Theories of graphene. Reinhold Egger Heinrich-Heine-Universität Düsseldorf Kolloquium, Hamburg

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1 Theories of graphene Reinhold Egger Heinrich-Heine-Universität Düsseldorf Kolloquium, Hamburg

2 Graphene monolayer Mother of all-carbon materials (fullerenes, nanotubes, graphite): made of benzene rings stripped of H atoms Reviews: Geim & Novoselov, Nature Materials 007 Castro Neto et al., Rev. Mod. Phys. 009

3 Discovery and fabrication Graphite: invention of the pencil in Writing process probably already produces monolayers Graphene monolayers: for the first time fabricated and prepared by mechanical exfoliation in 004 Novoselov et al., Science 004 Repeated peeling of graphite with adhesive tape gives graphite flakes, some of them are monolayers! Then deposit on Si wafer with thin SiO top layer of (precise!) thickness 300 nm Study in optical microscope: monolayers now produce characteristic interference fringes Simple & cheap scheme: easy to make, hard to see

4 Graphene monolayers Experimental proof for monolayer: quantum Hall measurements, Raman spectroscopy Novoselov et al., Nature 005, Zhang et al., Nature 005 DEG with exceptional properties Surface state: probe by STM/AFM/SEM Spontaneous crumpling (Mermin Wagner theorem!) of monolayer Meyer et al., Nature 007 Electronic transport: massless Dirac fermion quasiparticles, ballistic transport up to room temperature possible with mean free path ~3 μm on BN substrates Geim et al. arxiv:

5 Explosion of graphene research Nobel Prize 010: Geim & Novoselov Papers with graphene in title Source: ISI Web of Science, April 011

6 Graphene: Tight binding description Basis contains two atoms; nearest-neighbor hopping connects different sublattices: Pseudospin Wallace, Phys. Rev a 3d, d 0. 14nm

7 Dirac cone Exactly two independent corner points K, K in first Brillouin zone. Band structure: valence and conduction bands touch at corner points (E=0), these are the Fermi points in undoped graphene Low energies: Dirac light cone dispersion Lorentz invariance emerges at low energies Deviations at higher energies: trigonal warping Eq v q q k K v 10 6 m / sec

8 Relativistic quantum mechanics Low energy continuum limit: only momenta close to K or K matter, two decoupled copies of massless relativistic Dirac-Weyl Hamiltonians Dirac spinor H K iv x, y) (, ) ( A B Pauli matrices in pseudospin space: ( x, y) Experimental confirmation for presence of massless Dirac fermions in graphene monolayers: cyclotron resonance and half-integer quantum Hall effect Novoselov et al., Nature 005, Zhang et al., Nature 005

9 Chirality (Helicity) Electron/hole state with energy q 1 Transforms under full rotation as Chirality (Helicity): i e e i / hˆ q / q q Projection of momentum operator on pseudospin direction is conserved quantity. For electrons: +1; for holes: -1 Different from usual chirality as eigenvalue of 5 (which does not exist in +1 dimensions) E v q tan q q q hˆ q q q q x y spinor q

10 Electron-hole symmetry Hamiltonian anticommutes with σ z Unitary transformation exchanges electron and hole states H z z H Electron and hole states are entangled via the Dirac spinor z

11 Zero gap semiconductor Density of states vanishes linearly at Dirac point Highest resistivity for Fermi level at Dirac point Novoselov et al., Nature 005

12 Klein tunneling Klein, Z. Phys. B 199 Katsnelson & Geim, Nature Physics 006 Dirac fermions can tunnel through high and wide barriers: Klein paradox Conservation of pseudospin in the absence of short-range disorder (chirality effect) Hard to see in highenergy physics Experimental evidence in graphene n-p-n devices Williams et al., Science 007 Drawback: confinement by gating problematic!

