Size‐Dependent Electric Conductivity in Semiconducting thin Wires

A quantum theory for electric conductivity in a thin rectangular‐shaped wire is developed taking into account the wave character of an electron. It is shown that the quasi‐one‐dimensional (QOD) gas behaves somewhat like the strictly‐one‐dimensional (SOD) gas when the wire is ultra‐thin, so that the spacing between the quantized levels is quite large, and all the electrons are assumed to be in the lowest quantized level. In the ultra‐thin limit (UTL), the ratio of longitudinal resistivity to bulk resistivity is shown to be proportional to λD2/A, where λD = h/(2m*kBT)1/2 is the de‐Broglie wavelength of an electron of effective mass m* at temperature T, and A is the area of the cross‐section of the wire. The transverse resistivity ratio depends strongly on the scattering parameters and is inversely proportional to the area for wires of square cross‐section.