Phase diagram of ferroelectrics with tricritical and Lifshitz points at coupling between polar and antipolar fluctuations

Available experimental data about static and dynamic critical behaviors of ${\text{Sn}}_2{\text{P}}_2{\text{S}}_6$-type ferroelectrics and $({\mathrm{Pb}}_y{\mathrm{Sn}}_{1-y}{)}_2{\mathrm{P}}_2{\left({\mathrm{Se}}_x{\mathrm{S}}_{1-x}\right)}_6$ mixed crystals with a line of tricritical points and a line of Lifshitz points on the $T-x-y$ phase diagram, which meet at the tricritical Lifshitz point, are described in a combined Blume-Capel anisotropic next-nearest-neighbor Ising model. Such spin-1 Ising models with anisotropic competing first- and second-neighbor interactions is applied for the considered ferroelectrics with mixed displacive versus order/disorder character of phase transitions within the framework of a microscopic model with three-well total energy surface for ferroelectric distortion that was earlier built in an ab initio effective Hamiltonian approach. It was found that below the temperature of the tricritical Lifshitz point, the “chaotic” state accompanied by the coexistence of ferroelectric, metastable paraelectric, and modulated phases is expected. In addition to the frustration of polar fluctuations near the Brillouin zone center, in ${\mathrm{Sn}}_2{\mathrm{P}}_2{\mathrm{S}}_6$ crystals the antipolar fluctuations also strongly develop in the paraelectric phase on cooling to the continuous phase-transition temperature $T_0$. Here, the critical behavior can be described as a crossover between Ising and XY universality classes, which is expected near bicritical points with coupled polar and antipolar order parameters and competing instabilities in $q$ space.

Figure 1

(a) The crystal structure of $P{2}_1/n$ paraelectric phase of ${\mathrm{Sn}}_2{\mathrm{P}}_2{\mathrm{S}}_6$ crystals [27]. The crystallographic $b$ axis coincides with the $2_1$ symmetry axis and the (010) plane with the glide plane $n$. The inversion center is placed in the middle of the $P-P$ bonds of (${\mathrm{P}}_2{\mathrm{S}}_6{)}^{4-}$ anions. The positions of the tin cations (large spheres) in the paraelectric and ferroelectric phases are shown by different colors. Two sublattices are presented by two formula units in a monoclinic elementary cell. (b) Schematic representation (for tin ions only) of eigenvectors of the lowest energy polar $B_u$(39 ${\mathrm{cm}}^{-1}$) and the fully symmetrical $A_g$ (41 ${\mathrm{cm}}^{-1}$) long wave modes [28]. (c) Frozen-phonon energy surface (in eV) for a linear combination of $A_g$ and $B_u$ mode amplitudes for ${\mathrm{Sn}}_2{\mathrm{P}}_2{\mathrm{S}}_6$ within LDA calculations [28]. Black circles denote the positions of the global ferroelectric minima. (d) Presentation of the BC-ANNNI model which describes an array of pseudospins related to ${\mathrm{Sn}}_2{\mathrm{P}}_2{\mathrm{S}}_6$ formula units in two sublattices and flipped in the three-well potential with crystal field strength $\delta$ (energy difference between central and side wells). $J_1$—NN interaction within sublattices, $J_2$—NNN interaction between sublattices.

Figure 2

(a) Front and (b) side views on calculated $t-\lambda -\mathrm{\Delta }$ diagram. Solid lines show paraelectric-ferroelectric (orange) and paraelectric-modulated (blue) borders, Lifshitz point (LP) lines (red circles), tricritical point (TCP) lines (green squares), and end point (EP) lines (brown triangles). Thin lines indicate the second-order phase transitions while thick lines are the first-order ones. Letters F, P, and M correspond to ferroelectric, paraelectric, and modulated phases. The coexistence of ferroelectric and metastable paraelectric ($\mathrm{F}+\mathrm{P}$') and ferroelectric, metastable paraelectric, and commensurate modulated phases ($\mathrm{F}+\mathrm{M}$'$+\mathrm{P}$') below the dashed lines are also shown.

