We investigated the effect of out-of-plane crumpling on the mechanical response of graphene membranes. In our experiments, stress was applied to graphene membranes using pressurized gas while the strain state was monitored through two complementary techniques: interferometric profilometry and Raman spectroscopy. By comparing the data obtained through these two techniques, we determined the geometric hidden area which quantifies the crumpling strength. While the devices with hidden area $\sim0~\%$ obeyed linear mechanics with biaxial stiffness $428\pm10$ N/m, specimens with hidden area in the range $0.5-1.0~\%$ were found to obey an anomalous Hooke's law with an exponent $\sim0.1$.
We report angle-resolved electron transmission measurements through freestanding graphene sheets in the energy range of 18 to 30 eV above the Fermi level. The measurements are carried out in a low-energy electron point source microscope, which allows simultaneously probing the transmission for a large angular range. The characteristics of low-energy electron transmission through graphene depend on its electronic structure above the vacuum level. The experimental technique described here allows mapping the unoccupied band structure of freestanding two-dimensional materials as a function of ener…
In this study, we first show that the argon flow during epitaxial graphene growth is an important parameter to control the quality of the buffer and the graphene layer. Atomic force microscopy (AFM) and low-energy electron diffraction (LEED) measurements reveal that the decomposition of the SiC substrate strongly depends on the Ar mass flow rate while pressure and temperature are kept constant. Our data are interpreted by a model based on the competition of the SiC decomposition rate, controlled by the Ar flow, with a uniform graphene buffer layer formation under the equilibrium process at the…
We perform a femtosecond-THz optical pump-probe spectroscopy to investigate the cooling dynamics of hot carriers in quasi-free standing bilayer epitaxial graphene. We observed longer decay time constants, in the range of 2.6 to 6.4 ps, compared to previous studies on monolayer graphene, which increase nonlinearly with excitation intensity. The increased relaxation times are due to the decoupling of the graphene layer from the SiC substrate after hydrogen intercalation which increases the distance between graphene and substrate. Furthermore, our measurements do not show that the supercollision …
Freestanding graphene membranes were functionalized with SnO2 nanoparticles. A detailed procedure providing uniform coverage and chemical synthesis is presented. Elemental composition was determined using scanning electron microscopy combined with energy dispersive X-ray analysis. A technique called electrostatic-manipulation scanning tunneling microscopy was used to probe the electromechanical properties of functionalized freestanding graphene samples. We found ten times larger movement perpendicular to the plane compared to pristine freestanding graphene, and propose a nanoparticle encapsula…
In this study, we describe a new experimental approach based on constant-current scanning tunneling spectroscopy to controllably and reversibly pull freestanding graphene membranes up to 35 nm from their equilibrium height. In addition, we present scanning tunneling microscopy (STM) images of freestanding graphene membranes with atomic resolution. Atomic-scale corrugation amplitudes 20 times larger than the STM electronic corrugation for graphene on a substrate were observed. The freestanding graphene membrane responds to a local attractive force created at the STM tip as a highly-conductive y…
Knowledge of and control over the curvature of ripples in freestanding graphene are desirable for fabricating and designing flexible electronic devices, and recent progress in these pursuits has been achieved using several advanced techniques such as scanning tunneling microscopy. The electrostatic forces induced through a bias voltage (or gate voltage) were used to manipulate the interaction of freestanding graphene with a tip (substrate). Such forces can cause large movements and sudden changes in curvature through mirror buckling. Here we explore an alternative mechanism, thermal load, to c…
We report subnanometer, high-bandwidth measurements of the out-of-plane (vertical) motion of atoms in freestanding graphene using scanning tunneling microscopy. By tracking the vertical position over a long time period, a 1000-fold increase in the ability to measure space-time dynamics of atomically thin membranes is achieved over the current state-of-the-art imaging technologies. We observe that the vertical motion of a graphene membrane exhibits rare long-scale excursions characterized by both anomalous mean-squared displacements and Cauchy-Lorentz power law jump distributions.
