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What will be the conductivity of pure silicon crystal at $300K$ temperature? If electron hole pairs per \[c{m^3}\] is \[1.072 \times {10^{10}}\] at this temperature, \[{\mu _n} = 1350c{m^2}/volt.\operatorname{s} \] and \[{\mu _n} = 480c{m^2}/volt.\operatorname{s} \]
A) \[3.14 \times {10^{ - 6}}mho/cm\]
B) \[3 \times {10^{ - 6}}mho/cm\]
C) \[{10^{ - 6}}mho/cm\]
D) \[{10^6}mho/cm\]

Last updated date: 29th Feb 2024
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Hint: Pure Silicon and pure Germanium are two main examples of semiconductors. In which electron hole pairs are in the same concentration. If we add some impurity to the pure semiconductor, the conductivity is increased by time. These impurities are of trivalent and pentavalent elements.

Formula used:
The conductivity of semiconductor is given by
\[\sigma = {N_n}{\mu _n}\]
Where \[\sigma \] is conductivity, \[{N_n}\] is the number of electrons/holes and \[{\mu _n}\] is the mobility.
And when electron hole pairs show conductivity that depends only on the impurity added.
\[\sigma = {N_n}e{\mu _n} + {N_h}e{\mu _h}\]
Where \[e\] is the charge on both electrons and holes. \[{\mu _n}\] and \[{\mu _h}\] are the mobility of electrons and holes respectively.

Complete step by step solution:
The conductivity of semiconductors increases with the mobility of charge carriers. So we only consider mobility and number of charge carriers not on the temperature. Now according to the question, we have temperature \[T = 300K\], number of electrons/holes \[ =
{N_n} = {N_h} = 1.072 \times {10^{10}}\], mobility of electrons \[{\mu _n} =
1350c{m^2}/volt.\operatorname{s} \], mobility of holes \[{\mu _h} =
480c{m^2}/volt.\operatorname{s} \]. We have to find the conductivity \[\sigma \] of pure silicon crystal.
Now we know that, the conductivity of any semiconductor is –
\[\Rightarrow \sigma = {N_n}e{\mu _n} + {N_h}e{\mu _h}\]
We know that \[{N_n} = {N_h} = N\]
So, substituting these values in the above equation. We get-
   \Rightarrow \sigma = Ne({\mu _n} + {\mu _h}) \\
   \Rightarrow \sigma = 1.072 \times {10^{10}} \times 1.6 \times {10^{ - 19}}\left( {1350 + 480} \right) \\
   \Rightarrow \sigma = 1.072 \times {10^{10}} \times 1.6 \times {10^{ - 19}}\left( {1830} \right) \\
   \Rightarrow \sigma = 1.072 \times {10^{10}} \times 1.6 \times {10^{ - 19}} \times 1830 \\
   \Rightarrow \sigma = 1.072 \times 1.6 \times 1830 \times {10^{ - 9}} \\
   \Rightarrow \sigma = 1.072 \times 1.6 \times 1830 \times {10^{ - 9}} \\
   \Rightarrow \sigma = 1.7152 \times 1830 \times {10^{ - 9}} \\
   \Rightarrow \sigma = 3138.816 \times {10^{ - 9}} \\
   \Rightarrow \sigma = 3.138816 \times {10^{ - 6}} \\
   \Rightarrow \sigma = 3.14 \times {10^{ - 6}}mho/cm
Hence the conductivity of pure silicon at room temperature is \[\sigma = 3.14 \times {10^{ - 6}}mho/cm\].

Thus, Option A is correct.

Additional information:
Semiconductor is the device that is used in electric appliances. There are two types of semiconductor.
i) n-type semiconductor
ii) P-type semiconductor
n-type semiconductor can make by the doping of impurity of pentavalent element i.e.
(Phosphorous). and p-type semiconductor can make by the doping of impurity of trivalent element
i.e. (Boron). If impurity is added to the pure semiconductors then it is called doped. And this process is called doping. The main motive of adding impurities is increasing the conductivity of pure semiconductors.

Note: The electron and hole concentration is equal in any pure semiconductor. It can be increased sixteen times by adding impurity. By which electron and hole concentration is increased, depends on impurity added. If the impurity added is pentavalent, the free electron in the semiconductor increases. And if the impurity is of trivalent element, holes in the semiconductor increased. And the most important thing is that the conductivity of a semiconductor is increased with the increase of temperature. And $300K$ is considered as room temperature for a semiconductor.