Past exam of the condensed matter physics major option course of the University of Oxford / 2019 / 3 / b / Solution

For a nonmagnetic sample bordering air, the normal-incidence Fresnel equations giveAt short infrared wavelengths the carrier response becomes small and $R$ approaches the background value $R_{\infty }$. Extrapolating the short-wavelength trend gives about $0.29$, henceThe leftmost point at $4\mu \mathrm{m}$ is about $0.27$ and has not quite reached that asymptote. As a consistency check, using the minimum below and $R(4\mu \mathrm{m})\simeq 0.27$ gives ${ϵ}_{\infty }\simeq 11$ without treating that point as the limiting value.

The minimum is near ${\lambda }_{\min }\simeq 13\mu \mathrm{m}$. For weak damping, zero optical reflectance means refractive-index matching, ${ϵ}_r=1$, rather than ${ϵ}_r=0$. Thususing $N=5.4\times 10^{24}{\mathrm{m}}^{-3}$. Given graphical uncertainty, about $0.08m_e$ is an appropriate estimate. This is a carrier effective mass from a reflectivity minimum, not an independent determination of the bare electron mass.

Just beyond the minimum, $\mathrm{Re}{ϵ}_r$ passes through zero near ${\lambda }_p={\lambda }_{\min }/\sqrt{1-1/{ϵ}_{\infty }}\simeq 13.6\mu \mathrm{m}$. At longer wavelengths it becomes negative and reflection rises sharply toward the conducting regime. Finite damping rounds the edge, prevents a strictly exact zero minimum, and keeps the plotted long-wavelength reflectivity below unity.