Dispersion of absorption and refractive index of PbTe and Pb1−xEuxTe (x<0.05) below and above the fundamental gap

Abstract

We report extensive experimental and theoretical studies of the frequency dependence of the absorption constant α(ω) and of the index of refraction n(ω) in PbTe and its pseudobinary alloy ${\mathrm{Pb}}_{1\mathrm{-}\mathit{x}}$ ${\mathrm{Eu}}_{\mathit{x}}$Te. Mid-infrared transmission experiments on epitaxial layers of ${\mathrm{Pb}}_{1\mathrm{-}\mathit{x}}$ ${\mathrm{Eu}}_{\mathit{x}}$Te (0≤x≤0.0475, thickness ≊5 μm) on ${\mathrm{BaF}}_2$ substrate were performed in the frequency range 1000–5000 ${\mathrm{cm}}^{\mathrm{-}1}$ at temperatures from 5 to 300 K. The absolute values of the transmission as a function of frequency can be evaluated to yield information on both α(ω) and n(ω), due to the presence of interference fringes below the fundamental energy gap and due to the drop of transmission above the energy gap. For α(ω) a model calculation is preferred based on the nonparabolic Dimmock model for the energy-momentum relationship ε(k) of electrons and holes. For the oscillator strength, interband matrix elements ${\mathit{P}}_{\mathrm{\parallel }}$ and ${\mathit{P}}_{\mathrm{\perp }}$ are used that are in agreement with the values obtained from magneto-optical studies. In the procedure to fit the transmission spectra four parameters are used: energy gap, oscillator strength, damping parameter, and a background index of refraction. With these parameters the functional dependence α(ω) is calculated, and n(ω) is derived using a Kramers-Kronig transformation. The results show that nonparabolicity is quite important to reproduce α(ω), and that via Kramers-Kronig transformation it also substantially influences n(ω). Through the causality relation, even for frequencies below that corresponding to the energy gap, n(ω) is influenced by the shape of ε(k) above the gap. With increasing Eu content the energy gap increases and the extremum in n(ω) close to ${\mathrm{\epsilon }}_{\mathit{g}}$ shifts to higher energies. The enhancement over the background refraction index, which is determined by higher interband transitions, becomes weaker.