KIAS-SNU Winter School,
Feb. 29, 2011; 휘닉스 파크
Observation of Quantum Hall Effect in Graphene
Pohang University of Science and Technology
(POSTECH)
Quantum Transport and Superconductivity Laboratory
이후종
Introduction; Integer Quantum-Hall Effect
- in 2DEG, monolayer graphene, bilayer graphene
- edge conducting states
Observed QHE in Graphene
Overview
Observed QHE in Graphene
- Half-integer quantum-Hall effect in graphene
- Edge state equilibrium
- Lifting the degeneracy in high magnetic fields
- Fractional quantum Hall effect in high-mobility graphene in high fields
Summary
Carbon Allotropes : in Diverse Dimensions
Two dimension Three dimension
One dimensionZero dimension
Patterning
Functionality
1a
2a
Graphene
2.46 A°
‘A’ sublattice
‘B’ sublattice
graphene lattice SP2 covalent hybrid
orbital of a carbon atom
π
σ
2.46 A
(real space)
Two equivalent sublattices
Two atoms per unit cell
π-orbital
σ-bond
Band Structure and Low-Energy Dispersion
FE kυ=r
h
KM
Γ
K’ EF
• Dirac cone
• Linear dispersion at zero energy
- Massless relativistic Dirac fermions
- But moving with Fermi velocity
- Carrier type and density are easily controlled by gatingFυ
Chirality or Pseudospin in Graphene Lattice
KK’ kx
ky
kx
EE
electron-like
pseudospin or chirality
Chirality – momentum-locked phase value of a carrier in graphene when the carrier
moves along a Dirac cone (or the sublattice index)
hole-like
Carrier Mobility
VBGSiO2 300nm
Si
negraphene
E
υµ =
j E neσ υ= =
}
I11-2-V10-9
I = 20 nA, L=W=7 µm
22 22
2
2
@ =0,
= ( )2
F FF
F mfp
T
e Ev e v ne g E
ve n
τ τσ τ
π π
σ πτ
= =
= =
h h
hl
Boltzmann theory;
natural graphite
Graphene Preparation – Mechanical Exfoliation
exfoliated graphene on Si sub
1
32
1
10 µm
thin graphite on tape transferring graphene onto Si substrate
Graphene Preparation – Mechanical Exfoliation
10 µm 10 µm
10 µm
10 µm
Introduction; Integer Quantum-Hall Effect
- in 2DEG, monolayer graphene, bilayer graphene
- edge conducting states
Observed QHE in Graphene
Overview
Observed QHE in Graphene
- Half-integer quantum-Hall effect in graphene
- Edge state equilibrium
- Lifting the degeneracy in high magnetic fields
- Fractional quantum Hall effect in high-mobility graphene in high fields
Summary
Classical Hall Effect
1H
H
ne
R Bσ = = − σH is linearly dependent on n with a slope of e/B
Picture by Dr. Dong Su Kim
(Integer) Quantum-Hall Effect
- LLs - 2D carriers in high-enough H field and low T
HH=0 Fermi edge
VSD
S D
(Integer) Quantum-Hall Effect
VSD
VL
von Klitzing, 8.., PRL, 1980
S D
VH
; 2DEG
Halperin, PRB 25, 2185 (1982)
- Dissipationless and chiral edge state carries the current at QH plateaus
- Arising as Landau levels are pushed up by the confining edge potential
von Klitzing, Dorda, and Pepper,
PRL 45, 494 (1980)
Quantum-Hall Edge States
EF
ν=1
BeB
=h
l
Halperin, PRB 25, 2185 (1982)
von Klitzing, Dorda, and Pepper,
PRL 45, 494 (1980)
Quantum-Hall Edge States
ν=2
EF
Halperin, PRB 25, 2185 (1982)
von Klitzing, Dorda, and Pepper,
PRL 45, 494 (1980)
Quantum-Hall Edge States
ν=3
EF
Integer QHE in 2D Electron Gas
6
2-folddegeneracy
1( )
2n cE nω= +h
1 22− 1− 0
0n= 11−2−
nEstrong H
S D
VS
VH
2
xy
e
hσ ν= 0, 2, 4, ...ν = ± ±
0-2-4 2 4
filling factor ν (=eB/h)
2DEG
2
4
-2
-4
0
6
-6
Unconventional QHE in Monolayer Graphene
2DEG4
610 Monolayer
1 22− 1− 0
2-folddegeneracy
1( )
2n cE nω= +h
0n = 11−2−
nE
Landau level
4-folddegeneracy
11− 02− nE
0n = 11−2− 2
2
2n FE e B nυ= ± h
Integer QHE
2 0, 2, 4, ...nν = = ± ±
0-2-4 2 4
filling factor ν (=eB/h)
2DEG
2
4
-2
-4
0
-6
Half integer QHE
4 degenerate zero mode
14( ) 2, 6, 10, ...
