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Purpose Static transport properties of QHE systems are established. How about dynamical properties ? /16 Development of THz spectroscopy Anomalous QHE in graphene (Komiyama et al, PRL 2004) (Sumikura et al, JJAP, 2007) (Ikebe, Shimano, APL, 2008) (Novoselov et al, Nature 2005; Zhang et al, Nature 2005) (Sadowski et al, PRL 2006) The focus is optical properties of QHE systems: ●Cyclotron emission in graphene ・・・ xx ●Faraday rotations in QHE systems ・・・ xy (Morimoto, Hatsugai, Aoki PRB 2007) (to be published) 3 B
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東京大学青木研究室 D1
森本高裕
2009 年 7 月 10 日筑波大学
Optical Hall conductivityin ordinary and graphene QHE
systems
Morimoto, Hatsugai, Aoki arXiv:0904.2438
10 μm(Geim et al, Nature Mat. 2007)
Electronic structure of graphene
/16
xx
xy
(Novoselov et al, Nature 2005; Zhang et al, Nature 2005)
Massless Dirac quasiparticles
2
Dirac QHE
AB
Tight binding approx.
Effective Hamiltonian
PurposeStatic transport properties of QHE systems are established.
How about dynamical properties ?
/16
Development of THz spectroscopy
Anomalous QHE in graphene
(Komiyama et al, PRL 2004)(Sumikura et al, JJAP, 2007)(Ikebe, Shimano, APL, 2008)
(Novoselov et al, Nature 2005; Zhang et al, Nature 2005)(Sadowski et al, PRL 2006)
The focus is optical properties of QHE systems:●Cyclotron emission in graphene ・・・
xx
●Faraday rotations in QHE systems ・・・ xy
(Morimoto, Hatsugai, Aoki
PRB 2007) (to be published)
3
B
THz spectroscopy of 2DEG
Faraday rotation
(Sumikura et al, JJAP, 2007)
Ellipticity
Resonance structure at cyclotron energy
(Ikebe, Shimano, APL, 2008)
4 /16
● For ordinary 2DEG, Faraday rotation measurement for THz
● Optical (ac) Hall conductivity xy ( for ordinary QHE systems
So far only treated with Drude form(O'Connell et al, PRB 1982)
● xy () calculated with Kubo formula (Exact diagonalisation)
(Sumikura et al, JJAP, 2007; Ikebe, Shimano, APL, 2008)
ac Hall effect xy ()
for graphene QHE systems
/16
(Sum
ikur
a et
al,
JJAP,
20
07)
5
Effects of localization
How about for optical xy () ?
/16
Various range of impurities Short range : charged centersLong range : ripples of graphene
optical xy () :Exact diagonalization (ED) forlong-ranged random potentials
(Aoki & Ando 1980)
localization length
DOS
Effects of localization was significant for static Hall coductivity xy ()
2DEG
6
In clean limit…
●ac Hall conductivity from Kubo formula●How does dc Hall plateau structure evolve into ac region?
Hall step structure in the clean limitHow about with disorder? Is it robust?
Clean ordinary QHE system
/16
resonance structure
step structure
7
Static Hall conductivity and Localization
(K. Nomura et al, PRL, 2008)
/16
Scaling behavior of Thouless energy
Localization length
impu
rity
Robust n=0 Anderson transition 8
Formalism
●Diagonalization for randomly placed impurities(H0+V)
9 Landau levels retained~ 5000 configurations
Optical Hall conductivity from Kubo formula for T=0
/16
Free Dirac Hamiltonian +B Impurity potential whose range d ~ magnetic length
Strength of disorder
(Landau level broadening)
9
Optical conductivity for graphene QHE
/16=0.5
=0.2
01
-12
12
01
Step structure in both static and optical region
Plateau structure remains up to ac region (at least resonace?)
10
Results for Usual QHE system
/16
=0.2
01
12
DOS does not broaden uniformly for LLs
=0.7
Step structure in both static and optical region
Plateau structures seem to be more robust than in graphene.Difference of universarity classes
11
Disord
er
Plateau in xy () (ordinary QHE)
/16
ac step structure as a remnant of QHE remain for moderate disorders
12
= 0
= 0.9c
= 1.5c
Disord
er
= 0.2
= 0.4c
Plateau in xy () (graphene QHE)
/16
ac step structure as a remnant of QHE remain for moderate disorders
13
Resolution ~ 1 mrad in Ikebe, Shimano, APL, 2008)
14
Estimation of Faraday rotation
Faraday rotation ~ fine structure constant:
“ seen as a rotation”
Faraday rotation ∝ optical Hall conductivity
(O`Connell et al, PRB 1982)
exp quite feasible!
n0: air, ns: substrate
Step structure cause jumps of Faraday rotation by
(Nair et al, Science 2008)
/1614
Kubo formula, Localization, Robust step
Robust Hall step structure from ED calculation Localization and delocalization physics as in dc Hall conductivity? /16
resonance structure
step structure
15
(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
Summary – ac Hall effects
/16
01
-12
12
□Future problems● honeycomb lattice
calculation●dynamical scaling
arguments of xy ()
● 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
16