Electronic properties in moiré superlattice 科研費新学術領域研究「原子層科学」第３回全体会議 　東京大学　2014年8月7日 Electronic properties in moiré superlattice in rotationally stacked atomic layers Mikito Koshino (Tohoku University) Collaborators: Pilkyung Moon (NYU Shanghai) 1

Outline Study the electronic properties of moiré suprlattice using the effective continuum model 1) Rotationally stacked graphene + graphene 2) Graphene + h-BN 3) double wall carbon nanotubes From http://www2.chem.umd.edu/groups/wang/ 2

Rotationally stacked bilayer graphene Moiré period: Emerging in epitaxially grown graphene samples Berger et al., Science 312, 1191 (2006) Haas et al., PRL 100, 125504 (2008)

Effective continuum model θ = 5.09° d = 0 Local structure AA 　　non-rotated bilayer 　　with shift d BA d AB d d is position-dependent: R… rotation by q

Effective continuum model Interlayer coupling in uniform d Non-rotated bilayer A1 B1 A2 B2 Interlayer coupling in moire bilayer

Effective model for misoriented graphene-graphene monolayer 1 Moon and Koshino,  Phys. Rev. B 87, 205404 (2013) monolayer 2 Interlayer interaction: : Moire reciprocal vectors the only parameter: See also: J. Lopes dos Santos, N. Peres, and A. Castro Neto, PRL 99, 256802 (2007), R. Bistritzer and A. MacDonald, PNAS 108, 12233 (2011), M. Kindermann and P. First,PRB 83, 045425 (2011).

Band structure of TBG Moon and Koshino, Phys. Rev. B 87, 205404 (2013) LM u0 Folded band width 1/LM See also: S. Shallcross, S. Sharma, E. Kandelaki, and O. Pankratov, Phys. Rev. B 81, 165105 (2010). E. Morell, J. Correa, P. Vargas, M. Pacheco, and Z. Barticevic, Phys. Rev. B 82, 121407 (2010). tight-binding effective continuum

Graphene + h-BN (hexagonal Boron-Nitride) C B N lattice constant mismatch Moire structure (even when q = 0) 1 1+e C. R. Dean et al., Nat. Nanotechnol. 5, 722 (2010). e = 1.8%

Same strategy graphene-graphene graphene+hBN rotation R expansion M Moon and Koshino,  arXiv:1406.0668v1 graphene-graphene graphene+hBN rotation R expansion M Displacement vector Moire reciprocal vector [ : graphene’s reciprocal vector]

Modeling Graphene + h-BN Moon and Koshino,  arXiv:1406.0668v1 E (eV) -1.40 3.34 B N C Graphene… Metal h-BN… Insulator EF EF C B N Interlayer interaction (same as graphene-graphene) Eliminate h-BN bases (2nd order perturbation) Effective potenial on graphene Other effective models of graphene + hBN: M. Kindermann, B. Uchoa, and D. Miller, PRB 86, 115415 (2012). J. Wallbank, A. Patel, M. Mucha-Kruczynski, A. Geim, and V. Fal'ko, PRB 87, 245408 (2013). J. Jung, A. Raoux, Z. Qiao, and A. H. MacDonald, arXiv:1312.7723 (2013).

Effective model for Graphene + h-BN Moon and Koshino,  arXiv:1406.0668v1 Effective potenial on graphene Scalar potential Dirac mass Vector potential

Band structure of graphene + hBN Moon and Koshino,  arXiv:1406.0668v1 hBN G Band structure Experiments B. Hunt, et al, Science 340, 1427 (2013). --- Gap opening at mini-Dirac points G. Yu, et al.,arXiv:1404.3856 (2014).

Band structure of graphene + hBN Moon and Koshino,  arXiv:1406.0668v1 hBN G Band structure Spectrum in B-field K, K’ --- Gap opening at mini-Dirac points --- Valley splitting at mini-Dirac points … due to inversion symmetry breaking

Bilayer graphene + hBN … inversion symmetry strongly broken Moon and Koshino,  arXiv:1406.0668v1 hBN G Band structure Spectrum in B-field K, K’ --- Gap opening at mini-Dirac points and Dirac points --- Valley splitting everywhere (almost no correlation between K and K’) … inversion symmetry strongly broken

Inversion symmetry breaking? Moon and Koshino,  arXiv:1406.0668v1 Monolayer graphene + hBN hBN G G + effective potential V effective potential: … not inversion symmetric Bilayer graphene + hBN hBN G G + effective potential G Strong inversion symmetry breaking

Experiments Bilayer graphene + hBN Monolayer graphene + hBN C. R. Dean, et al Nature 497, 598 (2013) Monolayer graphene + hBN B. Hunt, et al, Science 340, 1427 (2013) See also, L. A. Ponomarenko, et al., Nature 497, 594 (2013).

Same strategy graphene-graphene graphene+hBN DWNT rotation R Moon and Koshino,  arXiv:1406.0668v1 graphene-graphene graphene+hBN DWNT rotation R expansion M rotation R + stretch M Displacement vector Moire reciprocal vector Koshino, Moon, Son In preparation [ : graphene’s reciprocal vector]

Summary --- Unified picture for the electronic properties of misoriented atomic layers: 1) Graphene + graphene 2) Graphene + h-BN 3) double wall carbon nanotube --- Develop the effective continuum model in the same theoretical basis --- Fractal spectrum and Quantum Hall effect in B-field [for 1) and 2)] 18