered graphene is free of contamination and exhibits a very similar intrinsic rippling as has been found for lithographically contacted flakes. Finally, we demonstrate a current induced closing of the previously found phonon gap appearing in scanning tunneling spectroscopy experiments, strongl…
We have mapped the Dirac point in exfoliated monolayer and bilayer graphene using spatially resolved scanning tunneling spectroscopy (STS) measurements at low temperature. The Dirac point shifts in energy at different locations in graphene. However, a cross correlation with the topography shows no correlation indicating that topographic features such as ripples are not the primary source of the variation. Rather, we attribute the shift of the Dirac point to random charged impurities located near the graphene. Our findings emphasize the need to advance exfoliated graphene sample preparation to …
We set up an evolutionary algorithm combined with density functional tight-binding (DFTB) calculations to investigate hydrogen adsorption on flat graphene and graphene monolayers curved over substrate steps. During the evolution, candidates for the new generations are created by adsorption of an additional hydrogen atom to the stable configurations of the previous generation, where a mutation mechanism is also incorporated. Afterwards a two-stage selection procedure is employed. Selected candidates act as the parents of the next generation. In curved graphene, the evolution follows a similar p…
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…
