Question

Pure silicon contains approximately \(1.0 \times 10^{16}\) free electrons per cubic meter.

(a) Referring to the table above, calculate the mean free time? for silicon at room temperature.

(b) Your answer in part (a) is much greater than the mean free time for copper. Why, then, does pure silicon have such a high resistivity compared to copper?

 

Answer 1

Concepts and reason

The concept required to solve the given problem is mean free time. Calculate the mean free time by noting the resistivity from the table provided in the question.

Fundamentals

The mean free time of a molecule in a fluid is defined as the average time between collisions. The mean free time is given as, \(T_{\lambda}=\frac{m}{n e^{2} \rho}\)

Here, \(\rho\) is the resistivity, and \(m, n\), and \(e\) is the mass, number density, and charge.

 

(a) The mean free time is given by, \(T_{\lambda}=\frac{m}{n e^{2} \rho}\)

For silicon, substitute \(2300 \Omega \cdot \mathrm{m}\) for \(\rho, 9.1 \times 10^{-31} \mathrm{~kg}\) for \(m, 1.0 \times 10^{16}\) electrons \(/ \mathrm{m}^{3}\) for \(n\) and \(1.6 \times 10^{-19} \mathrm{C}\) for \(e\) in the above equation.

 

 

$$ \begin{array}{c} T_{\lambda}=\frac{\left(9.1 \times 10^{-31} \mathrm{~kg}\right)}{\left(1.0 \times 10^{16} \text { electrons } / \mathrm{m}^{3}\right)\left(1.6 \times 10^{-19} \mathrm{C}\right)^{2}(2300 \Omega \cdot \mathrm{m})} \\ =1.55 \times 10^{-12} \mathrm{~s} \end{array} $$

 

 

The mean free time is directly proportional to the mass and inversely proportional to the resistivity, charge, and number density.

 

(b) The formula for mean free time is, \(T_{\lambda}=\frac{m}{n e^{2} \rho}\)

This shows that the mean free time is inversely proportional to the number density as well as the resistivity of the material. When calculated the mean free time of silicon was found to be greater than that for copper. This is because of the number density and not the resistivity of the material. Despite the fact that the resistivity of silicon is greater than that for copper the mean free time of silicon is greater than that of copper because the number density of silicon is very less as compared to the number density of copper. Also, the resistivity depends on number density as, \(\rho=\frac{E}{n e v d}\)

Here, \(E\) is the electric field, \(v_{d}\) is the drift velocity, \(n\) is the number density and \(e\) is the charge on an electron. Since resistivity is inversely proportional to number density, a higher number density of copper means it has low resistivity and a low number density of silicon means it has higher resistivity. Thus, the number of free electrons in copper is much larger than in pure silicon and the density of free electrons in silicon is smaller. A smaller density of current carriers means a higher resistivity.

The resistivity is inversely proportional to the number density.

Part a The mean free time for silicon will be \(1.55 \times 10^{-12} \mathrm{~s}\).

Part b The number of free electrons in copper is much larger than in pure silicon and the density of free electrons in silicon is smaller. A smaller density of current carriers means a higher resistivity.