Scattering mechanisms In general, carriers will exhibit ballistic conduction when L \le \lambda_{\rm MFP} where L is the length of the active part of the device (e.g., a channel in a
MOSFET). \lambda_{\rm MFP} is the mean free path for the carrier which can be given by
Matthiessen's rule, written here for electrons: :\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 • \lambda_\mathrm{el-el} is the electron-electron scattering length, • \lambda_\mathrm{ap} is the acoustic phonon (emission and absorption) scattering length, • \lambda_\mathrm{op,ems} is the optical phonon emission scattering length, • \lambda_\mathrm{op,abs} is the optical phonon absorption scattering length, • \lambda_\mathrm{impurity} is the electron-impurity scattering length, • \lambda_\mathrm{defect} is the electron-defect scattering length, • and \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. s E_{\rm F_A} and 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 :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
gs = 2, due to
spin degeneracy,
e is the electron charge,
h is the
Planck constant, E_{\rm F_A} and 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 :G = \frac{I}{V}, and the voltage separation between the Fermi levels is approximately eV = E_{\rm F_A}-E_{\rm F_B}, it follows that :G = G_0MT, with G_0=\frac{2e^2}{h} where
M is the number of modes in the transmission channel and spin is included. 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, :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). There is, however, a generalization of the Landauer–Büttiker formalism of transport applicable to time-dependent problems in the presence of
dissipation. ==Importance==