Reproducible dry and wet transfer techniques were developed to improve the transfer of large-area monolayer graphene grown on copper foils by chemical vapor deposition (CVD). The techniques reported here allow transfer onto three different classes of substrates: substrates covered with shallow depressions, perforated substrates, and flat substrates. A novel dry transfer technique was used to make graphene-sealed microchambers without trapping liquid inside. The dry transfer technique utilizes a polydimethylsiloxane frame that attaches to the poly(methyl methacrylate) spun over the graphene film, and the monolayer graphene was transferred onto shallow depressions with 300 nm depth. The improved wet transfer onto perforated substrates with 2.7 μm diameter holes yields 98% coverage of holes covered with continuous films, allowing the ready use of Raman spectroscopy and transmission electron microscopy to study the intrinsic properties of CVD-grown monolayer graphene. Additionally, monolayer graphene transferred onto flat substrates has fewer cracks and tears, as well as lower sheet resistance than previous transfer techniques. Monolayer graphene films transferred onto glass had a sheet resistance of ∼980 Ω/sq and a transmittance of 97.6%. These transfer techniques open up possibilities for the fabrication of various graphene devices with unique configurations and enhanced performance.

We have discovered that our form for the strained positions of the carbon atoms in the graphene lattice was incomplete. In this Erratum, we show how the complete treatment changes our conclusions. In particular, the second of our two conclusions below is not true:(1) To correctly describe the shift in the positions of the Dirac points from the reference (flat, undeformed) state to the strained state in terms of a strain-induced vector potential, the corrections arising from the deformation of the lattice ( A latt = A K i − A p , where A K i and A p are defined in Eqs. (4) of the original paper) should be taken into consideration in addition to the changes in the nearest-neighbor hoppings [e.g., Eqs. (4) or Fig. 2].(2) A nonuniform (yet smooth so that the effective mass approximation is still meaningful) strain distribution endows A latt with a position dependence which leads to a correction to the pseudomagnetic field Figure 3 showed the effect of this correction.The second conclusion is incorrect because, although the corrections A latt are finite and, in general, have a position dependence, their rotational is identically zero and, thus, so is their contribution to the pseudomagnetic field. This has been pointed out recently by de Juan et al. 1 Below, we elaborate on that and on the reason why Fig. 3 apparently shows a nonzero B K i when it should have been zero by construction. We trust the details will benefit the reader.Our original form for the strained nearest-neighbor vectors was incomplete. Under the Cauchy-Born hypothesis, the position of the ith atom in the deformed configuration R i is given with reference to the undeformed one r i in terms of the deformation field u(r),The electronic dispersion is affected by changes in the nearestneighbor vectors δ 1,2,3 , which, on account of (1), are given approximately bywhere ∇u is the Jacobian of the displacement field known as the displacement gradient tensor,whereω is the rotation tensor and˜ is the linear strain tensor, which is only one part of the full (Lagrange) strain tensor given by = 1 2 (∇u + ∇u + ∇u ∇u) =˜ + 1 2 (∇u ∇u). Instead of using Eq. (2), in our original paper, we mistakenly took the strained nearest-neighbor vector to be δ i (r) (1 + ) · δ i , a result that is only true in special cases. In fact, even δ i (r) (1 +˜ ) · δ i is only valid if the deformation does not involve local rotation (ω = 0).When the correct expansion for the strained position of the atoms is used, it becomes apparent that the lattice corrections cannot contribute to the pseudomagnetic field. Upon expansion around a corner of the Brillouin zone of the undeformed lattice K , the lattice corrections to the vector potential can be cast as 1(3)Since the above is a total derivative, it cannot contribute to the pseudomagnetic field because ∇ × ∇φ ≡ 0. This can also be verified by direct inspection of Eqs. (4) of the original paper, which remain valid in form if the replacement → ∇u is made.Having clarified and established the correct expansion of the nearest-neighbor vector an...

