Question

In an intrinsic semiconductor the energy gap ${E_g}$ is $1.2eV$. Its hole mobility is much smaller than electron mobility and independent of temperature. What is the ratio between conductivity at $600K$ and that at $300K$? Assume that the temperature dependence of intrinsic carrier concentration ${n_i}$ is given by,
${n_i} = {n_o}\exp \left( { - \dfrac{{{E_g}}}{{2{k_B}T}}} \right)$
Where ${n_o}$ is a constant.

Answer 1

a semiconductor with no external or internal impurities is known as an intrinsic semiconductor. An intrinsic semiconductor, since devoid of impurities, has thermally generated charge carriers.

Complete step by step solution:
Semiconductors are materials which have their value of conductivity between that of conductors and insulators. Conductors have very high values of conductivity and insulators have very low values of conductivity. Now, semiconductors are materials which have intermediate values of conductivity, not too high nor too low.
In conductors, the reason for high conductivity is zero energy gap between valence band and conduction band. And in insulators, the reason for low conductivity is the very high energy gap between their valence band and conduction band. Whereas, in semiconductors, the energy gap between their valence band and conduction band exists, but is achievable after giving a certain amount of energy.
This energy can be given via different modes like heat, light, mechanical energy. And that is the reason for thermal sensitivity of semiconductors. Now semiconductors are of two types – Intrinsic Semiconductors and Extrinsic Semiconductors.
When the semiconductors have impurities embedded into them, those semiconductors are known as Extrinsic Semiconductors. Extrinsic Semiconductors depend on the presence of holes and electrons in addition to temperature. But when the semiconductors have zero impurities embedded into them, those conductors are known as intrinsic semiconductors. These semiconductors depend completely on the temperature for their conductivity. This dependence is given by,
${n_i} = {n_o}{e^{\dfrac{{ - {E_g}}}{{{k_B}T}}}}$
Where, ${n_i} =$Density of intrinsic charge carriers,
${n_o} =$Constant,
${E_g} =$Energy gap,
${k_B} =$Boltzmann’s Constant$= 8.62 \times {10^{ - 5}}eV/K$
$T =$Temperature.
Now, ${n_i}$ signifies the conductivity of the intrinsic semiconductor. Let ${\left( {{n_i}} \right)_{300}}$ be the conductivity of semiconductor at $300K$ and ${\left( {{n_i}} \right)_{600}}$ be the conductivity of semiconductor at $600K$.
So,
${\left( {{n_i}} \right)_{300}} = {n_o}{e^{ - \dfrac{{{E_g}}}{{2{k_B} \times 300}}}}$, and
${\left( {{n_i}} \right)_{600}} = {n_o}{e^{ - \dfrac{{{E_g}}}{{2{k_B} \times 600}}}}$
Dividing the second equation by first one,
$\Rightarrow \dfrac{{{{\left( {{n_i}} \right)}_{600}}}}{{{{\left( {{n_i}} \right)}_{300}}}} = \dfrac{{{n_o}{e^{ - \dfrac{{{E_g}}}{{2{k_B} \times 600}}}}}}{{{n_o}{e^{ - \dfrac{{{E_g}}}{{2{k_B} \times 300}}}}}}$
$\Rightarrow \dfrac{{{{\left( {{n_i}} \right)}_{600}}}}{{{{\left( {{n_i}} \right)}_{300}}}} = {e^{ - \dfrac{{{E_g}}}{{2{k_B}}}\left[ {\dfrac{1}{{600}} - \dfrac{1}{{300}}} \right]}}$
$\Rightarrow \dfrac{{{{\left( {{n_i}} \right)}_{600}}}}{{{{\left( {{n_i}} \right)}_{300}}}} = {e^{\dfrac{{{E_g}}}{{2{k_B}}}\left[ {\dfrac{1}{{300}} - \dfrac{1}{{600}}} \right]}}$
$\Rightarrow \dfrac{{{{\left( {{n_i}} \right)}_{600}}}}{{{{\left( {{n_i}} \right)}_{300}}}} = {e^{\dfrac{{1.2}}{{2 \times 8.62 \times {{10}^{ - 5}}}}\left[ {\dfrac{{2 - 1}}{{600}}} \right]}}$
$\Rightarrow \dfrac{{{{\left( {{n_i}} \right)}_{600}}}}{{{{\left( {{n_i}} \right)}_{300}}}} = {e^{\dfrac{{1.2}}{{2 \times 8.62 \times {{10}^{ - 5}}}} \times \dfrac{1}{{600}}}}$
$\Rightarrow \dfrac{{{{\left( {{n_i}} \right)}_{600}}}}{{{{\left( {{n_i}} \right)}_{300}}}} = {e^{\left( {11.6} \right)}}$
$\therefore \dfrac{{{{\left( {{n_i}} \right)}_{600}}}}{{{{\left( {{n_i}} \right)}_{300}}}} = 1.09 \times {10^5}$

Note: In extrinsic semiconductors, the type of charge determines the type of semiconductor. If the charge carriers are positively charged holes, the semiconductor is known as a P-type semiconductor. And if the charge carriers are negatively charged electrons, the semiconductor is known as an N-type semiconductor. And when both these semiconductors are joined end to end, the device formed is known as a Diode.