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Mobilities of electrons and holes in a sample of intrinsic germanium at room temperature are $0.36{{m}^{2}}{{V}^{-1}}{{s}^{-1}}$ and $0.17{{m}^{2}}{{V}^{-1}}{{s}^{-1}}$. The electron and hole densities are each equal to $2.5\times {{10}^{19}}{{m}^{3}}$. The electrical conductivity of germanium is
$\begin{align}
  & \left( A \right)4.24S{{m}^{-1}} \\
 & \left( B \right)2.12S{{m}^{-1}} \\
 & \left( C \right)1.09S{{m}^{-1}} \\
 & \left( D \right)0.47S{{m}^{-1}} \\
\end{align}$

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Last updated date: 18th Jun 2024
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Answer
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Hint: An intrinsic can be described as a pure type of semiconductor.Given the mobilities of electrons and holes. Also given that the electron and hole densities are each equal. Thus the electrical conductivity is the product of the algebraic sum of the product of mobility and density of electron and holes and the magnitude of charge. Thus by substituting the values we will get the electrical conductivity of germanium.

Complete step-by-step solution
Given that,
Mobilities of electrons, ${{\mu }_{e}}=0.36{{m}^{2}}{{V}^{-1}}{{s}^{-1}}$
Mobilities of holes, ${{\mu }_{h}}=0.17{{m}^{2}}{{V}^{-1}}{{s}^{-1}}$
Also given that the electron and hole densities are each equal to $2.5\times {{10}^{19}}{{m}^{3}}$.Hence,
As we know conductivity,
$\sigma =e\left( {{\mu }_{e}}{{n}_{e}}+{{\mu }_{n}}{{n}_{n}} \right)$
$\Rightarrow \sigma =1.6\times {{10}^{-19}}\left[ 0.36\times 2.5\times {{10}^{19}}+0.17\times 2.5\times {{10}^{19}} \right]$
$\therefore \sigma =2.12S{{m}^{-1}}$
So the electrical conductivities of germanium is $2.12S{{m}^{-1}}$.
Additional information: Semiconductor materials at 0K have basically the same structure as insulators-a filled valence band separated from an empty conduction band by a bandgap containing no allowed energy states. This process is called the doping of semiconductors. Thus there are two types of semiconductors, n-type (mostly electrons) and p-type (mostly holes). Generally, pentavalent impurities are the dopants in an extrinsic semiconductor.

Note: An intrinsic can be described as a pure type of semiconductor. In an intrinsic semiconductor, the holes and electrons are generally equal. Trivalent impurities are the dopant in an intrinsic semiconductor. The electrical conductivity generally depends on the temperature of the material. In such material, there is no change in carriers at 0K, since the valence band is filled with electrons and the conduction band is empty. At higher temperatures, electron-hole pairs are excited thermally across the bandgap to the conduction band.