スライド 1

Quantum Hall transition in graphene with
correlated bond disorder
-- Unconventional Hall transition at n=0 Landau level --
Outlines
1.
2.
3.
4.
5.
T. Kawarabayshi (Toho University)
Y. Hatsugai (University of Tsukuba)
H. Aoki (University of Tokyo)
ArXiv:0904.1927
Landau levels in graphene
Roles of disorder and Numerical model
Density of states
Hall conductivity
Summary
Characteristic band structure and Landau levels of graphene
 Dirac cones (K and K’)
at E=0 (Fermi energy)
K’
K
2D Honeycomb Lattice
n=0 (E=0) Landau level
 Landau levels around K and K’
E n   2 eB  v F | n |
2
-3,-2,-1
n=0
n=1,2,3,…
n=0 Landau level
Novoselov et al. Nature 2005
 Essential to
anomalous quantum Hall effect
 xy 
e
2
h
( 2 m  1) , m  0 ,  1,  2 ,  (per spin)
Zheng & Ando (2002)
 Robustness
the index theorem for Dirac fermions
Criticality: Dirac fermions +
random potential (long-range)
 Sensitivity
mixing of two valleys (K and K’)
Koshino, Ando (2007)
Schweitzer, Markos(2008)
Ostrovsky et al. (2008)
Nomura et al. (2008)
Roles of Disorder
Key concepts
(A) Chiral symmetry (UHU-1 = -H)
(B) Mixing of two valleys (K and K’)
Random bonds Yes
Random magnetic fields Yes
Random potential No
Short-range disorder
Long-range disorder
Yes
No
How these properties show up in the Landau level structure ?
2D Honeycomb Lattice Model
+ Spatially Correlated Disorder
To control (B) the mixing between K and K’
t
f
Systematic study on
the correlation dependence
Chiral symmetric , n=0 (E=0)
a
~1.42Å
An intrinsic disorder in graphene
Ripples
Disorder in transfer integrals
A.H. Castro Neto et al. Rev. Mod. Phys. (2009)
A model with disordered transfer integrals
t (r )  t   t (r )
Chiral symmetry
P ( t )  exp(  ( t ) / 2 ) /
2
 t ( r1 ) t ( r2 )   exp(  | r1  r2 | / 4 )
2
2
2
2
2
2
Gaussian with 
Correlation length 
A typical landscape (/a=5)
e2
e1
 t (r ) / 
Ly=Ny|2e2-e1|
Region with
large t(r)
Region with
Small t(r)
Lx=Nx|e1|
Density of states: correlation dependence
n=0
 /a >1
n=-2 n=-1

n=1 n=2
 (E )  
1
Im G r , r ( E  i  )
r
Anomaly for n=0
Landau level
/t = 0.115 f/f0=1/50 /t = 0.000625 Nx=5000, Ny=100
The Green function Method
Schweitzer, Kramer, MacKinnon (1984)
Hall conductivity xy in terms of Chern number CE
 xy ( E F ) 
e
2
h
C EF 
e
2
h
M
C
l 1
EF
CM
Thouless, Hohmoto, Nightingale, den Nijs (1982)
Aoki, Ando (1986), Niu, Thouless, Wu (1985)
l
Sum over many filled Landau bands
E ~ 0, weak fields
Contributions mostly cancel out
Accurate numerical method for Cl
C1
Hatsugai (2004,2005)
Fukui, Hatsugai, Suzuki (2005)
/a=1.5
n=1
Nx=Ny=10
300 samples
n=0
Unconventional
n=0 Hall transition
for /a >1
/a=0
T.K., Y. Hatsugai, H.Aoki,
ArXiv:0904.1927
n=-1
E/t
Hall conductance xy (Chern Number CE ) as a function of E
f/f0=1/50
n=1
/a=1.5
n=0
/a=0
E/t
Transition at E=0 is sensitive to the range of bond disorder
Nx=5000, Ny=100
n=-1
/t = 0.000625
Nx=Ny=10, 300 samples
/t = 0.115
Size-independent
/a=1.5
Nx=Ny=5
Nx=Ny=10
n=1
n=0
n=-1
E/t
Additional potential disorder
[-w/2, w/2]
n=0
n=-1
w=0
w=0.4
insensitive
Breakdown of chiral symmetry
sensitive
Other disorder with chiral symmetry
Disordered magnetic fields
P (f )  exp(  (f ) / 2 f ) /
2
2
ff(r)
2  f
2
f ( r1 )f ( r2 )   f exp(  | r1  r2 | / 4 f )
2
/a
2
2
Anomaly at the n=0 level
f/f = 0.237 < 1
small disorder
large disorder
f/f = 2.37 > 1
f/f0=1/41
/t = 0.000625
Nx=5000, Ny=82
/a
Summary
The n=0 Landau level : anomalously robust
against long-range bond disorder
Chiral symmetry, Absence of scattering between K and K’
No broadening for /a > 1
Consistent with the index theorem
Scale of ripples in graphene : 10 ~ 15 nm >> a
 No broadening of n=0 level by
bond disorder by ripples
(Meyer, Geim et al, Nature 2007)
 Other disorder (ex. potential disorder)
should be responsible for the
broadening of n=0 level
 Possibility to observe this anomaly at n=0 in clean graphene
without potential disorder from substrates
Classical Hall transition ?
Ostrovsky et al. (2008)
Other disorder without chiral symmery
Potential disorder V(r)
P (V )  exp(  V / 2 S ) /
2
2
2 S
2
V ( r1 )V ( r2 )   S exp(  | r1  r2 | / 4 S )
2
S/t = 0.288
f/f0=1/41
/t = 0.000625
Nx=5000, Ny=82
No anomaly
2
2

Download Report