AlN is a direct band$-$gap semiconductor with a wide bandgap of $6\mathrm{.}026$eV at $300$K$.$

The empirical constants for AlN are: $=1\mathrm{.}799$ x $10-3$ eV$/$K and $=1462$ K$.$

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$($a$)$ The energy bandgap separates the permitted electron energy level between valance and conduction bands.

$($i$)$ What is the energy state at absolute $0$K$?[5$ marks$]$

$($ii$)$ What is the energy state at $300$K$?[5$ marks$]$

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$($b$)$ Calculate the energy bandgap of AlN at $0$K$.[10$ marks$]$

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$($c$)$ Calculate the minimum frequency and the respective wavelength required by a photon to transfer a valance electron to the conduction band at $300$K in vacuum. State the electromagnetic spectrum of this photon.$[10$ marks$]$