We study the effect of anisotropy (strain) on dynamical gap generation in graphene. We work with a low energy effective theory obtained from a tight-binding Hamiltonian expanded around the Dirac points in momentum space. We use a non-perturbative Schwinger-Dyson approach and calculate a coupled set of five momentum dependent dressing functions. Our results show that the critical coupling depends only weakly on the anisotropy parameter, and increases with greater anisotropy.
Owing to its array of unique properties, graphene is a promising material for a wide variety of applications. Being two-dimensional, the properties of graphene are also easily tuned via proximity to other materials. In this work, we investigate the possibility of inducing electrical and optical anisotropy in graphene by interfacing it with other anisotropic carbon systems, including nanoporous graphene and arrays of graphene nanoribbons. We find that such materials do indeed induce such anisotropy in graphene, while also preserving the unique properties offered by graphene's Dirac band structu…
We report on fundamental aspects of spin dynamics in graphene interfaced with transition metal dichalcogenides (TMDCs). By using realistic models derived from first principles we compute the spin lifetime anisotropy, defined as the ratio of lifetimes for spins pointing out of the graphene plane to those pointing in the plane. In the presence of strong intervalley scattering the anisotropy can reach unprecedented values of tens to hundreds, while it reduces to 1/2 for weak disorder. This behavior is mediated by spin-valley locking, which is strong in TMDCs and is imprinted onto graphene. Such g…
We show theoretically how constant-energy maps of the angle-resolved photoemission intensity can be used to test wave function symmetry in graphene. For monolayer graphene, we demonstrate that the observed anisotropy of ARPES spectra is a manifestation of what has been recently branded as electronic chirality. For bilayer graphene, we show that the anisotropy of the constant-energy maps may be used to extract information about the magnitude and sign of interlayer coupling parameters and about symmetry breaking inflicted on a bilayer by the underlying substrate.
The independent predictions of edge ferromagnetism and the Quantum Spin Hall phase in graphene have inspired the quest of other two dimensional honeycomb systems, such as silicene, germanene, stanene, iridiates, and organometallic lattices, as well as artificial superlattices, all of them with electronic properties analogous to those of graphene, but much larger spin orbit coupling. Here we study the interplay of ferromagnetic order and spin orbit interactions at the zigzag edges of these graphene-like systems. We find an in-plane magnetic anisotropy that opens a gap in the otherwise conductin…
We study the effect of anisotropy on phase transitions in graphene. We work with an low energy effective field theory which is strongly coupled, and solve a coupled set of Schwinger-Dyson equations. We show that the effect of anisotropy is to reduce the critical coupling.
We report strongly enhanced perpendicular magnetic anisotropy (PMA) of Co films by graphene coating from both first-principles and experiments. Our calculations show that graphene can dramatically boost the surface anisotropy of Co films up to twice the value of its pristine counterpart and can extend the out-of-plane effective anisotropy up to unprecedented thickness of 25~Å. These findings are supported by our experiments on graphene coating on Co films grown on Ir substrate. Furthermore, we report layer-resolved and orbital-hybridization-resolved anisotropy analysis which help understanding…
This article is a continuation of our previous studies of the frictional anisotropy of metal nanoparticles on the surface of a graphene substrate for other temperature conditions. The friction force acting on palladium nanoparticles on a graphene sheet in various lateral directions is investigated using classical molecular dynamics modelling. Anisotropy is studied at high sliding speeds of nanoparticles consisting of 10000 atoms on the surface of graphene. The effect of incommensurability and short-range order of the contact surfaces of nanoparticles, as well as the graphene deformation lead t…
Stone-Wales (SW) defects, analogous to dislocations in crystals, play an important role in mechanical behavior of $sp^2$-bonded carbon based materials. Here, we show using first-principles calculations that a marked anisotropy in the interaction among the SW defects has interesting consequences when such defects are present near the edges of a graphene nano-ribbon: depending on their orientation with respect to edge, they result in compressive or tensile stress, and the former is responsible to depression or warping of the graphene nano-ribbon. Such warping results in delocalization of electro…
We report on the properties of incompressible states of Dirac fermions in graphene in the presence of anisotropic interactions and a quantizing magnetic field. We introduce the necessary formalism to incorporate the anisotropy in the system. The incopmpressible state in graphene is found to survive the anisotropy upto a critical value of the anisotropy parameter. The anisotropy also introduces two branches in the collective excitations of the corresponding Laughlin state. It strongly influences the short-range behavior of the pair-correlation functions in the incompressible ground state.
