2009年7月10日
筑波大学
Optical Hall conductivity
in ordinary and graphene QHE
systems
東京大学
青木研究室D1
森本高裕
Morimoto, Hatsugai, Aoki
arXiv:0904.2438
Electronic structure of graphene
Effective
Hamiltonian
Tight binding approx.
A
B
Massless
Dirac
quasiparticles
Dirac QHE
sxy
rxx
10 μm
(Geim et al, Nature Mat. 2007)
(Novoselov et al, Nature 2005;
Zhang et al, Nature 2005)
2 /16
Purpose
B
Static transport properties of QHE
systems are established.
How about dynamical properties ?
Development of
THz spectroscopy
(Komiyama et al, PRL 2004)
(Sumikura et al, JJAP, 2007)
(Ikebe, Shimano, APL, 2008)
Anomalous QHE
in graphene
(Novoselov et al, Nature 2005;
Zhang et al, Nature 2005)
(Sadowski et al, PRL 2006)
The focus is optical properties of QHE systems:
(Morimoto, Hatsugai, Aoki
●Cyclotron emission in graphene
・・・ sxx
PRB 2007)
3 /16
●Faraday rotations in QHE systems ・・・ sxy
(to be published)
THz spectroscopy of 2DEG
(Ikebe, Shimano, APL, 2008)
Resonance structure at
cyclotron energy
Ellipticity
Faraday
rotation
(Sumikura et al,
JJAP, 2007)
4 /16
ac Hall effect
sxy (w)
● For ordinary 2DEG,
Faraday rotation measurement for THz w
(Sumikura et al, JJAP, 2007;
Ikebe, Shimano, APL, 2008)
● Optical (ac) Hall conductivity sxy (w)
for ordinary QHE systems
So far only treated with Drude form
(Sumikura et al, JJAP, 2007)
(O'Connell et al, PRB 1982)
for graphene QHE systems
● sxy (w) calculated with Kubo formula
5 /16
(Exact diagonalisation)
Effects of localization
2DEG
Effects of localization was significant
for static Hall coductivity sxy (w=0)
(Aoki & Ando 1980)
localization
length
DOS
How about for optical sxy (w) ?
Various range of impurities 
Short range : charged centers
Long range : ripples of graphene
optical sxy (w) :
Exact diagonalization (ED) for
long-ranged random potentials
6 /16
In clean limit…
●ac Hall conductivity from Kubo formula
●How does dc Hall plateau structure evolve into ac region?
Clean ordinary
QHE system
step
structure
resonance
structure
Hall step structure in the clean limit
How about with disorder?
Is it robust?
7 /16
Static Hall conductivity and Localization
Scaling behavior of
Thouless energy
impurity
Localization length
(K. Nomura et al, PRL, 2008)
Robust n=0
Anderson transition
8 /16
Formalism
●Diagonalization for
randomly placed impurities
(H0+V)
9 Landau levels retained
～5000 configurations
Strength of disorder G:
(Landau level broadening)
Free Dirac Hamiltonian +B
Impurity potential
whose range d ~ magnetic length
Optical Hall conductivity from Kubo formula for T=0
9 /16
Optical conductivity for graphene QHE
-12
01
12
G=0.2
Step structure in both static
and optical region
01
G=0.5
Plateau structure
remains up to ac region
(at least resonace?)
10 /16
Results for Usual QHE system
Step structure in both static
and optical region
12
01
G=0.2
DOS does not broaden
uniformly for LLs
G=0.7
Plateau structures seem to be more
robust than in graphene.
Difference of universarity classes
11 /16
Plateau in sxy (w) (ordinary QHE)
ac step structure
a remnant of QHE
remain for moder
disorders
12 /16
Plateau in sxy (w) (graphene QHE)
w = 1.5wc
G = 0.2
w = 0.9wc
w = 0.4wc
w=0
ac step structure as
a remnant of QHE
remain for moderate
disorders
13 /16
Estimation of Faraday rotation
Faraday rotation ∝ optical Hall conductivity
(O`Connell et al, PRB 1982)
(Nair et al, Science 2008)
Step structure cause jumps of Faraday rotation by
n0: air, ns: substrate
exp quite feasible!
Faraday rotation ~ fine structure constant:
“a seen as a rotation”
Resolution ~ 1 mrad in Ikebe,
Shimano, APL, 2008)
14 /16
Kubo formula, Localization, Robust step
(Aoki & Ando 1980)
Main contribution comes
from transitions between
extended states
Extended states
reside in the
center of LL as in
the clean sample
Contribution from
extended states
reproduce the
clean limit result
step
structure
resonance
structure
Robust Hall step structure from ED
calculation
 Localization and delocalization
physics as in dc Hall conductivity?
15 /16
Summary – ac Hall effects
● step structures in optical Hall condcutivity
 ac Hall effect
● effects of localization and robustness of plateau
structures
● estimated the magnitude of Faraday rotation and
experimentally feasible
□Future problems
● honeycomb lattice
calculation
●dynamical scaling
arguments of sxy (w)
-12
01
12
16 /16