Skin Effect Impact on Semiconductor Wires

Currently, our analysis has assumed a linear and constant resistance for a semiconductor wire, which is generally accurate for most semiconductor circuits. However, at elevated frequencies, the skin effect introduces a frequency-dependent resistance.

High-frequency currents exhibit a tendency to concentrate on the conductor's surface, with current density decreasing exponentially with depth. The skin depth, denoted as d, is defined as the depth at which the current diminishes to e⁻¹ of its nominal value, and is expressed as:

Given the signal frequency (f) and the permeability of the surrounding dielectric (U), which typically approximates the permeability of free space (m = 4p ´ 10-7 H/m), the skin depth for Aluminum at 1 GHz is calculated to be 2.6 mm.

The impact of the skin effect can be approximated by assuming a uniform current distribution within an outer shell of the conductor, with a thickness denoted as d, as illustrated in the provided figure for a rectangular wire. Consequently, the effective cross-sectional area of the wire is approximated to 

we obtain the following expression for the resistance (per unit length) at high frequencies (f > fs):

The increased resistance at higher frequencies may cause an extra attenuation and hence distortion of the signal being transmitted over the wire. To determine the on-set of the skin-effect, we can find the frequency fs where the skin depth is equal to half the largest dimension (W or H) of the conductor. Below fs the whole wire is conducting current, and the resistance is equal to (constant) low-frequency resistance of the wire. From Eq. (4.6), we find the value of fs: