Electronic properties of corrugated graphene: the Heisenberg principle and wormhole geometry in the solid state

Adopting a purely two-dimensional relativistic equation for graphene's carriers contradicts the Heisenberg uncertainty principle since it requires setting the off-the-surface coordinate of a three-dimensional wavefunction to zero. Here we present a theoretical framework for describing graphene's massless relativistic carriers in accordance with this most fundamental of all quantum principles. A gradual confining procedure is used to restrict the dynamics onto a surface and normal to the surface parts, and in the process the embedding of this surface into the three-dimensional world is accounted for. As a result an invariant geometric potential arises in the surface part which scales linearly with the mean curvature and shifts the Fermi energy of the material proportional to bending. Strain induced modification of the electronic properties or 'straintronics' is clearly an important field of study in graphene. This opens an avenue to producing electronic devices: micro- and nano-electromechanical systems (MEMS and NEMS), where the electronic properties are controlled by geometric means and no additional alteration of graphene is necessary. The appearance of this geometric potential also provides us with clues as to how quantum dynamics looks in the curved space-time of general relativity. In this context we explore a two-dimensional cross-section of the wormhole geometry, realized with graphene as a solid state thought experiment.