Question $3(20$ marks$)$

$($a$)$ Explain why the amount of free electrons and holes is equal in an intrinsic semiconductor.

$[4$ marks$]$

$($b$)$ For an intrinsic silicon at $300$ K $,$ determine:

$($i$)$ the conductivity.

$($ii$)$ the resistivity.

$($c$)$ A silicon sample is doped with $\backslash (8\backslash$times $10^\{15\}\backslash$mathrm$\{$~cm$\}^\{-3\}\backslash )$ of phosphorous atoms and $\backslash (3\backslash$times $10^\{15\}\backslash )\backslash (\backslash$mathrm$\{$cm$\}^\{-3\}\backslash )$ of boron atoms. Given that the temperature is $300$ K $,$ answer the following questions:

$($i$)$ State the type of semiconductor of this doped silicon sample and justify your answer.

$($ii$)$ Find the net dopant concentration.

$($iii$)$ Find the free electron concentration of the doped sample.