Quantum Spin Hall Effect in Graphene - APS Link Manager ...

PRL 95, 226801 (2005)
PHYSICAL REVIEW LETTERS week ending
25 NOVEMBER 2005
with a ferromagnetic contact) will be split, with the up
(down) spins transported to the top (bottom) contacts,
generating a measurable spin Hall voltage.
The magnitude of so may be estimated by treating the
microsopic SO interaction
@
V SO
4m 2 s rV p (7)
c2 in first order degenerate perturbation theory. We thus
evaluate the expectation value of (8) in the basis of states
given in (1) treating r as a constant. A full evaluation
depends on the detailed form of the Bloch functions. However
a simple estimate can be made in the ‘‘first star’’
P
p
approximation: u K;K 0 ; A;B r p exp iK p r d = 3 .
Here K p are the crystal momenta at the three corners of the
Brillouin zone equivalent to K or K 0 , and d is the a basis
vector from a hexagon center to an A or B sublattice site.
We find that the matrix elements have precisely the structure
(3), and using the Coulomb interaction V r e 2 =r we
estimate 2 so 4 2 e 2 @ 2 = 3m 2 c 2 a 3 2:4K. This is a
crude estimate, but it is comparable to the SO splittings
quoted in the graphite literature [8].
The Rashba interaction due to a perpendicular electric
field E z may be estimated as R @v F eE z = 4mc 2 .For
E z 50 V=300 nm [3] this gives R 0:5 mK. This is
smaller than so because E z is weaker than the atomic
scale field. The Rashba term due to interaction with a
substrate is more difficult to estimate, though since it is
presumably a weak Van der Waals interaction, this too can
be expected to be smaller than so.
This estimate of so ignores the effect of electronelectron
interactions. The long range Coulomb interaction
may substantially increase the energy gap. To leading order
the SO potential is renormalized by the diagram shown in
Fig. 3, which physically represents the interaction of electrons
with the exchange potential induced by so. This is
similar in spirit to the gap renormalizations in 1D Luttinger
liquids and leads to a logarithmically divergent correction
to so. The divergence is due to the long range 1=r
Coulomb interaction, which persists in graphene even
accounting for screening [20]. The divergent corrections
to so as well as similar corrections to @v F can be summed
using the renormalization group (RG) [20]. Introducing the
dimensionless Coulomb interaction g e 2 =@v F we integrate
out the high energy degrees of freedom with energy
between and e ‘ . To leading order in g the RG flow
equations are
dg=d‘ g 2 =4; d so =d‘ g so =2: (8)
These equations can be integrated, and at energy scale ",
so "
0 so 1 g 0 =4 log 0 =" 2 . Here g 0 and 0 so are
the interactions at cutoff scale
0 . The renormalized gap is
determined by
R R so so so . Using an effective interaction
g 0 0:74 [21] and
0
2eVthis leads to 2 R so
15 K.
σ τ s
z z z
FIG. 3. Feynman diagram describing the renormalization of
the SO potential by the Coulomb interaction. The solid line
represents the electron propagator and the wavy line is the
Coulomb interaction.
In summary, we have shown that the ground state of a
single plane of graphene exhibits a QSH effect, and has a
nontrivial topological order that is robust against small
perturbations. The QSH phase should be observable by
studying low temperature charge transport and spin injection
in samples of graphene with sufficient size and purity
to allow the bulk energy gap to manifest itself. It would
also be of interest to find other materials with stronger SO
coupling which exhibit this effect, as well as possible
three-dimensional generalizations.
We thank J. Kikkawa and S. Murakami for helpful
discussions. This work was supported by the NSF under
MRSEC Grant No. DMR-00-79909 and the DOE under
Grant No. DE-FG02-ER-0145118.
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