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In an intrinsic semiconductor the energy gap $\mathrm{E}_{\mathrm{g}}$ is $1.2 \mathrm{eV}$. Its hole mobility is much smaller than electron mobility and independent of temperature. What is the ratio between conductivity at $600 \mathrm{~K}$ and that at $300 \mathrm{~K}$ ? Assume that the temperature dependence of intrinsic carrier concentration $n_{i}$ is given by $n_{i}=n_{0} \exp \left[-\frac{E_{g}}{2 k_{B} T}\right]$where, $n_{0}$ is constant.
In an intrinsic semiconductor the energy gap $\mathrm{E}_{\mathrm{g}}$ is $1.2 \mathrm{eV}$. Its hole mobility is much smaller than electron mobility and independent of temperature. What is the ratio between conductivity at $600 \mathrm{~K}$ and that at $300 \mathrm{~K}$ ? Assume that the temperature dependence of intrinsic carrier concentration $n_{i}$ is given by
$n_{i}=n_{0} \exp \left[-\frac{E_{g}}{2 k_{B} T}\right]$
where, $n_{0}$ is constant.
CBSE | Class 12 | NCERT | Physics | Semiconductor Electronics: Materials, Devices and Simple Circuits
In an intrinsic semiconductor the energy gap $\mathrm{E}_{\mathrm{g}}$ is $1.2 \mathrm{eV}$. Its hole mobility is much smaller than electron mobility and independent of temperature. What is the ratio between conductivity at $600 \mathrm{~K}$ and that at $300 \mathrm{~K}$ ? Assume that the temperature dependence of intrinsic carrier concentration $n_{i}$ is given by
$n_{i}=n_{0} \exp \left[-\frac{E_{g}}{2 k_{B} T}\right]$
where, $n_{0}$ is constant.

Energy gap in an intrinsic semiconductor is given as $E_{g}=1.2 \mathrm{eV}$

The temperature dependence of the intrinsic carrier-concentration is given by the relation,

$n_{i}=n_{0} \exp \left[-\frac{E_{g}}{2 k_{B} T}\right]$

Here, $k_{B}$ is the Boltzmann constant $=8.62 \times 10^{-5} \mathrm{eV} / \mathrm{K}$

$\mathrm{T}$ is the temperature

$n_{0}$ is a constant

Initial temperature, $\mathrm{T}_{1}=300 \mathrm{~K}$

The intrinsic carrier concentration at the temperature $300 \mathrm{~K}$ can be expressed as,

$n_{i 1}=n_{0} \exp \left[-\frac{E_{g}}{2 k_{B} \times 300}\right]–(1)$

Final temperature, $\mathrm{T}_{2}=600 \mathrm{~K}$

The intrinsic carrier concentration at the temperature $600 \mathrm{~K}$ can be written as

$n_{i 2}=n_{0} \exp \left[-\frac{E_{g}}{2 k_{B} \times 600}\right]—(2)$

The ratio of conductivity at 600K to conductivity at 300K is the same as the ratio of carrier concentration at both temperatures.

$\frac{n_{i 2}}{n_{i 1}}=\frac{n_{0} \exp \left[-\frac{E_{g}}{2 k_{B} \times 00}\right]}{n_{0} \exp \left[-\frac{E_{g}}{2 K_{B} \times 300}\right]}$

$=\exp \frac{E_{q}}{2 k_{B}}\left[\frac{1}{300}-\frac{1}{600}\right]$

$=\exp \frac{1.2}{2 \times 8.62 \times 10^{-5}}\left[\frac{1}{300}-\frac{1}{600}\right]$

$=\exp \frac{1.2}{17.24 \times 10^{-5}}\left[\frac{2-1}{600}\right]$

$=\exp (11.6)=1.09 \times 10^{5}$

There the ratio between the conductivities is $1.09 \times 10^{5}$.

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