Temperature dependence of the refractive-index dispersion in amorphous germanium at elevated temperatures

Abstract

Refractive-index dispersion in amorphous germanium was measured in the vicinity of the optical gap (0.2-1.2 eV) and for the first time at elevated temperatures (25\ifmmode^\circ\else\textdegree\fi{}C-400\ifmmode^\circ\else\textdegree\fi{}C). The results at each temperature follow the dispersion of a single oscillator, whose resonance energy and strength decrease linearly with temperature. However, the magnitude of these variations differs markedly (e.g., by a factor of 2 or more, respectively) from temperature derivatives of band gaps or of electron density. It is apparent that in small gap materials, such as amorphous germanium, temperature dependences derived from dispersion do not reflect the temperature dependence of specific interband transitions (within the framework of the single-oscillator model). A new model is developed which accounts for the temperature dependence of the dispersion by replacing the single oscillator with two other interband electronic oscillators. The stronger of these two, which represents the majority of interband transitions, is located at the maximum of the optical transition strength. Assuming that its oscillator strength is close to unity, and that the temperature derivative of its resonance energy is identical with that of the Penn gap, the parameters of the second oscillator are obtained by fitting to the data. This is a weak oscillator located closely above the optical gap thus representing the low-energy portion of the ${\ensuremath{\epsilon}}_{2}$ spectrum. Moreover, the temperature derivative of its resonance energy acquires a value similar to that of the optical gap. In this model the dispersion is sensitive to the location of the weak oscillator due to the low value of its resonance energy and its proximity to the spectral region of interest. The dispersion is similarly sensitive to the location of the low-energy portion of ${\ensuremath{\epsilon}}_{2}$, as obtained by considering its moments. Thus in crystalline germanium, where the weak oscillator and the onset of strong interband absorption lie at higher energies than in amorphous germanium, a smaller temperature derivative of the single-oscillator energy is expected. This prediction is confirmed by experiment. It is concluded that using band-gap derivatives, the two-oscillator model may be utilized to calculate the temperature dependence of the dispersion or vice versa.