The Electrical Properties of Solids — Question 6

Question 6

Parallel conducting layers add conductance rather than resistivity

Two bonded films carry current along their common length L and width w. Thicknesses are $t_1=100\,\mathrm{nm}$ and $t_2=200\,\mathrm{nm}$; conductivities are $\sigma_1=2.0\times10^6$ and $\sigma_2=.50\times10^6\,\mathrm{S/m}$. Interfaces and contacts are ideal.

1. Find the sheet resistance.

2. Find the fraction of total current in layer 1.

Question 6 - Solution

Parallel conducting layers add conductance rather than resistivity

(a) Method and calculation

$$G_i=\sigma_iwt_i/L\Rightarrow G=(w/L)(\sigma_1t_1+\sigma_2t_2).$$$$\boxed{R_\square=1/(\sigma_1t_1+\sigma_2t_2)=1/(.20+.10)=3.333\,\Omega}.$$

(b) Interpretation and further calculation

$$I_1/I=G_1/(G_1+G_2).$$$$\boxed{I_1/I=.20/.30=2/3}.$$

Physical check

The layers experience the same longitudinal voltage, so their currents add. A square of any side length has the stated resistance within the uniform-film model.