Quantum paraelectricity of ${\mathrm{BaFe}}_{12}{\mathrm{O}}_{19}$

Using path-integral Monte Carlo sampling and a lattice Wannier function based effective Hamiltonian obtained from first principles, we show that the quantum fluctuations of the nuclei play a central role in the paraelectric phase of ${\text{BaFe}}_{12}{\text{O}}_{19}$ at low temperatures $(T$'s). Contribution from the geometrical frustrations, on the other hand, is negligible. The $T$ dependence of the dielectric function and the displacements of the Fe ions reported in previous experiments were reproduced. The nature of the quantum paraelectrics (QPEs) was assigned to order-disorder or displacive, using the relative magnitudes of the on-site/intersite energies of the electric dipoles, and the quantum fluctuations of the nuclei in the path-integral samplings. This study provides a unified atomic-level picture of QPEs, which could be used to understand different experiments in which new QPE materials were found.

Figure 1

(a) Crystal structure of ${\text{BaFe}}_{12}{\text{O}}_{19}$. Green balls denote Ba ions, yellow balls denote Fe ions, and red balls denote O ions. (b) Phonon spectrum of ${\text{BaFe}}_{12}{\text{O}}_{19}$ calculated by the first-principles calculations using $\text{PBE}+\text{U}$. (c) A comparison between the potential energy surface (PES) obtained from our model Hamiltonian (solid lines) and from DFT (markers) calculations. PESs along different high-symmetry $\mathbf{k}$-points were denoted by different markers, and the results calculated along the same $\mathbf{k}$-points by the model and DFT were represented by the same color.

Figure 2

(a) Schematic of the redefined lattice (blue lines) used in our model. Parts (b) and (c) show the short-range intersite coupling contained in our model. For clarity, we show only the TBP Fe-O polyhedrons whose centers are located on the 2b site (the lattice site of our re-defined lattice). (d) Schematic of the lattice Wannier function.

Figure 3

(a) Susceptibility vs temperature calculated by CMC and PIMC simulations with strain along the $c$-axis, ${\eta }_c=0.15\%$. The red dashed line is fitted by the Barrett formula. Inset of (a): Compare between PIMC results (purple circles) and previous experiments (black dots); the experimental data are extracted from Shen et al.'s paper [11]. (b) The inverse of the susceptibility from CMC (blue squares), PIMC (purple circles), and a linear fitting to the CMC results (dashed line). (c) The thermal and quantum contribution to the fluctuations vs $T$.

Figure 4

Susceptibility with changed nearest-neighbor short-range intersite coupling $J_1$. The circles denote that $J_1$ is not changed so that the symmetry and geometry frustration are preserved. To compare, one can also change one $J_1$ of the three bonds of the triangular lattice, as schematically shown on the upper right corner. The red line denotes the changed bonds. Squares denote the results when $J_1^{\prime }=0.5J_1$ and triangles denote the results when $J_1^{\prime }=1.5J_1$.

Figure 5

(a) Distributions of the amplitude $u$ of the lattice Wannier functions at 4 and 295 K. (b) $u$-dependence of the total and on-site energies with M-FE dipole configuration.

Figure 6

Parts (a) and (c) show the $u$-dependence of the total and on-site energies for the case of $E_{\text{on-site}}>E_{\text{inter}}$ and $E_{\text{on-site}}<E_{\text{inter}}$, respectively. Parts (b) and (d) show the distributions of $u$ obtained by performing PIMC simulations (4 K with $T{N}_{\text{time-slice}}=600)$ for systems shown in (a) and (c), respectively, with different effective mass. In panel (b), the effective mass $m=10$, 55, and 130 corresponds to the case of displacive QPE, order-disorder QPE, and ordered FE, respectively. In panel (d), the effective mass $m=10$ and 55 corresponds to the case of displacive QPE and ordered FE. We renormalize the maxima of all distributions to the same value for clarity.