Bloch, Landau, and Dirac: Hofstadter s Butterfly in Graphene Philip Kim Physics Department, Columbia University
Acknowledgment Prof. Cory Dean (now at CUNY) Lei Wang Patrick Maher Fereshte Ghahari Carlos Forsythe Prof. Jim Hone Prof. Ken Shepard Theory: P. Moon & M. Koshino (Tohoku) hbn samples: T. Taniguchi & K. Watanabe (NIMS) Funding:
Bloch Waves: Periodic Structure & Band Filling Zeitschrift für Physik, 52, 555 (1929) Felix Bloch Periodic Lattice e A 0 : unit cell volume a k Block Waves:
Bloch Waves: Periodic Structure & Band Filling Zeitschrift für Physik, 52, 555 (1929) Felix Bloch Periodic Lattice e A 0 : unit cell volume e F a n(e) k Block Waves: Band Filling factor MacDonald (1983)
Landau Levels: Quantization of Cyclotron Orbits 2-D Density of States Zeitschrift für Physik, 64, 629 (1930) 2-dimensional electron systems Lev Landau Free electron under magnetic field e F Energy and orbit are quantized: Each Landau orbit contains magnetic flux quanta e Massively degenerated energy level Landau level filling fraction:
Harper s Equation: Competition of Two Length Scales Proc. Phys. Soc. Lond. A 68 879 (1955) Tight binding on 2D Square lattice with magnetic field a Harper s Equation Two competing length scales: a : lattice periodicity l B : magnetic periodicity
Commensuration / Incommensuration of Two Length Scales Spirograph a / l B = p/q a l B
Hofstadter s Butterfly Harper s Equation 1 When b=p/q, where p, q are coprimes, each LL splits into q sub-bands that are p-fold degenerate f (in unit of f 0 ) Energy bands develop fractal structure when magnetic length is of order the periodic unit cell 0 1 energy (in unit of band width)
Energy Gaps in the Butterfly: Wannier Diagram Hofstadter s Energy Spectrum 1 1 Wannier, Phys. Status Solidi. 88, 757 (1978) Tracing Gaps in f and n t : slope, s : offset e /W 1/5 1/4 1/3 1/2 0 1 0 1/4 1/3 1/2 1 n 0 : # of state per unit cell f : magnetic flux in unit cell n : electron density Diophantine equation for gaps
Streda Formula and TKNN Integers What is the physical meaning of the integers s and t? Band Filling factor Quantum Hall Conductance
Experimental Challenges Bo (Tesla) 1 Obvious technical challenge: e /W 10000 1000 a = 1nm 0 1 100 10 1 Disorder, temperature Hofstadter (1976) 0.1 0 20 40 60 80 100 120 140 160 unit cell (nm)
Experimental Search For Butterfly -Schlosser et al, Semicond. Sci. Technol. (1996) Albrecht et al, PRL. (2001); Geisler et al, PRL (2004) Unit cell limited to ~40-100 nm limited field and density range accessible, weak perturbation Do not observe fully quantized mingaps in fractal spectrum
Electrons in Graphene: Effective Dirac Fermions Graphene, ultimate 2-d conducting system Band structure of graphene Novoselov et al. (2004) Effective Dirac Equations H eff v 0 F kx ik y k ik x 0 y v F k K K E k y k x Paul Dirac DiVincenzo and Mele, PRB (1984); Semenov, PRL (1984)
Graphene: Under Magnetic Fields Energy Energy (ev) E N = 3 N = 2 0.2 N = 1 0.1 k x ' k y ' N = 0 DOS N = -1 0-0.1 N = -2 N = -3-0.2 0 10 20 Quantum Hall Effect B (T) Quantization Condition Novoselov et al (2005) Zhang et al (2005)
Hexa Boron Nitride: Polymorphic Graphene a 0 = 0.246 nm a 0 = 0.250 nm graphene Boron Nitride Comparison of h-bn and SiO 2 Band Gap Dielectric Constant Optical Phonon Energy Structure BN 5.5 ev ~4 >150 mev Layered crystal SiO2 8.9 ev 3.9 59 mev Amorphous
Stacking graphene on hbn Polymer coating/cleaving/peeling Dean et al. Nature Nano (2009) Micro-manipulated Deposition Co-lamination techniques Submicron size precision Atomically smooth interface Remove polymer Anealing Mobility > 100,000 cm 2 V -1 s -1
Graphen/hBN Moire Pattern 5 o 10 o 30 o 20 o 15 o
Moire pattern in Graphene on hbn: a new route to Hofstadter s butterfly? Graphene on BN exhibits clear Moiré pattern q=5.7 o q=2.0 o q=0.56 o 9.0 nm 13.4 nm Xue et al, Nature Mater (2011); Decker et al Nano Lett (2011) Minigap formation near the Dirac point due to Moire superlattice momentum
