Past exam of the condensed matter physics major option course of the University of Oxford / 2015 / 7 / b / Solution

The sharp solid curve is the $4.2$ K spectrum; the broadened dashed curve is the $77$ K spectrum. At low temperature, a strong discrete exciton transition appears just below the continuum edge, with weaker higher members converging toward the band gap. Heating broadens those resonances through scattering, reduces their spectral resolution, and shifts the band edge to lower photon energy. The lower-energy broad dashed maximum is therefore not evidence for a second unrelated low-temperature band gap.

Read approximate energies from the native figure: the main low-temperature peak is near $1.610$ eV and a weaker feature is near $1.617$–$1.618$ eV, with the continuum edge around $1.620$ eV. A three-dimensional hydrogenic series $E_n=E_g-R^{\ast }/n^2$ then givesFor example, $E_2-E_1=3R^{\ast }/4$ predicts about $7.5$ meV, consistent with those feature spacings. These are graphical estimates at roughly meV precision, not extra digits extracted from the line widths. The strong direct interband onset and subgap series support a direct-gap Wannier-Mott interpretation. The continuum absorption scale is of order $10^6$ m${}^{-1}$, corresponding to an optical absorption length of order a micrometre. At $77$ K, $k_{B}T\simeq 6.6$ meV is already comparable with the inferred binding scale, explaining why a sharp Rydberg series is poorly resolved; at $4.2$ K, $k_{B}T\simeq 0.36$ meV.

With $m_e^{\ast }=m_h^{\ast }=0.2m_e$, the reduced mass is ${\mu }^{\ast }=0.1m_e$. In the screened hydrogen model,It follows thatA radius of several nanometres is consistent with an extended exciton in an ordinary semiconductor lattice. The relative permittivity inferred here is the effective dielectric screening parameter of that model; identifying the material chemically or determining a full dielectric spectrum is not possible from these data alone.