Session E17: Strained Graphene

Quasi-bound states in strained graphene

In this work, we explore the possibility of manipulating electronic states in graphene nanostructures by mechanical means. Specifically, we use molecular dynamics and tight-binding models to access the electronic and transport properties of strained graphene nanobubbles and graphene kirigami. We establish that low energy electrons can be confined in the arms of the kirigami and within the nanobubbles; under different load conditions the coupling between confined states and continuous states is modified creating different conductance line-shapes.   

Probing Graphene on Micropatterned Strain Arrays

The generation of enormous pseudo-magnetic fields in graphene by the application of carefully designed strain profiles opens a promising window onto an otherwise unattainable high-field regime. A central challenge in exploiting graphene's access to this regime is the difficulty of controllably generating the required strain profiles. Here, we report the tunable fabrication of nano-scale strain feature arrays, as well as optical, mechanical, and other measurements of graphene deposited on substrates prepared by this method. We describe the signatures of strain present in our measurements, and discuss the potential for further experimental exploration of this system.   

Current and Noise Saturation in Graphene Superlattice.

One of the merits of graphene is that the Fermi level can be easily tuned by electrical gating, which render charge carriers n type or p type, or even insulating around the Dirac point (DP). By aligning graphene on top of Boron Nitride (BN), the presence of graphene superlattice makes transport properties even more versatile owning to the emergence of secondary Dirac points (SDPs). Here we present a study of high electric field performance of graphene superlattice obtained from epitaxial approach. By using microwave cavity, noise produced from graphene by joule heating is recorded up to 5GHz. Current and noise saturation are observed and investigated. Depending on Fermi energy, saturation can be attributed to intrinsic optical or remote surface polar phonon scattering at a doping far away from DP, while no saturation are found around DP. Moreover, noise saturation is identified around Fermi energy between DP and SDP, which can be attributed to the influence of van Hove singularity arising from the superlattice. Lastly, saturation due to the bias induced shift of DP, or so called Dirac fermion pinch-off, is well observed by local top gate technique.   

Valley polarization in graphene with out-of-plane deformations

At low energy, the energy dispersion of graphene shows a conical valley structure with the conduction and valence bands touching at the Dirac points. The existence of two inequivalent Dirac points in the Brillouin zone, thus two valleys, suggests they may be used as new degrees of freedom to carry information. Several schemes based on different mechanisms have been advanced to achieve valley separation in this material, however the proposed setups remain challenging for experimental observation. In this work we investigate graphene with out-of-plane deformations- one of the most naturally occurring and practically realizable settings, as a candidate system to produce valley polarization. Local strains produced by the deformations serve as scattering potentials for electronic states. A second-order Born approximation calculation based on the continuum model reveals the existence of valley polarization and its dependence on the geometrical parameters of the deformations. We characterize the efficiency of valley filtering for different geometries and energies and discuss their implementation in currently available experimental setups.   

Tuning transport properties on graphene multiterminal structures by mechanical deformations

The realization of mechanical strain on graphene structures is viewed as a promise route to tune electronic and transport properties such as changing energy band-gaps and promoting localization of states. Using continuum models, mechanical deformations are described by effective gauge fields, mirrored as pseudomagnetic fields that may reach quite high values. Interesting symmetry features are developed due to out of plane deformations on graphene; lift sublattice symmetry was predicted and observed in centrosymmetric bumps and strained nanobubbles [1]. Here we discuss the effects of Gaussian-like strain on a hexagonal graphene flake connected to three leads, modeled as perfect graphene nanoribbons. The Green function formalism is used within a tight-binding approximation. For this particular deformation sharp resonant states are achieved depending on the strained structure details. We also study a fold-strained structure [2] in which the three leads are deformed extending up to the very center of the hexagonal flake. We show that conductance suppressions can be controlled by the strain intensity and important transport features are modeled by the electronic band structure of the leads. [1]RC-Bastos et al., Phys. Rev. B 91, 125408 (2015).[2]HLim et al., Nat. Commun. 6, 8601(2015).   

