Merons in strained ${\mathrm{PbZr}}_{0.2}{\mathrm{Ti}}_{0.8}{\mathrm{O}}_3$ thin films: Insights from phase-field simulations

Materials exhibiting nontrivial topological spin textures known as merons attract considerable interest for their fascinating underlying physics and promising device applications. The ${\mathrm{PbTiO}}_3$ family of compounds is known to host ferroelectric, ferroelastic, and piezoelectric polar orders and possess topological domain structures that are sensitive to external conditions. Here, we examine the polar states in ${\mathrm{PbZr}}_{0.2}{\mathrm{Ti}}_{0.8}{\mathrm{O}}_3$ (PZT) thin films and explore the influence of film thickness and epitaxial strain via phase-field modeling under short-circuit (sc) and open-circuit (oc) boundary conditions. In this paper, we uncovered four distinct meron states in ultrathin films with tensile strains. Under the sc boundary condition, there is an intermediate meron bubble texture comprising a twisted Néel-like and Bloch-like meron texture with a distinct helicity and a meron bubble pair texture combining two intermediate merons along the $z$ axis. Under the oc boundary condition, there is an antimeron bubble texture with negative winding numbers and an achiral topology, and a trimeron bubble texture with two antimeron bubbles and one meron bubble. These simulation results show that strain and film thickness have a major impact on the polarization strength and topology to dictate the formation and transition of meron phases in PZT thin films. The insights obtained in this paper may help elucidate and design materials with tailored and tunable microscopic and mesoscopic polar topological structures.

Figure 1

Sketches of the dipolar textures for skyrmions vs merons. (a) The Néel-like skyrmion with $N_{\text{sk}}=+1,m=+1,\text{and}\gamma =0$. (b) The antiskyrmion with $N_{\text{sk}}=-1,m=-1,\text{and}\gamma =0$. (c) The Bloch-like skyrmion with $N_{\text{sk}}=+1,m=+1,\text{and}\gamma =\pi /2$. (d) The antiskyrmion with $N_{\text{sk}}=-1,m=-1,\text{and}\gamma =\pi /2$. Here, $N_{\text{sk}}$ is topological charge, $m$ is topological vorticity (winding number), and $\gamma$ is helicity. The Bloch-like skyrmion in (c) has right-handed chirality. (e) and (f) are Néel-like meron and antimeron, with $N_{\text{sk}}=+\frac{1}{2},m=+1,\text{and}\gamma =0$ and $N_{\text{sk}}=-\frac{1}{2},m=-1,\text{and}\gamma =0$, respectively. (g) and (h) are Bloch-like meron and antimeron, $N_{\text{sk}}=+\frac{1}{2},m=+1,\text{and}\gamma =\pi /2$ and $N_{\text{sk}}=-\frac{1}{2},m=-1,\text{and}\gamma =\pi /2$, respectively. The topological character of the meron is half that of the skyrmion with topological charge $N_{\text{sk}}=\frac{1}{2}$.

Figure 2

Phase diagram in the parameter space defined by the thickness and epitaxial strain of PZT film under (a) short-circuit (sc) and (b) open-circuit (oc) electrical boundary conditions, simulated by the phase-field modeling approach. In (a), there are eight phases: $c$ zigzag domain ($c$ zigzag), $c$ stripe, $a_1$ ($a_2$) stripe domain insert in $c$ domain ($c/a$ stripe), $c$ domain or $a_1$ ($a_2$) domain like a funnel at the across section ($c/a$ funnel), bowl-shaped $c$ domain insert $a_1$ ($a_2$) domain ($c/a$ bowl), bobber-shaped meron bubble pair (meron pair), intermediate meron bubble (I-meron), and $a_1$ and $a_2$ stripe domain ($a_1/a_2$). In (b), there are six phases: labyrinth phase (labyrinth), $c+$ domain ($c+$), flux closure, trimeron bubble, antimeron bubble, and $a_1$ and $a_2$ stripe domain ($a_1/a_2$). The different colors show distinct textures, and at the boundaries of color areas, phase mixing occurs. Paths 1 and 3 are along the varied strains at 8 and 14 u.c., respectively. Paths 2 and 4 are along the different thicknesses under 0.6 and 0.2% strains, respectively. The white area in (a) and (b) designates nonferroelectric state as shown in Fig. S1(b1) in the Supplemental Material [50]. (c) Two polar textures of I-meron and meron pair under sc boundary conditions. (d) Two polar textures of antimeron and trimeron under oc boundary conditions.

Figure 3

(a)–(d) Polarization vector and (e)–(h) Pontryagin density maps for the $c$ stripe, $c/a$ funnel, I-meron, and meron pair phases under varied strain for 8 u.c. ($\sim 3.2$ nm) thin film along path 1 defined in Fig. 2. The black and white arrows show the in-plane polarization and the curl of polarization ($P$), respectively. These maps depict the spatial distribution of polarization in the cross-section (001) plane, showcasing the variation of the topological textures under different conditions.

Figure 4

(a)–(d) Polarization vector and (e)–(h) Pontryagin density maps for the labyrinth, trimeron, antimeron, and $a_1/a_2$ topological textures under varied strains for 14 u.c. ($\sim 5.6$ nm) thin film along path 3 defined in Fig. 2. The black and white arrows show the in-plane polarization and the curl of polarization ($P$), respectively.

Figure 5

Simulated polar meron bubble pair for 8 u.c. PZT thin film under 1% strain. (a) The topological texture on the top layer (left side) with the cross-sectional view along $q$ and $m$ positions (right side). Inset shows images of two types of I-meron. (b) and (c) A zoom-in view of the meron with in-plane and across the plane at $q$ and $m$ positions, on the top, middle, and bottom layers, respectively. Red area has up $P_{\text{op}}$ and blue area has down $P_{\text{op}}$. The black arrows indicate $P_{\text{ip}}$. (d) and (e) The converged and diverged diabolo-shaped meron pair. The arrow and area indicate the polar vector and cross-sections of the corresponding isosurfaces ($P_{\text{op}}=0$).