2nν = + = ± ± ±
2
6
10
-2
-6
-10
0-4-8 4 8
filling factor ν (=eB/h)
Monolayer
42
xy
e
hσ ν=
Due to the special status of the ν=0 Landau level : half of its
states are hole states, and the other half are electron states.
Unconventional QHE in Bilayer Graphene
1 22− 1− 0
2-folddegeneracy
1( )
2n cE nω= +h
0n = 11−2−
nE
4-folddegeneracy
8-folddegeneracy
( 1)n cE n nω= ± −h
1 22− 1− 0 nE
0,1n =
21−2− 3
2DEG4
610 Monolayer
12Bilayer
Landau level
4-folddegeneracy
11− 02− nE
0n = 11−2− 2
2
DOS
2n FE e B nυ= ± h
2
xy
e
hσ ν=
Integer QHE
0-2-4 2 4
filling factor ν (=eB/h)
2DEG
2
4
-2
-4
0
-6
2 0, 2, 4, ...nν = = ± ±
Half integer QHE
4 degenerate zero mode
-10
2
6
10
-2
-6
0-4-8 4 8
filling factor ν (=eB/h)
Monolayer
2+2
14( ) 2, 6, 10, ...
2nν = + = ± ± ±
Integer QHE
8 degenerate zero mode
4 4, 8, 12, ...nν = = ± ± ±
4
8
-4
-8
-12
0
0-4-8 4 8
filling factor ν (=eB/h)
4+2
Bilayer
0n =except for
Introduction; Integer Quantum-Hall Effect
- in 2DEG, monolayer graphene, bilayer graphene
- edge conducting states
Observed QHE in Graphene
Overview
Observed QHE in Graphene
- Half-integer quantum-Hall effect in graphene
- Edge state equilibrium
- Lifting the degeneracy in high magnetic fields
- Fractional quantum Hall effect in high-mobility graphene in high fields
Summary
Half-integer QHE in Monolayer Graphene
Novoselov et al., Nature 438, 197 (2005)
Zhang et al., Nature 438, 201 (2005)
610 14
-2
-6-10
(K) 420 (T)LLE H∆ =
Room Temperature QHE
0n= 11−2− 2
LLE∆
Novoselov et al., Science 315, 1379 (2007)
2n FE e B nυ= ± h
11− 02−nE2
xp
yp
E
Massive Dirac Fermions
Unconventional QHE in Bilayer Graphene
4-folddegeneracy
8-folddegeneracy
1 22− 1− 0 nE
0,1n=
21−2− 3
DOS
Introduction; Integer Quantum-Hall Effect
- in 2DEG, monolayer graphene, bilayer graphene
- edge conducting states
Observed QHE in Graphene
Overview
Observed QHE in Graphene
- Half-integer quantum-Hall effect in graphene
- Edge state equilibrium
- Lifting the degeneracy in high magnetic fields
- Fractional quantum Hall effect in high-mobility graphene in high fields
Summary
Quantum-Hall Conduction in Bi-polar Junction
H
VBG(V)
VLG (V)
(Two-terminal Studies)
Iin
Iout
H
[ Abanin & Levitov, Science 317, 641 (2007) ]
[ Williams, DiCarlo, Marcus,
Science 317, 638 (2007) ]
ν2
ν1
- For the incident current , only a fraction of transmits.
Quantum-Hall Conduction in p-n-p Junction
(Two-terminal Studies)
[ Ozyilmaz et al,
PRL 99, 166804 (2007) ]
..
