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Condensed Matter Physics At Low Dimensions Philip Kim Department of Physics Columbia University

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Condensed Matter Physics ~10 23 electrons ~10 23 ions

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SP 2 Carbon: 0-Dimension to 3-Dimension Atomic orbital sp 2 Fullerenes (C 60 ) Carbon Nanotubes GraphiteGraphene 0D1D2D3D Benzene molecule

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x E(k 2D ) kxkx kyky Electronic Band Structure of Graphene Band structure of graphene (Wallace 1947) Zero effective mass particles moving with a constant speed v F = c/300 2D Brillouin Zone kxkx kyky K K kx'kx' ky'ky' E empty filled A B

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Single Layer Graphene: Pseudo Spinor A sublattice: p z orbitals B sublattice : p z orbitals Spinor Representation Pseudo spin Graphene Lattice Structures Superposition: Two inequivalent lattice sites! [ ] Spin X

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Dirac Fermions in Graphene : Helicity K KK momentum pseudo spin E kxkx kyky E xx yy E yy xx G. Semenoff PRL (1984) Effective Dirac Equations k = tan -1 (k y / k x ) = e ik r. eiei 1

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V xy Hall Effect I V xx + + + + - - - - B Quantized Cyclotron Orbit R xy R xx s s R xx = V xx / I R xy = V xy / I = B/en Quantum Hall Effect: Klitzing (1980) Hall (1879)

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Quantum Hall Effect in Graphene (2005) Quantization: 4 (n + ) R xy = __ _ e h 2 2 1 spin (2) X pseudo-spin (2) pseudo-spin rotation kx'kx' ky'ky' E

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Klein Tunneling (1928) Step Potential problem V>m: transmission via negative energy states x V 0 Klein result: barrier sharpness ~Compton wavelength x V 0 x V 0 E filed

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Klein Tunneling and Pseudo spin Chiral tunneling in graphene pn junctions Katsnelson et al. (2006) n n p n p p V BG > 0 V TG < 0 V BG < 0 V TG > 0 graphene electrode 1 m 20 nm Young et al. (2009) |n 2 | (10 12 cm -2 ) G osc (e 2 /h) -1 0 +1 Magnetic field modulation of FP

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Spin and Electron Interaction Triplet Singlet Pseudo Spin Quantum Hall bilayer Valley spin Exchange Interaction:

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SU(4) Quantum Hall Ferromagnet in Graphene Valley spin Spin X SU(4) < K K K Yang, Das Sarma and MacDonal, PRB (2006); K kxkx kyky E Magnetic Wave Function Under magnetic fields: pseudospin = valley spin Degree of freedom: Spin (1/2), Valleys Charge Density Wave Kekule Distortion Anti FerroMagnetic FerroMagnetic Possible SU(4) Quantum Hall Ferromagnetism at the Neutrality

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Spin & pseudo spins: many body physics in graphene 5 m Dean et al. Nature Physics (2011) SU(4) hierarchical Fractional Quantum Hall Effect Spin and Pseudospin Ferromagnetic Quantum Hall Effect Spin Skyrmion and Valley Skyrmions Mobility > 300,000 cm 2 /Vsec

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Phase Transitions Among Fractional Quantum Hall States Maher*, Wang* et al. submitted Bilayer Graphene Encapsulated with top & bottom gate mobility > 10 6 cm/Vsec xx (S) 2/3 4/3 5/3 7/3 8/3 10/3 11/3 20 mK Bilayer graphene: Fractional Quantum Hall effect R xx (k ) 2/3 123456 5/3 8/3 E-field tunable FQHE Phase Transitions in Lowest Landau Levels

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X X M C B N graphene hexa-BN Metal-Chalcogenide M = Ta, Nb, Mo, W, Eu X = S, Se, Te, Bi 2 Sr 2 CaCu 2 O 8-x Charge Transfer Bechgaard Salt (TMTSF) 2 PF 6 Lead Halide Layered Organic Semiconducting materials: WSe 2, NbS 2, MoS 2, Complex-metallic compounds : TaSe 2, TaS 2, Magnetic materials: EuS 2, EuSe 2, Superconducting: NbSe 2, Bi 2 Sr 2 CaCu 2 O 8-x, ZrNCl, A B A C A Assembly of Various 2D Systems

