1. Transport properties of mesoscopic graphene Björn Trauzettel Journées du graphène Laboratoire de Physique des Solides Orsay, 22-23 Mai 2007 Collaborators: Carlo Beenakker, Yaroslav Blanter, Alberto Morpurgo, Adam Rycerz, Misha Titov, Jakub Tworzydlo

2. Outline Brief introduction Transport in graphene as scattering problem Conductance/conductivity and shot noise Photon-assisted transport in graphene Summary and outlook

3. Honeycomb lattice real lattice (2 atoms per unit cell) 1. Brillouin zone

4. Tight binding model or pseudospin structure Eigenstates:

5. Solution to Schrödinger equation conduction and valence band touch each other at six discrete points: the corner points of the 1.BZ (K points)

6. Effective Hamiltonian  Dirac equation Dirac equation in 2D for mass-less particles with effective Hamiltonian Low energy expansion: … similar for the other K-point

7. Outline Brief introduction Transport in graphene as scattering problem Conductance/conductivity and shot noise Photon-assisted transport in graphene Summary and outlook

8. Schematic of strip of graphene ™ different boundary conditions in y-direction ™ voltage source drives current through strip ™ gate electrode changes carrier concentration

9. Problem I: How to model the leads? ™ electrostatic potential shifts Dirac points of different regions ™ large number of propagating modes in leads ™ zero parameter model for leads for V   

10. Problem II: boundary conditions (iii) infinite mass confinement (i) armchair edge (ii) zigzag edge (mixes the two valleys; metallic or semi-conducting) (one valley physics; couples k x and k y ) (one valley physics; smooth on scale of lattice spacing) Brey, Fertig PRB 73, 235411 (2006) Berry, Mondragon Proc. R. Soc. Lond. (1987) see also: Peres, Castro Neto, Guinea PRB 73, 241403 (2006)

11. Experimental feasibility Geim, Novoselov Nature Materials 6, 183 (2007)

12. Underlying wave equation kinetic term boundary term (infinite mass confinement) gate voltage term Ansatz: in leads: in graphene:

13. Scattering state ansatz Dirac equation (first order differential equation) ” continuity of wave function at x=0 and x=L ” determines t n and r n  transmission T n =|t n | 2

14. Solution of transport problem Transmission coefficient (at Dirac point): phase  depends on boundary conditions for propagating modes in leads In the limit |V  |  (infinite number of propagating modes in leads):

15. Transmission through barrier send L   ; W   ; keep W/L = const.  transmission remains finite In contrast: Schrödinger case  transmission T n  0 for k lead  

16. Outline Brief introduction Transport in graphene as scattering problem Conductance/conductivity and shot noise Photon-assisted transport in graphene Summary and outlook

17. Conductivity: influence of b.c. infinite mass confinement metallic armchair edge universal limit: W/L  1 Landauer formula: conductivity: at Dirac point (in universal regime): conductance proportional to 1/L

18. Conductivity: V gate dependence Tworzydlo, et al. PRL 96, 246802 (2006) Experiment: Novoselov, et al. Nature 438, 197 (2005) Possible explanations: charged Coulomb impurities Nomura, MacDonald PRL 98, 076602 (2007) strong (unitary) scatterers Ostrovsky, Gornyi, Mirlin PRB 74, 235443 (2006) Our theory:

19. Alternative data (Delft group) Delft data:Our theory: conductivity vs. conductance: H. Heersche et al., Nature 446, 56 (2007)

20. Current noise Average current: Current fluctuations: We are interested in the zero frequency and zero temperature limit.  shot noise

21. Shot noise: effect of b.c. Fano factor: metallic armchair edge infinite mass confinement universal limit: W/L  1 Tworzydlo, Trauzettel, Titov, Rycerz, Beenakker, PRL 96, 246802 (2006)

22. Maximum Fano factor ” sub-Poissonian noise ” universal Fano factor 1/3 for W/L  1 same Fano factor as for disordered quantum wire Beenakker, Büttiker, PRB 46, 1889 (1992); Nagaev, Phys. Lett. A 169, 103 (1992) unaffected by different boundary conditions & scaling system size to infinity

23. Sweeping through Dirac point ‘normal’ tunneling (CB  CB):Klein tunneling (CB  VB): directly at the Dirac point: transport through evanescent modes resembles diffusive transport

24. How good is the model for leads? Schomerus, cond-mat/0611209 If graphene sample biased close to Dirac point ” difference between GGG and NGN junctions is only quantitative GGG NGN see also: Blanter, Martin, cond-mat/0612577

25. Experimental situation I Arrhenius plot: E gap  28meV for ribbon of graphene with length of 1  m and width of 20nm Chen, Lin, Rooks, Avouris cond-mat/0701599 Similar results: Han, Oezyilmaz, Zhang, Kim cond-mat/0702511

26. Experimental situation II Miao, Wijeratne, Coskun, Zhang, Lau cond-mat/0703052

27. Outline Brief introduction Transport in graphene as scattering problem Conductance/conductivity and shot noise Photon-assisted transport in graphene Summary and outlook

28. Motivation: Zitterbewegung superposition of positive and negative energy solution current operator with interference terms electron-likehole-like

29. Zitterbewegung in current operator Katsnelson EPJB 51, 157 (2006) Zitterbewegung contribution to current (due to interference of e-like and h-like solutions to Dirac equation)

30. Can Zitterbewegung explain the previous shot noise result? Answer: I don’t think so. Question: Why not? In the ballistic transport problem, the wave function is either of electron-type or of hole-type, but not a superposition of the two! no interference term in ballistic transport calculation

31. How to generate the desired state Trauzettel, Blanter, Morpurgo, PRB 75, 035305 (2007)

32. Transport properties The current oscillates due to applied ac signal and not due to an intrinsic zitterbewegung frequency. Differential conductance (in dc limit) can be used to probe energy dependence of transmission

33. Summary ballistic transport in graphene contains unexpected physics: conductance scales pseudo-diffusive  1/L conductivity has minimum at Dirac point shot noise has maximum at Dirac point universal Fano factor 1/3 if W/L  1 photon-assisted transport in graphene

34. Aim: spin qubits in graphene quantum dots Trauzettel, Bulaev, Loss, Burkard, Nature Phys. 3, 192 (2007)

35. Why is it difficult to form spin qubits in graphene? Problem (i): It is difficult to create a tunable quantum dot in graphene. (Graphene is a gapless semiconductor.  Klein paradox) Problem (ii): It is difficult to get rid of the valley degeneracy. This is absolutely crucial to do two-qubit operations using Heisenberg exchange coupling.

36. Solutions to confinement problem generate a gap by suitable boundary conditions Silvestrov, Efetov PRL 2007 Trauzettel et al. Nature Phys. 2007 magnetic confinement De Martino, Dell’Anna, Egger PRL 2007 biased bilayer graphene Nilsson et al. cond-mat/0607343

37. Illustration of degeneracy problem One K-point only:Two degenerate K-points: based on Pauli principle

38. Solution to both problems K point K’ point ribbon of graphene with semiconducting armchair boundary conditions  K-K’ degeneracy is lifted for all modes Brey, Fertig PRB 2006

39. Emergence of a gap bulk graphene with local gates ribbon of graphene (with suitable boundaries)  local gating allows us to form true bound states

40. Calculation of bound states solve transcendental equation for  appropriate energy window

41. Energy bands for single dot

42. Energy bands for double dot

43. Long-distance coupling ideal system for fault-tolerant quantum computing ” low error rate due to weak decoherence ” high error threshold due to long-range coupling