Transport Measurement Graphene with Moire Superlattice UHV AFM (Ishigami group) l T = 14.6 nm ~ 30 V ~ 30 V k Zone folding and mini-gap formation Dirac point 1 mm Moiré l AFM = 15.5 nm Second Dirac point E
B (Tesla) B (Tesla) Abnormal Landau Fan Diagram in Bilayer on hbn Special Samples with Large Moire Unit Cell Low Magnetic field regime I V xy I V xx R xx (kw) T=300 1.7 mk K T=300 1.7 mk K R xy (kw) V G (Volts) V G (Volts)
B (T) B (T) How to Read Normal Landau Fan Diagram? Landau Fan Diagram for typical graphene 0 5 R xx R xx (kw) 5 n =2 n =4 10 n =-10 R xx (kw) 1 n =-6-5 -4-3 n =-2-2 -1 0 1 n (cm -2 ) 0 0 0 5 R xy R xy (kw) 15 10 n =-6-5 -4-3 -2 n =2-2 -1 0 1 n (cm -2 ) 0
B (Tesla) Quantum Hall Effect in Graphene Moire R XX (kw) R XY (kw) Quantum Hall-like Transport Landau level filling factor R xx V xx I T=300 mk R xx (kw) Quantum Hall conductance 2.0 1.5 B=18T R XX 16 12 8 R XY 4 1.0 0.5 0-4 -8-12 V G (Volts) 0.0-6 -5-4 -3-2 -1 0 1 2 3 4 5 6 filling fraction (n) -16
Quantum Hall Effect with Two Integer Numbers n/n 0 : density per unit cell; f : flux per unit cell (t, s) R xx (kw) 0 15 (-4, 1) (-4, 2) (-3, 2) (-2, 1) (-1, 1) (-2, 2) (-1, 2) (3, 0) (4, 0) Quantization of xx and xy (-2, -2) (-4, 0) (-3, 1) (-8, 0) (8, 0) (-12, 0) (12, 0) (-16, 0) (16, 0)
B (T) Quantum Hall Effect with Two Integer Numbers B (T) t = -8 t = -4 R XX (kw) t =-6 t =-4 t =-2 t =0 t = -8 t = -4 t =-4 t =-2 R XY (kw) t =-6 t =0 t = -12 t =-8 t = -16 t = -12 t =-8 t = -16 R XX R XY s =-4 V g (V) s =-4 V g (V) s =-2 s = 0 s =-2 s = 0
B (T) Quantum Hall Effect with Two Integer Numbers B (T) t =-6 R XX (kw) t =-6 R XY (kw) t =-8 t =-8 R XX R XY V g (V) s =-2 V g (V) s =-2
R XY (kw) B (T) R XY (kw) Quantum Hall Effect with Two Integer Numbers 5 4 (1/6)h/e 2 t =-6 R XY (kw) 3 t =-8 2 1 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 B (Tesla) t =-6, s =-2 4 3 (1/8)h/e 2 R XY V g (V) s =-2 2 1 0 0 1 2 3 4 5 6 7 8 9 10 11 B (Tesla) t =-8, s =-2
Recursive QHE near the Fractal Bands Higher quality sample with lower disorder σ xy 1/2 Hall conductivity across Fractal Band 1/2 1/3 1/4 1/5 1/3 100 30 Recursive QHE! 20 At the Fractal Bands Sign reversal of xy xx (e 2 /h) 50 10 0-10 xy (e 2 /h ) Large enhancement of xx 0 0.0 0.2 0.4 0.6 0.8-20 f / f 0
Summary and Outlook Graphene on hbn with high quality interface created Moire pattern with supper lattice modulation Quantum Hall conductance are determined by two TKNN integers. Anomalous Hall conductance at the fractal bands Open Questions: Elementary excitation of the fractal gaps? Role of interactions, Hofstadter Butterfly in FQHE? Energy spectroscopy is needed for the next step
Fractal Gaps: Energy Scales Fractal Gap Size 83 K Large odd integer gap indicates (fractal) quantum Hall ferromagentism!!
SU(4) Quantum Hall Ferromagnet in Graphene K K E k y k x Magnetic Wave Function X Spin Valley spin q < SU(4) y K yk yk y K Degree of freedom: Spin (1/2), Valleys Under magnetic fields: pseudospin = valley spin Yang, Das Sarma and MacDonal, PRB (2006); Possible SU(4) Quantum Hall Ferromagnetism at the Neutrality FerroMagnetic Anti FerroMagnetic Kekule Distortion Charge Density Wave
Nature of Quantum Hall Ferromagnetism in Graphene Partial list of references Electron Interaction Fractal Spectrum
Graphene Franctional Quantum Hall Effect Dean et al. Nature Physics (2011) 5 mm Fractional Quantum Hall Gaps
B (T) Fractional Quantum Hall Effect in Moire Superlattice Single layer graphene on hbn @ 20 mk up to 35 T Hall Conductivity xy (ms) -1 0 1 Fractional Quantum Hall Effect 2/3 R xx R xy 4/3 5/3 7/3 V g (V) 4/3 5/3 7/3 8/3 10/3 (n) 13/3 xx (e 2 /h)
Acknowledgment Prof. Cory Dean (now at CUNY) Lei Wang Patrick Maher Fereshte Ghahari Carlos Forsythe Prof. Jim Hone Prof. Ken Shepard Theory: P. Moon & M. Koshino (Tohoku) hbn samples: T. Taniguchi & K. Watanabe (NIMS) Funding: