PHT.301 Physics of Semiconductor Devices

## Electrical conductivity of silicon

To calculate the temperature dependence of the conductivity of silicon, first the Fermi energy is calculated at each temperature from the charge neutrality condition. Once the Fermi energy is known, the carrier densities $n$ and $p$ can be calculated from the formulas,

$$n=N_c\left(\frac{T}{300}\right)^{3/2}\exp\left(\frac{E_F-E_c}{k_BT}\right)\qquad\text{and}\qquad p=N_v\left(\frac{T}{300}\right)^{3/2}\exp\left(\frac{E_v-E_F}{k_BT}\right).$$

The conductivity of a semiconductor is given by the expression,

$$\sigma = ne\mu_e+pe\mu_h,$$

where $\mu_e$ is the electron mobility, $\mu_h$ is the hole mobility, and $e$ is the elementary charge. Generally the mobilities decrease with increasing temperature and doping concentration. Because of this, the conductivity decreases in the extrinsic regime. In the intrinsic regime at higher temperatures, the carrier concentrations increase more rapidly with temperature than the mobility decreases so the conductivity increases in the high temperature regime. At low temperatures, the conductivity decreases because the carriers freeze out. The plot below uses models for electron and hole mobilities for silicon that are valid for temperatures in the range 250 K - 500 K and dopant concentrations in the range $10^{13}\text{ cm}^{-3}\text{ - }10^{20}\text{ cm}^{-3}$.

 $N_d$ = 1/cm³ $N_a$ = 1/cm³
 $\sigma$ [Ω-1 cm-1] T [K]

The donor and acceptor concentrations are added together to determine the dopant concentration that is used to calculate the mobilities. The mobility model is only specified to be correct in the temperature range 250 K - 500 K which is plotted in black. The gray lines show the conductivity when the mobility model is used outside this range. The results are only qualitatively correct in these regions.

 $T$ [K]   $\sigma$ [Ω-1 cm-1]