We derive the analytical expressions for the thermal Casimir-Polder energy and force between a spheroidal nanoparticle above a semi-infinite material and a graphene covered interface. We analyze in detail the Casimir-Polder force between a gold nanoparticle and a single sheet of pristine graphene focusing on the impact of anisotropy. We show that the effect of anisotropy, i.e. the shape and orientation of the spheroidal nanoparticle, has a much larger influence on the force than the tunability of graphene. The effect of tuning and anisotropy both add up such that we observe a force which is be…
Monolayer graphene and bilayer graphene have strikingly different properties. One such difference is the shape of the Fermi surface. Although anisotropic band structures can be detected in optical measurements, they have so far been difficult to detect in transport experiments on twodimensional materials. Here we describe a ballistic transport experiment using high-quality graphene that revealed Fermi surface anisotropy in the magnetoresistance. The shape of the Fermi surface is closely related with the cyclotron orbit in real space. Electron trajectories in samples with triangular lattices of…
Friction force acting on silver and nickel nanoparticles sheared on a graphene sheet in different lateral directions is investigated using classical molecular dynamics simulations. The results reveal the existence of frictional anisotropy for both metals. In most cases, the maximum value of the friction force is about two times larger than the minimum one. The form of dependencies of instantaneous values of the friction force components on the corresponding lateral components of the position of the centre of mass of the nanoislands strongly depends on sliding direction, varying between the saw…
A graphene quantum dot under intense ac field and static low magnetic field is investigated. From a tight-binding perspective, applying a Fourier-Floquet transformation and renormalization process, we observe that graphene -intrinsically anisotropic- reveals field polarization signatures in the quasi-density of states. For the ac field polarized along the armchair direction, the dressed electronic structure shows an emergent property: an ac field induced quantum ring. This is inferred by the orientation-dependent formation of a miniband of energy states periodically modulated with increasing m…
A transformation of the band structure in bilayer graphene (BLG) with relatively shifted layers has been studied in the framework of the tight-binding model. BLG is demonstrated to remain a zero-gap material in the whole range of experimentally attainable shifts, but the positions of contact points between the conduction and valence bands depend substantially on the shift direction. The shift results in a considerable anisotropy of the band spectrum, which is, in turn, responsible for a substantial (10-20%) anisotropy of the conductivity in BLG. A possibility of using this anisotropy in high-s…
Thermal conductance of graphene nanoribbons (GNRs) with the width varying from 0.5 to 35 nm is systematically investigated using nonequilibrium Green's function method. Anisotropic thermal conductance is observed with the room temperature thermal conductance of zigzag GNRs up to ~ 30% larger than that of armchair GNRs. At room temperature, the anisotropy is found to disappear until the width is larger than 100 nm. This intrinsic anisotropy originate from different boundary condition at ribbon edges, and can be used to tune thermal conductance, which have important implications for the applicat…
We determine the spin-lifetime anisotropy of spin-polarized carriers in graphene. In contrast to prior approaches, our method does not require large out-of-plane magnetic fields and thus it is reliable for both low- and high-carrier densities. We first determine the in-plane spin lifetime by conventional spin precession measurements with magnetic fields perpendicular to the graphene plane. Then, to evaluate the out-of-plane spin lifetime, we implement spin precession measurements under oblique magnetic fields that generate an out-of-plane spin population. We find that the spin-lifetime anisotr…
We report the first observation of a large spin lifetime anisotropy in bilayer graphene (BLG) fully encapsulated between hexagonal boron nitride. We characterize the out-of-plane ($τ_\perp$) and in-plane ($τ_\parallel$) spin lifetimes by oblique Hanle spin precession. At 75~K and the charge neutrality point (CNP) we observe a strong anisotropy of $τ_\perp/τ_\parallel$ = 8 $\pm$ 2. This value is comparable to graphene/TMD heterostructures, whereas our high quality BLG provides with $τ_\perp$ up to 9~ns, a more than two orders of magnitude larger spin lifetime. The anisotropy decreases to 3.5 $\…
