Dispersion of absorption and refractive index of PbTe and Pb1-xEuxTe (x

Abstract

We report extensive experimental and theoretical studies of the frequency dependence of the absorption constant \ensuremath{\alpha}(\ensuremath{\omega}) and of the index of refraction n(\ensuremath{\omega}) in PbTe and its pseudobinary alloy ${\mathrm{Pb}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Eu}}_{\mathit{x}}$Te. Mid-infrared transmission experiments on epitaxial layers of ${\mathrm{Pb}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Eu}}_{\mathit{x}}$Te (0\ensuremath{\le}x\ensuremath{\le}0.0475, thickness \ensuremath{\approxeq}5 \ensuremath{\mu}m) on ${\mathrm{BaF}}_{2}$ substrate were performed in the frequency range 1000--5000 ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$ 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 \ensuremath{\alpha}(\ensuremath{\omega}) and n(\ensuremath{\omega}), due to the presence of interference fringes below the fundamental energy gap and due to the drop of transmission above the energy gap. For \ensuremath{\alpha}(\ensuremath{\omega}) a model calculation is preferred based on the nonparabolic Dimmock model for the energy-momentum relationship \ensuremath{\epsilon}(k) of electrons and holes. For the oscillator strength, interband matrix elements ${\mathit{P}}_{\mathrm{\ensuremath{\parallel}}}$ and ${\mathit{P}}_{\mathrm{\ensuremath{\perp}}}$ 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 \ensuremath{\alpha}(\ensuremath{\omega}) is calculated, and n(\ensuremath{\omega}) is derived using a Kramers-Kronig transformation. The results show that nonparabolicity is quite important to reproduce \ensuremath{\alpha}(\ensuremath{\omega}), and that via Kramers-Kronig transformation it also substantially influences n(\ensuremath{\omega}). Through the causality relation, even for frequencies below that corresponding to the energy gap, n(\ensuremath{\omega}) is influenced by the shape of \ensuremath{\epsilon}(k) above the gap. With increasing Eu content the energy gap increases and the extremum in n(\ensuremath{\omega}) close to ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{g}}$ shifts to higher energies. The enhancement over the background refraction index, which is determined by higher interband transitions, becomes weaker.For Eu contents x\ensuremath{\approxeq}0.05 the oscillator strength increases by about 10% in comparison to PbTe. In addition, the damping parameters increase from 3 to 40 ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$ at T=5 K.