A technique to locally generate mechanical vibrations in freestanding graphene using scanning tunneling microscopy (STM) is presented. The frequency of the mechanical vibrations is tuned over nearly four decades and is centered around 10 Hz. The amplitude of the vibrations also changes over nearly three decades centered on 1 nm. The oscillating motion is generated in two ways: first, by scanning the STM tip on the surface and second, by scanning the bias voltage on the STM tip. The frequency and amplitude of the displaced freestanding graphene is quantitatively transformed to the frequency and…
Systematic displacement measurements of freestanding graphene as a function of applied bias voltage and tunneling current setpoint using scanning tunneling microscopy (STM) are presented. When the bias voltage is increased the graphene approaches the STM tip, while, on the other hand, when the tunneling current is increased the graphene contracts from the STM tip. To understand the role of the bias voltage, we quantitatively model the attractive force between the tip and the sample using electrostatics. For the tunneling current, we qualitatively model the contraction of the graphene using ent…
Freestanding graphene membranes were successfully functionalized with platinum nanoparticles (Pt NPs). High-resolution transmission electron microscopy revealed a homogeneous distribution of single-crystal Pt NPs that tend to exhibit a preferred orientation. Unexpectedly, the NPs were also found to be partially exposed to the vacuum with the top Pt surface raised above the graphene substrate, as deduced from atomic-scale scanning tunneling microscopy images and detailed molecular dynamics simulations. Local strain accumulation during the growth process is thought to be the origin of the NP sel…
Single-crystal carbon nanomaterials have led to great advances in nanotechnology. The first single-crystal carbon nanomaterial, fullerene, was fabricated in a zero-dimensional form. One-dimensional carbon nanotubes and two-dimensional graphene have since followed and continue to provide further impetus to this field. In this study, we fabricated designed three-dimensional (3D) single-crystal carbon architectures by using silicon carbide templates. For this method, a designed 3D SiC structure was transformed into a 3D freestanding single-crystal carbon structure that retained the original SiC s…
We report experimentally and theoretically the behavior of freestanding graphene subject to bombardment of energetic ions, investigating the ability of large-scale patterning of freestanding graphene with nanometer sized features by focused ion beam technology. A precise control over the He+ and Ga+ irradiation offered by focused ion beam techniques enables to investigate the interaction of the energetic particles and graphene suspended with no support and allows determining sputter yields of the 2D lattice. We find strong dependency of the 2D sputter yield on the species and kinetic energy of…
Intrinsic ripples in freestanding graphene have been exceedingly difficult to study. Individual ripple geometry was recently imaged using scanning tunneling microscopy, but these measurements are limited to static configurations. Thermally-activated flexural phonon modes should generate dynamic changes in curvature. Here we show how to track the vertical movement of a one-square-angstrom region of freestanding graphene using scanning tunneling microscopy, thereby allowing measurement of the out-of-plane time trajectory and fluctuations over long time periods. We also present a model from elast…
We apply scanning tunneling spectroscopy to determine the bandgaps of mono-, bi- and trilayer MoS$_2$ grown on a graphene single crystal on Ir(111). Besides the typical scanning tunneling spectroscopy at constant height, we employ two additional spectroscopic methods giving extra sensitivity and qualitative insight into the $k$-vector of the tunneling electrons. Employing this comprehensive set of spectroscopic methods in tandem, we deduce a bandgap of $2.53\pm0.08$ eV for the monolayer. This is close to the predicted values for freestanding MoS$_2$ and larger than is measured for MoS$_2$ on o…
Recent theoretical considerations have demonstrated that freestanding graphene doped with alkali metals (AC x ) supports strong Dirac and weak acoustic plasmons. Here we show that when AC x is deposited on a metallic surface, the intense Coulomb screening completely washes out these collective modes. However, even small increase of separation between AC x and metallic surface causes recovery of AC x plasmonic properties and especially the enhancement of acoustic plasmons intensities not present in the freestanding case. We further provide the physical background of these intriguing phenomena. …