Due to its potential connection with nematicity, electronic anisotropy has been the subject of intense research effort on a wide variety of material platforms. The emergence of spatial anisotropy not only offers a characterization of material properties of metallic phases, which cannot be accessed via conventional transport techniques, but it also provides a unique window into the interplay between Coulomb interaction and broken symmetry underlying the electronic order. In this work, we utilize a new scheme of angle-resolved transport measurement (ARTM) to characterize electron anisotropy in m…
We report the discovery of a strong and tunable spin lifetime anisotropy with excellent spin lifetimes up to 7.8 ns in dual-gated bilayer graphene. Remarkably, this realizes the manipulation of spins in graphene by electrically-controlled spin-orbit fields, which is unexpected due to graphene's weak intrinsic spin-orbit coupling. We utilize both the in-plane magnetic field Hanle precession and oblique Hanle precession measurements to directly compare the lifetimes of out-of-plane vs. in-plane spins. We find that near the charge neutrality point, the application of a perpendicular electric fiel…
Inducing sizable spin--orbit interactions in graphene by proximity effect is establishing as a successful route to harnessing two-dimensional Dirac fermions for spintronics. Semiconducting transition metal dichalcogenides (TMDs) are an ideal complement to graphene because of their strong intrinsic spin--orbit coupling (SOC) and spin/valley-selective light absorption, which allows all-optical spin injection into graphene. In this study, we present a microscopic theory of spin dynamics in weakly disordered graphene samples subject to uniform proximity-induced SOC as realized in graphene/TMD bila…
Room-temperature spin-lifetime anisotropy exceeding 60 in bilayer graphene spin valves proximity coupled to WSe$_2$
nd Institute of Physics and JARA-FIT, RWTH Aachen University（亚琛工业大学第二物理研究所和JARA-FIT）；Peter Grünberg Institute (PGI-9), Forschungszentrum Jülich（于利希研究中心彼得·格林贝格研究所（PGI-9））
We study spin-transport in bilayer-graphene (BLG), spin-orbit coupled to a tungsten di sulfide (WS$_2$) substrate, and measure a record spin lifetime anisotropy ~40-70, i.e. ratio between the out-of-plane $τ_{\perp}$ and in-plane spin relaxation time $τ_{||}$. We control the injection and detection of in-plane and out-of-plane spins via the shape-anisotropy of the ferromagnetic electrodes. We estimate $τ_{\perp}$ ~ 1-2 ns via Hanle measurements at high perpendicular magnetic fields and via a new tool we develop: Oblique Spin Valve measurements. Using Hanle spin-precession experiments we find a…
Dirac carriers in graphene are commonly characterized by a pseudospin degree of freedom, arising from the degeneracy of the two inequivalent sublattices. The inherent chirality of the quasiparticles leads to a topologically non-trivial band structure, where the in-plane component of sublattice spin and momentum are intertwined. Equivalently, sublattice imbalance is intimately connected with angular momentum, inducing a torque of opposite sign at each Dirac point. In this work we develop an intuitive picture that associates sublattice spin and winding number with angular momentum. We develop a …
We investigate the polarization dependence of the carrier excitation and relaxation in epitaxial multilayer graphene. Degenerate pump-probe experiments with a temporal resolution of 30 fs are performed for different rotation angles of the pump-pulse polarization with respect to the polarization of the probe pulse. A pronounced dependence of the pump-induced transmission on this angle is found. It reflects a strong anisotropy of the pump-induced occupation of photogenerated carriers in momentum space even though the band structure is isotropic. Within 150 fs after excitation an isotropic carrie…
We investigate the intraband nonlinear dynamics in doped bilayer graphene in the presence of strong, linearly-polarized, in-plane terahertz fields. We perform degenerate pump-probe experiments with 3.4 THz fields on doped bilayer graphene at low temperature (12 K) and find that when the pump is co-polarized with the probe beam, the differential pump-probe signal is almost double that found in the cross-polarized case. We show that the origin of this pump-induced anisotropy is the difference in the average electron effective mass in the probe direction when carriers are displaced in k-space by …
Dirac electrons in graphene are to lowest order spin 1/2 particles, owing to the orbital symmetries at the Fermi level. However, anisotropic corrections in the $g$-factor appear due to the intricate spin-valley-orbit coupling of chiral electrons. We resolve experimentally the $g$-factor along the three orthogonal directions in a large-scale graphene sample. We employ a Hall bar structure with an external magnetic field of arbitrary direction, and extract the effective $g$-tensor via resistively-detected electron spin resonance. We employ a theoretical perturbative approach to identify the intr…