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Andreev Reflection btw graphene/NbSe 2 1.5 K 2.5 K 3.5 K 4.5 K 5.5 K 6.5 K 6.8 K 7.0 K 7.2 K 7.5 K Andreev Reflections between NbSe 2 & Graphene Efetov et al. (2014) NbSe 2 T c = 7 K H c2 = 4.5 T graphene 5 m Tomasch Oscillations Andreev Reflection Andreev Reflection into QH edge states are more efficient! Superconductivity and QHE

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Vertical & Lateral Channels - Al contact to MoS 2 for electron injection - Pd contact to WSe 2 for hole injection Lateral Transport in Channels C. Lee et al, submitted Atomically Thin vdW p-n junction Forward Lateral and vertical electron band alignment Interlayer recombination by inelastic tunneling process Gate Tunable Diode Characteristic

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Graphene Materials and Applications Large-Scale CVD Graphene + Graphene Nanoplatelet Composites Cars, Aerospace Appliations Semi- conductors Ultrafast Transistors, RFIC, Photo/Bio/Gas Sensors Transparent Electrodes Flexible/Transparent Electrodes/Touch Panels Printable Inks Conductive Ink, EMI shields Energy Electrodes Super Cap./Solar Cells Secondary Batteries Fuel Cells Heat Dissipation LED Lights, BLU ECU, PC Gas Barriers Gas barriers fo Displays, Solar Cells Images: Royal Swedish Academy Courtesy: B. H. Hong

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CERN Electro-Positron Collider Relativistic QM: High Energy Physics Conclusions Kim Lab @ Columbia in City of New York Quasi Relativistic QM: Low Energy Physics Dirac Equation:. Majorana Equation: ??

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Acknowledgement Past Members Melinda Han (Ph.D. 2010, Frontier of Science Fellow, Columbia University) Meninder S. Purewal (Ph.D. 2008) Josh Small (Ph.D. 2006) Yuanbo Zhang (Ph.D. 2006, Professor, Fundan University) Yuri Zuev (Ph.D. 2011, IBM Fishkill) Kirill Bolotin (Assistant Professor, Department of Physics, Vanderbilt University) Byung Hee Hong (Associate Professor, Department of Chemistry, Seoul National University) Pablo Jarillo-Herrero (Assistant Professor, Department of Physics, MIT) Keunsoo Kim (Assistant Professor, Department of Physics, Sejong University) Namdong Kim (Research Scientist, POSTECH) Barbaros Oezyilmaz (Assistant Professor, Department of Physics, National University of Singapore) Current Members Yue Zhao Mitsuhide Takekoshi Andrea Young Dmitri Efetov Fereshte Ghahari Patrick Maher Young-Jun Yu (jointly with GRL, POSTECH) Vikram Deshpande (jointly with Hone group) Paul Cadden-Zimansky (Columbia Frontier of Science Fellow) Chenguang Lu (jointly with Hone and Herman Collaborating Students/postodcs Cory Dean, Inanc Meric, Lei Wang, Sebastian Sorgenfrei, Kevin Knox, Nayung Jung, Seok Ju Kang, Jun Yan, Yanwen Tan, Kevin Knox Collaborators Horst Stormer, Aron Pinczuk, Tony Heinz, Abhay Pasupathy, Latha Venkataraman Louis Brus, George Flynn, Colin Nuckolls, Jim Hone, Ken Shepard, Louis Campos, Rick Osgood T. Taniguchi, K, Watanabe Andre Geim, Kostya Novoselov, Sanka Das Sarma Kim group and friends (2011) Funding: Amelia Barreiro Chul-ho Lee (jointly with Nuckolls group) Jean-Damien Pillet Jayakanth Ravichandran Adam Wei Tsen (jointly with Pasupathy group) Dmitri Efetov Fereshte Ghahari Patrick Maher Carlos Forsythe Giselle Elbaz (jointly with Brus group) Austin Cheng Frank Zhao Xiaomeng Liu Collaborations: Brus, Dean, Heinz, Hone, Nuckolls, Shepard