Question

The intrinsic conductivity of germanium at \[{{27}^{\text{o}}}\text{C}\] is \[2.13\text{ mho/m}\] and mobilities of electrons and holes are \[\text{0}\text{.38}\] and \[\text{0}\text{.18 }{{\text{m}}^{2}}\text{/Vs}\] respectively. To find the density of charge carriers.

Answer 1

Hint: In intrinsic semiconductors, electron concentration is equal to the hole concentration.

Formula used:
For intrinsic semiconductor, the electrical conductivity \[\sigma \] is given by
\[\sigma =(ne{{\mu }_{e}}+pe{{\mu }_{h}})\]
 Where \[{{\mu }_{e}}\] and \[{{\mu }_{h}}\] denote the mobilities of electrons and holes respectively; n and p is the number of electrons and holes in the semiconductor, and e is the charge of an electron or a hole.
In intrinsic semiconductors, electron concentration is equal to the hole concentration,
So, \[n=p={{n}_{i}}\],
Thus,
\[\sigma ={{n}_{i}}e({{\mu }_{e}}+{{\mu }_{h}})\]
Where \[{{n}_{i}}\] denotes the density of charge carriers.
The density of charge carriers is
\[{{n}_{i}}=\dfrac{\sigma }{e({{\mu }_{e}}+{{\mu }_{h}})}\]

Complete step by step solution:
The intrinsic conductivity of germanium at \[{{27}^{\text{o}}}\text{C}\],\[\sigma =2.13\text{ mho/m}\]
Mobility of electrons, \[{{\mu }_{e}}=\text{0}\text{.38 }{{\text{m}}^{2}}\text{/Vs}\]
Mobility of holes, \[{{\mu }_{h}}=\text{0}\text{.18 }{{\text{m}}^{2}}\text{/Vs}\]
Charge on an electron/hole, \[e=\text{1}\text{.6}\times \text{1}{{\text{0}}^{-19}}\text{ C}\]
Substituting the values in the formula:
\[
   {{n}_{i}}=\dfrac{\sigma }{e({{\mu }_{e}}+{{\mu }_{h}})} \\
  {{n}_{i}}=\dfrac{2.13\text{ mho/m}}{(1.6\times {{10}^{-19}}\text{ C)}(0.38\text{ }{{\text{m}}^{2}}/\text{Vs}+0.18\text{ }{{\text{m}}^{2}}/\text{Vs})} \\
  {{n}_{i}}=\dfrac{2.13\text{ mho/m}}{(1.6\times {{10}^{-19}}\text{ C)}(0.56{{\text{m}}^{2}}/\text{Vs})} \\
 {{n}_{i}}=\dfrac{2.13\text{ mho/m}}{(8.96\times {{10}^{-20}}\text{ }{{\text{m}}^{2}}/\text{ohm})} \\
  {{n}_{i}}=2.37\times {{10}^{19}}/{{\text{m}}^{3}} \\
\]

The density of charge carriers is \[2.37\times {{10}^{19}}/{{\text{m}}^{3}}\]

Additional information:
In intrinsic semiconductors such as germanium, a valence electron breaks its covalent bond if it gets sufficient thermal energy from photons of suitable frequency and becomes free. The valency left behind serves as a hole. The hole has charge equal and opposite to that of an electron.
As the charge carriers are created due to the breaking of the covalent bond, the concentration of electrons (n) becomes equal to the concentration of holes (p), that is, \[n=p={{n}_{i}}\], where \[{{n}_{i}}\] is called the intrinsic concentration or the density of carrier charges. The electron and holes are called intrinsic charge carriers. The value of \[{{n}_{i}}\] depends on the temperature of the semiconductor.
The carrier concentration in extrinsic semiconductors depends on the donor concentration, and the electron concentration is not equal to the hole concentration.

Note: The electron and holes are called intrinsic charge carriers. The value of \[{{n}_{i}}\] depends on the temperature of the semiconductor, that is, the density of charge carriers is \[2.37\times {{10}^{19}}/{{\text{m}}^{3}}\] only at \[{{27}^{\text{o}}}\text{C}\].