Condensed Matter Physics At Low Dimensions Philip Kim Department of Physics Columbia University Condensed Matter Physics ~1023 electrons ~1023 ions SP2 Carbon: 0-Dimension to 3-Dimension p Benzene molecule 0D Fullerenes (C60) s Atomic orbital sp2 1D Carbon Nanotubes 2D Graphene 3D Graphite Electronic Band Structure of Graphene Band structure of graphene (Wallace 1947) ˆ  t H E(k2D) AB  An E Bm  c.c ( n,m)  E  vF k  empty filled ky kx' K’ K x kx ky' kx ky 2D Brillouin Zone Zero effective mass particles moving with a constant speed vF = c/300 Single Layer Graphene: Pseudo Spinor Graphene Lattice Structures Spinor Representation ‘A’ sublattice: pz orbitals A ‘B’ sublattice : pz orbitals B Superposition: [ A  eiq B ] Cs X Two inequivalent lattice sites! Spin q Pseudo spin Dirac Fermions in Graphene : “Helicity” ˆ  t H E  An E Bm  c.c ( n,m) momentum pseudo spin E ky K’ ky kx kx K ky kx Effective Dirac Equations H eff   v  0 k x  ik y    F  k x  ik y 0  G. Semenoff PRL (1984)    vF s  k k = eik.r 1 2 1    iq e   qk = tan-1(ky / kx) k Hall (1879) Hall Effect Rxx = Vxx / I Rxy = Vxy / I = B/en B I + - + Vxx + + Rxy Quantum Hall Effect: Klitzing (1980) - Rxx Vxy Quantized Cyclotron Orbit s s Quantum Hall Effect in Graphene (2005) E kx' Quantization: 1 __ _ Rxy = 4 (n + ) e 2 h -1 2 ky' spin (2) X pseudo-spin (2) pseudo-spin rotation Klein Tunneling (1928) Step Potential problem V>m: transmission via negative energy states V T 1 0 x T e V 0 V 0 x x  p m 2 E Klein result: barrier sharpness ~Compton wavelength Klein Tunneling and Pseudo spin Chiral tunneling in graphene pn junctions Katsnelson et al. (2006) VBG > 0 VTG < 0 graphene n 2 Magnetic field modulation of FP n p 1 mm +1 Gosc (e2/h) p n 0 p B (T) electrode -1 VBG < 0 VTG > 0 0 1 20 nm Young et al. (2009) 2 |n2| (1012 cm-2) 3 4 Spin ½ and Electron Interaction “Triplet”  Exchange Interaction: 1 e2  X     dr dr  i* (r ) i (r) *j (r) j (r )  si , s j 2 i, j | r  r |  Pseudo Spin “Singlet”   c1   c2  “Quantum Hall bilayer” “Valley spin” … SU(4) Quantum Hall Ferromagnet in Graphene K’ K’ < SU(4) Magnetic Wave Function E X ky q kx Spin Degree of freedom: Spin (1/2), Valleys Under magnetic fields: pseudospin = valley spin Valley spin K K K’ K’ Yang, Das Sarma and MacDonal, PRB (2006); Possible SU(4) Quantum Hall Ferromagnetism at the Neutrality FerroMagnetic Anti FerroMagnetic Kekule Distortion Charge Density Wave Spin & pseudo spins: many body physics in graphene Dean et al. Nature Physics (2011) 5 mm Mobility > 300,000 cm2/Vsec • SU(4) hierarchical Fractional Quantum Hall Effect • Spin and Pseudospin Ferromagnetic Quantum Hall Effect • Spin Skyrmion and Valley Skyrmions Phase Transitions Among Fractional Quantum Hall States Bilayer Graphene Encapsulated with top & bottom gate sxx (S) mobility > 106 cm/Vsec Phase Transitions in Lowest Landau Levels Bilayer graphene: Fractional Quantum Hall effect 70 12000 Rxx () 8000 6000 4000 50 2/3 40 4/3 1 30 5/3 7/3 2 20 8/3 3 2000 0 60 Rxx Rxy 0.5 1.0 Vg (Volts) Maher*, Wang* et al. submitted 10/3 11/3 4 1.5 10 2.0 0 Rxy (k) B=18T SiGate 20 mK = +40V 10000 