Strain, bending rigidity, and adhesion are interwoven in determining how graphene responds when pulled across a substrate. Using Raman spectroscopy of circular, graphene-sealed microchambers under variable external pressure, we demonstrate that graphene is not firmly anchored to the substrate when pulled. Instead, as the suspended graphene is pushed into the chamber under pressure, the supported graphene outside the microchamber is stretched and slides, pulling in an annulus. Analyzing Raman G band line scans with a continuum model extended to include sliding, we extract the pressure dependent sliding friction between the SiO2 substrate and mono-, bi-, and trilayer graphene. The sliding friction for trilayer graphene is directly proportional to the applied load, but the friction for monolayer and bilayer graphene is inversely proportional to the strain in the graphene, which is in violation of Amontons' law. We attribute this behavior to the high surface conformation enabled by the low bending rigidity and strong adhesion of few layer graphene.

Analysis of the strain-induced pseudomagnetic fields generated in graphene nanobulges under three different substrate scenarios shows that, in addition to the shape, the graphene-substrate interaction can crucially determine the overall distribution and magnitude of strain and those fields, in and outside the bulge region. We utilize a combination of classical molecular dynamics, continuum mechanics, and tight-binding electronic structure calculations as an unbiased means of studying pressure-induced deformations and the resulting pseudomagnetic field distribution in graphene nanobubbles of various geometries. The geometry is defined by inflating graphene against a rigid aperture of a specified shape in the substrate. The interplay among substrate aperture geometry, lattice orientation, internal gas pressure, and substrate type is analyzed in view of the prospect of using strain-engineered graphene nanostructures capable of confining and/or guiding electrons at low energies. Except in highly anisotropic geometries, the magnitude of the pseudomagnetic field is generally significant only near the boundaries of the aperture and rapidly decays towards the center of the bubble because under gas pressure at the scales considered here there is considerable bending at the edges and the central region of the nanobubble displays nearly isotropic strain. When the deflection conditions lead to sharp bends at the edges of the bubble, curvature and the tilting of the pz orbitals cannot be ignored and contributes substantially to the total field. The strong and localized nature of the pseudomagnetic field at the boundaries and its polarity-changing profile can be exploited as a means of trapping electrons inside the bubble region or of guiding them in channel-like geometries defined by nano-blister edges. However, we establish that slippage of graphene against the substrate is an important factor in determining the degree of concentration of PMFs in or around the bulge since it can lead to considerable softening of the strain gradients there. The nature of the substrate emerges thus as a decisive factor determining the effectiveness of nanoscale pseudomagnetic field tailoring in graphene.PACS numbers: 81.05.ue, 73.22.Pr, 71.15.Pd, 61.48.Gh Since the discovery of a facile method for its isolation, graphene 1 , the simplest two-dimensional crystal, has attracted intense attention not only for its unusual physical properties 2-5 , but also for its potential as the basic building block for a wealth of device applications. There exist key limitations that appear to restrict the application of graphene for all-carbon electronic circuits: one such limitation is that graphene, in its pristine form, is well known to be a semi-metal with no band gap 3 . A highly active field of study has recently emerged based on the idea of applying mechanical strain to modify the intrinsic response of electrons to external fields in graphene [6][7][8] . This includes the strain-induced generation of spectral * Electronic address: zenanqi@bu.edu † E...

The electronic implications of strain in graphene can be captured at low energies by means of pseudovector potentials which can give rise to pseudomagnetic fields. These strain-induced vector potentials arise from the local perturbation to the electronic hopping amplitudes in a tight-binding framework. Here we complete the standard description of the strain-induced vector potential, which accounts only for the hopping perturbation, with the explicit inclusion of the lattice deformations or, equivalently, the deformation of the Brillouin zone. These corrections are linear in strain and are different at each of the strained, inequivalent Dirac points, and hence are equally necessary to identify the precise magnitude of the vector potential. This effect can be relevant in scenarios of inhomogeneous strain profiles, where electronic motion depends on the amount of overlap among the local Fermi surfaces. In particular, it affects the pseudomagnetic field distribution induced by inhomogeneous strain configurations, and can lead to new opportunities in tailoring the optimal strain fields for certain desired functionalities.Comment: Errata for version