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, the quantum spin Hall effect and topological
insulators. This year's work builds on the work discussed in our 2007 report.
Our previous work has achieved a high profile as a result of two exciting
recent experiments, which have confirmed aspects of our theory. First, the two
dimensional quantum spin Hall state has been observed in a transport experiment
on HgCdTe quantum well structures by the group of Laurens Molenkamp in
Wuerzburg. This experiment establishes the existence of the electronic state
which we originally proposed in graphene, and was rated by Science as on of the
top 10 breakthroughs of the year. In addition, the three dimensional version
of this state, which we dubbed the strong topological insulator has been
observed by Zahid Hasan's group from Princeton using angle resolved
photoemission spectroscopy (ARPES) on the surface of the semiconducting alloy
BiSb. In our 2007 report we described our prediction that BiSb is a topological
insulator - a prediction that was confirmed in less than a year. These
experimental breakthroughs open the door to a wide variety of future
experimental and theoretical endeavors, and could be characterized as the
dawning of a new field. Topological insulators will be the subject of a
workshop at the Kavli Institute for Theoretical Physics in Santa Barbara in
December 2008.
This years work builds on aspects of topological insulators. The specific
accomplishments include:
1. We developed a detailed theory of the surface states of BiSb
In our previous work we identified the topological class of the BiSb crystal,
and made a general prediction about the topological nature of the surface
states (which was confirmed experimentally). In the present work, we
considered a detailed tight binding model of the BiSb band structure and
computed the surface states of the 111 surface explicitly. This calculation
provided a concrete theoretical confirmation of our topological prediction as
well as a specific prediction for the structure of the surface Fermi surface,
which was in accordance with the Princeton experiment.
In addition, this work led to two new "model independent" ideas, which in
addition to BiSb are relavent to a wider class of materials. We showed spatial
symmetries further constrain the topological classification of crystal band
structures. Specifically, we explored the consequences of inversion symmetry
and mirror symmetry. The first result was that mirror symmetric crystals are
characterized by additional topological "parity invariants". These invariants
further constrain the structure of the surface states by determining which time
reversal invariant momenta are inside the surface fermi surface and which are
outside. This allowed us to make additional specific predictions for the
surface state structure of the 100 and 110 surfaces without doing the full
calculation.
A second result was that the presence of mirror symmetry defines a new integer
"mirror Chern number", which further constrains the structure of surface
states. We explored the role of the mirror Chern number in BiSb and in pure
Bi. We showed that it's value is related to the structure of a three
dimensional Dirac point in pure bismuth, and defines the "mirror chirality" of
that point - a previously unexplored parameter in the k.p theory of pure
bismuth. These considerations explained a subtle disagreement between our
tight binding calculation and previous first principle calculations on the
surface states of bismuth. We showed that the value of the mirror Chern number
could be deduced using spin polarized ARPES.
2. Building on our prediction of the mirror Chern number, we provided
theoretical guidance for new set of spin polarized ARPES experiments performed
by the Hasan group. These experiments confirm the value of the Mirror Chern
number predicted by our theory in BiSb, and also probe the surface state
structure in pure Sb. This work is currently under review at Science.
3. We developed a theory of the Josephson effect mediated by the edge states
of 2D quantum spin Hall insulator.
We showed that the interface between a magnetic and superconducting region of
the edge states binds a Majorana fermion. For a junction between two
superconductors, this leads to a "fractional Josephson effect", where the
Josephson current has a 4pi rather than a 2pi periodicity as a function of the
phase difference . This effect is deeply related to the Non Abelian statistics
of the Majorana fermions, and provides a possible experimental route to
observing those elusive excitations. This work is presently under review at
Physical Review Letters.
