QHE QSHE

Oleksii Shevtsov
Commisariat à l’Energie Atomique, INAC-SPSMS-GT
Grenoble, France
GDR Réunion: Physique Mésoscopique Quantique
Aussois, December 8, 2011

Outline
Introduction to graphene
Main features of topological insulators
Quantum spin Hall phase in graphene
Crossover between QHE and QSHE
Conclusions
B
QHE
vs
QSHE

Introduction
Graphene:
one atom thick planar sp2-bonded sheet of carbon atoms
honeycomb crystal lattice
Low-energy theory:
‣ nearest-neighbors tight-binding
model with one orbital per site
‣ two sublattices: A and B – sites
A B
(pseudospin)
‣ hopping parameter, 2.7 eV
Bulk spectrum:

Topological insulators
Bulk insulators with gapless edge excitations – bulk-edge correspondence
Edge states are robust to a broad class of possible perturbations
Properties of a system change abruptly – topological phase transition
Possible candidates:
‣ Chiral qauntum Hall edge channels:
1. strong magnetic field
2. broken time-reversal symmetry
3. electrons with different projections of spin
propagate in the same direction (without Zeeman)
(2DEG nanowires, graphene)
B
QHE
‣ Helical spin Hall edge channels:
QSHE
1. no magnetic field
2. preserved time-reversal symmetry
3. electrons with different projections of spin
propagate in opposite directions
(CdTe/HgCdTe quantum wells, graphene with spin-orbit coupling)

QSH phase in graphene
PRL 95, 226801 (2005)
Graphene with SO coupling – topological insulator
W=56
edge
states
But intrinsic SO interaction is very weak, ~10 mK
PRB 74, 155426 (2006)
Recent proposal to induce stronger SO (~70-210 K)
by depositing In or Tl adatoms PRX 1, 021001 (2011)

Random distribution of adatoms
2
3
2.5
0
T j0
2
1.5
1
1 3
2
QSHE
0.5
0
0 0.05 0.1 0.15 0.2
n ad
λ so

Scaling laws for QSH phase
QSHE
Perfect robustness to disorder: the effective Quantum Spin Hall (QSH) phase is totally
homogeneous with a renormalized SO coupling constant
Shevtsov et al., arXiv:1109.5568

Crossover between QHE and QSHE
Time-reversal symmetry is broken!
What happens to the spin, which propagated in the “wrong” direction?
vs
edge
states
Bulk:
B
QHE
– magnetic length
But do we have states inside the
gap?
QSHE

Crossover between QHE and QSHE
The answer is – YES!
width,
SO gap, 0.1
For energies inside the gap, electrons with different spins propagate on the
opposite edges

Crossover between QHE and QSHE
Current density plot in a nanoribbon
Spin – down
Spin – up
I, [a.u.]
1.0
0.5
0.0
Spins appear on the opposite edges – quantum spin Hall effect

Crossover between QHE and QSHE
Current density plot in a nanoribbon
Spin – down
Spin – up
I, [a.u.]
1.0
0.5
0.0
Spin down gets localized as Fermi energy aligns with the gap

Crossover between QHE and QSHE
Current density plot in a nanoribbon
Spin – down
Spin – up
I, [a.u.]
1.0
0.5
0.0
Both spins appear on the same edges – quantum Hall effect

Transmission
Crossover between QHE and QSHE
Transmission though a 4-terminal cross with armchair edges
QSHE
QHE
B
03
01
02
QHE
Fermi energy
QSHE
Intermediate region: finite-size effects

Conclusions
Graphene is a good candidate for investigating new
topological effects (easy to fabricate)
In the presence of both magnetic field and SO
coupling we observe features of QSHE and QHE by just
tuning the Fermi energy. At the transition point, the spin
which has to change an edge is getting localized.
B
QHE
vs
QSHE

This work was done in collaboration with
Xavier Waintal
David Carpentier
Pierre Carmier
Christoph Groth

Graphene ribbons
Two types of regular edges:
zigzag
armchair
Armchair nanoribbons:
‣ Semiconductor, N=3p or 3p+1
‣ Semimetal, N=3p+2
p – integer
1 2 N

Full TB spectrum of AGNR

Landau levels in the presence of
varying mass

The KNIT package: Toward a universal mesoscopic tool box
A UNIQUE TOOL FOR QUANTUM TRANSPORT
- Arbitrary geometry
- Arbitrary internal degrees of
freedom (s/p/d,e/h,up/down,…)
-Arbitrary electrodes (contacts)
- 1D, 2D, 3D, 4D,…
-Inovative algorithm for both global (conductance, shot noise, …) and
local (spin torque, local density of states,…) quantities.
ONLINE AND OPEN SOURCE WITH FULL DOCUMENTATION.
http://inac.cea.fr/Pisp/xavier.waintal/KNIT