Coexistence of the Phonon and Relaxation Soft Modes in the Terahertz Dielectric Response of Tetragonal ${\mathrm{BaTiO}}_3$

The dielectric response to infrared waves polarized along the tetragonal axis of a ferroelectric single-domain crystal of ${\mathrm{BaTiO}}_3$ was determined by time-domain THz spectroscopy and Fourier-transform infrared reflectivity techniques. In addition to the three well-known polar lattice modes, the experiment shows an additional mode of the relaxation type in the THz spectral region, which accounts for the Curie-Weiss behavior of the $c$-axis dielectric constant. A comparison of experimental results with ab initio based effective-Hamiltonian simulations allows us to elucidate its relation to the order-disorder model of Comes, Lambert, and Guinier [Solid State Commun. 6, 715 (1968)].

Figure 1

Frequency dependence of real (a) and imaginary (b) part of complex permittivity ${ϵ}_c$ in the tetragonal phase of ${\mathrm{BaTiO}}_3$ as obtained by THz spectroscopy technique. Continuous lines comes from the Debye model [Eq. (1)] adjusted to the data. Inset: Same data in Cole-Cole representation.

Figure 2

Temperature dependence of (a) Debye mode frequency and (b) inverse static permittivity. Full symbols are from the THz experiment, open symbols are derived from MD simulations described in the text.

Figure 3

Dielectric response of $A_1$-symmetry modes of tetragonal ${\mathrm{BaTiO}}_3$. From top to bottom: reflectivity spectra (a), imaginary (b), and real (c) part of the dielectric function, respectively. Continuous lines are obtained from the simultaneous fitting of THz and far-IR data to the model of Eqs. (2, 3, 4). For clarity, far-IR data are shown for highest and lowest temperatures only. Maxima in the loss spectrum correspond to the Debye ($D$), Last ($L$), Slater ($S$), and Axe ($A$) modes.

Figure 4

Dipole moment $p_z$ of an arbitrarily chosen unit cell as a function of time in MD simulations at $T=T_c-10\mathrm{K}$. The inset shows the 8 possible off-center Ti sites. The Ti ion mostly fluctuates around and among the 4 sites preferred by the molecular field (full circles, $p_z>0$). Occasionally, it also hops towards the excited site (dashed circles, $p_z<0$). These hops correspond to the few negative spikes encountered on the $p_z$ trajectory.