Van-der-Waals heterostructures have become a paradigm for designing new materials and devices, in which specific functionalities can be tailored by combining the properties of the individual 2D layers. A single layer of transition metal dichalcogenide (TMD) is an excellent complement to graphene (Gr), since the high quality of charge and spin transport in Gr is enriched with the large spin-orbit coupling of the TMD via proximity effect. The controllable spin-valley coupling makes these heterostructures particularly attractive for spintronic and opto-valleytronic applications. In this work, we …
Spin lifetime anisotropy in graphene induced by the SiO2 interface
Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and BIST（加泰罗尼亚纳米科学和纳米技术研究所（ICN2））；University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；Hefei University of Technology（合肥工业大学）；University of California, Santa Cruz（加州大学圣克鲁兹分校）；Montana State University（蒙大拿州立大学）
Anisotropic electronic transport is a possible route towards nanoscale circuitry design, particularly in two-dimensional materials. Proposals to introduce such a feature in patterned graphene have to date relied on large-scale structural inhomogeneities. Here we theoretically explore how a random, yet homogeneous, distribution of zigzag-edged triangular perforations can generate spatial anisotropies in both charge and spin transport. Anisotropic electronic transport is found to persist under considerable disordering of the perforation edges, suggesting its viability under realistic experimenta…
In the present study, the anisotropic resistivity of the monolayer graphene has been obtained in semiclassical regime beyond the Dirac point approximation. In particular, detailed investigations were made on the dependence of conductivity on the Fermi energy. At low energies, in the vicinity of the Dirac points, band energy of the monolayer graphene is isotropic at the Fermi level. Meanwhile, at the intermediate Fermi energies anisotropic effects such as trigonal warping is expected to be the origin of the anisotropic resistivity. However, besides the band anisotropy there also exists an other…
We investigated thermal conductivity of free-standing reduced graphene oxide films subjected to a high-temperature treatment of up to 1000 C. It was found that the high-temperature annealing dramatically increased the in-plane thermal conductivity, K, of the films from 3 W/mK to 61 W/mK at room temperature. The cross-plane thermal conductivity, Kc, revealed an interesting opposite trend of decreasing to a very small value of 0.09 W/mK in the reduced graphene oxide films annealed at 1000 C. The obtained films demonstrated an exceptionally strong anisotropy of the thermal conductivity, K/Kc ~ 67…
We study the role of long-range electron-electron interactions in a system of two-dimensional anisotropic Dirac fermions, which naturally appear in uniaxially strained graphene, graphene in external potentials, some strongly anisotropic topological insulators, and engineered anisotropic graphene structures. We find that while for small interactions and anisotropy the system restores the conventional isotropic Dirac liquid behavior, strong enough anisotropy can lead to the formation of a quasi-one dimensional electronic phase with dominant charge order (anisotropic excitonic insulator).
We study the effect of anisotropy of the Rashba coupling on the extrinsic spin Hall effect due to spin-orbit active adatoms on graphene. In addition to the intrinsic spin-orbit coupling, a generalized anisotropic Rashba coupling arising from the breakdown of both mirror and hexagonal symmetries of pristine graphene is considered. We find that Rashba anisotropy can strongly modify the dependence of the spin Hall angle on carrier concentration. Our model provides a simple and general description of the skew scattering mechanism due to the spin-orbit coupling that is induced by proximity to large…
Using scanning tunneling microscopy (STM) and Fourier Transform STM (FT-STM), we have studied a point defect in an epitaxial graphene sample grown on silicon carbide substrate. This analysis allows us to extract the quasiparticle energy dispersion, and to give a first experimental proof of the validity of Fermi liquid theory in graphene for a wide range of energies from -800 meV to +800 meV. We also find evidence of a strong threefold anisotropy in the standing waves generated by the defect. We discuss possible relations between this anisotropy, the chirality of the electrons, and the asymmetr…
Large spin-orbital proximity effects have been predicted in graphene interfaced with a transition metal dichalcogenide layer. Whereas clear evidence for an enhanced spin-orbit coupling has been found at large carrier densities, the type of spin-orbit coupling and its relaxation mechanism remained unknown. We show for the first time an increased spin-orbit coupling close to the charge neutrality point in graphene, where topological states are expected to appear. Single layer graphene encapsulated between the transition metal dichalcogenide WSe$_2$ and hBN is found to exhibit exceptional quality…
