Past exam of the quantum condensed matter course of the University of Cambridge / 2008 / Section B / B1

For a single carrier type with unit Hall factor, the magnitude of the Hall voltage is $V_H=IB/(net)$, where $t$ is the slab thickness parallel to the magnetic field. ThereforeThe longitudinal electrical resistivity is $\rho =Rwt/L=1000(2\times 10^{-3})(0.5\times 10^{-3})/(5\times 10^{-3})=0.20\Omega \mathrm{m}$. Using $1/\rho =ne\mu$, the electron mobility magnitude isWithout the Hall polarity these data do not identify the carrier charge sign. The inference assumes one dominant carrier population; electron-hole compensation would require a multicarrier analysis.

Put $N=N_x{N}_y{N}_z$ primitive cells in a periodic box. In each reciprocal-space direction, the allowed Bloch wavevectors are spaced by $2\pi /L_i$, giving $N_i$ independent values in the first Brillouin zone. There are consequently $N$ wavevectors per band and, including two electron spin states at each wavevector,The total and per-atom counts are different; a factor $N$ cannot remain in the per-atom count for this monatomic lattice.

In independent-electron band theory, a partially occupied band permits low-energy changes of occupancy and gives a metal. Filled bands separated from empty bands by a finite band gap give a band insulator at zero temperature. An odd number of electrons per primitive cell cannot fill spin-degenerate bands, whereas an even number can; even filling alone does not ensure an insulator if bands overlap. Magnetic symmetry breaking and strong correlations, such as a Mott insulator, go beyond this simple counting criterion.

A plasmon is a quantum of a collective longitudinal oscillation of electron density against the positive background. A small charge displacement creates an electric restoring field; Newton's law then gives the plasma angular frequency. Assuming approximately one conduction electron per atom and spacing $d=3\text{A˚}$ gives $n\sim d^{-3}=3.7\times 10^{28}{\mathrm{m}}^{-3}$. HenceThe ordinary frequency is about $1.7\times 10^{15}\mathrm{Hz}$, with energy $\hslash {\omega }_p\sim 7\mathrm{eV}$. A different conduction-electron count per atom changes this estimate by its square root; dielectric and band-structure corrections also modify the simplest free-electron result.