13 Klein tunneling: single-particle theory neglect many-body effects and physical spin H 0 iv V ( x) V ( x) V 0, 0, Local barrier of width D and height V 0, electrons are converted into holes underneath the barrier Assume no KK scattering & smooth potential (same for both sublattices) Plane wave solutions for Dirac spinors for the three regions Matching conditions: imposed only for wave functions (not their derivatives!) at the interfaces 0 x D otherwise

14 Transmission probability Result (for small E): T cos 1 cos q Dsin x qx k F cos Barrier fully transparent for normal incidence or under resonance condition q x D N, N 0, 1,,... Monolayer Bilayer

15 Universal minimal conductivity Measured conductivity: linear increase with density away from Dirac point ne Minimal conductivity universal 4 min e / h How to explain? Novoselov et al., Nature 005

16 Brief excursion: Disorder in graphene Linear conductivity increase with density: charged impurities important (from substrate) Nomura & MacDonald, PRL 006 Minimal conductivity at Dirac point unexpected from standard localization theory for D conductor. Theoretical possibilities: ZERO Aleiner & Efetov, PRL 006, Altland, PRL 006 INFINITE Beenakker et al., PRL 007 INTERMEDIATE VALUE SCBA min 1 4e h

17 Conductivity at the Dirac point Subtle issue: Kubo result depends on order of limits (dc vs clean) 0, 1/ 0 If dc limit taken first: If clean limit first: Similar, but different. In any case: Finite conductivity without disorder min min 1 4 h 4 8 h Certain disorder types are predicted to enhance conductivity beyond those values e e Ryu et al., PRB 007 Beenakker et al., PRL 007

18 Orbital magnetic field (perpendicular) De Martino, Dell Anna & Egger, PRL 007 Dirac Hamiltonian from minimal substitution: A A v i ea x, y E B B equivalent to pair of decoupled Schrödinger equations: v i ea ev B E 0 Energies come in plus-minus pairs (chiral Hamiltonian), zero energy is special z Magnetic fields localize electrons: no Klein paradox! Important to establish quantum Hall effect z

19 Magnetic barrier Square barrier: with kinematic incidence angle (using Landau gauge) q k cos Gauge invariant velocity: B0, x B( x, y) 0, x q x y k F F sin v cos v sin D / D / edb 0 / q y is conserved

20 Incoming scattering state (from left) Left of the barrier: Under the barrier: Right of the barrier: with emergence angle in i x iq i x iq left e re e e x x 1 1 B B y l k B F B B y l k barrier l x l q D l k i l x l q D c B F B F / / / ) ( / ) ( 1 eb 0 c l B 1 i x iq x x right e e q q t x cos x k F q

21 Perfect reflection regime Transmission/reflection probability T t, R r 1T Relation between emergence and incidence angle from q y conservation sin sin D No solution, i.e. perfect reflection, for low energy or wide barrier k l B / D 1 allows to confine Dirac fermions F k l F B

22 Transmission probability angular plot of transmission probability T() (away from the perfect reflection regime) k l F B 3.7 d D /

23 Cyclotron frequency Now consider homogeneous field Magnetic lengthscale l B c eb Only other scale in the problem is Fermi velocity, gives cyclotron frequency c v / l B B c B Nonrelativistic Schrödinger case: For B=10T, cyclotron frequency for Dirac fermions corresponds to K, but only 10 K for Schrödinger fermions! Implies negligible Zeeman effects for Dirac case

24 Solution for homogeneous field Landau gauge: ikx x, y e 1D harmonic oscillator bosonic creation/annihilation operators [ b, b ] 1 l k Dirac equation in magnetic field then takes the form 0 b E c b 0 b b 1 1 B y l B

25 Relativistic Landau levels Solution for E=0: b ground state wavefunction of 1D harmonic oscillator All other solutions (N=1,,3, ) then constructed from zero mode: relativistic Landau levels N, N 1 N Experimental confirmation: Shubnikov-de Haas, IR spectroscopy, scanning tunneling spectroscopy E 0 N, e c N / N H N

26 Half-integer quantum Hall effect Zero-energy Landau level state is responsible for unusual quantization rule of the integer quantum Hall effect Gusynin & Sharapov, PRL e Hall conductivity: xy N h Connects to interesting mathematics for Dirac fermions: Index theorem Number of zero-energy states is a topological invariant, depends only on total flux through the system (even with inhomogeneities in magnetic field) Half-integer quantum Hall effect extremely robust!

