Electron Optics and Valley Hall Effect in Undulated Graphene

Electron optics is the systematic use of electro-magnetic (EM) fields to control electron motions and forms the basis of many technological applications such as particle accelerators, tokamak reactors, or electron microscopes. The same principles have been applied to 2D materials to create Hall effects or Veselago-lens. However, the small length-scale of nanomaterials have made it challenging to create more complex “optical” designs to truly achieve 2D electron optics. Graphene offers a unique opportunity as its strain gradients produce pseudo-electromagnetic fields to guide its electron motion. Based on this concept, we demonstrate the use of substrate topography to impart desirable strain patterns on graphene to induce useful pseudo-EM fields. We derive the quasi-classical equation of motion for Dirac Fermions in a pseudo-EM field in graphene. Based on the trajectory analysis, we design sculpted substrates to realize various “optical devices” such as a converging lens or a collimator, and further propose a setup to achieve valley Hall effect solely through substrate patterning, without any external fields, to be used in valleytronics applications. Finally, we discuss how the predicted strain/pseudo-EM field patterns can be experimentally sustained by typical substrates and generalized to other 2D materials.