In neutral graphene, two prominent cusps known as Kohn anomalies are found in the phonon dispersion of the highest optical phonon at $q=Γ$ (LO branch) and $q=K$ (TO branch), reflecting a significant electron-phonon coupling to undoped Dirac electrons. In this work, high-resolution electron energy loss spectroscopy is used to measure the phonon dispersion around the $Γ$ point in quasi-freestanding graphene epitaxially grown on Pt(111). The Kohn anomaly for the LO phonon is observed at finite momentum $q\sim2k_F$ from $Γ$, with a shape in excellent agreement with the theory and consistent with k…
At room temperature, micron-sized sheets of freestanding graphene are in constant motion even in the presence of an applied bias voltage. We quantify the out-of-plane movement by collecting the displacement current using a nearby small-area metal electrode and present a Langevin model for the motion coupled to a circuit containing diodes. The system reaches thermal equilibrium and the rates of heat, work, and entropy production tend quickly to zero. However, there is power generated by graphene which is equal to the power dissipated by the load resistor. The exact power formula is similar to N…
Lateral heterostructures of two-dimensional materials may exhibit various intriguing emergent properties. Yet when specified to the orientationally aligned heterojunctions of zigzag graphene and hexagonal boron nitride (hBN) nanoribbons, realizations of the high expectations on their properties encounter two standing hurtles. First, the rapid accumulation of strain energy prevents large- scale fabrication. Secondly, the pronounced half-metallicity predicted for freestanding graphene nanoribbons is severely suppressed. By properly tailoring orientational misalignment between zigzag graphene and…
In search for a new generation of spintronics hardware, material candidates for room temperature quantum spin Hall effect (QSHE) have become a contemporary focus of investigation. Inspired by the original proposal for QSHE in graphene, several heterostructures have been synthesized, aiming at a hexagonal monolayer of heavier group IV elements in order to promote the QSHE bulk gap via increased spin-orbit coupling. So far, however, the monolayer/substrate coupling, which can manifest itself in strain, deformation, and hybridization, has proven to be detrimental to the aspired QSHE conditions fo…
A theory of the nonlinear plasma waves in graphene is developed in the nonperturbative regime. The influence of strong electric fields on the position and linewidth of plasma resonances in the far-infrared transmission experiments, as well as on the wavelength and the propagation length in the scanning near-field optical microscopy experiments is studied. The theory shows that the fields of order of a few to a few tens of kV/cm should lead to a red shift and broadening of plasma resonances in the first type and to a reduction of the wavelength and the propagation length in the second type of e…
We calculate the Casimir interaction of two freestanding graphene samples under uniaxial strain. Our approach fully takes retardation and dispersion into account and is based on quantum field theoretical expressions for conductivities in terms of the polarization operator. The force shows a rather weak dependence on the realistic values of strain, changing just by a few percent in its maximum as compared to the non-strained case.
Pristine, undoped graphene has a constant absorption of 2.3 % across the visible to near-infrared (VIS-NIR) region of the electromagnetic spectrum. Under certain conditions, such as nanostructuring and intense gating, graphene can interact more robustly with VIS-NIR light and exhibit a large nonlinear optical response. Here, we explore the optical properties of graphene/LaAlO$_3$/SrTiO$_3$ nanostructures, where nanojunctions formed at the LaAlO$_3$/SrTiO$_3$ interface enable large (~10$^8$ V/m) electric fields to be applied to graphene over a scale of ~10 nm. Upon illumination with ultrafast V…
The nonlinear frequencies of pre-stressed graphene-based structures, such as flat graphene sheets and carbon nanotubes, are calculated. These structures are modeled with a nonlinear hyperelastic shell model. The model is calibrated with quantum mechanics data and is valid for high strains. Analytical solutions of the natural frequencies of various plates are obtained for the Canham bending model by assuming infinitesimal strains. These solutions are used for the verification of the numerical results. The performance of the model is illustrated by means of several examples. Modal analysis is pe…
Intense efforts have been made in recent years to realize nonlinear optical interactions at the single-photon level. Much of this work has focused on achieving strong third-order nonlinearities, such as by using single atoms or other quantum emitters while the possibility of achieving strong second-order nonlinearities remains unexplored. Here, we describe a novel technique to realize such nonlinearities using graphene, exploiting the strong per-photon fields associated with tightly confined graphene plasmons in combination with spatially nonlocal nonlinear optical interactions. We show that i…
