Strain-induced incommensurate phases in hexagonal manganites

An incommensurate phase refers to a solid state in which the period of a superstructure is incommensurable with its primitive unit cell. It was recently shown that an incommensurate phase, which displays a single chiral modulation of six domain variants, could be induced by applying an in-plane strain to a hexagonal manganite. Here we combine Landau theory description of thermodynamics and the phase-field method to investigate and understand the formation of the incommensurate phase in hexagonal manganites. It is shown that the equilibrium wavelength of the incommensurate phase is determined by both the temperature and the magnitude of the applied strain, and a temperature-strain phase diagram is constructed for graphically displaying the temperature and strain conditions for the stability of the incommensurate phase. Temporal evolution of domain structures reveals that the applied strain not only produces the force pulling the vortices and antivortices in opposite directions, but also results in the creation and annihilation of vortex-antivortex pairs.

Figure 1

Domain patterns on the basal plane of $h\text{−}\mathrm{REMn}{\mathrm{O}}_3$. (a) and (b) Vortex domains from (a) a phase-field simulation and (b) an optical microscope. (c) and (d) Single-chirality striped domains from (c) a phase-field simulation and (d) atomic force microscope scanning on a chemically etched sample.

Figure 2

Comparison between the XY model and YMO. (a) and (b) Energy landscapes in the order parameter space for (a) the XY model and (b) YMO at 0 K. The magnitude and phase of the order parameter are labelled by $Q$ and ϕ, respectively. (c) and (d) Distribution of the order parameter along the modulation direction for (c) the XY model and (d) YMO. In (c), λ denotes the wavelength of the modulation.

Figure 3

Profiles of the order parameters $Q$ and ϕ in $h\text{−}\mathrm{YMO}$ at different temperatures and under different strains. (a)–(c) Profiles of $Q$ and ϕ with $\xi =0.75\mathrm{eV}/\mathrm{nm}$ at (a) 1195, (b) 1170, and (c) 1100 K. (d)–(f) Profiles of $Q$ and ϕ with $\xi =2.83\mathrm{eV}/\mathrm{nm}$ at (d) 1195, (e) 1170, and (f) 1100 K.

Figure 4

Equilibrium wavelength of the IC phase. (a) Free energy as a function of wavelength for the XY model and YMO. The pink line denotes the free energy of the XY model at 1100 K with $\xi =2.39\mathrm{eV}/\mathrm{nm}$. The blue line, cyan line, and green line are the results of YMO at 1100 K with $\xi =2.39$, 1.64, and 0.90 eV/nm, respectively, which represent the energy profiles of the IC phase, intermediate state (IM), and C phase, respectively. (b) Equilibrium wavelength as a function of ξ for the XY model and YMO at different temperatures.

Figure 5

Temperature-Lifshitz invariant coefficient ($T$-ξ) phase diagram of $h\text{−}\mathrm{YMO}$. In the high-symmetry (HS) phase, all the order parameters are equal to zero. The solid lines are from the phase-field method, and the dashed lines are from Eq. (20).

Figure 6

Temporal evolution of the domain structures on the basal plane in $h\text{−}\mathrm{YMO}$ at 1199 K. (a) Initial domain structure. (b) Intermediate domain structure with $\xi =1.49\mathrm{eV}/\mathrm{nm}$ for the upper half, and $\xi =0.00\mathrm{eV}/\mathrm{nm}$ for the lower half. (c) Final domain structure with $\xi =1.49\mathrm{eV}/\mathrm{nm}$ for the upper half, and $\xi =0.00\mathrm{eV}/\mathrm{nm}$ for the lower half. (d) Final domain structure with $\xi =2.98\mathrm{eV}/\mathrm{nm}$ for the upper half, and $\xi =0.00\mathrm{eV}/\mathrm{nm}$ for the lower half. The colors are assigned based on the nearest C domains.

Figure 7

Creation and annihilation of vortex-antivortex pairs under the effect of the applied strain. (a)–(d) Zoomed-in snapshots from a phase-field simulation. (a) and (b) show the creation of the vortex-antivortex pair (3− and 4+) near two bubble-like domains. The circled out region in (a) indicates the position where the vortex-antivortex pair is created. (b)–(d) demonstrate the annihilation of the vortex-antivortex pair (2+ and 3–). (e)–(g) Schematics for the creation of a vortex-antivortex pair from a domain wall.