Editing Quantum wire (section)

== Quantum effects ==
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 (physics)|quantization]] is that the classical formula for calculating the [[electrical resistance]] of a wire,
: <math>R = \rho \frac{l}{A},</math>
is not valid for quantum wires (where <math>\rho</math> is the material's [[resistivity]], <math>l</math> is the length, and <math>A</math> 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 <math>2e^2/h</math>, where <math>e</math> is the [[electron charge]] and <math>h</math> is the [[Planck constant]]. The factor of two arises from [[Spin (physics)|spin]] degeneracy. A single [[ballistic transport|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.<ref>S. Datta, ''Electronic Transport in Mesoscopic Systems'', Cambridge University Press, 1995, {{ISBN|0-521-59943-1}}.</ref>

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 (solid-state physics)|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, [[semiconductor]]s can show clear conductance quantization for large wire transverse dimensions (~100&nbsp;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 (physics)|quantization]] corresponding to the lowest [[energy state]]s 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.