Fold assisted transport in graphene systems

Sasaki pointed out that a constant uniaxial strain applied along the zigzag direction in graphene causes localized states, akin to edge states in nanoribbons[1]. These states are dispersionless and can carry ballistic transport. Recent experiments reported the presence of ballistic channels in graphene grown on SiC characterized with STM spectroscopy[2, 3]. In this work, we show that out-of plane deformations in the form of folds produce states as those predicted by Sasaki. Using tight-binding calculations and recursive Green’s function methods, we obtain conductance, density of states (DOS), local density of states, and band structure (BS) for graphene nanoribbons with zigzag termination. Regions with enhanced DOS are identified in the deformed area corresponding to states in new flattened bands in the BS and new ballistic channels in the conductance. Adjusting the fold parameters, desired properties of these states can be tailored. Our results show that folds could serve as pathways for electronic transport and open the possibility of circuitry design within a simple graphene membrane. [1]Sasaki et al., J. Phys. Soc. Jpn. 75 (2006). [2]Baringhaus et al., Nature 506 (2014). [3]Palacio et al., Nano Lett. 15 (2015).   

Helium adsorption potential near mechanically deformed graphene

Mechanical strain modifies the van der Waals interactions of neutral adatoms near two-dimensional materials like graphene and commonly used parameters for helium interacting with carbon do not capture these effects. Using the polarization function of strained graphene, we have compared the long-distance Lifshitz dispersion force with an effective potential computed from the sum of two-body interactions. The resulting optimized many-body adsorption potential exhibits an anisotropic minimum that is displaced higher above the graphene sheet as unixaxial strain is increased. The competing energy scales introduced by strain open up the possibility of mechanically tuning novel anisotropic adsorbed superfluid phases of helium on graphene.   

Elastic Solitons and Vortices in Bilayer Graphene

Using electron microscopy Alden et al [PNAS 110, 11256 (2013)] find that in strained bilayer graphene regions of stable AB and BA stacking are separated by domain walls. Vortex-like defects are also observed at the intersections of three domain walls; scanning transmission electron microscopy reveals that the vortex cores have an unstable AA stacking. We develop a continuum elasticity model that describes the relative displacement of the two layers of graphene. In addition to the usual gradient energy cost we posit a nonlinear potential for the displacement field that favors AB and BA stackings and disfavors the AA stacking. In our model the domain walls appear as soliton solutions to a double sine-Gordon equation. We find that the ratio of the width of domain walls with tensile strain to those with shear strain is $\sim \!\!1.63$, in excellent agreement with observations. We study the stability and oscillatory modes of the domain walls motivated by experimental observations that show the domain walls undergoing damped oscillations. Estimates of the vortex core size based on our model are in agreement with the experiment. A homotopy analysis of bilayer graphene vortices shows that they carry non-abelian topological charge.   

Graphene on a curved substrate with a controllable curvature: Device fabrication and transport measurements

In monolayer graphene, the local deviation of carbon positions from the perfect lattice has been predicted to lead to a pseudo magnetic field with measurable effects. A striking confirmation of this effect is the observation of Landau levels that are attributed to a pseudo magnetic field in excess of ~300 T in graphene nanobubbles. However, typical experimental methods of generating such local deviations in graphene rely on strain accompanied by a surface curvature. Whether a surface curvature alone can produce measurable effects in graphene has not been explored experimentally. It is therefore of interest to study graphene in a system that decouples strain from surface curvature. Of particular interest is its response to an external magnetic field. We developed a grayscale electron beam lithography technique for preparing PMMA substructures with a continuously variable radius of curvature from $\sim$100 nm to $\sim$1 $\mu$m. Magnetoelectrical transport measurements on exfoliated graphene supported by these substructures are being carried out. The flexibility of this process may be further exploited in the study of the bilayer and trilayer graphene systems. We will also study hybrid structures of 2D superconductors and graphene.   

Graphene with out-of-plane deformations and external magnetic fields

Microscopic measurements in corrugated graphene have helped to elucidate how its electronic properties are affected in deformed regions, confirming the pseudomagnetic description for strained graphene. As the pseudofields reverse sign at each valley, an enhancement of the local density of states (LDOS) was found with broken sublattice symmetry in strained regions[1,2]. As a consequence, these systems have been proposed as a viable way to obtain valley filters. In graphene ribbons with a centrosymmetric deformation, the LDOS modification was predicted to be accompanied with a decrease in conductance, pointing to the presence of confined states [3]. To further characterize these states, we investigate the effect of strain in the presence of an external magnetic field, using a tight-binding model and recursive Green’s function techniques. Anomalous conductance and DOS features are consistent with interpretation in terms of Fano resonances appearing for bound states in the continuum. We discuss the robustness of the sublattice asymmetry against the magnetic field and the possibility of exploring the valley separation in chosen spatial regions. [1]Mashoff et al. NanoLetters 10 (2010). [2]Schneider et al. 91 (2015). [3]Carrillo-Bastos et al. PRB 90 (2014).   

