Graphene and topological insulators: Teaching electrons new tricks

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The periodic potential imposed by the crystal lattice allows new effective Hamiltonians for electrons to be generated which may be qualitatively different than the Schrodinger equation for a free electron at low energies. I will discuss two striking recent examples. In graphene, the basis of two identical atoms in the honeycomb lattice leads to a massless Dirac Hamiltonian with an emergent spin-1/2 “pseudospin” spinor coupled to momentum. In the three-dimensional topological insulators, strong spin-orbit coupling leads to a Hamiltonian which is topologically distinct from the free-electron case, which gives rise to emergent metallic states at the boundaries (surfaces) of the material. These surface states also have a Dirac effective Hamiltonian, but with momentum coupled to the intrinsic electron spin. I will discuss how these materials are made, and how they are used to enable the study of Dirac electrons in the laboratory.