D12009710Optical Hall conductivityin ordinary and graphene QHE systemsMorimoto, Hatsugai, Aoki arXiv:0904.2438

(Geim et al, Nature Mat. 2007)Electronic structure of graphene/16*Dirac QHETight binding approx.Effective Hamiltonian

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Purpose/16(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 sxxFaraday rotations in QHE systems sxy(Morimoto, Hatsugai, Aoki PRB 2007) (to be published)*

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THz spectroscopy of 2DEGFaraday rotation (Sumikura et al, JJAP, 2007)EllipticityResonance structure at cyclotron energy(Ikebe, Shimano, APL, 2008)*/16

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For ordinary 2DEG, Faraday rotation measurement for THz w Optical (ac) Hall conductivity sxy (w) for ordinary QHE systemsSo far only treated with Drude form(O'Connell et al, PRB 1982) sxy (w) calculated with Kubo formula (Exact diagonalisation)(Sumikura et al, JJAP, 2007; Ikebe, Shimano, APL, 2008)ac Hall effect sxy (w) for graphene QHE systems/16(Sumikura et al, JJAP, 2007)*

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Effects of localizationHow about for optical sxy (w) ?/16Various range of impurities Short range : charged centersLong range : ripples of grapheneoptical sxy (w) :Exact diagonalization (ED) forlong-ranged random potentialsEffects of localization was significant for static Hall coductivity sxy (w=0)2DEG*

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In clean limitac Hall conductivity from Kubo formulaHow 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/16resonance structurestep structure*

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Static Hall conductivity and Localization(K. Nomura et al, PRL, 2008)/16impurityRobust n=0 Anderson transition*

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FormalismDiagonalization for randomly placed impurities(H0+V)9 Landau levels retained5000 configurationsOptical Hall conductivity from Kubo formula for T=0/16*

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Optical conductivity for graphene QHE/16G=0.501Step structure in both static and optical regionPlateau structure remains up to ac region (at least resonace?)*

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Results for Usual QHE system/16G=0.20112DOS does not broaden uniformly for LLs G=0.7Step structure in both static and optical regionPlateau structures seem to be more robust than in graphene.Difference of universarity classes*

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DisorderPlateau in sxy (w) (ordinary QHE)/16ac step structure as a remnant of QHE remain for moderate disorders*

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w = 0w = 0.9wcw = 1.5wcDisorderw = 0.4wcPlateau in sxy (w) (graphene QHE)/16ac step structure as a remnant of QHE remain for moderate disorders*

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*Estimation of Faraday rotationFaraday rotation ~ fine structure constant:a seen as a rotationFaraday rotation optical Hall conductivityexp quite feasible!n0: air, ns: substrateStep structure cause jumps of Faraday rotation by/16*

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Kubo formula, Localization, Robust stepRobust Hall step structure from ED calculation Localization and delocalization physics as in dc Hall conductivity?/16resonance structurestep structure*(Aoki & Ando 1980)Main contribution comes from transitions between extended statesExtended states reside in the center of LL as in the clean sampleContribution from extended states reproduce the clean limit result

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Summary ac Hall effects/16Future problems honeycomb lattice calculationdynamical scaling arguments of sxy (w)

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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