Towards Quantum Strain Engineering in Ultra-short Ballistic Graphene Devices

We report our progress towards combining ultra-short and clean suspended graphene transistors [1] with mechanical breakjunction (MCBJ) instrumentation to create a widely tunable quantum strain engineering platform for graphene electronics. We first present data from our electromigrated ultra-short ($\sim$ 10 to 50 nm) suspended graphene transistors, which show ballistic transport [1], and demonstrate that we can tailor the length and shape of the suspended graphene by adjusting the electromigration parameters. We observe Fabry-P\'{e}rot interferences, which correspond to coherent transport not only in the suspended graphene channel, but also in the suspended graphene contacts connecting to the channel. We then describe the MCBJ assembly we use to bend the substrate in-situ at low temperature. Using long suspended cantilever contacts, this instrumentation allows strains up to $\sim$ 10\% for device lengths $\sim$ 10 nm. The assembly is hosted in a cryostat operating down to 0.3 K and in magnetic fields up to 9 T. We finally report on our progress towards the application of large uniaxial strain for graphene strain transistors, and graphene NEMS with tunable frequencies approaching the THz range. [1] V. Tayari \textit{et al.}, Nano. Lett. \textbf{15}, 1 (2015)   

Local Probes of Strain Texture and Individual Atomic Dopant Sites in Monolayer MoS$_{2}$

The 2D semiconductor MoS$_2$ is an optically active material uniquely responsive to local perturbations. As an atomically thin membrane with exceptional strength, it can embed wide band gap variations overlapping the visible light spectrum when subjected to biaxial strain, where the modified electronic potential emanating from point-induced tensile strain perturbations mimics the Coulomb potential in a mesoscopic atom. We have realized this ``artificial atom'' concept via monolayer nanoindentation, and demonstrate that a synthetic superlattice of these building blocks forms an optoelectronic crystal capable of broadband light absorption and efficient funneling of photogenerated excitons to points of maximum strain at the artificial-atom nuclei. We also investigate the effects of individual atomic dopant sites through STM/STS, and visualize the atomic-scale local band structure changes. The modification of 2D semiconductors through methods such as strain texturing and doping connects to applications in next generation optoelectronics and photovoltaics.   

Effects of Strain on CVD-Grown Few-Layered Terrace Structures of MoS$_{2}$

In this report, we used CVD-grown terrace MoS$_{2}$ layers to study how the number and size of the layers affected the physical properties under uniaxial and biaxial tensile strain. Interestingly, we observed significant shifts in both the Raman in-plane mode (as high as -5.2 cm$^{-1})$ and photoluminescence (PL) energy (as high as -88 meV) for the few-layered MoS$_{2}$ under approximately 1.5{\%} applied uniaxial tensile strain. The observed results were compared to monolayers and few-layers of MoS$_{2}$ previously reported. We also observed slippage between the layers which resulted in a hysteresis of the Raman and PL spectra during further applications of strain. Through DFT calculations, we contended that this random layer slippage was due to defects present in CVD-grown materials. This work demonstrates that the properties of CVD-grown few-layered MoS$_{2}$ studied here can be tuned under strain as well as, if not better than, it's exfoliated monolayered counterpart.   

First-Principles Computation of Graphene's Phonon Anharmonicity

Here we use density-functional theory to compute an interatomic potential for graphene, including anharmonicities up to at least fourth order. We generate all group-theoretically allowed terms within a hexagon via the recently-developed slave mode expansion. This expands the potential in terms of the normal modes of overlapping hexagons, while obeying the space group symmetry and homogeneity of free space. We further introduce the notion of cooperative modes, which combine strain and $q=0$ phonons to yield the same pure mode amplitude on each hexagon. Within the cooperative subspace, cooperative modes allow for arbitrarily-precise meshing to directly compute energies, or calculation of the anharmonic coefficients via finite-difference. We demonstrate the power of our approach in the context of strained graphene, which is known to have a novel strain-driven soft mode at the $K$-point. We identify the dominant anharmonic terms which drive the soft $K$ mode, and study the role of finite temperatures using molecular dynamics and Monte-Carlo simulations.