Ballistic conduction

The mean free path of a particle can be described as the average length that the particle can travel freely, i.e., before a collision, which could change its momentum. The mean free path can be increased by reducing the number of impurities in a crystal or by lowering its temperature. Ballistic transport is observed when the mean free path of the particle is (much) longer than the dimension of the medium through which the particle travels. The particle alters its motion only upon collision with the walls. In the case of a wire suspended in air/vacuum the surface of the wire plays the role of the box reflecting the electrons and preventing them from exiting toward the empty space/open air. This is because there is an energy to be paid to extract the electron from the medium (work function).

Ballistic conduction is typically observed in quasi-1D structures, such as carbon nanotubes or silicon nanowires, because of extreme size quantization effects in these materials. Ballistic conduction is not limited to electrons (or holes) but can also apply to phonons. It is theoretically possible for ballistic conduction to be extended to other quasi-particles, but this has not been experimentally verified. For a specific example, ballistic transport can be observed in a metal nanowire: due to the small size of the wire (nanometer-scale or 10⁻⁹ meters scale) and the mean free path which can be longer than that in a metal.

Ballistic conduction differs from superconductivity due to 1) a finite, non-zero resistance and 2) the absence of the Meissner effect in the material. The presence of resistance implies that the heat is dissipated in the leads outside of the "ballistic" conductor, where inelastic scattering effects can take place.

Theory

Scattering mechanisms

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

${\displaystyle \frac{1}{{\lambda }_{\mathrm{M}\mathrm{F}\mathrm{P}}}=\frac{1}{{\lambda }_{\mathrm{e}\mathrm{l}-\mathrm{e}\mathrm{l}}}+\frac{1}{{\lambda }_{\mathrm{a}\mathrm{p}}}+\frac{1}{{\lambda }_{\mathrm{o}\mathrm{p},\mathrm{e}\mathrm{m}\mathrm{s}}}+\frac{1}{{\lambda }_{\mathrm{o}\mathrm{p},\mathrm{a}\mathrm{b}\mathrm{s}}}+\frac{1}{{\lambda }_{\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{u}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{y}}}+\frac{1}{{\lambda }_{\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{e}\mathrm{c}\mathrm{t}}}+\frac{1}{{\lambda }_{\mathrm{b}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}\mathrm{a}\mathrm{r}\mathrm{y}}}}$

where

• ${\displaystyle {\lambda }_{\mathrm{e}\mathrm{l}-\mathrm{e}\mathrm{l}}}$ is the electron-electron scattering length,
• ${\displaystyle {\lambda }_{\mathrm{a}\mathrm{p}}}$ is the acoustic phonon (emission and absorption) scattering length,
• ${\displaystyle {\lambda }_{\mathrm{o}\mathrm{p},\mathrm{e}\mathrm{m}\mathrm{s}}}$ is the optical phonon emission scattering length,
• ${\displaystyle {\lambda }_{\mathrm{o}\mathrm{p},\mathrm{a}\mathrm{b}\mathrm{s}}}$ is the optical phonon absorption scattering length,
• ${\displaystyle {\lambda }_{\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{u}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{y}}}$ is the electron-impurity scattering length,
• ${\displaystyle {\lambda }_{\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{e}\mathrm{c}\mathrm{t}}}$ is the electron-defect scattering length,
• and ${\displaystyle {\lambda }_{\mathrm{b}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}\mathrm{a}\mathrm{r}\mathrm{y}}}$ 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.

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

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

where g_s = 2, due to spin degeneracy, e is the electron charge, h is the Planck constant, ${\displaystyle E_{{\mathrm{F}}_{\mathrm{A}}}}$ and ${\displaystyle E_{{\mathrm{F}}_{\mathrm{B}}}}$ 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 ${\displaystyle eV=E_{{\mathrm{F}}_{\mathrm{A}}}-E_{{\mathrm{F}}_{\mathrm{B}}}}$, 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,

${\displaystyle T\approx \frac{M}{M_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{c}\mathrm{t}}}}$.

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). There is, however, a generalization of the Landauer–Büttiker formalism of transport applicable to time-dependent problems in the presence of dissipation.