Question

Read the following statements carefully :
$Y:$ The resistivity of a semiconductor decreases with increases of temperature.
$Z:$ In a conducting solid, the rate of collision between free electrons and ions increases with increase of temperature.
Select the correct statement from the following

• A. $Y$ is true but $Z$ is false
• B. $Y$ is false but $Z$ is true
• C. Both $Y$ and $Z$ are true
• D. Both $Y$ and $Z$ are false

Answer 1

The correct option is C Both $Y$ and $Z$ are true

The variation of resistivity with temperature is given by

$\rho ={\rho }_o(1+\alpha \Delta T)$

Where,
$\alpha \to -ve$ for semiconductors

So, as $T\uparrow \rho \downarrow$

The resistivity of semiconductor decreases with increase in temperature i.e. $\alpha$ for semiconductors is $-ve$ and high.

Thus, the statement $Y$ is true.

Now, as temperature increases relaxation time $\tau$ decreases.

$\Rightarrow T\uparrow \tau \downarrow$

On increasing temperature, relaxation time will decrease and hence rate of collision between free electrons will increase in the conducting solid.

Thus, the statement $Z$ is also true.

Why this question ? $\left(i\right)$ $\alpha$ for semiconductors is $-ve$ and high. $\left(ii\right)$ Though at ordinary temperature the value of $n$ (no. of free electrons per unit volume) for these materials is very small as compared to metals, but increases very rapidly with rise in temperature (this happens due to breaking of covalent bonds). Though $T$ decreases but factor of $n$ dominates. Therefore, the resistance $R=\frac{ml}{n{e}^2\tau A}$ goes on decreasing with increase in temperature.