1. Quantum Interference in Multiwall Carbon Nanotubes Christoph Strunk Universität Regensburg Coworkers and Acknowledgements: B. Stojetz, Ch. Hagen, Ch. Hendlmeier (Regensburg) L. Forró, E. Ljubovic (Lausanne) A. Bachtold, M. Buitelaar, Ch. Schönenberger (Basel) K. Richter, G. Cuniberti (Regensburg) R. Schäfer (Karlsruhe)

2. multiwalled carbon nanotubes S. Ijima, Nature 354, 56 (1991) 26 nm

3. Outline Introduction: Electronic structure of carbon nanotubes Quantum interference Changing the electron density Coulomb blockade Perspectives

4. sp 2 -hybridization leads to planar carbon sheets 2D electronic bandstructure determined by p-orbitals  -bands touch at K-points kxkx kyky E **  K’ K  kxkx kyky Graphene: a single sheet of graphite

5. wrapping graphene to nanotubes: x y wrapping vector R determines: chirality (real space)allowed k-vectors (k-space) RARA RBRB

6. Density of states kxkx kyky K K’ K Metallic behavior Semicond. behavior

7. are MWNTs ballistic conductors at 300 K? Frank, et al., Science 280, 1744 (1998) G (2e²/h) z-position (nm) Conductance changes in units of 2e²/h !

8. Weak localization and universal conductance fluctuations (UCF) signatures of coherent backscattering in disordered quantum wires: r r’r’ AiAi AjAj r =r’ A + =A - Closed loop of time reversed paths: enhanced backscattering probability! Magnetic field breaks time-reversal symmetry: coherent backscattering suppressed by magnetic field: negative magnetoresistance near B=0 reproducible fluctuation pattern specific for impurity configuration: “magneto-fingerprints”

9. Weak localization and universal conductance fluctuations (UCF) signatures of coherent backscattering in disordered quantum wires: r r’r’ AiAi AjAj r =r’  Closed loop of time reversed paths: enhanced backscattering probability! Magnetic field breaks time-reversal symmetry: coherent backscattering suppressed by magnetic field: negative magnetoresistance near B=0 reproducible fluctuation pattern specific for impurity configuration: “magneto-fingerprints”  A + =A -

10. A. Bachtold et al., ‘98

13. Similar results obtained by many other groups: Leuven, IBM, Stuttgart, Helsinki …..

14. How to confirm the presence of elastic scattering ? 200 nm Au contact Al gate (native oxide) MWNT Induce drastic change of electron density by gate electrode (distance 2-3 nm) Change number of current carrying subbands Tune electrochemical potential through charge neutrality point Induce transition between quasi-1dim and strictly 1dim transport ? k E EFEF Doping state of MWNTs Effect on weak localization ? Effects of Coulomb interaction ?

15. Gate sweep 2 -3-201 10 15 20 25 R (k  ) U Gate (V) 1.7 K 5 K 10 K 15 K 20 K 40 K low temperatures universal conductance fluctuations (UCFs) (curves shifted) high temperatures shallow minimum in conductance

16. Universal conductance fluctuations Ensemble averaging of conductance fluctuations  G if L < l  ll Interference of many diffusion paths lead to aperiodic fluctuation pattern in the conductance: vary interference pattern by applying electric or magnetic fields determine phase coherence length l  at different temperatures l  > tube diameter (28 nm) l  < tube length (400 nm)

17. Magnetoresistance at different gate voltages magnetic field B perpendicular to tube axis magnetoresistance traces taken at various gate voltages (arrows) select different members within statistical ensemble of magneto-fingerprints T = 1.7 K

18. Ensemble averaging average weak localization peak survives averaging UCFs averaged out partially, but not completely T = 1.7 K (curves shifted) Stojetz et al., New J. Phys. ‘04

19. Weak localization conductance correction due to weak localization: Fitting WL-theory to data: T (K) l  (nm) 1.7 150 20 80 40 50 1.7 K 20 K 40 K effective width W~diameter/2 required origin: flux-cancellation effects ?

20. Phase coherence length diamonds: UCF measurement triangles: weak localization line: prediction for electron- electron dephasing ~T -1/3 elastic mfp: 14 nm Good agreement of l  from WL and UCFs Substantiation of diffusive transport picture Further experiments required to identify origin of disorder  :UCF  :WL

21. Measure a larger statistical ensemble: shallow conductance minimum at 300K emerging fluctuation pattern at lower T decrease of correlation voltage V c

22. Crossover to Coulomb blockade at lowest T : decrease of average conductance Resonant transmission of single channels?

23. T=30 mK disordered MWNT with irregular Coulomb diamonds: typical capacitances: C Gate ~ 55 aF C   ~ 800 aF charging energy E c ~ 100  eV ~ 1.2 K

24. broad zero bias anomalies remain at higher T: T = 3 K T = 10 K estimated subband spacing ~ 25 meV gate lever arm  E F /U Gate ~ 1/10

25. T = 10 K Magnetoconductance shows pronounced gate dependence:

26. Open questions Source of disorder - extrinsic or intrinsic ? Strength of disorder? Effect of Coulomb blockage and number of channels on the shape of the WL-peak? Gate dependence of Aharonov-Bohm effect in parallel magnetic field? B