quantum hall effects other - kitp online talks
TRANSCRIPT
KITP, Dec 2008
Other quantum Hall effects
• Part 1: Quantum Valley Hall
– No interactions
– Valley = “emergent” spin
– “Edge” states
Ivar Martin, Ya. M. Blanter, A. F. Morpurgo, PRL 100, 036804 (2008)
• Part 2: Spontaneous Hall effect
– Interactions – yes (weak coupling)
– Broken Continuous and Ising symmetries
Ivar Martin and C. D. Batista, PRL 101, 156402 (2008)
K K’
KITP, Dec 2008
Single layer graphene
H =
0 vF( p
xip
y) 0 0
vF( p
x+ ip
y) 0 0 0
0 0 0 vF( p
xip
y)
0 0 vF( p
x+ ip
y) 0
= px x z + py y
Valley K
Valley K’
Linear (Dirac) dispersion
K
K’
KITP, Dec 2008
Graphene with “mass” EA – EB=2m 0 (G. Semenoff 1984)
H =
m vF( p
xip
y) 0 0
vF( p
x+ ip
y) m 0 0
0 0 m vF( p
xip
y)
0 0 vF( p
xip
y) m
= px x z + py y +m z g
Valley K
Valley K’
How about quantum Hall?
There is quantum Valley Hall effect! BUT! 1. is suspicious 2. Need sublattice imbalance, EA-EB=2m
g
px py
KITP, Dec 2008
Quantum spin hall effect (Kane + Mele, 2005)
• Another possibility (T-invariant) for the mass term:
Still, dispersion
But instead of valley quantum Hall obtain Quantum Spin Hall
T-inv – spin-orbit
KITP, Dec 2008
Quantum spin Hall effect: (Molenkamp, SC Zhang, et al, 2007)
Quantized conductance!
I: trivial insulator
II, III, IV: quantum spin hall insulators I & II – (20 x 13) um2
III – (1 x 1) um2 & IV (1 x 0.5) um2
M. Konig et al Science 318, 766 (2007)
KITP, Dec 2008
What about valley Hall effect?
Possible to observe in the conventional Bilayer graphene!
1. Can generate mass term 2. Can create a smooth “edge”
KITP, Dec 2008
K
K’
Bilayer graphene
v
F= 3ta / 2
t
Bilayer, one of the valleys (e.g. K):
H =
0 vF( p
xip
y) 0 0
vF( p
x+ ip
y) 0 t 0
0 t 0 vF( p
xip
y)
0 0 vF( p
xip
y) 0
Layer 1
Layer 2
vF
2
t
0 ( px
ipy)2
( px+ ip
y)2 0
| E |< t
E ± p2
Interlayer coupling Quadratic dispersion!
KITP, Dec 2008
Biased graphene bilayers
Field-induced gap in the spectrum McCann ‘06
H =v
F
2
t
( px
ipy)2
( px+ ip
y)2
g( px, p
y)i
BLG
KITP, Dec 2008
Topology of insulating BLG
H =v
F
2
t
( px
ipy)2
( px+ ip
y)2
g( px, p
y)i
g = ( py
2p
x
2 ,2 pxp
y, )
Quantum Valley Hall effect?
xy=
e2
4 hdp
xdp
yg [
px
gp
y
g] =e2
hsgn( ) z
Mass !
Biased bilayer:
Single layer with mass:
g
g
Quantized Valley Conductance, similar to charge and spin Hall effects
px py
px py
p
g
KITP, Dec 2008
How about the quantum Valley Hall effect? Is it observable?
Edge states
QH edge states, Halperin ‘82
OK: any disorder (V, Vso, S)
OK: V and Vso
X: S - magnetic
OK: any, Unless it scatters
X: K K’
.
.
K
K’
QHE (charge) QSHE (spin) QVHE (valley)
..
..
.
.
KITP, Dec 2008
Observation of Valley Hall “Internal” edges
K
K’
..
..
V(x) > 0 V
x
V(x) > 0 V
x
0 0
V(x) < 0
X X X X
Usual edge states are destroyed by disorder. Not the internal ones!
KITP, Dec 2008
Topological confinement
Equations of motion for V(x)
(x)u + (x+ p
y)2
v = u
(x)v + (x
py)2
u = v
Martin, Blanter, Morpurgo ‘07
KITP, Dec 2008
Valleytronics
Proposals to use the valley degree of freedom for classical or quantum manipulaton (intervalley relaxation can be slow)
Use zig-zag ribbon edge state or Aharanov-Bohm (Rycerz, Tworzydlo, Beenakker ’07, Recher et al ’07)
Both require ideal interfaces: Hard with the existing technology
+V
top gate
-V
top gate
K K’
Alternative: use INTERNAL valley-polarized states:
I
K’ K
Valley filter:
+V -V
KITP, Dec 2008
Valley Valve
Direction of propagation of K-valley states
+V -V
-V +V
Iin
K I I
0
-V +V
-V +V
Vtr I
in
K
Iout
= Iin
0 0
KITP, Dec 2008
Open questions
• Interesting phenomena in more complicated structures? Aharonov-Bohm, non-ocal transport,
+ Superconductivity, + Magnetism?
• Interesting physics upon inclusion of interactions? Luttinger liquid?
• Valley Hall vs. Spin Hall: Spin <~> Valley
KITP, Dec 2008
Spontaneous CHARGE quantum Hall effect
S1
S2
S3
Hopping in a texture Hopping in an orbital field
H = t ci† cj
<ij>
JH Sii
ici† ci
ci† itinerant electrons
Si local moments (classical)
S1i[S2 S3]
= / 2
KITP, Dec 2008
Weak-coupling instability
• Chiral spin ordering:
• Ordering with Q1 & Q2 & Q3 fully gaps Fermi surface • for classical spins “tetrahedral” order is the best
At n = 1.5 el/site
KITP, Dec 2008
Answers and Questions
Band structure Observable: Hall conductivity (mean field Hubbard)
Questions: - role of fluctuations? - topological defects (Z2 vortices, domain walls)? - topological doping, superconductivity? - connection to quantum limit (S >>1 S~1)?
L = [ r + 2 (q Q )2 ] | S |2 u | S |4 +v[Sa i(Sb Sc )]
2