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1 Physics of Moiré Pattern in Atomic Scale Pilkyung Moon New York University Shanghai

2 Physics Solid State Physics Atomic species Periodicity of atoms determine material properties

3 Period of atoms Material properties Long period period electrical magnetic Short period optical Period : nm (1 nm = 10-9 m) Compression / elongation : < 1% limit variability!!

4 Moiré Crystal Interference of periodic patterns lattice period difference in lattice period difference in lattice orientation

5 Moiré Crystal conventional materials: a0 ~ nm a0 = nm moiré crystal: Graphene LM ~ nm (*) * tunable with interlayer registry LM a0 = nm hexagonal BN LM = (1 + ε ) a0 2 ε + 2 (1 + ε )(1 cos θ ) (ε: lattice mismatch, θ: lattice misorientation)

6 Angle Dependence (Absorption Spectrum) Theory Moon and Koshino, Phys. Rev. B 87, (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) Energy (ev) Rotation angle θ (degree)

7 Wide Spectral Range III-V materials, composite 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)

8 Moiré Crystal in Magnetic field magnetic field How the spectrum looks like?

9 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)

10 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

11 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

12 Condition to Observe Hofstadter Butterfly The two scale are quite different!! lattice period (~ 0.1 nm) magnetic length B (T) l B (nm) ,

Hofstadter Butterfly by High Magnetic Field B ~

1 nm)] Energy (in unit of band width) 730 T,

ISSP, Japan 45 T, continuous, NHMFL, USA 1 2,800 T

(bunker), spectrum VNIIEF, obtained Russia by

13 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)

14 Hofstadter Butterfly by Large Lattice B 0 (Tesla) feasible magnetic field Lattice period (nm) ordinary crystalline lattice

Hofstadter Butterfly by Large Lattice Moiré Superlattice

= 0 θ = 4 θ = 8 Φ / Φ 0 1 (strong field regime) Φ / Φ 0

MacDonald, Phys. Rev. B 84, 035440 (2011).

15 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, (2011). Moon and Koshino, Phys. Rev. B 85, (2012). Hofstadter butterfly at moderate B!

16 θ Hofstadter Butterfly by Moiré Superlattice = 9.43 nearly monolayer s Landau levels Moon and Koshino, Phys. Rev. B 85, θ = 2.65 n=0-1 -2

17 Hofstadter Butterfly by Moiré Superlattice θ = 9.43 Moon and Koshino, Phys. Rev. B 85, θ = 2.65 fractal bands subgeneration bands n= subgeneration levels

18 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).

19 Conclusion Der Schmetterlingsjäger (The butterfly hunter) by Carl Spitzweg (1840), Butterfly and Chinese wisteriaflowers by Xü Xi (970)

20 Thank you for your attention Absorption spectra Optical dichroism