Electromagnetic excitations of a small gyrotropic sphere

Theoretical expressions for the microwave power absorbed in a small spherical magnetoplasma are obtained using a perturbation expansion in powers of a dimensionless parameter proportional to the square of the radius. There are four linearly independent orientations of the microwave electric and magnetic fields (${\overrightarrow{\mathrm{E}}}_1$ and ${\overrightarrow{\mathrm{B}}}_1$) relative to the dc magnetic field ${\overrightarrow{\mathrm{B}}}_0$: transverse magnetic (${\overrightarrow{\mathrm{B}}}_1\perp {\overrightarrow{\mathrm{B}}}_0$), longitudinal magnetic (${\overrightarrow{\mathrm{B}}}_1\parallel {\overrightarrow{\mathrm{B}}}_0$), transverse electric (${\overrightarrow{\mathrm{E}}}_1\perp {\overrightarrow{\mathrm{B}}}_0$), and longitudinal electric (${\overrightarrow{\mathrm{E}}}_1\parallel {\overrightarrow{\mathrm{B}}}_0$). The theory includes the effects of electron inertia, displacement current, and the averages over the energy-dependent electron relaxation time. If the relaxation time is energy independent, in the limit of very small radius (Rayleigh limit), the result for the transverse electric case becomes the well-known formula for plasma-shifted cyclotron resonance. The results are applied to explain the microwave experiments on small spheres of $n$-type InSb by Evans, Furdyna, and Galeener. The calculation accounts quantitatively for (a) the position and shape of the plasma-shifted cyclotron resonance, (b) the fact that the longitudinal electric absorption is negligible, (c) the position, shape, and size dependence of the transverse magnetic resonance, and (d) the broad absorption shoulder observed in the longitudinal magnetic case. It is proposed that with this theory it is now possible to measure accurately in a single microwave experiment all four macroscopic electrical parameters of a given sample: mobility, carrier effective mass, lattice dielectric constant, and carrier density.