hBN-caged graphene Marcin Mucha-Kruczynski, Xi Chen, John Wallbank, Vladimir Falko Andre Geim & National Graphene Institute @Manchester Klaus Ensslin group @ETH • • Lifshitz transition in gapped BLG Magnetic (Brown-Zak) minibands due to moiré superlattice in monolayer and bilayer graphene Bilayer graphene with a gap Oostinga, et al - Nature Mat 7, 151 (2008) Zhang, et al - Nature 459, 820 (2009) 1eV  hBN allows for better quality and larger Ez T. Ohta et al – Science 313, 951 (2006) (Rotenberg’s group at Berkeley NL) u  Ez d 3 1 skew inter-layer 3  3a ~ v   ~ 0.1v 3 AB hopping 2    p x  ip y  pei 0 v  A  12 u v3  ~     1 0  B   v3  2 u v H  ~   1 0 v  2 u  1 A      v 1   0  1 2 u  B   McCann, VF ‐ PRL 96, 086805 (2006) Gapped BLG: intricate band features due to trigonal warping F 1 2 u u p~ v 2 u v3  1 ~ 14meV v 8  1 12 ( u1 )2  2 u1 vv3 F 1 2  u  1 12 ( u1 )2  2 u1 vv3  F   12 u Lifshitz transition in metals • Topology of the Fermi surface changes, DoS diverges • Cyclotron orbits in magnetic field change circulation • Magnetic breakdown - field mixes disconnected parts of Fermi surfaces, at δp~1/λB. u p3 ~ v 2 F 1 2 u  1 12 ( u1 )2  2 u1 vv3 F 1 2 Ilya Lifshitz 1917‐1982 Kharkov/Moscow u  1 12 ( u1 )2  2 u1 vv3  F   12 u Lifshitz transition, magnetic breakdown, and phase transitions between QHFM states Varlet, Ihn, Ensslin - ETH Mucha-Kruczynski, VF  Dirac point generates a 4-fold degenerate ε=0 Landau level McClure ‐ PR 104, 666 (1956)    2n  m ~ 0 . 05 m e  m  0.035me  0   1   ,    0 0 v B     e p  i  c A, rot A  Bl z   p x  ip y ;    p x  ip y descending/raising operators in LL orbitals    c n(n  1) 8-fold degenerate ε=0 Landau level, which splits when inversion symmetry is broken. McCann, VF ‐ PRL 96, 086805 (2006)  12 u   v   3 H  0    v  v3 v    1 0  2 u v v  12 u  1   1  0 1 2 u 0 E [eV ] u  0.08eV valley K valley K ' B [Tesla] u  0.08eV 6-fold (2 x spin and 3 x orbital) degenerate LL at small magnetic fields valley K E [eV ] ν = -3 B [Tesla] spin polarised (ferromagnetic) QHE state ν = -6 unpolarised QHE state magnetic breakdown E [eV ] B [Tesla] E [eV ] ν = 0,-1,-2 ferromagnetic and normal QHE Landau level crossing Polarised ν = -3,-5 QHFM gaps vanish and ν = -4 undergoes ferromagnetic transition. ν = -3,-5 ν =-4,-6 QHE B [Tesla] Varlet, Ihn, Ensslin - ETH Mucha-Kruczynski, VF Highly oriented hBN-graphene heterostructures Geim (Manchester) highly oriented graphene-BN: Jarillo-Herrero (MIT) heterostructure with new electronic properties highly oriented BLG-hBN heterostructures Kim (Harvard) & Hone (Columbia) Due to a separation between layers larger than distance between atoms within the layers, moiré perturbation is dominated by the simplest spatial harmonics Lopes dos Santos, Peres, Castro Neto - PRL 99, 256802 (2007) Lopes dos Santos, Peres, Castro Neto - arXiv:1202.1088 (2012) Bistritzer, MacDonald - PRB 81, 245412 (2010) Kindermann, Uchoa, Miller - Phys. Rev. B 86, 115415 (2012)   4     3a  ˆ b0  bG  bBN  1 (1 )R   0  3 | b0 | b   2  2 4a lattice mismatch 1.8% for G/hBN misalignment <20  b0  b1    b2  b5 '  ''  b3  b4 Effective low-energy ‘Dirac theory’ for electrons Phenomenological approach to classify generic miniband structure caused by a moiré perturbation.  