E-field tunable FQHE 2/3 5/3 8/3 1 2 3 4 5 6 Rxx (k) Assembly of Various 2D Systems Charge Transfer Bechgaard Salt graphene Metal-Chalcogenide Bi2Sr2CaCu2O8-x C (TMTSF)2PF6 hexa-BN X Lead Halide Layered Organic M B X N M = Ta, Nb, Mo, W, Eu … X = S, Se, Te, … Semiconducting materials: WSe2, NbS2, MoS2, … Complex-metallic compounds : TaSe2, TaS2, … Magnetic materials: EuS2, EuSe2 ,… Superconducting: NbSe2, Bi2Sr2CaCu2O8-x, ZrNCl,… A C A B A Andreev Reflections – between NbSe2 & Graphene Efetov et al. (2014) Superconductivity and QHE NbSe2 graphene Tc = 7 K Hc2 = 4.5 T 5 mm Andreev Reflection btw graphene/NbSe2 Andreev Reflection (dI/dV)/(dI/dV(10K)) 1.08 Tomasch Oscillations 1.06 1.5 K 1.04 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 1.02 1.00 0.98 0.96 -40 -20 0 VSD(mV) 20 40 Andreev Reflection into QH edge states are more efficient! Atomically Thin vdW p-n junction C. Lee et al, submitted Vertical & Lateral Channels - Al contact to MoS2 for electron injection - Pd contact to WSe2 for hole injection Gate Tunable Diode Characteristic Lateral Transport in Channels Lateral and vertical electron band alignment -4 10 -5 WSe2 -6 MoS2 10 10 Vds = 0.5 V Ids (A) -7 10 -8 10 -9 10 -10 10 -11 Forward 10 -12 10 -60 -40 -20 0 Vg (V) 20 40 60 Interlayer recombination by inelastic tunneling process Graphene Materials and Applications Flexible/Transparent Electrodes/Touch Panels Printable Inks Transparent Electrodes Semiconductors Ultrafast Transistors, RFIC, Photo/Bio/Gas Sensors Large-Scale CVD Graphene + Graphene Nanoplatelet Composites Conductive Ink, EMI shields Gas Barriers Gas barriers fo Displays, Solar Cells Heat Dissipation Energy Electrodes Composites LED Lights, BLU ECU, PC … Super Cap./Solar Cells Secondary Batteries Fuel Cells Images: Royal Swedish Academy Cars, Aerospace Appliations Courtesy: B. H. Hong Conclusions Relativistic QM: High Energy Physics CERN Electro-Positron Collider Quasi Relativistic QM: Low Energy Physics Kim Lab @ Columbia in City of New York . Equation: Dirac Majorana Equation: ?? Acknowledgement Amelia Barreiro Chul-ho Lee (jointly with Nuckolls group) Jean-Damien Pillet Current Members Students/postodcs Jayakanth RavichandranCollaborating Yue Zhao Cory Dean, Inanc Meric, Lei Wang, Adam Wei Tsen (jointly with Pasupathy Mitsuhide Takekoshi Sebastian Sorgenfrei,group) Kevin Knox, Nayung Andrea Young Jung, Seok Ju Kang, Jun Yan, Yanwen Tan, Dmitri Efetov Dmitri Efetov Kevin Knox Fereshte Ghahari 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 Patrick Maher Carlos Forsythe Giselle Elbaz (jointly with Brus group) Collaborators Austin Cheng Horst Stormer, Aron Pinczuk, Tony Heinz, Abhay Pasupathy, Latha Venkataraman Frank Zhao Louis Brus, George Xiaomeng LiuFlynn, Colin Nuckolls, Jim Hone, Ken Shepard, Louis Campos, Rick Osgood T. Taniguchi, K, Watanabe Andre Geim, Kostya Novoselov, Sanka Das Sarma 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) Kim 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) Collaborations: Brus, Dean, Heinz, Hone, Nuckolls, Shepard Funding: group and friends (2011)