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44776-AC10
Electronic Properties of Graphene
In the past year, this grant has supported a graduate student who has been working on the theory of graphene and the quantum spin Hall effect. This work builds on the PI's earlier work which establishes the quantum spin Hall state as a new state of matter, dubbed a "topological insulator" which in principle exists at low temperature in graphene. Our recent work focuses on elucidating the properties that this state of matter has, and on establishing a general framework for determining whether or not a given material exhibits this state. We have also studied generalizations of this effect applicable to three dimensional materials. Specific work includes:
1. We developed a new and simple method for evaluating the Z_2 topological invariants characterizing a band structure.
The insulating state we predicted in graphene is distinguished from an ordinary insulator by a topological invariant characterizing its bandstructure. In our original formulation, this invariant was a rather complicated function of the wavefunctions of the energy bands. In our new work, we showed that for a crystal with inversion symmetry the topological invariant is related to the parity of the occupied energy bands evaluated at the four time reversal invariant momenta, k in the Brillouin zone, which satisfy -k = k + G for a reciprocal lattice vector G. The invariant is simply given by the product over all of the parities characterizing pairs of Kramers degenerate occupied states. This can easily be determined from published bandstructures of materials.
This paper provided a simple proof of the fact that graphene is in the nontrivial topological class. In addition, this method makes it easy to determine whether other materials could be in this class, and we have predicted a number of other materials to be topological insulators, including the alloy Bi_(1-x) Sb_x along with HgTe and grey tin under uniaxial strain. These three dimensional materials are more robust topological insulators than graphene because their stronger spin orbit interactions lead to a larger energy gap.
This paper also established the connection between the Z_2 invariants characterizing the bulk bandstructure and the nontrivial properties of the surface (or edge in two dimensions) predicted for topological insulators. In particular, we showed that the Z_2 invariants determine how the Kramers degenerate states at the time reversal momenta in the surface bandstructure are connected to one another. This allowed us to make specific predictions regarding the surface state band structures of materials such as Bi_(1-x) Sb_x. These properties can be probed directly using angle resolved photoemission spectroscopy. We also made predictions for the behavior of electronic transport at the surface of topological insulators.
2. We explored the non trivial properties of the surface states of topological insulators and showed that in proximity to a superconductor they lead to states which may be useful for topological quantum computation.
The central prediction for topological insulators is that they have "spin filtered" surface states (or edge states in two dimensions). These states are related to the edge states that occur in the quantum Hall effect, and can only exist at the interface between a topological insulator and the vacuum. In recent work we showed that when a topological insulator is placed in contact with a superconductor, the superconducting proximity effect leads to a two dimensional surface state which has a highly non trivial topological order. This new state is similar in many regards to a chiral p_x + i p_y superconductor. However, unlike the chiral superconductors, this state does not violate time reversal symmetry. We showed that vortices in this superconducting state are associated with Majorana zero modes, which obey non-Abelian statistics. This provides a new direction for the realization of these non trivial states which have not yet been observed, but have previously been predicted to exist in exotic fractional quantum Hall states (such as filling 5/2 and 12/5) and in the superconductor Sr_2 Ru O_4.
States with non Abelian statistics are the foundation for proposals for topological quantum computation. If these states can be adiabatically manipulated, then fault tolerant quantum operations can be implemented. In our paper we showed how a new type of Josephson junction, where the contact between two ordinary superconductors is mediated by a topological insulator (a "S-TI-S junction"), could provide a means for manipulating the non-Abelian Majorana particles.
