Past exam of the quantum condensed matter course of the University of Cambridge / 2007 / Section B / B1

The mobile electrons in a metal screen an incident optical field. In the weak-damping Drude model, $\epsilon (\omega )\simeq {\epsilon }_{\infty }-{\omega }_p^2/{\omega }^2$. Below the screened plasma frequency, the real dielectric response is negative, the wave is evanescent inside the metal, and a clean surface is highly reflective. Above the plasma edge, the response can become positive and propagating waves are allowed, explaining the qualitative transmission window at sufficiently high, often ultraviolet, frequencies.

This is a free-carrier argument, not a statement that every metal is transparent throughout the ultraviolet. Interband absorption, damping and sample thickness can still produce strong absorption above the plasma edge.

The hydrogenic donor state radius is the Bohr radius rescaled by screening and effective mass:The donor-state overlap estimate compares the mean spacing $n_D^{-1/3}$ to this radius. Taking the spacing as $a_B^{\ast }$ gives $n_D\sim (a_B^{\ast }{)}^{-3}=1.32\times 10^{25}{\mathrm{m}}^{-3}=1.32\times 10^{19}\mathrm{c}{\mathrm{m}}^{-3}$. Taking it as the orbital diameter gives a factor eight less, $1.65\times 10^{18}\mathrm{c}{\mathrm{m}}^{-3}$. Thus a geometric estimate is roughly $10^{18}$–$10^{19}\mathrm{c}{\mathrm{m}}^{-3}$, with its contact convention stated. The exponential tails do not have a sharp overlap boundary, and this estimate should not be identified with a precise metal-insulator transition threshold.

Let the missing electron occupy the Bloch state of crystal momentum $\hslash {\mathbf{k}}_e$ and energy $E_e({\mathbf{k}}_e)$. Relative to the filled band, define ${\mathbf{k}}_h=-{\mathbf{k}}_e$ modulo a reciprocal-lattice vector and $E_h({\mathbf{k}}_h)=C-E_e(-{\mathbf{k}}_h)$, with a fixed reference $C$. ThenThe velocity equality follows by differentiating the hole energy with respect to ${\mathbf{k}}_h$; two minus signs cancel. The signed band effective mass is therefore opposite, becoming a positive hole mass near an electron-band maximum. The hole electric charge is $+e$ when the electron charge is $-e$. These momenta are crystal momenta, and the electron mass being compared is the signed band mass, not the bare electron mass. The positive hole current equals the current lost by removing that electron.