We theoretically examine the role of Kerr nonlinearities for graphene plasmonics in nanostructures, specifically in nanoribbons. The nonlinear Kerr interaction is included semiclassically in the intraband approximation. The resulting electromagnetic problem is solved numerically by self-consistent iteration with linear steps using a real-space discretization. We derive a simple approximation for the resonance shifts in general graphene nanostructures, and obtain excellent agreement with numerics for moderately high field strengths. Near plasmonic resonances the nonlinearities are strongly enha…
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…
The high flexibility, impermeability and strength of graphene membranes are key properties that can enable the next generation of nanomechanical sensors. However, for capacitive pressure sensors the sensitivity offered by a single suspended graphene membrane is too small to compete with commercial sensors. Here, we realize highly sensitive capacitive pressure sensors consisting of arrays of nearly ten thousand small, freestanding double-layer graphene membranes. We fabricate large arrays of small diameter membranes using a procedure that maintains the superior material and mechanical propertie…
Vertical stacking of monolayers via van der Waals assembly is an emerging field that opens promising routes toward engineering physical properties of two-dimensional (2D) materials. Industrial exploitation of these engineering heterostructures as robust functional materials still requires bounding their measured properties so to enhance theoretical tractability and assist in experimental designs. Specifically, the short-range attractive van der Waals forces are responsible for the adhesion of chemically inert components and are recognized to play a dominant role in the functionality of these s…
Graphene is a two-dimensional material with strongly nonlinear electrodynamics and optical properties. We present some of our recent theoretical results on the quantum and non-perturbative quasi-classical theories of nonlinear effects in graphene, influence of substrates on graphene nonlinearities, plasma oscillations in graphene in the nonlinear regime and other effects.
We analyze intrinsic nonlinearities in two-dimensional polaritonic materials interacting with an optical wave. Focusing on the case of graphene, we show that the second-order nonlinear optical conductivity due to carrier density fluctuations associated with the excitation of a plasmon polariton is closely related to the ponderomotive force due to the oscillating optical field. This relation is first established through an elegant thermodynamic approach for a Drude-like plasma, in the frequency range where intraband scattering is the dominant contribution to conductivity. Subsequently, we exten…
Graphene is an atomically thin two-dimensional material demonstrating strong optical nonlinearities including harmonics generation, four wave mixing, Kerr and other nonlinear effects. In this paper we theoretically analyze the optical heterodyne detection (OHD) technique of measuring the optical Kerr effect (OKE) in two-dimensional crystals and show how to relate the quantities measured in such experiments with components of the third-order conductivity tensor $σ^{(3)}_{αβγδ}(ω_1,ω_2,ω_3)$ of the two-dimensional crystal. Using results of a recently developed quantum theory of the third-order n…
We present a novel methodology to synthesize two-dimensional (2D) lateral heterostructures of graphene and MoS2 sheets with molecular carbon nanomembranes (CNMs), which is based on electron beam induced stitching. Monolayers of graphene and MoS2 were grown by chemical vapor deposition (CVD) on copper and SiO2 substrates, respectively, transferred onto gold/mica substrates and patterned by electron beam lithography or photolithography. Self-assembled monolayers (SAMs) of aromatic thiols were grown on the gold film in the areas where the 2D materials were not present. An irradiation with a low e…
Nonlinearities are inherent to the dynamics of two-dimensional materials. Phenomena like intermodal coupling already arise at amplitudes of only a few nanometers, and a range of unexplored effects still awaits to be harnessed. Here, we demonstrate a route for generating mechanical frequency combs in graphene resonators undergoing symmetry-breaking forces. We use electrostatic force to break the membrane's out-of-plane symmetry and tune its resonance frequency towards a two-to-one internal resonance, thus achieving strong coupling between two of its mechanical modes. When increasing the drive l…
We present a general formulation to calculate the dynamic optical conductivity, beyond the linear response regime, of any electronic system whose quasiparticle dispersion is described by a two band model. Our phenomenological model is based on the optical Bloch equations. In the steady state regime it yields an analytic solution for the population inversion and the interband coherence, which are nonlinear in the optical field intensity, including finite doping and temperature effects. We explicitly show that the optical nonlinearities are controlled by a single dimensionless parameter which is…
