Conductivity Of Doped Semiconductor

In a heavily doped n-type semiconductor, the donor density vastly exceeds the intrinsic concentration, so essentially every donor contributes one electron and the electron density equals the donor density, (n_e \approx N_d). This assumes the donors are fully ionised at room temperature, which is realistic for shallow donors in silicon since their binding energy is far smaller than the thermal energy available. The hole contribution is utterly negligible because the law of mass action drives the minority hole density extremely low, many orders of magnitude beneath the electron density, so it can be dropped from the conductivity sum without measurable error. Conductivity is then governed by electrons alone: (\sigma = n_e,e,\mu_e). Substituting (n_e = 5 \times 10^{22},\text{m}^{-3}), (e = 1.6 \times 10^{-19},\text{C}) and (\mu_e = 0.135,\text{m}^2\text{V}^{-1}\text{s}^{-1}) gives (\sigma = (5 \times 10^{22})(1.6 \times 10^{-19})(0.135) = 1080,\text{S m}^{-1}). The value 540 halves the donor density without justification. The value 108 misplaces a power of ten. The value 2160 wrongly doubles the carrier density by also counting holes. As a final check, this conductivity is enormously larger than the intrinsic silicon value of about (4 \times 10^{-4},\text{S m}^{-1}), which is exactly why controlled doping is used to tailor semiconductor conductivity by many orders of magnitude.