Question

A semiconductor having electron and hole mobilities \[{\mu _{\text{n}}}\] and \[{\mu _{\text{p}}}\] respectively. If its intrinsic carrier density is \[{n_{\text{i}}}\] then what will be the value of hole concentration \[P\] for which the conductivity will be minimum at a given temperature?
(A) \[{n_{\text{i}}}\sqrt {\dfrac{{{\mu _{\text{n}}}}}{{{\mu _{\text{p}}}}}} \]
(B) \[{n_{\text{p}}}\sqrt {\dfrac{{{\mu _{\text{n}}}}}{{{\mu _{\text{p}}}}}} \]
(C) \[{n_{\text{i}}}\sqrt {\dfrac{{{\mu _{\text{p}}}}}{{{\mu _{\text{n}}}}}} \]
(D) \[{n_{\text{p}}}\sqrt {\dfrac{{{\mu _{\text{p}}}}}{{{\mu _{\text{n}}}}}} \]

Answer 1

Hint: First of all, we will find the total conductor of the semiconductor. After that we will differentiate and for minimum value the derivative is zero. Then we will manipulate accordingly to obtain the result.

Complete step by step answer:
In the given question, we are supplied with the following data:
A semiconductor has electron mobilities represented by \[{\mu _{\text{n}}}\] .
It also has hole mobilities \[{\mu _{\text{p}}}\] .
The intrinsic carrier density is given as \[{n_{\text{i}}}\] .
We are asked to find the hole concentration \[P\] for which the conductivity will be minimum at a given temperature.
To begin with, we will find look into the total conductivity of the semiconductor which is given by:
\[\sigma = {n_{\text{e}}}e{\mu _{\text{n}}} + {n_{\text{p}}}e{\mu _{\text{p}}}\] …… (1)
Where,
\[\sigma \] indicates total conductivity of the semiconductor.
\[{n_{\text{e}}}\] indicates the concentration of electrons.
\[e\] indicates the charge.
\[{\mu _{\text{n}}}\] indicates the mobilities of electrons.
\[{n_{\text{p}}}\] indicates the concentration of holes.
\[{\mu _{\text{p}}}\] indicates the mobilities of holes.
From equation (1), we can write:
\[\sigma = e\left( {{n_{\text{e}}}{\mu _{\text{n}}} + {n_{\text{p}}}{\mu _{\text{p}}}} \right)\] …… (2)
Again, we have a condition for intrinsic semiconductor, which is given by:
$
{n_{\text{e}}}{n_{\text{p}}} = n_{\text{i}}^2 \\
\Rightarrow{n_{\text{e}}} = \dfrac{{n_{\text{i}}^2}}{{{n_{\text{p}}}}} \\
$
Now, we use the above value in the equation (2) and we get:
\[\sigma = e\left( {\dfrac{{n_{\text{i}}^2}}{{{n_{\text{p}}}}} \times {\mu _{\text{n}}} + {n_{\text{p}}}{\mu _{\text{p}}}} \right)\] …… (3)
Again, we have, if the conductivity to be minimum the differentiation of conductivity with respect to the number of holes should be zero.
Mathematically, we can write:
\[\dfrac{{d\sigma }}{{d{n_{\text{p}}}}} = 0\]
Now, we do the operation:
$
\dfrac{{d\sigma }}{{d{n_{\text{p}}}}} = \dfrac{d}{{d{n_{\text{p}}}}}\left[ {e\left( {\dfrac{{n_{\text{i}}^2}}{{{n_{\text{p}}}}} \times {\mu _{\text{n}}} + {n_{\text{p}}}{\mu _{\text{p}}}} \right)} \right] \\
\Rightarrow\dfrac{{d\sigma }}{{d{n_{\text{p}}}}} = \dfrac{d}{{d{n_{\text{p}}}}}\left[ {e\left( {\dfrac{{n_{\text{i}}^2}}{{{n_{\text{p}}}}} \times {\mu _{\text{n}}}} \right)} \right] + \dfrac{d}{{d{n_{\text{p}}}}}\left( {e{n_{\text{p}}}{\mu _{\text{p}}}} \right) \\
\Rightarrow 0 = e\left[ { - \dfrac{{n_{\text{i}}^2}}{{n_{\text{p}}^2}} \times {\mu _{\text{n}}} + {\mu _{\text{p}}}} \right] \\
\Rightarrow{\mu _{\text{p}}} = \dfrac{{n_{\text{i}}^2}}{{n_{\text{p}}^2}} \times {\mu _{\text{n}}} \\
$
Again, we further manipulate the above expression:
$
\dfrac{{{\mu _{\text{p}}}}}{{{\mu _{\text{n}}}}} = \dfrac{{n_{\text{i}}^2}}{{n_{\text{p}}^2}} \\
\Rightarrow\sqrt {\dfrac{{{\mu _{\text{n}}}}}{{{\mu _{\text{p}}}}}} =
\dfrac{{{n_{\text{p}}}}}{{{n_{\text{i}}}}} \\
\therefore{n_{\text{p}}} = {n_{\text{i}}}\sqrt {\dfrac{{{\mu _{\text{n}}}}}{{{\mu _{\text{p}}}}}} \\
$

Hence, the value of hole concentration \[P\] for which the conductivity will be minimum at a given temperature is \[{n_{\text{i}}}\sqrt {\dfrac{{{\mu _{\text{n}}}}}{{{\mu _{\text{p}}}}}} \].Thus,the correct option is A.

Additional information:
An intrinsic(pure) semiconductor is a pure semiconductor without any noticeable dopant species present, often referred to as an undoped semiconductor. Therefore, instead of the quantity of impurities, the number of charge carriers is determined by the properties of the substance itself. An extrinsic semiconductor is a semiconductor doped with a particular impurity that is capable of deeply altering its electrical characteristics, making it suitable for electronic (diode, capacitor, etc.) or optoelectronic (light emitter and detector) applications.

Note:Since, we are asked for the minimum value so the first derivative is always zero. However, in an intrinsic semiconductor the electrical conductivity depends on the temperature only. The examples of intrinsic semiconductors are germanium and silicon.