1. Graphene Christian Mendl February 6, 2008 MPQ Theory Group Seminar

2. Mother of all Graphitic Forms Graphene is a 2D building material for carbon materials of all other dimensionalities. It can be wrapped up into 0D buckyballs, rolled into 1D nanotubes or stacked into 3D graphite. Graphite Graphene 0.3 nm

3. One-atom-thick single crystals: the thinnest material you will ever see. a) Graphene visualized by atomic-force microscopy. The folded region exhibiting a relative height of ≈4Å clearly indicates that it is a single layer. b) A graphene sheet freely suspended on a micron-size metallic scaffold. c) scanning-electron micrograph of a relatively large graphene crystal, which shows that most of the crystal’s faces are zigzag and armchair edges as indicated by blue and red lines and illustrated in the inset. 1D transport along zigzag edges and edge-related magnetism are expected to attract significant attention. a) Novoselov, K. S. et al. Two- dimensional atomic crystals. Proc. Natl Acad. Sci. USA 102, 10451- 10453 (2005) c) T.J. Booth, K.S.N, P. Blake & A.K.G. From: A.K. Geim and K.S. Novoselov: The Rise of Graphene b) Meyer, J.C. et al. Nature 446, 60- 63 (March 2007) Landau and Peierls: strictly two-dimensional crystals are thermodynamically unstable and cannot exist… Landau, L. D. Zur Theorie der Phasenumwandlungen II. Phys. Z. Sowjetunion, 11, 26-35 (1937)

4. Micromechanical Cleavage Graphene becomes visible in an optical microscope if placed on top of a Si wafer with a carefully chosen thickness of SiO 2, owing to a feeble interference-like contrast with respect to an empty wafer.

5. Tight-Binding Model Tight-binding description for π-orbitals of carbon: V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte. AC conductivity of graphene: from tight-binding model to 2+1-dimensional quantum electrodynamics. International Journal of Modern Physics B, 21, No.27:4611-4658, 2007

6. Effective Hamiltonian: Dirac Equation Conduction and valence band touch each other at six discrete points: the corner points of the 1.BZ (K points) „Relativistic“ condensed matter physics: condensed matter analogue of (2+1)-dimensional quantum electrodynamics Effective speed of light: v F ≈10 6 m/s

7. b) Room-temperature quantum Hall effect: quasiparticles in graphene are massless and exhibit little scattering even under ambient conditions Ballistic Electron Transport a) The rapid decrease in resistivity ρ with adding charge carriers indicates high electron mobility (in this case, μ≈5000cm 2 /Vs and does not noticeably change with temperature up to 300K). Andre Geim et al. (University of Manchester), Graphene Speed Record, Physics News Update Number 854 #2, January 23, 2008: μ≈200 000 cm 2 /Vs

8. Chiral Quantum Hall Effects a) The hallmark of massless Dirac fermions is QHE plateaux in σ xy at half integers of 4e 2 /h b) Anomalous QHE for massive Dirac fermions in bilayer graphene is more subtle (red curve): σ xy exhibits the standard QHE sequence with plateaux at all integer N of 4e 2 /h except for N=0. The zero-N plateau can be recovered after chemical doping, which shifts the neutrality point to high V g so that an asymmetry gap (≈0.1eV in this case) is opened by the electric field effect (green curve) c-e) Different types of Landau quantization in graphene. The sequence of Landau levels in the density of states D is described by E N ∝ √N for massless Dirac fermions in single-layer graphene (c) and by E N ∝ √N(N −1) N for massive Dirac fermions in bilayer graphene (d). The standard LL sequence E N ∝ (N+½) N is expected to recover if an electronic gap is opened in the bilayer (e).

9. Spin Qubits in Graphene Quantum Dots Spin-orbit coupling is weak in carbon (low atomic weight) → spin decoherence due to spin- orbit coupling should be weak Natural carbon consists predominantly of the zero-spin isotope 12 C → spin decoherence due to hyperfine interaction of electron spin with surrounding nuclear spins should be weak Björn Trauzettel, Denis V. Bulaev, Daniel Loss, and Guido Burkard: Spin qubits in graphene quantum dots. Nature Phys., 3:192, 2007 Graphene double quantum dot: ribbon of graphene (grey) with semiconducting armchair edges (white). Confinement is achieved by tuning the voltages applied to the “barrier” gates (blue) to appropriate values such that bound states exist. Additional gates (red) allow to shift the energy levels of the dots. Virtual hopping of electrons through barrier 2 (thickness d) gives rise to a tunable exchange coupling J between two electron spins localized in the left and the right dot.

10. Idea: create ribbon of graphene with semiconducting armchair boundary conditions:  Valley degeneracy is lifted for all modes (necessary to do two-qubit operations using Heisenberg exchange coupling)  Generates energy gap → solves the quantum dot confinement problem (Klein paradox!) Björn Trauzettel, Denis V. Bulaev, Daniel Loss, and Guido Burkard: Spin qubits in graphene quantum dots. Nature Phys., 3:192, 2007 Energy bands for single and double dot case

11. Exchange Coupling Exchange coupling based on Pauli principle: with singlet-triplet splitting (t is the tunneling matrix element and U the on-site Coulomb energy) Can estimate t for the ground state: Room for tuning!

12. Long Distance Coupling Triple quantum dot setup. Dot 1 and dot 3 are strongly coupled via cotunnelling processes through the valence bands of barrier 2, barrier 3, and dot 2. The center dot 2 is decoupled by detuning. The energy levels are chosen such that ∆ε 2 ≪ ∆ε 1. The triple dot example illustrates that in a line of quantum dots, it is possible to strongly couple any two of them and decouple the others by detuning. This is a unique feature of graphene and cannot be achieved in semiconductors such as GaAs that have a much larger gap. Tunnel coupling via Klein tunneling through the valence band!

13. Conclusions 2D → conceptually new material Extraordinary crystal and electronic properties Opens a door for testing QED phenomena experimentally Promising candidate for classical and quantum computing (high mobility at room-temperature, long distance coupling)