11 2
Our Graphene p-n-p Junction (4-terminal Studies)
EF
top gate
S
D
1
2
1
half-integer QHE
EF
H= 0EF
EF
Ki and Lee, PRB 79, 195327 (2009)
half-integer QHE
Longitudinal QH Resistance
H = 10 T
VL
PRL 99, 166804 (2007)
VLG(V)- Consistent with two-terminal results
- Inversion symmetry
- Zero RL ; full transmission of edge states when ν1=ν2- Fractionally quantized RL ; partial transmission of edge states for ν1=ν2
reflection of a certain portion of edge states
Ki and Lee, PRB 79, 195327 (2009)
Diagonal QH Resistances
VD
RD ;
- No inversion symmetry w.r.t. (ν1=0, ν2=0), but inversion symmetry between the two
VLG(V)
Introduction; Integer Quantum-Hall Effect
- in 2DEG, monolayer graphene, bilayer graphene
- edge conducting states
Observed QHE in Graphene
Overview
Observed QHE in Graphene
- Half-integer quantum-Hall effect in graphene
- Edge state equilibrium
- Lifting the degeneracy in high magnetic fields
- Fractional quantum Hall effect in high-mobility graphene in high fields
Summary
K
E
x
- Countercirculating edge states
- Spin is 100% polarized in each edge state
ν=+1
ν= -1ν= 0
ν=+2ν=+4∆E
∆E
∆E
∆En=0LL
n=+1LL
Level Splitting in High Fields
n=0LL
[기동근, 이후종, 물리학과 첨단기술July/August 2009]
magnetic-field-induced
spontaneous symmetry
breaking mediated by e–e
interactions
K’
x
bulk edge
QH ferromagnet
spin valley∆ > ∆
K
K’
E
x
bulk edge
w/o any symmetry
breaking
Real-spin symmetry
broken first
H
ν= -1
n=-1LL
1ν = valley splits
LL
K
K’
E
x
bulk edge
K
E
x
w/o any symmetry
breaking
Level Splitting in High Fields
n=0LL
ν=0
QH inuslator
- No edge state itself,
- Insulating - longitudinal resistance is thermally activated
K’
x
bulk edge spin valley∆ < ∆
0ν = valley splits
Pseudospin symmetry
broken first
LL
QH inuslator for E=0
Level Splitting in High Fields
K
E
K
K’
E
x
bulk edge
spin valley∆ < ∆(a)
(b)
magnetic-field-induced
spontaneous symmetry
breaking mediated by e–e
interactions
[기동근, 이후종, 물리학과 첨단기술July/August 2009]
Pseudospin symmetry broken first
QH ferromagnet for E=0
K’
x
bulk edge
K
K’
E
x
bulk edge
bulk edge
spin valley∆ > ∆
(c)
Incresing H field
w/o symmetry breaking
Real-spin symmetry broken first
Metallic Character of ν=0 State
H=30 T
For high HE
Abanin et al., PRL 98, 196806 (2007)
countercirculating edge
states with opposite spin
K
K’
x
bulk edge
Insulating Character of ν=0 State
2 2
xy
xy
xy xx
ρσ
ρ ρ=
+
K
E
Checkelsky, Li, and Ong, PRL 100, 206801 (2008)
20
-2
-6
ν=6
ν=0
ν=-2
2
K
K’
x
bulk edge
Level Splitting in High Fields
B=45 T
1
2
-1
-2
Y. Zhang et al., Phys. Rev. Lett. 96, 136806 (2006)
4-4
9 T 30 mK
25 T 1.4 K
30 T 1.4 K
37 T 1.4 K
42 T 1.4 K
45 T 1.4 K
25 T 1.4 K
1
2
4
6
-1-2
-4
0
Level Splitting in High Fields
K
K’
E
x
bulk edge
- ν=0 QH plateau ; resolved at B>11 T
- Many-body electron correlation within the LL - an alternative origin for the lifting
of the degeneracy at the Dirac point.
