6:35 PM - CT07.04.05
Computational Analysis of Critical Points in Temperature Dependent and Time Resolved Ellipsometry Spectra of Ge Using Digital Filtering
Carola Emminger1,Stefan Zollner1,Nuwanjula Samarasingha1,Farzin Abadizaman1,Jose Menendez2,Shirly Espinoza3,Steffen Richter4,Mateusz Rebarz3,Oliver Herrfurth5,Martin Zahradnik3,Rudiger Schmidt-Grund6,Jakob Andreasson3
New Mexico State University1,Arizona State University2,ELI Beamlines3,Linköping University4,Universität Leipzig5,TU Ilmenau6
Critical points (CPs) are structures in the dielectric function (DF) which are related to interband transitions and depend on temperature and doping. We analyze CPs in the DF of bulk Ge measured with static and time-resolved spectroscopic ellipsometry using a linear filter technique based on Gaussian kernels recently introduced by Le, Kim, Kim, and Aspnes , which combines interpolation, noise reduction, scale change, and differentiation.
Utilizing this linear filter method, we calculate the second derivatives of the complex DF with respect to energy of experimental data taken in the spectral range of the direct band gap E0 at various temperatures from 10 K to 718 K. The choice of the filter width is crucial to eliminate noise while preserving information at the same time and is defined according to the onset of white noise in the Fourier coefficients determined from a discrete Fourier transform of the data .
Applying a Levenberg-Marquardt algorithm, the second derivatives are fitted simultaneously with the imaginary part of the DF to a lineshape based on the Elliott-Tanguy theory  considering excitonic effects present at the direct band gap. The Elliott-Tanguy lineshape depends on the excitonic binding energy, the effective masses of the valence and conduction bands, the dipole matrix element, the threshold energy, and a broadening parameter. Effective masses of the conduction band and the heavy (hh) and light hole (lh) valence bands at cryogenic temperatures, as well as the matrix element are taken from literature. For the data set at 10 K, fitting only the band gap energy and the broadening parameters of the hh- and lh-bands provides reasonable fitting results for both the imaginary part and the second derivatives of the DF. For data above 10 K, the temperature dependence of the effective masses and the matrix element are considered. A red shift of the direct band gap energy with increasing temperature is found as well as an increase in broadening which can be fitted using a Bose-Einstein statistical factor taking into account electron-phonon interactions.
The same analysis method is applied to the E1 and E1+Δ1 CPs of Ge dependent on photo-excited charge carrier density and temperature obtained from femtosecond pump-probe ellipsometry measurements . Using a two-dimensional CP lineshape, the amplitude, excitonic phase angle, threshold energy, and broadening parameters are determined as functions of delay time. We find a distinctive change of the various parameters and a relaxation which starts at about 4 ps after the pump pulse. The decrease of the E1 and E1+Δ1 energies suggests a laser heating of about 20 K to 40 K, respectively. In the analysis of these data, we are especially interested in changes to the E1 and E1+Δ1 band gaps due to many-body effects (such as band gap renormalization and band filling), broadenings (due to screening of the electron-phonon interactions), spin-orbit coupling strengths, and excitonic phase angles as a function of the delay time between the pump and probe pulses.
 V. L. Le, T. J. Kim, Y. D. Kim, and D. E. Aspnes, J. Vac. Sci. & Technol. B 37, 052903 (2019)
 C. Tanguy, Phys. Rev. B 60, 10660 (1999)
 S. Espinoza, S. Richter, M. Rebarz, O. Herrfurth, R. Schmidt-Grund, J. Andreasson, and S. Zollner, Appl. Phys. Lett. 115, 052105 (2019)