The Rashba effect, a spin splitting in electronic band structure, attracts much attention for the potential applications in spintronics with no requirement of external magnetic field. Realizing one-dimensional (1D) Rashba system is a big challenge due to the difficulties of growing high-quality heavy-metal nanowires or introducing strong spin-orbit coupling (SOC) and broken inversion symmetry in flexible materials. Here, based on first-principles calculations, we propose a pathway to realize the Rashba spin-split by adsorbing Gd atom on zigzag graphene nanoribbons (Gd-ZGNR) and further investi…
A combination of theoretical modelling and experiments reveals the origin of the large perpendicular magnetic anisotropy (PMA) that appears in nanometer-thick epitaxial Co films intercalated between graphene (Gr) and a heavy metal (HM) substrate, as a function of the Co thickness. High quality epitaxial Gr/Co\n/HM(111) (HM=Pt,Ir) heterostructures are grown by intercalation below graphene, which acts as a surfactant that kinetically stabilizes the pseudomorphic growth of highly perfect Co face-centered tetragonal ($fct$) films, with a reduced number of stacking faults as the only structural def…
Twist- and gate-tunable proximity spin-orbit coupling, spin relaxation anisotropy, and charge-to-spin conversion in heterostructures of graphene and transition-metal dichalcogenides
The effect of Rashba spin-orbit interaction and anisotropic strain on the electronic, optical and thermodynamic properties of artificial graphene-like superlattice composed of InAs-GaAs quantum dots has been considered theoretically. The electronic energy dispersions have been obtained using Green's function formalism in combination with Fourier transformation to the reciprocal space and an exact diagonalization technique. We have observed a splitting of Dirac points and appearance of edditional Dirac-like points due to the Rashba spin-orbit interaction. Breaking of the hexagonal symmetry of t…
Valley-polarized quantum Hall states in graphene are described by a Heisenberg O(3) ferromagnet model, with the ordering type controlled by the strength and sign of valley anisotropy. A mechanism resulting from electron coupling to strain-induced gauge field, giving leading contribution to the anisotropy, is described in terms of an effective random magnetic field aligned with the ferromagnet z axis. We argue that such random field stabilizes the XY ferromagnet state, which is a coherent equal-weight mixture of the $K$ and $K'$ valley states. Other implications such as the Berezinskii-Kosterli…
The theoretical results are presented showing that strain-induced anisotropy of graphene spectrum gives rise to the valley currents under the illumination by normally incident light. The currents of the two graphene valleys are mutually compensated providing zero net electric current. The magnitude and direction of the valley currents are determined by the parameters of strain and light polarization. For not too high photon energy strain-induced valley current exceed that due to intrinsic warping of the graphene spectrum which suggests feasibility of strain-mediated valleytronics.
We compare the two-dimensional (2D) plasmon dispersion relations for monolayer graphene when the sample is doped with carriers in the conduction band and the temperature $T$ is zero with the case when the temperature is finite and there is no doping. Additionally, we have obtained the plasmon excitations when there is doping at finite temperature. The results were obtained in the random-phase approximation which employs energy electronic bands calculated using ab initio density functional theory. We found that in the undoped case the finite temperature results in appearance in the low-energy r…
Strain engineering has been recently recognized as an effective way to tailor the electrical properties of graphene. In the optical domain, effects such as strain-induced anisotropic absorption add an appealing functionality to graphene, opening the prospect for atomically thin optical elements. Indeed, graphene is currently one of the notable players in the intense drive towards bendable, thin, and portable electronic displays, where its intrinsically metallic, optically transparent, and mechanically robust nature are major advantages. Given that the intrinsic transparency of a graphene monol…
We compare different methods to measure the anisotropy of the spin-lifetime in graphene. In addition to out-of-plane rotation of the ferromagnetic electrodes and oblique spin precession, we present a Hanle experiment where the electron spins precess around either a magnetic field perpendicular to the graphene plane or around an in-plane field. In the latter case, electrons are subject to both in-plane and out-of-plane spin relaxation. To fit the data, we use a numerical simulation that can calculate precession with anisotropies in the spin-lifetimes under magnetic fields in any direction. Our …