Hmoire graphene sublattice graphene valley inversion symmetric inversion asymmetric eliminated by a gauge transformation inversion symmetric inversion asymmetric inv-asymm (honeycomb) ~ gap  24vb | u1u 0 gap  u0u~1 | 1 ~| u~ | vb Wallbank, Patel, Mucha-Kruczynski, Geim, VF - PRB 87, 245408 (2013) Chen, Wallbank, Patel, Mucha-Kruczyński, McCann, VF - PRB 89, 075401 (2014) vb Brown, PR 133, A1038 (1964); Zak, PR 134, A1602 & A1607 (1964)   0 , 0  p q h e Magnetic minibands at rational values of magnetic field flux per super-cell ‘Magnetic lattice’ with a q2 times bigger supercell and q2 times smaller Brillouin minizone. Each state in this Brillouin minizone is q times degenerate. Branded as ‘Hofstadter butterfly’ spectrum. ‘Magnetic lattice’ with a 9 times bigger supercell   GqM  { R , R  qm1a1  qm2 a2 }  GM Generations of Dirac electrons in Zak’s magnetic minibands in moiré superlattices Magnetic minibands at H Dirac   0 - gapped Dirac electrons p q    e  vmDP (k  c A)    12 u z 1 2 0 Chen, Wallbank, Patel, Mucha-Kruczyński, McCann, VF - PRB 89, 075401 (2014)  / 0 Quantum Hall ferromagnetism in moiré superlattices capacitance spectroscopy Yu, Gorbachev, Tu, Kretinin, Cao, Jalil, Withers, Ponomarenko, Chen, Piot, Potemski, Elias, Watanabe, Taniguchi, Grigorieva, Novoselov, VF, Geim, Mishchenko (2014) n0 n' 0 SU 4 n' 0 B1/1 B N 0 N ' 0 N ' 0 B1/1 B D N 0 Ec ν=0, νL=0 D B1/1 B ν=0, νL=1 B1/1 B Reverse Stoner transition n0 ~ Ec ν=-1, νL=0 ν=-1, νL=0 ν=-1, νL=-1 B1/1 B Highly oriented hBN-caged bilayer graphene BLG-hBN heterostructures Substrate affecting one layer produces inversion non-symmetric moiré superlattice potential. For encapsulated BLG, different misalignment between BLG and top/bottom hBN layers has the same effect (low-energy electronic properties are determined by the moiré pattern due to the better aligned hBN layer. Mucha‐Kruczynski, Wallbank, VF ‐ PRB 88, 205418 (2013) BLG-hBN heterostructures Mucha‐Kruczynski, Wallbank, VF ‐ PRB 88, 205418 (2013) BLG-hBN heterostructures Inversion symmetry is broken because moire perturbation is applied only to one layer: this promotes gaps at the 1st miniband edge. gap at the 1st miniband edge (VB) overlapping minibands (VB) Mucha‐Kruczynski, Wallbank, VF ‐ PRB 88, 205418 (2013) Zak’s minibands in BLG-hBN heterostructures BLG K sublattice B BLG K’ sublattice A’ Chen, Mucha‐Kruczynski, Wallbank, VF (2014) BLG K’ Zoom in to 0th and 1st LL MLG K Zoom in to 0th LL only BLG K’ hBN-caged graphene Marcin Mucha-Kruczynski, Xi Chen, John Wallbank, Vladimir Falko Andre Geim & National Graphene Institute @Manchester Klaus Ensslin group @ETH • • Lifshitz transition in gapped BLG Magnetic (Brown-Zak) minibands due to moiré superlattice in monolayer and bilayer graphene