Graphene and Quantum Hall (2+1)D Physics

The 4 th QMMRC-IPCMS Winter School 8 Feb 2011, ECC, Seoul, Korea Outline 2 Graphene and Quantum Hall (2+1)D Physics Lecture 1. Electronic structures of graphene and bilayer graphene Lecture 2. Electrons in graphitic systems with magnetic fields Young-Woo Son Korea Institute for Advanced Study, Seoul, Korea 3 QUANTUM MECHANICS TWO OUT OF MANY FOUNDERS Lecture 1 ElectronicStructures of Graphene and BilayerGraphene Schrödinger Equation Dirac Equation

QUANTUM MECHANICS FOR FREE PARTICLES QUANTUM MECHANICS CONSEQUENCE OF DIRAC S EQ. Schrödinger Equation Dirac Equation (Particle) (Anti-Particle) Dirac Equation (Particle) IF m = 0 (Anti-Particle) Carbon What is allotropes graphene? - Carbon allotropes 7 Why Carbon? 8 Graphene 21C? Intel CPU 1960~ 1947~ The first transistor This slides is inspired by T. Ohta at LBNL and Fritz-Haber-Institut.

Brief history of graphene - Early works 9 mc - Isolation of graphene? 10 Intercalated graphite as a route to graphene! L. M. Vicilis et al, Science 299, 1361 (2003). Graphene nano-pencil? (P. Kim@Columbia) mc - Isolation of graphene? 11 Exfoliated Graphene - Breakthrough 12 Micromechanical cleavage of bulk graphite up to 100 micrometer in size via adhesive tapes! Novoselov et al, Science 306, 666 (2004) 윤두희, 정현식 ( 서강대 ) A. K. Geim Group @ Menchester P. Kim Group @ Columbia K. S. Novoselov et al, Nature 438, 197 (2005) YZh Y. Zhang et al, Nature 438, 201 (2005) 02μm 0.2

BRIEF HISTORY OF GRAPHENE NOBEL PRIZE FEW FACTS ON GRAPHENE Density: 0.77 mg/m 2 Breaking strength: th 42 N/m (Hypothetical steel of graphene thickness~0.4 N/m) Theoretical RT mobility: 200,000000 cm 2 V -1 s -1 for n=10 12 /cm -2 Strongest materials ever measured : Young s modulus 1.0 Tpa Thinnest flexible membrane ever created Impermeable to gases (even atomic hydrogen) Record value for RT thermal conductivity of ~5000 W/mK Ballistic transport over micrometers at RT Current density two order of magnitude higher than that of Cu Room temperature Quantum Hall Effects Unique material showing something exotics at RT - Nature of bonds in graphene 15 - Real space: tight-binding Hamiltonian 16 sp 2 sp 3 A π-orbital σ-bond a 2 s 3 s 1 B s 2 a 1 Two sublattices - Bipartite system Hexagonal network of Carbon sp 2 bonding TEM image, Zettl group at UC Berkeley C. Girit et al. Science 323, 170 (2009) Nearest-neighbor tight-binding Hamiltonian for π-orbitals

- Energy spectrum 17 - Linear energy bands K Hexagonal BZ with two special Fermi points, Two inequivalent Dirac cones at K and K 18 - Real space 19 - Neutrino in your pencil? 20 A B (Pseudo) Spin Up t (Pseudo) Spin Down Two sublattices - Bipartite system zero mass p y Dirac equation with p x charged neutrino in your pencil? Relativistic particle : c=v F : effective speed of light m=0

Gap in graphene - Linear bands? 21 Gap in graphene - Linear bands? 22 Linear band? Is that true? Yes from QHE ( If including NNI t, ) MOST direct answer : ARPES measurement of suspended graphene does NOT published until now (now - I wrote in Dec 2008) ELECTRONS IN GRAPHENE DIRECT OBSERVATION Quasi-particle spectrum of graphene - IR measurements 24 Angle Resolved Photoemission Spectroscopy Graphene Typical Semiconductor (Epitaxial graphene on SiC(000-1) (Indium Nitride surface states) ) Z. Q. Li et al, Nature Phys 4, 532 (08) Strong renormalization of group velocities near Dirac points Sprinkle et al, Phys. Rev. Lett. 103, 226803 (2009) Colakerol et al, Phys. Rev. Lett. 97, 237601 (2006)

