Finite-temperature properties of the relaxor ${\mathrm{PbMg}}_{1/3}{\mathrm{Nb}}_{2/3}{\mathrm{O}}_3$ from atomistic simulations

An atomistic numerical scheme is developed and used to study the prototype of relaxor ferroelectrics, that is ${\mathrm{PbMg}}_{1/3}{\mathrm{Nb}}_{2/3}{\mathrm{O}}_3$ (PMN), at finite temperatures. This scheme not only reproduces known complex macroscopic properties of PMN, but also provides a deep microscopic insight into this puzzling system. In particular, relaxor properties of PMN are found to originate from the competition between (1) random electric fields arising from the alloying of Mg and Nb ions belonging to different columns of the Periodic Table within the same sublattice; (2) the simultaneous condensation of several off-center $\mathbf{k}$ points as a result of a specific short-range, antiferroelectriclike interaction between lead-centered dipoles; and (3) ferroelectriclike interactions. Such origins contrast with those recently proposed for the homovalent Ba(Zr,Ti)${\mathrm{O}}_3$ solid solution, despite the fact that these two materials have similar macroscopic properties—which therefore leads to a comprehensive understanding of relaxor ferroelectrics.

Figure 1

Predicted temperature dependence of several properties in PMN solid solutions. Panel (a) shows the dielectric susceptibility, and its inverse, when cooling down the disordered PMN system under no bias field. $T_B$ and $T_0$ correspond to the Burns temperature and the critical temperature extrapolated from the Curie-Weiss law at high temperatures, respectively. Panel (b) displays the $x$ component of the supercell average of the local mode, $〈\mathbf{u}〉$, (which is equal to its $y$ and $z$ components) when heating the system is under no field, after having cooled it under an electric field and then having removed this field at 10 K. The depolarizing temperature is $T_{\mathrm{depol}}=250\pm 50$ K and is indicated by a vertical dashed line. The left inset of panel (b) shows an example of the dipolar configuration when the field is removed at 10 K. The right inset of panel (b) shows the $x$ component of $〈\mathbf{u}〉$, but when the system is cooled under no field. The error bars in panels (a) and (b) are those arising from computing averages over 30 different chemical configurations. Panels (c) and (d) show the dielectric susceptibility (as well as its inverse) and the $x$ component of the supercell average of the local mode (which is identical to its $y$ and $z$ components), respectively, when random fields are switched off. The left inset of panel (d) displays the resulting dipolar configuration at 10 K, when random fields are turned off.

Figure 2

Snapshots of the local modes' patterns within a $(x,y)$ plane for different temperatures, panels (a)–(d), and temperature dependence of the average size of the polar nanoregions, panel (e), when disordered PMN is cooled under no field. In panels (a)–(d), the corresponding temperature is indicated on the top left and the red lines delimit the PNRs. The error bars in panel (e) are those resulting from incorporating 30 different chemical configurations into the computation of the averaged PNR size.

Figure 3

Properties associated with anticorrelations and off-center points in disordered PMN. Panels (a) and (b) show the ${\theta }_{x,x}\left(\mathbf{r}\right)$ correlation between the $x$ components of the local modes centered on lead atoms for the r vectors ling in the $(x,y)$ plane at 10 K for two different realizations of the disordered PMN system. Panels (c) and (d) display the temperature dependence of the square of the Fourier transform of the $x$ component of the local modes' configurations for different ${\mathbf{k}}_{max,i}$ points for the atomic configurations corresponding to panels (a) and (b), respectively. The three integers $n_x$, $n_y$, and $n_z$ indicated in the legends of panels (c) and (d) index the ${\mathbf{k}}_{max,i}$ points, that is such $k$ points are given by $\frac{2\pi }{18a_{\mathrm{lat}}}(n_x\mathbf{x}+n_y\mathbf{y}+n_z\mathbf{z})$ for our $18\times 18\times 18$ supercell. Note also that the square of the Fourier transform of the $x$ component of the local modes' configurations is invariant by inversion in the reciprocal space, i.e., $\frac{2\pi }{18a_{\mathrm{lat}}}(n_x\mathbf{x}+n_y\mathbf{y}+n_z\mathbf{z})$ and $\frac{2\pi }{18a_{\mathrm{lat}}}(-n_x\mathbf{x}-n_y\mathbf{y}-n_z\mathbf{z})$ have precisely the same value of the Fourier transform.