Physics of Moiré Pattern in Atomic Scale Pilkyung Moon New York University Shanghai
Physics Solid State Physics Atomic species Periodicity of atoms determine material properties
Period of atoms Material properties Long period period electrical magnetic Short period optical Period : 0.1-1 nm (1 nm = 10-9 m) Compression / elongation : < 1% limit variability!!
Moiré Crystal Interference of periodic patterns lattice period difference in lattice period difference in lattice orientation
Moiré Crystal conventional materials: a0 ~ 0.1-1 nm a0 = 0.246 nm moiré crystal: Graphene LM ~ 1-100 nm (*) * tunable with interlayer registry LM a0 = 0.250 nm hexagonal BN LM = (1 + ε ) a0 2 ε + 2 (1 + ε )(1 cos θ ) (ε: lattice mismatch, θ: lattice misorientation)
Angle Dependence (Absorption Spectrum) Theory Moon and Koshino, Phys. Rev. B 87, 205404 (2013). (σ twisted bilayer graphene σ Bernal ) Experiment R. W. Havener, Y. Liang, L. Brown, L. Yang, and J. Park, Nano Lett. 14, 3353 (2014). 4.5 4.0 Energy (ev) 3.5 3.0 2.5 2.0 1.5 10 15 20 25 30 Rotation angle θ (degree)
Wide Spectral Range III-V materials, composite 0.4 0.5 0.6 lattice constant (nm) 0.7 rotation angle (θ) Moiré crystal - from terahertz to UV - (Moon and Koshino) peak energy (ev) moiré crystal moiré lattice constant (nm)
Moiré Crystal in Magnetic field magnetic field How the spectrum looks like?
Electron in Periodic Lattice Electron in Magnetic Field e - Electron energy energy bands (finite widths) allowed energies How Nature forbidden energies e - Electron energy harmonizes the two? energy levels (discrete) allowed energies Felix Bloch Zeitschrift für Physik 52, 555 (1929) Lev Landau Zeitschrift für Physik 64, 629 (1930)*...(*age 22)
Hofstadter Butterfly e - Energy (in unit of band width) 0 Φ BA = Φ h/ e 0 1 D. R. Hofstadter, Phys. Rev. B 14, 2239 (1976) Φ Φ 0 B magnetic flux magnetic flux quantum magnetic field
当前无法显示此图像 当前无法显示此图像 当前无法显示此图像 Fractal (Self-Similar) Energy Spectrum 1 st generation 2 nd generation 4 th generation 3 rd generation
Condition to Observe Hofstadter Butterfly The two scale are quite different!! lattice period (~ 0.1 nm) magnetic length B (T) l B (nm) 1 25.7 10 8.11 10,000 0.257
Hofstadter Butterfly by High Magnetic Field B ~ O(10 4 T) for usual crystalline solids [L ~ O(0.1 nm)] Energy (in unit of band width) 730 T, pulsed, destructive (but not destroying the lab), ISSP, Japan 45 T, continuous, NHMFL, USA 1 2,800 T 0 ~80,000 T 2,800 T, pulsed, blowing off the lab (bunker), spectrum VNIIEF, obtained Russia by blowing off the facility image courtesy of Dr. Chuck Mielke (LANL)
Hofstadter Butterfly by Large Lattice B 0 (Tesla) feasible magnetic field Lattice period (nm) ordinary crystalline lattice
Hofstadter Butterfly by Large Lattice Moiré Superlattice superlattice by incoherent stacking of atomic lattices θ = 0 θ = 4 θ = 8 Φ / Φ 0 1 (strong field regime) Φ / Φ 0 0 (entire regime) R. Bistritzer and A. H. MacDonald, Phys. Rev. B 84, 035440 (2011). Moon and Koshino, Phys. Rev. B 85, 195458 (2012). Hofstadter butterfly at moderate B!
θ Hofstadter Butterfly by Moiré Superlattice = 9.43 nearly monolayer s Landau levels Moon and Koshino, Phys. Rev. B 85, 195458 +2 +1 θ = 2.65 n=0-1 -2
Hofstadter Butterfly by Moiré Superlattice θ = 9.43 Moon and Koshino, Phys. Rev. B 85, 195458 +2 +1 θ = 2.65 fractal bands subgeneration bands n=0-1 -2 subgeneration levels
Bilayer graphene / hbn C. R. Dean, L. Wang, P. Maher, C. Forsythe, F. Ghahari, Y. Gao, J. Katoch, M. Ishigami, P. Moon, M. Koshino, T. Taniguchi, K. Watanabe, K. L. Shepard, J. Hone, and P. Kim, Nature 497, 598 (2013). ( : theory) Monolayer graphene / hbn B. Hunt, J. D. Sanchez-Yamagishi, A. F. Young, M. Yankowitz, B. J. LeRoy, K. Watanabe, T. Taniguchi, P. Moon, M. Koshino, P. Jarillo-Herrero, and R. C. Ashoori, Science 340, 1427 (2013).
Conclusion Der Schmetterlingsjäger (The butterfly hunter) by Carl Spitzweg (1840), Butterfly and Chinese wisteriaflowers by Xü Xi (970)
Thank you for your attention Absorption spectra Optical dichroism pilkyung.moon@nyu.edu