Ballistic conduction
In mesoscopic physics, ballistic conduction is the transport of charge carriers in a medium, having negligible electrical resistivity caused by scattering. Without scattering, electrons simply obey Newton's second law of motion at non-relativistic speeds.
In general, the resistivity of a material exists because an electron, while moving inside a medium, is scattered by impurities, defects, thermal fluctuations of ions in a crystalline solid, or, generally, by any freely-moving atom/molecule composing a gas or liquid.
For a given particle, a mean free path can be described as being the average length that the electron 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 electron is longer than the dimension of the medium through which the electron travels. The electron 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.
For example, ballistic transport can be observed in a metal nanowire: this is simply because the wire is of the size of a nanometer and the mean free path can be longer than that in a metal.
Ballistic conduction is the unimpeded flow of charge, or energy-carrying particles, over relatively long distances in a material. Normally, transport of electrons is dominated by scattering events, which relax the carrier momentum in an effort to bring the conducting material to equilibrium. Thus, ballistic transport in a material is determined by how ballistically conductive that material is. Ballistic conduction differs from superconductivity due to the absence of the Meissner effect in the material. A ballistic conductor would stop conducting if the driving force is turned off, whereas in a superconductor current would continue to flow after the driving supply is disconnected.
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 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.
Theory
Scattering mechanisms
In general, carriers will exhibit ballistic conduction when where is the length of the active part of the device. is the mean free path for the carrier which can be given by Matthiessen's rule, written here for electrons:where
- is the electron-electron scattering length,
- is the acoustic phonon scattering length,
- is the optical phonon emission scattering length,
- is the optical phonon absorption scattering length,
- is the electron-impurity scattering length,
- is the electron-defect scattering length,
- and is the electron scattering length with the boundary.
s and.
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 on the right, the current from A to B, given by the Boltzmann transport equation, iswhere gs=2, due to spin degeneracy, e is the electron charge, h is the Planck's constant, and are the Fermi levels of A and B, M is the number of propagating modes in the channel, f' is the deviation from the equilibrium electron distribution, and T is the transmission probability. Based on the definition of conductance
and the voltage separation between the Fermi levels is approximately, it follows that
where M is the number of modes in the transmission channel and spin is included. 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,
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 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. There is, however, a generalization of the Landauer–Büttiker formalism of transport applicable to time-dependent problems in the presence of dissipation.
Importance
Ballistic conduction enables use of quantum mechanical properties of electron wave functions. Ballistic transport is coherent in wave mechanics terms. Phenomena like double-slit interference, spatial resonance could be exploited in electronic systems at nanoscale in systems including nanowires and nanotubes.The widely encountered phenomenon of electrical contact resistance or ECR, arises as an electric current flowing through a rough interface is restricted to a limited number of contact spots. The size and distribution of these contact spots is governed by the topological structures of the contacting surfaces forming the electrical contact. In particular, for surfaces with high fractal dimension contact spots may be very small. In such cases, when the radius of the contact spot is smaller than the mean free path of electrons , the resistance is dominated by the Sharvin mechanism, in which electrons travel ballistically through these micro-contacts with resistance that can be described by the following
This term, where and correspond to the specific resistivity of the two contacting surfaces, is known as Sharvin resistance. Electrical contacts resulting in ballistic electron conduction are known as Sharvin Contacts. When the radius of a contact spot is larger than the mean free path of electrons, the contact resistance can be treated classically.
Optical analogies
A comparison with light provides an analogy between ballistic and non-ballistic conduction.Ballistic electrons behave like light in a waveguide or a high-quality optical assembly. Non-ballistic electrons behave like light diffused in milk or reflected off a white wall or a piece of paper.
Electrons can be scattered several ways in a conductor. Electrons have several properties: wavelength, direction, phase, and spin orientation. Different materials have different scattering probabilities which cause different incoherence rates. Some kinds of scattering can only cause a change in electron direction, others can cause energy loss.
Consider a coherent source of electrons connected to a conductor. Over a limited distance, the electron wave function will remain coherent. You still can deterministically predict its behavior. After some greater distance, scattering causes each electron to have a slightly different phase and/or direction. But there is still almost no energy loss. Like monochromatic light passing through milk, electrons undergo elastic interactions. Information about the state of the electrons at the input is then lost. Transport becomes statistical and stochastic. From the resistance point of view, stochastic movement of electrons is useless even if they carry the same energy – they move thermally. If the electrons undergo inelastic interactions too, they lose energy and the result is a second mechanism of resistance. Electrons which undergo inelastic interaction are then similar to non-monochromatic light.
For correct usage of this analogy consideration of several facts is needed:
- photons are bosons and electrons are fermions;
- there is coulombic repulsion between electrons thus this analogy is good only for single-electron conduction because electron processes are strongly nonlinear and dependent on other electrons;
- it is more likely that an electron would lose more energy than a photon would, because of the electron's non-zero rest mass;
- electron interactions with the environment, each other, and other particles are generally stronger than interactions with and between photons.
Examples