- Weaker features of ν= 3 a hierarchy exists in lifting the degeneracy of LLs
strength of e–e interactions
2
, / , 1B
B
eeB ε
ε∝ = ≈l h
l
±
9 T 30 mK
11.5 T 30 mK
17.5 T 30 mK
-4
-60
Introduction; Integer Quantum-Hall Effect
- in 2DEG, monolayer graphene, bilayer graphene
- edge conducting states
Observed QHE in Graphene
Overview
Observed QHE in Graphene
- Half-integer quantum-Hall effect in graphene
- Edge state equilibrium
- Lifting the degeneracy in high magnetic fields
- Fractional quantum Hall effect in high-mobility graphene in high
fields
Summary
Enhancing Carrier Mobility - Suspending Graphene
- Graphene exfoliation and e-beam patterning electrodes
- Etching away SiO2 in HF solution (DI water:HF = 6:1)
- Immersing in DI water and IPA, and critical-point drying
- Ar/H2 annealing or current annealing to remove water or organic residue
gas
liquid supercritical fluid
Du, G., Andrei,
Nature Nanotechnology 3, 491 (2008)
Enhancing Mobility – Exfoliation on BN Substrates
Dean et al., Nature Nanotechnology 5, 722 (2010)
Substrate-supported geometry while retaining the quality of suspended graphene
- smooth surface, relatively free of dangling bonds and charge traps
- better match of the lattice constant (1.7% mismatch)
- large electrical band gap (~6 eV)
Mechanical Transfer
h-
Fractional QHE in High-mobility Graphene
- QH plateaus at ν=0, 1, 4 for H>25 T as interaction effects lifting the degeneracy
new integer plateaus outside the usual sequence
- Observable only when the energy scale > energy fluctuations induced by external sources
- High quality (ballistic transport and low carrier density) graphene is required to clarify ;
Role of correlations in the low density phases
Whether graphene can support an FQHE
± ±
X. Du, GG E. Y. Andrei, Nature 462, 192, (2009)
X. Du, GG E. Y. Andrei, Nature 462, 192, (2009)
Fractional QHE in High-mobility Graphene
T=1.2 K
FQHE;
- Strongly correlated fractional QH liquid in high H field minimize its energy for the filling factors
(with m and p integers)
- Electrons and magnetic flux quanta bind to form complex composite state with fractionally
charged quasiparticles for elementary excitations
- ν=1/3 plateau ; corresponds to composite particles of one electron and two flux lines, with
fractionally charged quasiparticles of q*=e/3 and an excitation gap, ∆1/3
- Insulating state at | ν |=0.1
2 1
m
pmν =
±
In low magnetic field
Fractional QHE in High-mobility Graphene
K. Bolotin,GPhilip Kim, Nature 462, 196 (2009)
10
6
ν=2
10
- QH plateaus even at 0.3 T2n FE e B nυ= ± h
ν=0.3
Magnetotransport in high magnetic fields
Fractional QHE in High-mobility Graphene
ν=1
ν=2
ν=-1
ν=-2
0.46v =0.68v =0.32v =
K. Bolotin,GPhilip Kim, Nature 462, 196 (2009)
− ν=0, 1 plateaus appear as e–e interactions among Dirac quasiparticles lift the pseudospin
and spin degeneracy of the zeroth Landau level
− ν=1/3 plateau persists up to 10 K for H=6 T, much robust than in 2DEG due to strong e-e
interaction for small ε (=1)
±
onset of an insulating
state at low density
ν=0
- B; not a FQH plateau
- Arising from two-term meas.
Insulating State in Graphene near Zero Density
n=0.17x1011 cm-2
K. Bolotin,GPhilip Kim, Nature 462, 196 (2009)
Insulating state near n=0 stems from the symmetry
breaking of the zeroth Landau level by e–e interactions.
µ∼30,000 cm2/Vs
Fractional QHE in Graphene – 4 Probe Meas.
- A complete lifting of the four-fold degeneracy in n = 0, 1 Landau levels
n=0
Fractional QHE in Graphene – 4 Probe Meas.
n=1n ν
1
2
Strongly interacting
electrons in a high H
2 1
m
pmν =
±
A system of weakly interacting CFs
consisting of an electron bound to
2p magnetic flux vortices
FQHE State
mapping
(i) all degeneracies are explicitly broken,
for example, by coupling to external fields;
(ii) only one of spin or valley isospin degeneracy
is broken, preserving an SU(2) symmetry in the
ν
Rxx
0 1-1
is broken, preserving an SU(2) symmetry in the
remaining degenerate space;
(iii) the full degeneracy is preserved, leading to
an emergent SU(4) symmetry in the combined
spin-isospin space
ν0 1-1
ν0 1-1
Introduction; Integer Quantum-Hall Effect
- in 2DEG, monolayer graphene, bilayer graphene
- edge conducting states
Observed QHE in Graphene
- Half-integer quantum-Hall effect in graphene
Overview
- Half-integer quantum-Hall effect in graphene
- Edge state equilibrium
- Lifting the degeneracy in high magnetic fields
- Fractional quantum Hall effect in high-mobility graphene in high fields
Summary
FQHE - suggesting the possibility of observing novel spin textures with no
analog in other single-layer quantum Hall systems