Quantum Hall transition in graphene with

1. Quantum Hall transition in graphene with correlated bond disorder -- Unconventional Hall transition at n=0 Landau level -- T. Kawarabayshi (Toho University) Y. Hatsugai (University of Tsukuba) H. Aoki (University of Tokyo) ArXiv:0904.1927 Outlines Landau levels in graphene Roles of disorder and Numerical model Density of states Hall conductivity Summary

2. Characteristic band structure and Landau levels of graphene • Dirac cones (K and K’) • at E=0 (Fermi energy) K’ • Landau levels around K and K’ K 2D Honeycomb Lattice -3,-2,-1 n=0 n=1,2,3,… n=0 (E=0) Landau level

3. n=0 Landau level Novoselov et al. Nature 2005 • Essential to • anomalous quantum Hall effect (per spin) Zheng & Ando (2002) • Robustness the index theorem for Dirac fermions Criticality: Dirac fermions + random potential (long-range) Nomura et al. (2008) • Sensitivity mixing of two valleys (K and K’) Koshino, Ando (2007) Schweitzer, Markos(2008) Ostrovsky et al. (2008)

4. Roles of Disorder Yes Random bonds Key concepts Random magnetic fields Yes Random potential No (A) Chiral symmetry (UHU-1 = -H) Yes Short-range disorder (B) Mixing of two valleys (K and K’) Long-range disorder No How these properties show up in the Landau level structure ? 2D Honeycomb Lattice Model + Spatially Correlated Disorder To control (B) the mixing between K and K’ t Systematic study on the correlation dependence f Chiral symmetric , n=0 (E=0) a ~1.42Å

5. An intrinsic disorder in graphene Ripples Disorder in transfer integrals A.H. Castro Neto et al. Rev. Mod. Phys. (2009) A model with disordered transfer integrals Chiral symmetry Gaussian with s Correlation length h A typical landscape (h/a=5) Region with large t(r) Ly=Ny|2e2-e1| Region with Small t(r) e1 e2 Lx=Nx|e1|

6. Density of states: correlation dependence n=0 h /a >1 Anomaly for n=0 Landau level n=-1 n=-2 n=1 n=2 Correlation length s/t = 0.115 f/f0=1/50 g/t = 0.000625 Nx=5000, Ny=100 The Green function Method Schweitzer, Kramer, MacKinnon (1984)

7. Hall conductivity sxyin terms of Chern number CE Thouless, Hohmoto, Nightingale, den Nijs (1982) Aoki, Ando (1986), Niu, Thouless, Wu (1985) Sum over many filled Landau bands EF CM E ~ 0, weak fields Contributions mostly cancel out Accurate numerical method for Cl C1 Hatsugai (2004,2005) Fukui, Hatsugai, Suzuki (2005) h/a=1.5 Unconventional n=0 Hall transition for h/a >1 n=1 h/a=0 n=0 Nx=Ny=10 300 samples T.K., Y. Hatsugai, H.Aoki, ArXiv:0904.1927 n=-1 E/t

8. Hall conductance sxy (Chern Number CE ) as a function of E s/t = 0.115 f/f0=1/50 n=1 h/a=1.5 g/t = 0.000625 n=0 Nx=Ny=10, 300 samples n=-1 h/a=0 Nx=5000, Ny=100 E/t Transition at E=0 is sensitive to the range of bond disorder

9. Size-independent Nx=Ny=5 h/a=1.5 Nx=Ny=10 n=1 n=0 n=-1 E/t Additional potential disorder [-w/2, w/2] n=0 n=-1 w=0 w=0.4 insensitive sensitive Breakdown of chiral symmetry

10. Other disorder with chiral symmetry f+df(r) Disordered magnetic fields h/a Anomaly at the n=0 level sf/f = 0.237 < 1 small disorder large disorder sf/f = 2.37 > 1 h/a f/f0=1/41 g/t = 0.000625 Nx=5000, Ny=82

11. Summary The n=0 Landau level : anomalously robust against long-range bond disorder Chiral symmetry, Absence of scattering between K and K’ No broadening for h/a > 1 Consistent with the index theorem Scale of ripples in graphene : 10 ~ 15 nm >> a • No broadening of n=0 level by • bond disorder by ripples • Other disorder (ex. potential disorder) • should be responsible for the • broadening of n=0 level (Meyer, Geim et al, Nature 2007) • Possibility to observe this anomaly at n=0 in clean graphene • without potential disorder from substrates Classical Hall transition ? Ostrovsky et al. (2008)

12. Other disorder without chiral symmery Potential disorder V(r) sS/t = 0.288 No anomaly f/f0=1/41 g/t = 0.000625 Nx=5000, Ny=82