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 quantized into a series of discrete values. One consequence of this quantization is that the classical formula for calculating the electrical resistance of a wire:
is not valid for quantum wires (where is the resistivity, is the length, and 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, where is the electron charge and 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.
Read more about this topic: Quantum Wire
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