If the diameter of a wire is sufficiently small,
electrons will experience
quantum confinement in the transverse direction. As a result, their transverse energy will be limited to a series of discrete values. One consequence of this
quantization is that the classical formula for calculating the
electrical resistance of a wire, : R = \rho \frac{l}{A}, is not valid for quantum wires (where \rho is the material's
resistivity, l is the length, and A is the cross-sectional area of the wire). Instead, an exact calculation of the transverse energies of the confined electrons has to be performed to calculate a wire's resistance. Following from the quantization of electron energy, the
electrical conductance (the inverse of the resistance) is found to be quantized in multiples of 2e^2/h, where e is the
electron charge and h is the
Planck constant. The factor of two arises from
spin degeneracy. A single
ballistic quantum channel (i.e. with no internal scattering) has a conductance equal to this
quantum of conductance. The conductance is lower than this value in the presence of internal scattering. The importance of the quantization is inversely proportional to the diameter of the
nanowire for a given material. From material to material, it is dependent on the electronic properties, especially on the
effective mass of the electrons. Physically, this means that it will depend on how conduction electrons interact with the atoms within a given material. In practice,
semiconductors can show clear conductance quantization for large wire transverse dimensions (~100 nm) because the electronic modes due to confinement are spatially extended. As a result, their Fermi wavelengths are large and thus they have low energy separations. This means that they can only be resolved at
cryogenic temperatures (within a few degrees of
absolute zero) where the thermal energy is lower than the inter-mode energy separation. For metals,
quantization corresponding to the lowest
energy states is only observed for atomic wires. Their corresponding wavelength being thus extremely small they have a very large energy separation which makes resistance quantization observable even at room temperature. == Carbon nanotubes ==