Atomistic treatment of depolarizing energy and field in ferroelectric nanostructures

An atomistic approach allowing an accurate and efficient treatment of depolarizing energy and field in any low-dimensional ferroelectric structure is developed. Application of this approach demonstrates the limits of the widely used continuum model (even) for simple test cases. Moreover, implementation of this approach within a first-principles-based model reveals an unusual phase transition—from a state exhibiting a spontaneous polarization to a phase associated with a toroid moment of polarization—in a ferroelectric nanodot for a critical value of the depolarizing field.

Figure 1

Depolarizing factors $⟨F_{xx}^{\left(D\right)}\left(l\right)⟩$, $⟨F_{yy}^{\left(D\right)}\left(l\right)⟩$, and $⟨F_{zz}^{\left(D\right)}\left(l\right)⟩$ obtained with our atomistic approach and normalized to $4\pi$ (that is the prediction of $F_{zz}$ in the continuum model) in the case of (001) films [(a)] and $2\pi$ (prediction of $F_{xx}$ in the continuum model) in the case of wires periodic along the $z$ axis [(b)] as a function of the layer or shell index $l$. The different shells of a wire are shown in the inset of part (b). The most inner layer or shell is indexed by 0. The different symbols correspond to different thicknesses.

Figure 2

Normalized depolarization energy as a function of $d∕L$ for quasi 0D, 1D, and 2D systems with stripe domains along with continuum model prediction (Refs. 13, 15) for $d⪡L$ and $d⪢L$ for $D=2$ systems. The inset shows the schematic representation of the chosen polarization pattern in (001) films, or wires periodic along $y$ or cubic dots.

Figure 3

(a) Dipole moments and (b) toroid moments of polarization $G$ in a PZT nanodot as a function of the screening coefficient $\beta$. Insets of (a) and (b) show the polarization pattern for $\beta =1$ (SC conditions) and $\beta =0$ (OC conditions), respectively.