Figure 3

Dependence of the phase transition temperature on the single-ion term in dimensionless $t-\mathrm{\Delta }$ coordinates calculated in the mean-field approximation on the BC model [37, 57]. Dashed and solid lines indicate second- and first-order transitions, respectively, that meet at the tricritical point.

Figure 4

(a) Experimental phase diagram in $t-\lambda -\mathrm{\Delta }$ coordinates for $({\mathrm{Pb}}_y{\mathrm{Sn}}_{1-y}{)}_2{\mathrm{P}}_2{\left({\mathrm{Se}}_x{\mathrm{S}}_{1-x}\right)}_6$ ferroelectrics and (b) front view of both experimental and calculated phase diagrams in temperature-concentration $T-x-y$ coordinates. Dashed and solid lines denote second-order and first-order phase transitions, respectively. Green and red spheres correspond to tricritical point lines and Lifshitz point lines, blue spheres to the calorimetric data [36, 56, 58, 59, 60, 61, 62].

Figure 5

Temperature dependence of (a) heat capacity according to experimental data [63] and (b) related to the phase transition excess heat capacity for $({\mathrm{Pb}}_y{\mathrm{Sn}}_{1-y}{)}_2{\mathrm{P}}_2{\mathrm{Se}}_6$ crystals.

Figure 6

Comparison of the anomalies for low-frequency dielectric susceptibility (according to data [21, 22, 64]) and excess heat capacity (according to data [63]) in the region of the phase transitions for $({\mathrm{Pb}}_y{\mathrm{Sn}}_{1-y}{)}_2{\mathrm{P}}_2{\mathrm{Se}}_6$ crystals.

Figure 7

(a) Determined by neutron scattering at 440 K transverse soft optical $TO\left(X\right)$, acoustic longitudinal $LA\left(XX\right)$ and transverse $TA\left(XY\right)$ phonon branches along $q_y$ direction of ${\mathrm{Sn}}_2{\mathrm{P}}_2{\mathrm{S}}_6$ crystal monoclinic Brillouin zone [46]; (b) temperature dependence of the dielectric susceptibility and its reciprocal from submillimeter soft optical mode contribution near the phase transition in ${\mathrm{Sn}}_2{\mathrm{P}}_2{\mathrm{S}}_6$ crystal [68]; (c) comparison of the inverse of the dielectric susceptibility temperature dependence in ${\mathrm{Sn}}_2{\mathrm{P}}_2{\mathrm{S}}_6$ crystal [69] for 20 MHz—1, 4 GHz—2, 27 GHz—3 and submillimeter soft optical mode contribution—4; (d) diffuse x-ray scattering at $T_0+2$ K in the paraelectric phase of ${\mathrm{Sn}}_2{\mathrm{P}}_2{\mathrm{S}}_6$ crystal in the plane (0KL) of the Brillouin zone [51].

Figure 8

(a) Dependence of the inverse of thermal diffusivity measured in the [100] crystallographic direction in the vicinity of continuous phase transition in ${\mathrm{Sn}}_2{\mathrm{P}}_2{\mathrm{S}}_6$ and ${\mathrm{Sn}}_2{\mathrm{P}}_2{\left({\mathrm{Se}}_{0.2}{\mathrm{S}}_{0.8}\right)}_6$ crystals [58, 59]. (b) Results of the fittings of the inverse of thermal diffusivity for ${\mathrm{Sn}}_2{\mathrm{P}}_2{\mathrm{S}}_6$ crystal, and (c) corresponding deviation curves for the fittings. The points correspond to experimental measurements, the continuous lines to the fittings to Eq. (6). Blue color corresponds to the fitting below the critical temperature, red to the ones above it.