- Energy spectrum 25 - Total Hamiltonian s=+ K - s= - K Hexagonal BZ with two special Fermi points, K with σ acts on sublattice A (B) and τ on (K - ) 26 - Gap generation in graphene - Consequences of massless Dirac fermions Linear Density of States 28 E E 0 0 k N 2D (E) Onsite energy difference Mixing between and K - - Mixing pseudo-spins - Mixing chiralities 2α 2β 27 Zhang et al, Novoselov et al (05)

Transport properties of graphene - Mobility of graphene 29 - Low energy dispersions Mobility of suspended graphene ~ 200,000 cm 2 /Vs Graphene Usual Semiconductor Observation of nearly ballistic transport regime/ FQHE X. Du et al, Nature Nanotech. 3, 491 (2008). K. I. Bolotin et al. SSC 146, 351 (2008) X. Du et al, Nature 462, 192 (2009). K. I. Bolotin et al. Nature 462, 196 (2009) - Pseudospin and chiral states 31 - Berry s phase Eigenfunctions : Spinor representation p y p y for a path of C at K+ and K- p x θ p p x θ p Pseudo-spin up (down): A (B) sublattice

- Chiral states: charged neutrino in your pencil Helicity it operator : 33 - Consequence of chirality Eigenstates : Conduction band : For a long-range disorder where a elastic scattering matrix element is σ x = -1/2 σ =+1/2 -e x p x < 0 p x > 0 : Complete absence of backscattering (pseudospin conservation) Klein paradox: M. I. Katsnelson et al,, Nature Phys. 2,, 620 (2006) Veselago lens: V. V. Cheianov et al, Science 315, 1252 (2007) 34 - Scattering mc - Tunneling 36 K - K Intra-valley scattering: small momentum transfer, lattice distortion, etc. Inter-valley scattering: large momentum transfer, short range atomic impurities, etc A. K. Geim & P. Kim Scientific American, Apr. 2008

- Klein paradox 37 - Klein tunneling 38 Graphene particle ~ 2 mc 2 antiparticle Klein paradox: Unimpeded penetration of relativistic particles through very high potential barriers. 2 Potential drop ~ 2mc over mc : ~10 8 V/Å Event horizon of Black hole Supercritical massive atoms O. Klein, Z. Phys. 53, 157 (1929) Katsnelson et al, Nature Phys. 2, 620 (2006) V. V. Cheianov et al, Science 315, 1252 (2007) C. Park, Y.-W. Son et al, Nature Phys. (2008), Phys. Rev. Lett. (2008), Nano Lett (2008) - Klein tunneling 39 Transport properties of graphene - Klein tunneling 40 Pabry-Ferot interference: observation of fb Berry s phase A. F. Young & P. Kim, Nature Phys. 5, 222 (2009). N. Stander, B. Huard, D. Goldhaber-Gordon, Phys. Rev. Lett. 102, 026807 (2009) Katsnelson et al, Nature Phys. 2, 620 (2006) V. V. Cheianov et al, Science 315, 1252 (2007)

- Opacity 41 - Opacity 42 graphene ω Si Science 320, 1308 (2008) Transmittance ~ 97.7% What is bilayer graphene? - Pseudospin and chiral states 44 Nearest neighbor hopping (t) between A and B sublattices Normal material Neutrino Graphene Bilayer Graphene Only single layer graphene has a linear dispersion. All others are massive, i.e., almost normal metals But, Pseudo-spin up (down): A (B) sublattice

Bilayer graphene : Massive chiral particles 45 Bilayer graphene : Massive chiral particles 46 H. Min et al, PRB 75, 155115 (07) Minimal i model: Two coupled single layer graphene with dimer couplings A1-B2 (Bernal stacking) Low energy effective Hamiltonian projected on a spinor space for the two layers of bilayer graphene (σ acts on layers): spin up (down) upper (lower) layer Single and bilayer graphene : Massless and massive chiral particles 47 Bilayer graphene : Massive chiral particles Single layer Graphene Pauli matrices on A and B sublattices Electric field A Sublattice B Sublattice MASSLESS CHIRAL - Bilayer Graphene Low energy effective Hamiltonian by integrating out high energy dimer part Pauli matrices on upper and lower layers Transverse electric field can generate energy gaps in spectrum!! Upper Layer Lower Layer MASSIVE CHIRAL

