$25.(20$ points$)$ Metal$-$Semiconductor junctions

$($a$)$ UNBIASED, CASE $\backslash$#$1$: On a graph paper $($or else using a ruler to be quantitatively accurate$)$ draw a band diagram of a metal$-$semiconductor junction with no bias applied. Assume a metal work function of $6$ V and a semiconductor electron affinity of $5$ V $,$ a semiconductor with bandgap of $2$ eV and negative Fermi potential of $-0\mathrm{.}5$ V $.$ Assume that the intrinsic energy lies at mid$-$gap. $($HINT: Recall that the Fermi potential times the electron charge is the energy between the intrinsic energy and the Fermi energy, AND that the sign of the Fermi energy is DEFINED to be negative for $\backslash ($ n $\backslash )-$type semiconductors and positive for $\backslash ($ p $\backslash )-$type$).$

$($b$)$ What is the semiconductor work function? Is it greater or less than the metal work function?

$($c$)$ Is this junction a Schottky contact or an ohmic contact?

$($c$\mathrm{.}2)$ BIASED, CASE $\backslash$#$1$: ONLY DO PART $($c$\mathrm{.}2)$ IF THE ANSWER TO PART $($c$)$ WAS "SCHOTTKY", OTHERWISE LEAVE BLANK. On a graph paper $($or else using a ruler to be quantitatively accurate$)$ draw a band diagram of this metal$-$semiconductor junction with forward bias of $0\mathrm{.}25$ V $.($Recall that forward bias means that a positive voltage is applied to the $"$p$-$like" side of the metal$-$semiconductor junction. Recall as well that for majority carrier devices, we do not bother drawing the quasi$-$Fermi energies for the minority carriers in the semiconductor.$)$

$($d$)$ UNBIASED, CASE $\backslash$#$2$: On a graph paper $($or else using a ruler to be quantitatively accurate$)$ draw a band diagram of a metal$-$semiconductor junction with no bias applied. Assume a metal work function of $6$ V and a semiconductor electron affinity of $4$ V $,$ a semiconductor with bandgap of $2$ eV and positive Fermi potential of $+0\mathrm{.}5$ V $.$ Assume that the intrinsic energy lies at mid$-$gap

$($e$)$ What is the semiconductor work function? Is it greater or less than the metal work function?

$($f$)$ Is this junction a Schottky contact or an ohmic contact?

$($f$\mathrm{.}2)$ BIASED, CASE $\backslash$#$2$: ONLY DO PART $($f$\mathrm{.}2)$ IF THE ANSWER TO PART $($f$)$ WAS "SCHOTTKY", OTHERWISE LEAVE BLANK: On a graph paper $($or else using a ruler to be quantitatively accurate$)$ draw a band diagram of this metal$-$semiconductor junction with reverse bias of $0\mathrm{.}25$ V $.($Recall that reverse bias means that a negative voltage is applied to the $"$p$-$like" side of the metal$-$semiconductor junction.$)$