Scattering mechanisms

In general, carriers will exhibit ballistic conduction when ${\displaystyle L\leq \lambda _{\rm {MFP}}}$ where ${\displaystyle L}$ is the length of the active part of the device (e.g., a channel in a MOSFET). ${\displaystyle \lambda _{\rm {MFP}}}$ is the mean free path for the carrier which can be given by Matthiessen's rule, written here for electrons:

${\displaystyle {\frac {1}{\lambda _{\mathrm {MFP} }}}={\frac {1}{\lambda _{\mathrm {el-el} }}}+{\frac {1}{\lambda _{\mathrm {ap} }}}+{\frac {1}{\lambda _{\mathrm {op,ems} }}}+{\frac {1}{\lambda _{\mathrm {op,abs} }}}+{\frac {1}{\lambda _{\mathrm {impurity} }}}+{\frac {1}{\lambda _{\mathrm {defect} }}}+{\frac {1}{\lambda _{\mathrm {boundary} }}}}$

where

• ${\displaystyle \lambda _{\mathrm {el-el} }}$ is the electron-electron scattering length,
• ${\displaystyle \lambda _{\mathrm {ap} }}$ is the acoustic phonon (emission and absorption) scattering length,
• ${\displaystyle \lambda _{\mathrm {op,ems} }}$ is the optical phonon emission scattering length,
• ${\displaystyle \lambda _{\mathrm {op,abs} }}$ is the optical phonon absorption scattering length,
• ${\displaystyle \lambda _{\mathrm {impurity} }}$ is the electron-impurity scattering length,
• ${\displaystyle \lambda _{\mathrm {defect} }}$ is the electron-defect scattering length,
• and ${\displaystyle \lambda _{\mathrm {boundary} }}$ is the electron scattering length with the boundary.

In terms of scattering mechanisms, optical phonon emission normally dominates, depending on the material and transport conditions. There are also other scattering mechanisms which apply to different carriers that are not considered here (e.g. remote interface phonon scattering, Umklapp scattering). To get these characteristic scattering rates, one would need to derive a Hamiltonian and solve Fermi's golden rule for the system in question.

A graphene nanoribbon field-effect transistor (GNR-FET). Here contacts A and B are at two different Fermi levels ${\displaystyle E_{\rm {F_{A}}}}$ and ${\displaystyle E_{\rm {F_{B}}}}$.

Landauer–Büttiker formalism

In 1957, Rolf Landauer proposed that conduction in a 1D system could be viewed as a transmission problem. For the 1D graphene nanoribbon field effect transistor (GNR-FET) on the right (where the channel is assumed to be ballistic), the current from A to B, given by the Boltzmann transport equation, is

${\displaystyle I_{\rm {AB}}={\frac {g_{\text{s}}e}{h}}\int _{E_{\rm {F_{B}}}}^{E_{\rm {F_{A}}}}M(E)f^{\prime }(E)T(E)dE}$,

where g_s = 2, due to spin degeneracy, e is the electron charge, h is the Planck constant, ${\displaystyle E_{\rm {F_{A}}}}$ and ${\displaystyle E_{\rm {F_{B}}}}$ are the Fermi levels of A and B, M(E) is the number of propagating modes in the channel, f′(E) is the deviation from the equilibrium electron distribution (perturbation), and T(E) is the transmission probability (T = 1 for ballistic). Based on the definition of conductance

${\displaystyle G={\frac {I}{V}}}$,

and the voltage separation between the Fermi levels is approximately ${\displaystyle eV=E_{\rm {F_{A}}}-E_{\rm {F_{B}}}}$, it follows that

${\displaystyle G=G_{0}MT}$, with ${\displaystyle G_{0}={\frac {2e^{2}}{h}}}$

where M is the number of modes in the transmission channel and spin is included. ${\displaystyle G_{0}}$ is known as the conductance quantum. The contacts have a multiplicity of modes due to their larger size in comparison to the channel. Conversely, the quantum confinement in the 1D GNR channel constricts the number of modes to carrier degeneracy and restrictions from the energy dispersion relationship and the Brillouin zone. For example, electrons in carbon nanotubes have two intervalley modes and two spin modes. Since the contacts and the GNR channel are connected by leads, the transmission probability is smaller at contacts A and B,

${\displaystyle T\approx {\frac {M}{M_{\rm {contact}}}}}$.

Thus the quantum conductance is approximately the same if measured at A and B or C and D.

The Landauer–Büttiker formalism holds as long as the carriers are coherent (which means the length of the active channel is less than the phase-breaking mean free path) and the transmission functions can be calculated from Schrödinger's equation or approximated by semiclassical approximations, like the WKB approximation. Therefore, even in the case of a perfect ballistic transport, there is a fundamental ballistic conductance which saturates the current of the device with a resistance of approximately 12.9 kΩ per mode (spin degeneracy included).^[2] There is, however, a generalization of the Landauer–Büttiker formalism of transport applicable to time-dependent problems in the presence of dissipation.^[3][4]

Carbon nanotubes and graphene nanoribbon

The dominant scattering mechanism at room temperature is that of electrons emitting optical phonons. If electrons don't scatter with enough phonons (for example if the scattering rate is low), the mean free path tends to be very long (${\displaystyle \lambda _{MFP}\approx 1{\mu }}$m). So a nanotube or graphene nanoribbon could be a good ballistic conductor if the electrons in transit don't scatter with too many phonons and if the device is about 100 nm long. Such a transport regime has been found to depend on the nanoribbon edge structure and the electron energy.^[7]

Silicon nanowires

It is often incorrectly thought that Si nanowires are quantum confined ballistic conductors. There are major differences between carbon nanotubes (which are hollow) and Si nanowires (which are solid). Nanowires are about 20–50 nm in diameter and are 3D solid while carbon nanotubes have diameters around the wavelength of the electrons (2–3 nm) and are essentially 1D conductors. However it is still possible to observe ballistic conduction in Si nanowires at very low temperatures (2–3 K).

Isotopically enriched diamond

Isotopically pure diamond can have a significantly higher thermal conductivity. See List of thermal conductivities.