1. 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

2. 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 2

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

4. 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

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

6. Effective model for misoriented graphene-graphene
monolayer 1
Moon and Koshino, 
Phys. Rev. B 87, (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, (2007),
R. Bistritzer and A. MacDonald, PNAS 108, (2011), M. Kindermann and P. First,PRB 83, (2011).

7. 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, (2010).
E. Morell, J. Correa, P. Vargas, M. Pacheco, and Z. Barticevic,
Phys. Rev. B 82, (2010).
tight-binding
effective continuum

8. 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%

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

10. Modeling Graphene + h-BN
Moon and Koshino, 
arXiv: v1
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, (2012).
J. Wallbank, A. Patel, M. Mucha-Kruczynski, A. Geim, and V. Fal'ko, PRB 87, (2013).
J. Jung, A. Raoux, Z. Qiao, and A. H. MacDonald, arXiv: (2013).

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

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

13. Band structure of graphene + hBN
Moon and Koshino, 
arXiv: v1
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

14. Bilayer graphene + hBN … inversion symmetry strongly broken
Moon and Koshino, 
arXiv: v1
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

15. Inversion symmetry breaking?
Moon and Koshino, 
arXiv: v1
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

16. 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).

17. Same strategy graphene-graphene graphene+hBN DWNT rotation R
Moon and Koshino, 
arXiv: v1
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]

18. 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