Past exam of the condensed matter physics major option course of the University of Oxford / 2015 / 5 / c

At zero magnetic field the two circular conductivities coincide, so an isotropic material has no magnetically induced circular birefringence. In the collisionless Drude model range the dielectric function is $ϵ={ϵ}_{\infty }-{\omega }_p^2/{\omega }^2$. A high carrier density gives a large plasma frequency, often placing the screened plasma edge ${\omega }_p/\sqrt{{ϵ}_{\infty }}$ in the infrared or visible range.

Below that edge the real dielectric function is negative and a transverse wave is evanescent in the bulk. The material is strongly reflecting, with penetration through a skin depth and losses set by finite $1/\tau$. Above the plasma edge propagation becomes possible, subject to absorption and interband transitions. Carrier density alone does not specify a complete optical spectrum: the statements assume the probe frequency relative to the plasma scale and the validity of the single-band Drude model.

At low density, the plasma frequency and free-carrier dielectric screening are smaller, so radiation can penetrate a suitable semiconductor sample. In a high field with ${\omega }_c\tau \gg 1$, the active circular component has a resolved cyclotron absorption line near $\omega ={\omega }_c$, with width of order $1/\tau$. The opposite circular component lacks that positive-frequency resonance.

Different complex refractive indices for the two components produce Faraday rotation and circular dichroism. Transmission with controlled circular polarization therefore isolates the cyclotron resonance and measures $m^{\ast }=e{B}_{\mathrm{res}}/\omega$. The low-density/high-field labels do not by themselves imply complete transparency at every frequency; one must still avoid the plasma stop bands, strong interband absorption and excessive sample thickness.

At high density the plasma scale remains large, but a strong magnetic field separates the two circular dielectric responses. Ignoring damping, the positive plasma cutoffs areBetween suitable cutoffs one circular wave can propagate while the other is excluded; reflection and transmission are strongly polarization dependent. Around the cyclotron frequency the active response is resonant, while at optical frequencies far above ${\omega }_c$ the splitting can be a relatively small correction to a large plasma edge.

There is also a low-frequency window: for the active component, $0<\omega \ll {\omega }_c$ gives $ϵ\simeq {\omega }_p^2/(\omega {\omega }_c)>0$ when losses are small. This supports a helicon-like mode even when $\omega \ll {\omega }_p$. That does not make an ordinary metal broadly transparent; the large index, boundary coupling, collisions and skin geometry still matter. In the common regime ${\omega }_c\ll {\omega }_p$, bulk cyclotron absorption remains hidden from straightforward transmission, motivating a surface-sensitive experiment. Thus the relevant comparisons are among $\omega$, ${\omega }_c$, ${\omega }_p$ and $1/\tau$, not carrier density and field in isolation.