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

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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 ?

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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

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(Sum

ikur

a et

al,

JJAP,

20

07)

5

Effects of localization

How about for optical xy () ?

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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

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resonance structure

step structure

7

Static Hall conductivity and Localization

(K. Nomura et al, PRL, 2008)

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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

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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

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=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)

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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)

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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

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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

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