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)

N = infinite (metals) EFEF Ve - Free-e - VB CB Spacing between levels becomes too small to be distinguished So it can be regarded as a band structure Fermi sea Free-e - As long as kinetic energy is sufficient free electron movement can change from lane to lane

a. Underlying mechanism for ballistic transport Bulk Cu Corresponding band structure CB VB Conduction electron paths in all directions within CB EFEF Nanowire (quantum wire) Nanodot (quantum dot) EFEF EFEF Discrete levels Sub-bands  Spacing increase

Band gap M EFEF temp

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 = G o = 12.9 (K  ) -1 = 2e 2 /h. c.Two conduction channels = 2G o = 4e 2 /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

Electrons are unable to enter into CNT (joule heating) High contact resistance

Thick contact barrier Thin contact barrier CNT + - Electric charges induce voltage at leads but no current flow in CNT (coulomb oscillation) tunneling Existing conduction electrons in CNT Existing electrons in CNTs exclude incoming electrons Coulomb blockage

Thin contact barrier CNT tunneling 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

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Ballistic transport in carbon nanotubes Electron conduction has no resistance and no heat generation and structures are defect-free.

e-e- Conductor (e.g. Cu) e-e- Transport delay by resistance

e-e- Resistance point (scattering) Resistance comes from thermal vibration of crystal lattice, electrons and impurities forward scattering backscattering mean free path relaxation time mean free path: no resistance Cu (mean free path) = 1  m

Size reduction to below 1  m in length scattering point Cu nanowire e-e- free path with no resistance

Nanowire on electrodes electrodes resistance at wire-electrode contact e-e- forward scattering backscattering

Fabry-Perot interference

When defects exist the ballistic transport is absent Fabry-perot interference defect No Fabry-perot interference

Defects in CNTs EFEF Scattering center Blocking of two conduction channels 此時碳管電阻值昇高

Mean free path in Cu is 1  m 1m1m Mean free path is ca nm in CNT Scattering centers

Conditions for ballistic transport a. Tube length  electron mean free paths (or no defects) b. Low contact resistance (low capacitance) +- High contact resistance (capacitor-like) CNT capacitor leads dielectric c. Gate voltage is not needed (different from that of quantum wire) gate High contact resistance (tunneling)Low contact resistance

Science, 280, 1744

R + - R Distance A A R + - B R + - C BC A general case

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

Single atom 電極 電子 How to transmit through a single atom Conduction through individual orbits

3 6 5

Ohmic conductor (linear I-V profile) metals Voltage Current

Non-linear I-V profile (non-ohmic conductor) voltage current Light bulb

semiconductors voltage current

Theory of coulomb blockage source drain nanotube If one transfers the charge Q from the source to the grain the change in the energy of the system is

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. the effective capacitance C the gate voltage V G source drain nanotube e-e- - + Polarization of leads Q = –CV G 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 V G, and the difference vanishes. It means that only at that values of the gate voltage resonant transfer is possible.