Download integer QHE in graphene

Effects of Interaction and Disorder in Quantum
Hall region of Dirac Fermions in 2D Graphene
Donna Sheng (CSUN)
In collaboration with:
Hao Wang (CSUN), L. Sheng (UH), Z.Y. Weng (Tsinghua)
F. D. M. Haldane (Princeton) and L. Balents (UCSB)
Supported by DOE and NSF
Introduction: Experiment observations of “halfinteger” and odd integer QHE in graphene
an atomic layer of carbon atoms
forming honeycomb lattice
K. S. Novoselov et al., Nature (2005)
Y. Zheng et al., Nature (2005)
Carbon nanotubes are a beautiful material.
Problem is how to make them in wafers
and with reproducible properties. You can't
distinguish between semiconducting
and metallic nanotubes. You've got
uncontrollable material. Graphene is essentially
the same material, but unrolled carbon
nanotubes.
One of applications is to make graphene
transistors of 10nm size.
Experiment observations of “half-integer” integer
QHE in graphene
K. S. Novoselov et al., Nature (2005)
Y. Zheng et al., Nature (2005) Columbia Univ.
Theoretical work using continuous model for Dirac fermions can
account/predicted such quantizations: Gusynin et al. Peres et al.,
Zheng and Ando
Relation with band structure of honeycomb lattice model?
Three regions of IQHE in the energy band
IQHE for Dirac fermions in the middle
D.N.Sheng, Phys. Rev. B73, 233406 (2006)
F2p/M
H  - t  C C e  h.c   w C C

iB

iAij
jA
i
i
i
H 
2V / il
F
B
(a0

a
0
)
Effect of disorder and phase diagram: PRB 73 (2006)
Phase diagram
Disorder splits one
extended level
at the center of
n=0 LL to 2 critical
points, leaving
n=0 region an
insulating phase
Experiment discovers n1 IQHE and
“n=0” insulating phase
Y. Zhang et. al., PRL 2006
Interaction has to be taken
into account to explain the
n=1IQHE---Pseudospin
Ferromagnet
delocalization of Dirac fermions at B=0 (a
different issue)
E_f=0
M
Transfer matrix calculation of the “finite size localization length”
for quasi-1D system with width M, it indicates “delocalization”
at Dirac point E_f=0 at E<1.0t
Experiment discovers n1 IQHE and
“n=0” insulating phase
Y. Zhang et. al., PRL 2006
Interaction has to be taken into
account to explain the n=1
IQHE---Pseudospin
Ferromagnet
More experiments by Z. Jiang et. al. on
activation gap of IQHE
For n1 , DE is proportional to e^2/ el
Interaction and pseudospin FM state:
Theoretical works
Nomura & MacDonald
Stoner criteria for pseudospin FM
Alicea & Fisher
lattice effect is relevant
Yang et al., Gusynin et al.
Toke & Jain, Goerbig et al.
continuum model, SU(2)*SU(2)
symmetry
Haldane’s Pseudo-Potential
gives rise to incompressible
state, SU(4) invariant
Our motivation:
detailed nature of quantum
phases, quantitative
behavior of systems
Competitions between:
Coulomb interaction,
lattice, and disorder
scattering effect based on
exact calculations
Exact diagonalization using lattice model
H  - t  C C e  h.c   w C C

iB

iAij
jA
i
i
i
Only keep states inside the
top-Landau level, large
lattice size and keep a
degeneracy of Ns around 20
Energy spectrum for pure system: PFM ordering
n= 3 XY plane PFM
n 1 Ising PFM
(zero-energy states)
A-sub,
B-sub,
CDW
No pseudospin conservation
for higher LL!!
the excitation energy gap (with double occupation)
|S_z|
W
the excitation gap scales with 1/Ne,
possibly extrapolates to zero
at large Ne limit
Directly look at the transport property instead of “gaps”
The destruction of odd IQHE is due to the mixing of
various Chern numbers
Chern number “IS” Hall conductance
D. J. Thouless et al 1982, J. E. Avron et al. 1883
2p
C

em  e
C ( m)
F
C is for
many-body
D.N. Sheng et al., PRL 2003;
Sheng, Balents, Wang
Xin et al. PRB (FQHE)
0
2p
(qx, qy) boundary phase
Just get y(q) at all nodes of mesh
of 100-1000 points,
overlap of y(q) at nearest points
The importance of mobility gap (activation gap of experiment):
from direct Hall conductance and Chern number calculations
the size of the
Mobility gap
States inside
mobility gap
finite size scaling confirms a finite transport gap at
large size limit (more data are coming)
Fluctuation of Chern numbers determine a mobility edge
e^2/ el
Example of comparing mobility gap with experiments for 1/3
FQHE: D. N. Sheng et al (2003) PRL, Xin et al. (2005) PRB
typing
Phase diagram for Odd IQHE states
Symmetry broken states (stripes and bubbles) in n=3
and n=4 Dirac LLs
-1.20
4.5
-1.25
3.0
-1.35
*
S0( q )
-1.40
1.5
-1.45
10
-1.50
4
6
Momentum J
8
10
12
5
10
-5
-10
p/
0
K (
2 p /a
x
)
(2
2
0
-5
y
0
5
K
-1.55
0.0
-10
b)
2
E(e /el)
-1.30
n=3(LL=2&3) Graphene, 12/24 systerm, a/b=0.74,q*=(0,±0.595)
Disorder-Caused Phase Transition
ust=0.2
ust=2.4
ust=3.2
0.9
4
*
S0( q )
*
S0( q )
*
S0( q )
2
2
10
5
10
16
18
20
22
24
( 2p
-5
5
/a )
-10
10
p/
0
2
4
6
8
10
12
14
16
18
20
22
24
24
24
22
22
22
22
20
20
20
20
18
18
18
18
16
16
16
16
14
14
14
14
12
12
12
12
10
10
10
8
8
8
6
6
6
4
4
4
2
2
2
0
0
0
6
4
2
0
5
-5
K
x
0
0
( 2p
/a )
-5
5
-10
10
p
(2
10
/b)
Ky
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.14
0.12
0.10
0.08
0.06
0.04
0.02
24
8
/b
2p
Ky
24
10
(
)
0.0
-10
0
2
4
6
8
10
12
14
16
18
20
22
0.14
0.12
0.10
0.08
0.06
0.04
0.02
14
0
0
10
0.14
0.12
0.10
0.08
0.06
0.04
0.02
12
0.11
10
K
x
0.10
8
0.09
6
0.08
4
0.07
2
0.06
0
0.05
0.11
0.10
0.09
0.08
0.07
0.06
0.05
5
-5
(2
-5
-10
y
0
K (2
5
x
p/a)
0
-10
b)
0
-5
K
0-10
24
24
24
22
22
20
20
18
18
16
16
14
14
12
12
10
10
8
8
6
6
4
4
2
2
0
0
3
*
S0(q )
4
2
1
0
1
2
3
ust ( e /el)
2
4
8/24 system, LL2+3, a/b=0.65, bubble phase
Structure factor:S0(q)
Correlation function:G(r)
8/24 system, LL2+3, a/b=0.86, stripe phase
Structure factor:S0(q)
Correlation function:G(r)
Summary:
The n1 IQHE state is an Ising and valley polarized PFM
The n3 IQHE state is a valley mixing, xy plane polarized PFM
Critical Wc, mobility gap, and quantum phase diagram
Symmetry broken states (stripes and bubbles) are predicted to be
The ground state in higher (n=3 and n=4) Dirac LLs
Open questions: Is there a Non-Abelian FQHE at n=1
and n=2 Dirac LLs?