Abstract
One of the enticing features common to most of the two-dimensional (2D) electronic systems that, in the wake of (and in parallel with) graphene, are currently at the forefront of materials science research is the ability to easily introduce a combination of planar deformations and bending in the system. Since the electronic properties are ultimately determined by the details of atomic orbital overlap, such mechanical manipulations translate into modified (or, at least, perturbed) electronic properties. Here, we present a general-purpose optimization framework for tailoring physical properties of 2D electronic systems by manipulating the state of local strain, allowing a one-step route from their design to experimental implementation. A definite example, chosen for its relevance in light of current experiments in graphene nanostructures, is the optimization of the experimental parameters that generate a prescribed spatial profile of pseudomagnetic fields (PMFs) in graphene. But the method is general enough to accommodate a multitude of possible experimental parameters and conditions whereby deformations can be imparted to the graphene lattice, and complies, by design, with grapheneʼs elastic equilibrium and elastic compatibility constraints. As a result, it efficiently answers the inverse problem of determining the optimal values of a set of external or control parameters (such as substrate topography, sample shape, load distribution, etc) that result in a graphene deformation whose associated PMF profile best matches a prescribed target. The ability to address this inverse problem in an expedited way is one key step for practical implementations of the concept of 2D systems with electronic properties strain-engineered to order. The general-purpose nature of this calculation strategy means that it can be easily applied to the optimization of other relevant physical quantities which directly depend on the local strain field, not just in graphene but in other 2D electronic membranes.
Highlights
With their intrinsic two-dimensionality, “electronic membranes” are pulled or pinched by atomic-scale tips [1, 2, 3], can be made to conform to the substrate topography [4, 5, 6], can be inflated as balloons [7], can be stretched [8] or bent [9], crumpled on demand [10], and so on
Among other features, it exhibits an unconventional contribution in the electron–phonon coupling leading to the emergence of so-called pseudomagnetic fields (PMF) [15, 16, 17, 18]
Even though A is not a magnetic vector potential, the actual dynamics has the same characteristics and the Dirac electrons in graphene react to static and non-uniform lattice deformations as though they were under the influence of an effective magnetic field, with all the consequences that a magnetic field brings to electronic motion, except that time-reversal symmetry is not broken and, A will have an opposite sign for the effective Hamiltonian at the time-reversal transformed K point
Summary
With their intrinsic two-dimensionality, “electronic membranes” are pulled or pinched by atomic-scale tips [1, 2, 3], can be made to conform to the substrate topography [4, 5, 6], can be inflated as balloons [7], can be stretched [8] or bent [9], crumpled on demand [10], and so on. Even though A is not a magnetic vector potential, the actual dynamics has the same characteristics and the Dirac electrons in graphene react to static and non-uniform lattice deformations as though they were under the influence of an effective magnetic field, with all the consequences that a magnetic field brings to electronic motion, except that time-reversal symmetry is not broken and, A will have an opposite sign for the effective Hamiltonian at the time-reversal transformed K point One such consequence is the modification of the electronic energy spectrum with the development of local Landau levels for certain lattice deformations. Interesting spacedependent Fermi velocities have been reported in recent experiments on strained graphene [21], bringing this other theoretical prediction [22, 23, 24] and implication of (a)
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