1. Physics of Graphene A. M. Tsvelik

2. Graphene – a sheet of carbon atoms The spectrum is well described by the tight- binding Hamiltonian on a hexagonal lattice:

3. Lattice effects: Ripples in graphene 2D membranes embedded in 3D space have a tendency to get crumpled. These dangerous fluctuations can be suppressed by an anharmonic coupling between bending and stretching modes. Result: the membranes can exist, but with strong height fluctuations. Monte Carlo simulations (Katsnelson et. al. (2007)): disordered state with weakly T-dependent correlation length (70A at 300K and 30A at 3500K). A typical snapshot of graphene at room temperature. The size of height fluctuations is comparable to the lattice size.

4. Crumpling of graphene sheet – the main source of disorder.

5. Dirac Hamiltonian for low energy states The Bloch functions A and B are peaked on the corresponding sublattices. They are conveniently joined in a vector V = c/300

6. Klein paradox – electrons go through potential barriers Penetration of particles through potential barriers. The transmission probability T is directionally- dependent. For high barriers (V >> E)

7. Hopes for applications - spintronics The transmission is sensitive to the barrier height V. If V’s are different for different spin orientations (magnetic gates) one can produce spin-polarized currents. This will allow to manipulate electron’s spin. One can also create electronic lenses.

8. Electronic lenses

9. Beam splitter for electrons (Falko, 2007)

10. Is it dirty? STM measurements of graphene (Martin et. Al. 2007) A color map of the spatial density variations in the graphene flake. Blue regions are holes and gold regions are electrons. The black contour – zero density. About 100 particles/puddle, k_Fl ~ 10. Histogram of the density distribution. The energy width is ~400K

11. They make it dirty, we make it clean! Angle Resolved Photoemission Spectroscopy (ARPES) study of the graphene spectrum done by T. Valla (BNL) on locally grown samples. The spectral width is smaller than in any material measured before. Clean substrates?

12. Hall effect ( Cho and Fuhrer (2007)) Conductivity as a function of the chemical potential.

13. Hall effect In the absence of disorder the Landau levels are Disorder broadens the levels and when the broadening or T exceed the Zeeman splitting they become 4-fold degenerate. Filling fractions  n + ½) for B < 9T. For 20T < B < 45T there are plateaus at  (interactions ?),  q – spin degeneracy is lifted.

14. Special Landau level n=0 Integer Quantum Hall effect measurements (Giesbers et.al. 2007) indicate that at B < 9T the n=0 Landau level is unusually narrow which increases the T range where Hall effect is seen. Why it is so narrow?

15. Zero mode and Index theorem Hamiltonian in one of the valleys. We neglect the Zeeman splitting. Vector potential parametrization: Eigenfunction with zero energy always exists, no matter how non-uniform the field is: where f(z) is a polynomial of power smaller than the magnetic flux.

16. Fractional Quantum Hall effect  1 state is pseudospin (valley) ferromagnet (McDonald et. al (2006), Haldane et. al. (2006))  state is the XY pseudospin magnet (Haldane et. al (2006)). FQHE at these fillings is the only effect observed so far where interactions play a role.

17. Interaction –The strongest interaction in graphene is Coulomb interaction: it breaks the Lorentz symmetry. It breaks the Lorentz invariance of the kinetic energy. It is predicted to make the velocity energy dependent (Aleiner et.al 2007): -fine structure constant

18. Conclusions There are possible technological applications related to directional and energy dependence of transmission in graphene. The problem #1 is manufacturing of clean samples. Most of the physics observed so far is a single particle one. Many-body effects are observed in FQHE in strong magnetic fields. The role of bending fluctuations is not very clear, the theory is not finalized. It is possible that further many-body effects will be observed in clean samples at low T. Get rid of high  substrate!

19. Clean or dirty? Resistor network model by Cheianov et. Al. (2007)