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Ballistic and quantum transports in carbon nanotubes Discrete energy levels in carbon nanotubes Two atoms two energy levels Three atoms three energy levels atoms CB VB Metallic (no band gap) Semiconductor Small band gap (0.1-3eV) Insulator (large band gap > 5 eV) Spacing between levels becomes too small to be distinguished So it can be regarded as a band structure CB Fermi sea Free-eFree-e- EF VB Ve- N = infinite As long as kinetic energy is sufficient free electron movement (metals) can change from lane to lane a. Underlying mechanism for ballistic transport Bulk Cu CB EF EF VB Corresponding band structure Conduction electron paths in all directions within CB Discrete levels Nanowire (quantum wire) Nanodot (quantum dot) Spacing increase EF Sub-bands M temp EF Band gap Science, 283, 52 (1999). Quantum transport in carbon nanotubes a. Metallic CNTs have two conduction bands (two conduction channels). b. one conduction channel of quantum unit = Go = 12.9 (K)-1 = 2e2/h. c. Two conduction channels = 2Go = 4e2/h = 6.5 (K)-1 If contact resistance is small and negligible then CNT resistance between contacts Should be 6.5K In practice, resistance exceeds 6.5 k and underlying Mechanism comes from A. Semiconducting tubes B. Huge contact resistance C. Defects in metallic CNTs High contact resistance Electrons are unable to enter into CNT (joule heating) Electric charges induce voltage at leads but no current flow in CNT (coulomb oscillation) + CNT Thick contact barrier Existing electrons in CNTs exclude incoming electrons tunneling Coulomb blockage CNT Thin contact barrier Existing conduction electrons in CNT Thin contact barrier tunneling CNT Gate voltage Gate voltage Gate voltage: electric field modulated chemical potential (energy levels) gate Two electrons occupying one level gate Modulation of energy levels by gate voltage Go Ballistic transport in carbon nanotubes Electron conduction has no resistance and no heat generation and structures are defect-free. Transport delay by resistance e- e- Conductor (e.g. Cu) Resistance comes from thermal vibration of crystal lattice, electrons and impurities Resistance point (scattering) e- mean free path relaxation time forward scattering backscattering mean free path: no resistance Cu (mean free path) = 1 m scattering point scattering point Size reduction to below 1 m in length free path with no resistance eCu nanowire electrodes backscattering eforward scattering forward scattering Nanowire on electrodes resistance at wire-electrode contact Fabry-Perot interference When defects exist the ballistic transport is absent Fabry-perot interference defect No Fabry-perot interference Defects in CNTs Scattering center EF Blocking of two conduction channels 此時碳管電阻值昇高 Mean free path in Cu is 1 m Scattering centers 1m Mean free path is ca. 100-300 nm in CNT Conditions for ballistic transport a. Tube length electron mean free paths (or no defects) b. Low contact resistance (low capacitance) dielectric leads + CNT - High contact resistance (capacitor-like) capacitor c. Gate voltage is not needed (different from that of quantum wire) High contact resistance (tunneling) gate Low contact resistance Science, 280, 1744 R A general case A B A C Distance B C - R + R + R + Ballistic transport effects a. No heat generation, because no electron-phonon interaction (i.e. no scattering by defects) b. Stepwise I-V profile (or quantum conduction) Conduction via individual atoms a. Nano-contact b. electro-sharpening of metal wire Diffusive conduction Quantum conduction c. Mechanical break junction 電極 電子 Single atom How to transmit through a single atom Conduction through individual orbits 3 6 5 Ohmic conductor (linear I-V profile) Current metals Voltage Non-linear I-V profile (non-ohmic conductor) current Light bulb voltage semiconductors current voltage Theory of coulomb blockage http://edu.ioffe.ru/register/?doc=galperin/l13pdf1.tex nanotube source drain If one transfers the charge Q from the source to the grain the change in the energy of the system is the effective capacitance C the gate voltage VG the first item is the work by the source of the gate voltage while the second is the energy of Coulomb repulsion at the grain. - Polarization of leads + enanotube source drain Q = –CVG So Q can be tuned by the gate voltage VG the charge is transferred by the electrons with the charge –e. Then, the energy as a function of the number n of electrons at the drain is the difference at certain values of VG, and the difference vanishes. It means that only at that values of the gate voltage resonant transfer is possible.