Polarization Fluctuations and the Optical-Absorption Edge in BaTiO3

Abstract

Results of optical absorption and electroabsorption (EA) measurements in the vicinity of the interband absorption edge are reported for top-seeded solution-grown crystals of BaTi${\mathrm{O}}_3$. In common with other perovskite oxides, the absorption edge in BaTi${\mathrm{O}}_3$ is found to display Urbach-rule behavior. The exponential absorption tail can be described between 20 and 450 °C by an effective temperature $T^{*}=T+T_0$, where $T_0=140\mathrm{}$ K, i.e., $\alpha \propto e^{\frac{\hslash \omega }{k{T}^{*}}}$. Although no uniquely defined band gap can be extracted from an exponential edge, we propose, on the basis of indirect arguments, that the room-temperature band gaps are 3.38 and 3.27 eV, respectively, for light polarized parallel and perpendicular to the ferroelectric $c$ axis. At high temperatures in the cubic phase, the band gap decreases at the rate -4.5×${10}^{-4\mathrm{}}$ eV/°C. EA measurements in the tetragonal phase show that an applied electric field along the $c$ axis shifts the entire Urbach edge rigidly upward in energy by an amount $\Delta \mathcal{E}$, which is proportional to the square of the total polarization $P$, spontaneous plus field-induced, i.e., $\Delta \mathcal{E}=\beta P^2$. The effect can be described by a temperature-independent band-edge polarization potential $\beta$ having the value ${\beta }_{11}=1.\mathrm{}16$ eV ${\mathrm{m}}^4$/${\mathrm{C}}^2$. The smaller ${\beta }_{12}$ coefficient could not be measured, because of photoconductivity and carrier-trapping effects. An anomalous increase in the band gap with decreasing temperature within 150°C of the Curie point is attributed to coupling between polarization fluctuations and the band edge. A simple thermodynamic model is shown to describe the temperature dependence of this fluctuation contribution with reasonable accuracy. The results suggest that the correlation volume $V_c$ is at most a weak function of temperature and that $V_c$ does not display critical behavior. This conclusion is consistent with several recent experiments in displacive ferroelectrics. The magnitude of the observed mean square polarization fluctuation contribution to the band-edge position ($\approx 15$ meV at $T=T_C$) can be understood using the simple fluctuation theory with the value $V_c\approx 4.5\times {10}^4$ ${\mathrm{\AA }}^3$ deduced previously from photoelastic constant measurements. It is also suggested that a mean square polarization fluctuation contribution to the band-edge position is present in the tetragonal phase below approximately 100 °C owing to the proximity of the tetragonal-orthorhombic transition. A fluctuation contribution of about 40 meV is indicated at room temperature.