# Optical Response of an Interacting Polaron Gas in Strongly Polar Crystals
## 4 Application to SrTiO\({}_{3}\)  
The approach described above is focused mainly on crystals with a high ratio \(\varepsilon_{0}/\varepsilon_{\infty}\) like strontium titanate, where it can reveal specific features related to the high polarizability. In the previous treatment of the many-polaron optical conductivity in doped SrTiO\({}_{3}\)[13], the pronounced peak for \(\hbar\Omega\sim 130\) meV at a relatively low temperature remains unexplained. It was suggested in [13] that it might be provided by other (non-polaron) mechanisms, for example, the small-polaron and mixed-polaron [35] channels for the optical response. As we show below, additional mechanisms are not necessary for the explanation of this 130-meV feature. The numeric results for the many-polaron optical conductivity are shown in Figure 1. In the computation, the following set of electron band and phonon material parameters is used. The conduction band shape is simulated by the analytic tight-binding fit as described in Ref. [36] and here in Appendix A. The optimal values for this analytic approximation are the diagonal matrix elements \(t_{\delta}\), \(t_{\pi}\) corresponding to the recent results of the microscopic calculation [37] using the GW method [38]: \(t_{\delta}=54.2\) meV, \(t_{\pi}=490.9\) meV, and the band splitting parameters from Ref. [36]\(\xi=18.8\) meV, \(D=2.2\) meV. The optical-phonon energies at the Brillouin zone center of SrTiO\({}_{3}\) are taken from the experimental data of Ref. [2], the same as described in Ref. [13]. Also, the direct TO-phonon optical response has been included in the figure in the same way as in Ref. [13]. It is represented by sharp peaks in the low-energy part of the optical conductivity spectrum. The calculated many-polaron optical conductivity in SrTiO\({}_{3}\) is compared with the experimental data of Ref. [2] for two temperatures: \(T=7\) K and \(T=300\) K and for several values of the carrier concentration. Also, the earlier calculation the optical conductivity [13] is reproduced in the figure. As can be seen from the low-temperature results shown in the left-hand panels of the figure, the 130-meV peak and the dip at \(\hbar\Omega\sim 200\) meV (corresponding to twice the highest-energy LO-phonon mode ion strontium titanate) experimentally observed at the low temperature and at relatively low concentrations are fairly well revealed in the calculated spectra of the optical conductivity. Figure 1: Many-polaron optical conductivity of \(n\)-doped SrTiO\({}_{3}\) in the mid-infrared frequency range for several values of the doping \(x\) and two temperatures (the parameters are indicated in the figure), corresponding to the experimental conditions of Ref. [2]. The calculated spectra (solid curves) are compared with the experimental data (dashed curves). Dotted curves show the results of Ref. [13]. The subgraphs correspond to the following cases: in (_a,b_) doping relative concentration is \(x=0.1\%\), for \(T=7\) K (_a_) and \(T=300\) K (_b_). The panels (_c,d_), (_e,f_), (_g,h_), describe the results for \(x=0.2\%\), \(x=0.9\%\) and \(x=2\%\), respectively. The obtained expression (24) for the retarded density-density Green's function gives us a transparent explanation of the shape of the optical conductivity spectrum, which is more complicated than in the absence of the dynamic screening. The Green's functions enter the memory function (19) with the arguments \(\left(\pm\Omega-\omega_{L,j}\right)\). Therefore the dynamically screened electron-electron interaction matrix element (22) contains poles, in particular, at \(\Omega=2\omega_{L,j}\left(\mathbf{q}\right)\), which result in dips of the optical conductivity at these frequencies. The most significant contribution to the dips comes from the highest LO-phonon energy \(\left.\hbar\omega_{L,3}\right|_{q=0}\approx 98\) meV. This feature is visible in both the measured and calculated optical conductivity spectra. The part of the spectrum below \(2\omega_{L,3}\) constitutes the aforesaid 130-meV peak.