27 QHE: experimental data Novoselov et al., Nature 005 observed even at room temperature!

28 Electron-electron interaction effects Emergent Lorentz invariance broken by Coulomb interaction Retardation negligible for Coulomb interactions Strength parametrized by effective fine structure constant Review: Kotov et al., arxiv: For density n, typical kinetic energy: Typical Coulomb energy: e Fine structure constant: r E C E k v e n e. v Strong-coupling regime of (+1)d QED E E C k n

29 General remarks Fine structure constant independent of density, no Wigner crystallization expected No screening of interactions (up to renormalization of dielectric constant) at Dirac point due to vanishing DoS In magnetic field, kinetic energy is quenched, and interactions dominate more (observation of fractional QHE) Fermi liquid quasi-particles well-defined? (here: B=0)

30 Weak-coupling results One-loop calculation yields logarithmic growth of Fermi velocity at low energies Corresponds to slow logarithmic RG flow at low energies Weak interactions are marginally irrelevant v k v1 ln 4 k Gonzalez et al., Nucl. Phys. B 1994 Cutoff scale (for validity of Dirac cone spectrum) Only very weak effects on minimal conductivity (cancellation of diagrams) Same conclusions from two-loop or RPA calculations for 1 Son, PRB 007 0

31 Quasi-particle lifetime Linear energy dependence of inverse quasiparticle lifetime close to Dirac point due to e-e interactions k, vk consistent with ARPES and STM experiments Signature of marginal Fermi liquid behavior near Dirac point For 1, more dramatic things may happen Spontaneous gap generation (excitonic insulator, chiral symmetry breaking)

32 Mesoscopic geometry: Quantum Dots Quantum dots containing N particles (on top of filled Dirac sea) in graphene: Magnetic confinement De Martino et al., PRL 007 Quasibound states in electrostatic potentials Finite-size flake or nanoribbon Silvestrov & Efetov, PRL 006 Here: circular dot with no out-current (infinite mass) boundary condition at r R 1 Single-particle states define artificial atom energy levels In contrast to atomic physics: interactions much stronger

33 Brown-Ravenhall disease Problem: unbounded negative-energy spectrum allows interaction to excite arbitrary numbers of electron-hole pairs Finite-size geometry: natural formation of energy gap. For sufficiently weak interactions: freeze the (inert) Dirac sea (Sucher s projection) Sucher, Phys. Rev Allows to use first quantization For Brown & Ravenhall, Proc. R. Acad. Sci this is justified here! Häusler & Egger, PRB 009

34 Hartree-Fock calculations for interacting particles in graphene dot Hartree Fock is very accurate despite of strong interactions Paananen, Egger & Siedentop, PRB 011 Carefully benchmarked against exact diagonalization for N= particles Here: for simplicity, single-valley spin-polarized version of graphene Sucher projection is (self-consistently) valid Results for up to 0 particles on top of filled Dirac sea: ground state energy, density profile, pair correlations, etc.

35 HF addition spectrum addition energy: N EN 1 EN 1 EN Magic numbers (artificial atom with N particles especially stable) are different and more pronounced with interactions! Measurable by Coulomb blockade spectroscopy

36 Wigner crystallization radial density profile pair correlations: spatial shells with sequence

37 Wigner crystallization Electrostatic energy starts to dominate over kinetic energy for 1 Particles maximize their distance and form crystal here ring-like arrangement Wigner crystallization favored in confined geometry (no Wigner crystal in bulk graphene!) Crossover from Fermi liquid regime to Wigner molecule as in standard DEG possible Egger, Häusler, Mak & Grabert, PRL 1999

38 (Some of the) topics left out 1D graphene nanoribbons : open boundary conditions in transverse direction Nanotubes correspond to periodic boundary conditions Superconductivity and proximity effect Coulomb impurity problem: supercritical regime accessible Strongly correlated phases, magnetic moments Bilayer graphene: massive Dirac fermions

39 More Dirac cones: Topological insulators Hasan & Kane, Rev. Mod. Phys D band insulators with strong spin orbit coupling can have conducting surface states, e.g. bismuth selenides or bismuth tellurides Surface state: odd number of Dirac spinors Another realization of relativistic quantum mechanics First proposed in 005 (Kane, Mele), first realized experimentally in 007 (Molenkamp group) / 008 (Hasan group)

40 Acknowledgements Thanks to my collaborators on graphene: Alessandro De Martino, Köln Wolfgang Häusler, Augsburg Luca Dell Anna, Padua Tarun Ghosh, Kanpur/India Tomi Paananen, Düsseldorf Heinz Siedentop, Math. Inst. LMU München THANK YOU FOR YOUR ATTENTION!