Renormalization of the Optical Response of Semiconductors by Electron-Phonon Interaction

Abstract

In the past five years enormous progress has been made in the ab initio calculations of the optical response of electrons in semiconductors. The calculations include the Coulomb interaction between the excited electron and the hole left behind, as well as local field effects. However, they are performed under the assumption that the atoms occupy fixed equilibrium positions and do not include effects of the interaction of the lattice vibrations with the electronic states (electron-phonon interaction). This interaction shifts and broadens the energies at which structure in the optical spectra is observed, the corresponding shifts being of the order of the accuracy claimed for the ab initio calculations. These shifts and broadenings can be calculated with various degrees of reliability using a number of semiempirical and ab initio techniques, but no full calculations of the optical spectra including electron-phonon interaction are available to date. This article discusses experimental and theoretical aspects of the renormalization of optical response functions by electron-phonon interaction, including both temperature and isotopic mass effects. Some of the theoretical techniques used can also be applied to analyze the renormalization of other response functions, such as the phonon spectral functions, the lattice parameters, and the elastic constants.

Comparison of some theoretical models for fittings of the temperature dependence of the fundamental energy gap in GaAs

“…(3), a value of ∆E(0) = 110 meV is obtained for GaAs. As a matter of fact, the values of ∆E(0) that are found in the literature for GaAs from estimates of isotopic substitution, linear extrapolation, and theoretical calculations are quite different [16,18,20]. According to our understanding, one important factor that must be taken into account for the analysis of these contrasting results is the lack of systematic experimental date for T ≥ Θ D , which hides a possible non-linear behavior of E g (T ) in the high temperature range.…”

“…For instance, using the Pässler-p model [9] to describe the temperature dependence of the energy gap of GaAs in the range from 2 K to 280 K, and the values Θ p = 226 K, α p = 0.347 meV/K and p = 2.51, Cardona et al [18] estimated ∆E(0) = 53 meV (-53 meV in the Cardona [16] notation) from the photoluminescence data measured by Grilli et al [26]. Using another set of experimental data with a different upper limit defining the domain of temperature (974 K ∼ 2.7Θ D ), Cardona et al [18] obtained ∆E(0) = 90 meV.…”

“…The values obtained here are therefore different from that founded in the literature obtained extrapolating the linear dependence of the gaps at high T to T = 0 K and that determined by the dependence of the gap on isotope mass [16,18,20]. As we mention before, we consider that the lack of experimental data of E g (T ) in the high-T range in the fit models of E g (T ) is one of the relevant factors to consider for 1 Table I -Fitting parameters of E g (T) and their respective incertitudes according to Viña (E gB (0), E B , Θ B , a B , α B ) [12] , Pässler-p (E gp (0), Θ p , a p , α p , p) [9] and Pässler-ρ (E gρ (0), Θ r , a ρ , α ρ , ρ) [13] models for the temperature ranges 12 K < T < ~192 K, 12 K< T < 280 K, and 12 K < T < 974 K for the group I experimental data and in the range 12 K < T < 957 K for group II.…”

“…At low temperatures, the change of the energy gap with the isotopic mass is proportional to M −1/2 , where M is the mean effective masses of the constituent atoms of the material [17]. In the high temperature range, the energy gap does not depend on M. For GaAs, due to its similarity with Ge and due to the fact that As has only one stable isotope, the effect of the substitution of the 69 Ga and 71 Ga isotopes on the energy gap can be described by [18]:…”

“…Using another set of experimental data with a different upper limit defining the domain of temperature (974 K ∼ 2.7Θ D ), Cardona et al [18] obtained ∆E(0) = 90 meV. As can be find out from these results, the values of ∆E(0) obtained for GaAs seem to be dependent on the upper limit of the domain of temperature used to perform the fits.…”