Question

Pure silicon at room temperature contains approximately$\mathbf{1}\mathbf{.}\mathbf{0}\mathbf{\times }{\mathbf{10}}^{\mathbf{16}}$ free electrons per cubic meter. (a) Referring to Table 25.1, calculate the mean free time for$\mathit{\tau }$ silicon at room temperature. (b) Your answer in part (a) is much greater than the mean free time for copper given in Example 25.11. Why, then, does pure silicon have such a high resistivity compared to copper?

Short answer

1. The mean time for$\tau$ silicon at room temperature is $1.55\times {10}^{-12}\mathrm{s}$.
2. Silicon has greater than the mean free for cooper.

Step-by-step solution

Define the ohm’s law, resistance (R) and power (P).

The average time between the collisions for the electron is known as mean free time $\mathit{\tau }$.

$\tau =\frac{m}{n{e}^2\rho }$

Where, is charge of the electron which is equal to $1.6\times {10}^{-19}C$, is the mass of electron which is equal to $9.11\times {10}^{-31}\mathrm{kg}$,$n$ is concentration and$\rho$ is resistivity.

Determine the mean free time.

The resistivity $\rho$of silicon is equal to $2300\mathrm{\Omega }·\mathrm{m}$and the

The mean free time for silicon is

$$\begin{gathered} \tau =\frac{m}{n{e}^2\rho } \\ =\frac{9.11\times {10}^{-31}}{\left(1.0\times {10}^{16}\right)\left(1.6\times {10}^{-19}\right)\left(2300\right)} \\ =1.55\times {10}^{-12}\mathrm{s} \end{gathered}$$

From the expression$\tau =\frac{m}{n{e}^2\rho }$ the mean free time is inversely proportional to concentration $n$.

$\tau \propto \frac{1}{n}$

As the silicon has less value$n$ than the copper. So, silicon has larger value.

Hence, mean free time$\tau$ for silicon at room temperature is$1.55\times {10}^{-12}s$ and silicon has greater than the mean free for cooper.