Ballistic conduction in single-walled carbon nanotubes

Single-walled carbon nanotubes in the fields of quantum mechanics and nanoelectronics, have the ability to conduct electricity.

When ballistic in nature conductance can be treated as if the electrons experience no scattering.

Conduction in single-walled carbon nanotubes is quantized due to their one-dimensionality and the number of allowed electronic states is limited, if compared to bulk graphite.

The nanotubes behave consequently as quantum wires and charge carriers are transmitted through discrete conduction channels.

This conduction mechanism can be either ballistic or diffusive in nature, or based on tunneling.

When ballistically conducted, the electrons travel through the nanotubes channel without experiencing scattering due to impurities, local defects or lattice vibrations.

As a result, the electrons encounter no resistance and no energy dissipation occurs in the conduction channel.

Assuming no scattering and ideal (transparent) contacts, the conductance of the one-dimensional system is given by G = G0NT, where T is the probability that an electron will be transmitted along the channel, N is the number of the channels available for transport, and G0 is the conductance quantum 2e2/h = (12.9kΩ)−1.

Thus there are two conducting channels and each band accommodates two electrons of opposite spin.

[2] In a non-ideal system, T in the Landauer formula is replaced by the sum of the transmission probabilities for each conduction channel.

When the resistance measured at the contacts is high, one can infer the presence of Coulomb blockade and Luttinger liquid behavior for different temperatures.

Low contact resistance is a prerequisite for investigating conduction phenomena in CNTs in the high transmission regime.

When the size of the CNT device scales with the electron coherence length, important in the ballistic conduction regime in CNTs becomes the interference pattern arising when measuring the differential conductance

[3] This pattern is due to the quantum interference of multiply reflected electrons in the CNT channel.

Phase coherent transport, electron interference, and localized states have been observed in the form of fluctuations in the conductance as a function of the Fermi energy.

Phase coherent electrons give rise to the observed interference effect at low temperatures.

CNT FETs exhibit four regimes of charge transport: Ohmic contacts ballistic require no scattering as the charge carriers are transported through the channel, i.e. the length of the CNT should be much smaller than the mean free path (L<< lm).

In semiconducting CNTs at room temperature and for low energies, the mean free path is determined by the electron scattering from acoustic phonons, which results in lm ≈ 0.5μm.

In order to satisfy the conditions for ballistic transport, one has to take care of the channel length and the properties of the contacts, while the geometry of the device could be any top-gated doped CNT FET.

Ballistic transport in a CNT FET takes place when the length of the conducting channel is much smaller than the mean free path of the charge carrier, lm.

Ohmic i.e. transparent contacts are most favorable for an optimized current flow in a FET.

In order to derive the current-voltage (I-V) characteristics for a ballistic CNT FET, one can start with Planck's postulate, which relates the energy of the i-th state to its frequency:

The total current for a many-state system is then the sum over the energy of each state multiplied by the occupation probability function, in this case the Fermi–Dirac statistics:

For a system with dense states, the discrete sum can be approximated by an integral:

gives the ballistic current dependence on the voltage in a CNT FET with ideal contacts.

Ideally, ballistic transport in CNT FETs requires no scattering from optical or acoustic phonons, however the analytical model yields only partial agreement with experimental data.

Thus, one needs to consider a mechanism, which would improve the agreement and recalibrate the definition of ballistic conduction in CNTs.

Partially ballistic transport is modeled to involve optical phonon scattering.

In order to understand the charge conduction in Schottky barrier CNT FETs, we need to study the band schemes under different bias conditions [4](Fig 2): Thus, the Schottky barrier CNT FET is effectively an ambipolar transistor, since the ON electron current is opposed by an OFF hole current, which flows at values smaller than the critical gate voltage value.

Above the critical gate voltage in the ON state, the electron current prevails and reaches a maximum at

Figure 1: a) Energy contour plot of the electronic band structure in CNTs.; b) Linear dependence of the electron energy on the wave vector in CNTs; c) Dispersion relation near the Fermi energy for a semiconducting CNT; d) Dispersion relation near the Fermi energy for a metallic CNT
Figure 2: Example of the band structure of a ballistic CNT FET. a) The net current through the channel is the difference between the electrons tunneling from the source and holes tunneling from the drain.; b) ON-state: the current is produced by the source electrons; c) OFF-state: hole leakage current induced by drain holes.