Q.A pure Si crystal having $5\times10^{28}$ atoms m$^{-3}$ is doped with 1 ppm concentration of antimony. If the concentration of holes in the doped crystal is found to be $4.5\times10^{9}$ m$^{-3}$, the concentration (in m$^{-3}$) of intrinsic charge carriers in the Si crystal is about ______.
(A) $1.2\times10^{15}$
(B) $1.5\times10^{16}$
(C) $3.0\times10^{15}$
(D) $2.0\times10^{16}$

False — heating a semiconductor INCREASES its conductance.

In a semiconductor, raising the temperature breaks more covalent bonds, generating additional free electron–hole pairs. The number of charge carriers increases rapidly with temperature, so the conductivity (and conductance) increases and the resistance decreases. This is opposite to metals, where resistance …

We use the given doping concentration to find the donor concentration, which approximates the electron concentration in the N-type semiconductor. Then, applying the mass action law ($n \cdot p = n_i^2$) with the given hole concentration, we calculate the intrinsic carrier concentration to be $\boxed{1.5 \times 10^{16} \text{ m}^{-3}}$.

The law of mass action, $n_en_h=n_i^2$, holds in a doped semiconductor just as in the intrinsic case, so $n_e$ can be found directly from the given $n_h$.

For pure (intrinsic) silicon at 300 K, $n_e = n_h = n_i = 1.5\times10^{16}\,\text{m}^{-3}$.

Doping with indium (a trivalent, acceptor-type impurity) makes the material p-type, raising the hole concentration to $n_h=4.5\times10^{22}\,\text{m}^{-3}$. In thermal equilibrium, the product of electron and hole concentrations remains fixed at the square of the intrinsic concentration (the law of mass action):

$n_e\,n_h = n_i^2$ …

The mass-action law states that $n_e \cdot n_h = n_i^2$ holds even after doping. Using the intrinsic carrier concentration $n_i = 1.5 \times 10^{16}\ \text{m}^{-3}$ and the new hole concentration $n_h = 4.5 \times 10^{22}\ \text{m}^{-3}$, we find $n_e = 5 \times 10^{9}\ \text{m}^{-3}$.

Plot $\rho$ (y-axis) against $T$ (x-axis): a smooth curve that is high at low $T$ and drops steeply (exponentially) as $T$ increases — semiconductors have a negative temperature coefficient of resistivity.

Concept. In a semiconductor the number density $n$ of free charge carriers is very small at low temperature. As the absolute temperature $T$ rises, thermal energy excites more electrons across the energy gap into the conduction band (and creates holes), so $n$ increases rapidly.

Why the resistivity falls. Since $\rho = \dfrac{m}{ne^2\tau}$, the strong increase in carrier density $n$ dominates over the small fall in relaxation time $\tau$, so $\rho$ decreases as $T$ increases. This is opposite to a metal, where $\rho$ rises with $T$.

The graph (description).

• x-axis: absolute temperature $T$ (in kelvin).
• y-axis: resistivity $\rho$. …

By the law of mass action, $n_en_h=n_i^2$ always holds in a semiconductor whether pure or doped; $n_e = n_i^2/n_h$ gives an answer matching option (b) once the given hole concentration is taken consistently with the intrinsic value.

For any semiconductor in thermal equilibrium (whether intrinsic or doped), the law of mass action states that the product of electron and hole concentrations is a constant, equal to the square of the intrinsic carrier concentration $n_i$:

$n_e\,n_h = n_i^2$

This holds true regardless of doping — doping only changes how that fixed product $n_i^2$ is shared between $n_e$ and $n_h$ (more holes means correspondingly fewer electrons, and vice versa), because each electron-hole pair created thermally is a fixed background process, while doping with indium (trivalent, an acceptor impurity) creates additional holes without a matching electron, sharply increasing $n_h$ and, by mass action, sharply suppressing $n_e$.

With $n_i = 1.5\times10^{16}\,m^{-3}$ (so $n_i^2 = 2.25\times10^{32}\,m^{-6}$):

$n_e = \dfrac{n_i^2}{n_h} = \dfrac{2.25\times10^{32}}{n_h}$

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A semiconductor's resistivity decreases as temperature increases.

In a semiconductor the number of free charge carriers (electrons in the conduction band and holes) is small at low temperature. Heating supplies energy that breaks covalent bonds and promotes many electrons across the small energy gap, sharply increasing carrier concentration n. Since conductivity σ ∝ n, σ rises and resistivity ρ = 1/σ …

Fewer carriers at lower temperature → higher resistance; semiconductors have a negative temperature coefficient.

In a semiconductor, charge carriers (electrons in the conduction band and holes) are produced by thermal excitation across the energy gap. Their number rises steeply with temperature.

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Heating an intrinsic semiconductor increases its conductivity.

In a pure (intrinsic) semiconductor at low temperature, very few electrons have enough energy to cross the energy gap into the conduction band, so there are few charge carriers and conductivity is low.

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