Pinched hysteresis loop in defect-free ferroelectric materials

In addition to the single polarization-versus-electric field hysteresis loop that is characteristic of ferroelectrics and the double hysteresis loop that is known to occur in antiferroelectrics, a third kind of polarization-versus-electric field function has been reported in several systems. This third kind is commonly termed the “pinched” loop due to its unusual shape, and is typically believed to originate from the pinning of domain walls interacting with defects. Here, using an atomistic effective Hamiltonian scheme, we demonstrate that such a belief has to be broadened since our simulations also yield pinched loops in defect-free ferroelectric materials, as a result of the occurrence of intermediate modulated phases exhibiting an inhomogeneous dipolar pattern leading to the coexistence of both ferroelectric and antiferroelectric orders.

Figure 1

Schematic of three different types of electrical polarization vs applied electric field loops. (a) Single loop, as found in ferroelectrics, (b) double hysteresis loop with zero remnant polarization, as characteristic of AFE compounds, and (c) pinched hysteresis loop with finite but small remnant polarization.

Figure 2

Temperature-composition phase diagram of disordered ${\mathrm{Bi}}_{1-x}{\mathrm{Nd}}_x{\mathrm{FeO}}_3$ solid solutions. The open symbols denote our predicted phase boundaries at varied compositions, and the solid lines are guides for the eyes based on the calculated data. The solid (orange) symbols denote experimental data [20, 24, 25]. The intermediate phases consist of a series of modulated structures energetically and structurally bridging the $R3c$ and the $Pnma$ phase. The numbers 1, 2, and 3 indicate the three temperatures considered in the present simulations for a Nd composition of 15%.

Figure 3

Studied phases of ${\mathrm{Bi}}_{1-x}{\mathrm{Nd}}_x{\mathrm{FeO}}_3$. (a) The ferroelectric $R3c$ phase. (b) The antiferroelectric orthorhombic $Pnma$ phase. (c) Complex phases with modulation of cation displacements (which results in the formation of a polarization and an AFE vector) and of oxygen octahedral distortions being along the pseudocubic [001] direction. To illustrate the structural difference, two opposite tiltings about the [001] axis are denoted by red and blue curled arrows, respectively. The purple arrows on Bi/Nd atoms indicate the local electric dipole, as averaged for each (001) layer. (d) Normalized component of the polarization along the [111] direction of all the occurring phases (the $R3c$ phase has a reference polarization of 1 while the polarization of the $Pnma$ phase is zero). The vesta code is used for the visualization [32].

Figure 4

Calculated $\mathit{P}-\mathit{E}$ loops of ${\mathrm{Bi}}_{1-x}{\mathrm{Nd}}_x{\mathrm{FeO}}_3$ having a 15% Nd composition, at three temperatures: (a) 300, (b) 700, and (c) 1100 K. The electric field is applied along the [111] direction, and the displayed polarization is the projected value in this direction. Solid symbols denote the $R3c$ phase, and open symbols denote complex phases. Arrows indicate how the electric field is varied in the simulated cycle.