(i) What is the occupation number of a state $i$ with energy $E_{i}$ according to the Fermi-Dirac statistics for a given chemical potential $\mu$ ?

(ii) Assuming that the energy $E$ is spin independent, what is the number $g_{s}$ of electrons which can occupy an energy level?

(iii) Consider a semi-infinite metal slab occupying $z \leqslant 0$ (and idealized to have infinite extent in the $x y$ plane) and a vacuum environment at $z>0$. An electron with momentum $\left(p_{x}, p_{y}, p_{z}\right)$ inside the slab will escape the metal in the $+z$ direction if it has a sufficiently large momentum $p_{z}$ to overcome a potential barrier $V_{0}$ relative to the Fermi energy $\epsilon_{\mathrm{F}}$, i.e. if

$$\frac{p_{z}^{2}}{2 m} \geqslant \epsilon_{\mathrm{F}}+V_{0}$$

where $m$ is the electron mass.

At fixed temperature $T$, some fraction of electrons will satisfy this condition, which results in a current density $j_{z}$ in the $+z$ direction (an electron having escaped the metal once is considered lost, never to return). Each electron escaping provides a contribution $\delta j_{z}=-e v_{z}$ to this current density, where $v_{z}$ is the velocity and $e$ the elementary charge.

(a) Briefly describe the Fermi-Dirac distribution as a function of energy in the limit $k_{\mathrm{B}} T \ll \epsilon_{\mathrm{F}}$, where $k_{\mathrm{B}}$ is the Boltzmann constant. What is the chemical potential $\mu$ in this limit?

(b) Assume that the electrons behave like an ideal, non-relativistic Fermi gas and that $k_{\mathrm{B}} T \ll V_{0}$ and $k_{\mathrm{B}} T \ll \epsilon_{\mathrm{F}}$. Calculate the current density $j_{z}$ associated with the electrons escaping the metal in the $+z$ direction. How could we easily increase the strength of the current?