$1.$ The density of states function in the conduction band of a semiconductor is given by:

$$

$4(2,)3/2$

$()=$

$3$

$($Eq$.2\mathrm{.}1)$

And the free electron concentration in the conduction band is given by:

$$

$=()()$

$$

$($Eq$.2\mathrm{.}2)$

For intrinsic Si at room temperature, the Fermi level is approximately $0\mathrm{.}56$ below the conduction band edge. When $,$ which is valid for intrinsic Si$,$ the Fermi$-$Dirac distribution function can be approximated to be:

$$

$1$

$()$

$()$

$$

$($Eq$.2\mathrm{.}3)$

Which leads to a simple analytical expression for the free electron carrier density in the conduction band:

$$

$()$

$=$

$$

$($Eq$.2\mathrm{.}4)$

Where the effective density of states in the conduction band, $,$ is:

$$

$23/2$

$=2(,$

$2)$

$$

$($Eq$.2\mathrm{.}5)$

$$

$,$

$$

Thedensity$-$of$-$stateseffectivemassforelectrons inSiis$=1\mathrm{.}080.$

$($a$)$ Derive $(2\mathrm{.}5)$ using $(2\mathrm{.}3)$ in $(2\mathrm{.}2).$

$($b$)$ Calculate and tabulate $$ of Si for $=250,275,300,325,350$

$$

$$