Transport properties of mesoscopic graphene

1. Transport properties of mesoscopic graphene Journées du graphène Laboratoire de Physique des Solides Orsay, 22-23 Mai 2007 Björn Trauzettel 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 1. Brillouin zone real lattice (2 atoms per unit cell)

4. Tight binding model Eigenstates: or pseudospin structure

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 with Low energy expansion: effective Hamiltonian Dirac equation in 2D for mass-less particles … 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 (i) armchair edge (mixes the two valleys; metallic or semi-conducting) Brey, Fertig PRB 73, 235411 (2006) (ii) zigzag edge (one valley physics; couples kx and ky) (iii) infinite mass confinement (one valley physics; smooth on scale of lattice spacing) 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 gate voltage term boundary term (infinite mass confinement) 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 tn and rn •  transmission Tn=|tn|2

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

15. Transmission through barrier • send L  ; W  ; • keep W/L = const. •  transmission remains finite In contrast: Schrödinger case  transmission Tn  0 for klead  

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

18. Conductivity: Vgate dependence Experiment: Our theory: Novoselov, et al. Nature 438, 197 (2005) Tworzydlo, et al.PRL 96, 246802 (2006) Possible explanations: charged Coulomb impuritiesNomura,MacDonaldPRL 98, 076602 (2007) strong (unitary) scatterersOstrovsky, Gornyi, MirlinPRB 74, 235443 (2006)

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

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

22. Maximum Fano factor unaffected by different boundary conditions & scaling system size to infinity • 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)

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? • If graphene sample biased • close to Dirac point • difference between GGG and NGN junctions is only quantitative GGG NGN Schomerus,cond-mat/0611209 see also: Blanter, Martin,cond-mat/0612577

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

26. Experimental situation II Miao, Wijeratne, Coskun, Zhang, Laucond-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-like hole-like

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

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, EfetovPRL 2007 Trauzettel et al. Nature Phys. 2007 biased bilayer graphene Nilsson et al.cond-mat/0607343 magnetic confinement De Martino, Dell’Anna, EggerPRL 2007

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

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

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 appropriate energy window solve transcendental equation for 

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