Strained Bilayer graphene bilayer graphene - Energy gap under perpendicular electric field Strained Bilayer graphene bilayer graphene - Energy gap under perpendicular electric field Min, McDonald et al, PRB 75, 155115 (2007) McCann, PRB 74, 161403 (R) (2006) Oostinga et al, Nature Mat. 7, 151 (2007) Zhang et al, Nature 459, 820 (2009) T. Ohta et al, Science 313, 951 (2006) Min, McDonald et al, PRB 75, 155115 (2007) McCann, PRB 74, 161403 (R) (2006) Bilayer graphene : Next nearest neighbor inter-layer hopping 51 Bilayer graphene : Next nearest neighbor inter-layer hopping 52 1.1 mev The next nearest neighbor hopping breaks a global U(1) symmetry into Z 3 (magnitude of k Di is about 0.4% of distance from Γ to K point)

Bilayer graphene in AB-stacking - First principles calculation with vdw corrections 53 54 Lecture 2 1.1 mev Electrons in Graphitic Systems with Magnetic Fields κ = 100 ( k ΓK) / ΓK x κ = 100 k / ΓK y y x Contour line interval =0.5 mev SEMI-CLASSICAL APPROACH 2 Dimensional Electron Gas in SEMI-CLASSICAL APPROACH 2 Dimensional Electron Gas in R H Drude Model : I I Y R L X R xx = R L Longitudinal Resistance R xy = R H Hall Resistance (Classical) Hall Effect (1879) Drude Model :

QUANMTUM MECHANICS 2 Dimensional Electron Gas in LANDAU QUANTIZATION 2 Dimensional Electron Gas in R H I I Y R L X R xx = R L Longitudinal Resistance R xy = R H Hall Resistance B C Shubnikov-de Hass effect (1930) LANDAU LEVELS 2 Dimensional Electron Gas in INTEGER QUANTUM HALL EFFECT BIG BIG BREAKTROUGH!! R L B C At least, one cyclotron orbit before scattering

INTEGER Quantum Hall Effect 2 Dimensional Electron Gas in INTEGER Quantum Hall Effect 2 Dimensional Electron Gas in R H I I I I Y R L X R xx = R L Longitudinal Resistance R xy = R H Hall Resistance Y R xx = R L Longitudinal Resistance R xy = R H Hall Resistance Fermi energy X Fractional Quantum Hall Effects (1982) 2 Dimensional Electron Gas in movie

- Pseudospin and chiral states 65 - Berry s phase Eigenfunctions : Spinor representation p y p y for a path of C at K+ and K- p x θ p p x θ p Pseudo-spin up (down): A (B) sublattice - Shubnikov-de Hass Oscillation and Berry s phase 67 LANDAU QUANTIZATION Graphene in Landau orbit near Fermi level Normal 2DEG:

LANDAU LEVELS Graphene in Normal 2DEG: Quasi-particle spectrum of graphene - Transport measurements : QHE In the presence of magnetic field, 70 E n = sgn( n ) v 2e n B F Cyclotron mass: m * = π v F n Novoselov et al (05), Zhang et al (05) Quasi-particle spectrum of graphene - Magneto-transporttransport 71 Evolution of Landau Levels 2 DEG vs. Graphene in n=3 n=2 n=1 n=0 n=3 n=2 n=1 Magneto-oscillation in tunneling conductance: v F =(1.070±0.006) x 10 6 m/s Graphene on SiC(000-1): rotational stacking fault 0 0 n=0 J. Stroscio group, Science 324, 924 (2009)

- Landau levels in perpendicular Parabolic band (normal metal) 73 Bilayer graphene : Massive chiral particles Normal Single layer Metal Graphene Linear band (single layer graphene) - Consequences of chiral massless Dirac fermions Half-integer Quantum Hall Effect (Room T)! (Manifestation of Berry s phase of pseudospin) 75 Zero energy states SINGLE LAYER GRAPHENE Zhang et al (05), Novoselov et al (05) Kim & Geim et al (07) Haldane (88), T. Ando (02)

Zero energy states GENERALIZATIONS Bilayer graphene : Massive chiral particles Normal Single layer Bilayer Metal Graphene Graphene J different states are zero energy states