東京大学 青木研究室 D1 森本高裕 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 conductivity in ordinary and graphene QHE systems Optical Hall conductivity in ordinary and graphene QHE systems Morimoto, Hatsugai, Aoki arXiv:

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 A B Tight binding approx. Effective Hamiltonian

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

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 (Sumikura et al, JJAP, 2007) 5

Effects of localization How about for optical  xy (  ) ? /16 Various range of impurities  Short range : charged centers Long range : ripples of graphene optical  xy (  ) : Exact diagonalization (ED) for long-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 limit  How 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 impurity Robust n=0 Anderson transition 8

Formalism ●Diagonalization for randomly placed impurities (H 0 +V) 9 Landau levels retained ～ 5000 configurations Optical Hall conductivity from Kubo formula for T=0 /16 Free Dirac Hamiltonian +BImpurity potential whose range d ~ magnetic length Strength of disorder  (Landau level broadening) 9

Optical conductivity for graphene QHE /16  =0.5  =0.2 0101 -1  2 1212 0101 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 0101 1212 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

Disorder Plateau in  xy (  ) (ordinary QHE) /16 ac step structure as a remnant of QHE remain for moderate disorders 12

 = 0  = 0.9  c  = 1.5  c Disorder  = 0.2  = 0.4  c 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! n 0 : air, n s : substrate Step structure cause jumps of Faraday rotation by (Nair et al, Science 2008) /16 14

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 0101 -1  2 1212 □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