We revisit the uniaxially strained graphene immersed in a uniform homogeneous magnetic field orthogonal to the layer in order to describe the time evolution of coherent states build from a semi-classical model. We consider the symmetric gauge vector potential to render the magnetic field, and we encode the tensile and compression deformations on an anisotropy parameter $ζ$. After solving the Dirac-like equation with an anisotropic Fermi velocity, we define a set of matrix ladder operators and construct electron coherent states as eigenstates of a matrix annihilation operator with complex eigen…
We report a significant Dzyaloshinskii-Moriya interaction (DMI) and perpendicular magnetic anisotropy (PMA) at interfaces comprising hexagonal boron nitride (h-BN) and Co. By comparing the behavior of these phenomena at graphene/Co and h-BN/Co interfaces, it is found that the DMI in latter increases as a function of Co thickness and beyond three monolayers stabilizes with one order of magnitude larger values compared to those at graphene/Co, where the DMI shows opposite decreasing behavior. At the same time, the PMA for both systems shows similar trends with larger values for graphene/Co and n…
Graphene monolayers are known to display domains of anisotropic friction with twofold symmetry and anisotropy exceeding 200 percent. This anisotropy has been thought to originate from periodic nanoscale ripples in the graphene sheet, which enhance puckering around a sliding asperity to a degree determined by the sliding direction. Here we demonstrate that these frictional domains derive not from structural features in the graphene, but from self-assembly of atmospheric adsorbates into a highly regular superlattice of stripes with period 4 to 6 nm. The stripes and resulting frictional domains a…
The spin lifetime anisotropy is an important quantity for investigating the spin relaxation mechanisms in graphene and in heterostructures of two-dimensional materials. We generalize the diffusive spin transport equations of oblique spin precession in a lateral spin valve with finite contact resistance. This yields a method to determine the spin lifetime anisotropy ratio ξ=τ$_{\perp}$/τ$_{\parallel}$, which is the ratio between lifetimes of spin polarized perpendicular and parallel to the graphene surface. By solving the steady-state Bloch equations, we show that the line-shape of the oblique …
Crystallographic alignment between two-dimensional crystals in van der Waals heterostructures brought a number of profound physical phenomena, including observation of Hofstadter butterfly and topological currents, and promising novel applications, such as resonant tunnelling transistors. Here, by probing the electronic density of states in graphene using graphene-hexagonal boron nitride tunnelling transistors, we demonstrate a structural transition of bilayer graphene from incommensurate twisted stacking state into a commensurate AB stacking due to a macroscopic graphene self-rotation. This s…
We demonstrate that charge density wave (CDW) phase transition occurs on the surface of electronically doped multilayer graphene when the Fermi level approaches the M points (also known as van Hove singularities where the density of states diverge) in the Brillouin zone of graphene band structure. The occurrence of such CDW phase transitions are supported by both the electrical transport measurement and optical measurements in electrostatically doped multilayer graphene. The CDW transition is accompanied with the sudden change of graphene channel resistance at T$_m$= 100K, as well as the split…
The simplest tight-binding model is used to study lattice effects on two properties of doped graphene: i) magnetic orbital susceptibility and ii) regular Friedel oscillations, both suppressed in the usual Dirac cone approximation. i) An exact expression for the tight-binding magnetic susceptibility is obtained, leading to orbital paramagnetism in graphene for a wide range of doping levels which is relevant when compared with other contributions. ii) Friedel oscillations in the coarse-grained charge response are considered numerically and analytically and an explicit expression for the response…
We studied the magneto-transport in SiO2 substrate-supported monolayer graphene and the quantum phase transitions that characterize the quantum Hall regime, using magnetic fields up to 28T and temperatures down to 4K. The analysis of the temperature dependence of the Hall and longitudinal resistivity reveals new non-universalities of the critical exponents of the plateau-insulator transition. These exponent depends on the type of disorder that governs the electrical transport, which knowledge is important for the design and fabrication of new graphene nano-devices.
We consider a simple spin-membrane model for rippling in graphene. The model exhibits transitions from a flat but rippled membrane to a buckled one. At high temperature the transition is second order but it is first order at low temperature for appropriate strength of the spin-spin coupling. Driving the system across the first order phase transition in nonequilibrium conditions that mimic interaction of the graphene membrane with a STM tip explains recent experiments. In particular, we observe a reversible behavior for small values of the STM current and an irreversible transition from flat ri…
