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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 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. s xy. r xx. 10 μ m. (Novoselov et al , Nature 2005; Zhang et al , Nature 2005). Electronic structure of graphene.

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東京大学 青木研究室 D1 森本高裕

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  1. 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

  2. sxy rxx 10 μm (Novoselov et al, Nature 2005; Zhang et al, Nature 2005) Electronic structure of graphene Effective Hamiltonian Tight binding approx. A B Massless Dirac quasiparticles Dirac QHE /16 (Geim et al, Nature Mat. 2007)

  3. B Purpose Static transport properties of QHE systems are established. How about dynamical properties ? Anomalous QHE in graphene Development of THz spectroscopy (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　　 ・・・ sxx ●Faraday rotations in QHE systems　・・・ sxy (Morimoto, Hatsugai, Aoki PRB 2007) /16 (to be published)

  4. THz spectroscopy of 2DEG (Ikebe, Shimano, APL, 2008) Resonance structure at cyclotron energy Faraday rotation Ellipticity /16 (Sumikura et al, JJAP, 2007)

  5. ac Hall effectsxy (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 (O'Connell et al, PRB 1982) for graphene QHE systems (Sumikura et al, JJAP, 2007) ● sxy(w) calculated with Kubo formula (Exact diagonalisation) /16

  6. Effects of localization (Aoki & Ando 1980) Effects of localization was significant for static Hall coductivity sxy(w=0) 2DEG localization length DOS How about for opticalsxy(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 /16

  7. In clean limit… ●ac Hall conductivity from Kubo formula ●How does dc Hall plateau structure evolve into ac region? Clean ordinary QHE system • Hall step structure in the clean limit • How about with disorder? Is it robust? step structure resonance structure /16

  8. Static Hall conductivity and Localization impurity Scaling behavior of Thouless energy Localization length (K. Nomura et al, PRL, 2008) Robust n=0 Anderson transition /16

  9. 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 /16

  10. Optical conductivity for graphene QHE -12 01 12 G=0.2 Step structure in both static and optical region Plateau structure remains up to ac region (at least resonace?) 01 G=0.5 /16

  11. 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 Plateau structures seem to be more robust than in graphene. Difference of universarity classes G=0.7 /16

  12. Plateau insxy(w) (ordinary QHE) ac step structure as a remnant of QHE remain for moderate disorders Disorder /16

  13. G = 0.2 Plateau insxy(w) (graphene QHE) w = 1.5wc w = 0.9wc w = 0.4wc w = 0 ac step structure as a remnant of QHE remain for moderate disorders Disorder /16

  14. (Nair et al, Science 2008) Estimation of Faraday rotation Faraday rotation ∝ optical Hall conductivity (O`Connell et al, PRB 1982) 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” 14 /16 Resolution ~ 1 mrad in Ikebe, Shimano, APL, 2008)

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

  16. Summary – ac Hall effects -12 ● 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 01 12 □Future problems ● honeycomb lattice calculation ●dynamical scaling arguments of sxy(w) /16

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