Introduction

Nanoscale ferroelectric materials attract broad fundamental interest and are of high demand across emerging technologies due to their unique properties stemming from switchable spontaneous polarization. Notably, ferroelectric nanostructures host a variety of topological polarization textures1,2,3,4,5,6,7,8, which are tunable by external fields or temperature variations. This versatility is a key factor in various applications, including highly sensitive sensors and information-storage devices, as well as innovative nonbinary, neuromorphic, and low-dissipation computing circuits9,10,11,12,13,14.

Currently, research on topological textures in ferroelectrics is primarily focused on thin films and superlattices1,8,15,16,17,18, where the in-plane misfit strain acts as a driving parameter for the system. Nanorods, or nanowires—long cylindrical nanoparticles with nanoscale diameters—form another distinct group of nanostructured ferroelectrics. Despite substantial experience in their fabrication dating back to the 90s19,20,21,22,23,24,25,26, they remain less explored, particularly in regard to polarization distribution. The axial geometry of nanorods suggests two primary polarization directions defined by the shape-induced anisotropy of the crystal lattice: along the nanorod axis or perpendicular to it. By analogy with magnetism27, we refer to them as easy-axis and easy-plane configurations of polarization, respectively. The interplay between them generates various topological states of polarization, including uniform easy-axis state, polarization domains, vortices, and helicoidal structures28,29,30. However, tuning this competition in practice to manipulate the emerging topological states presents a challenge.

In this work, we find that the surface tension naturally arising at the nanorod’s surface acts as a geometry-dependent driving parameter. In combination with temperature and external mechanical loads, surface tension effects can be used to manipulate and switch the topological polarization states in ferroelectric nanorods.

Surface tension, a prominent phenomenon in liquids, arises from the cohesive forces between surface molecules, driven by imbalanced intermolecular attractions31,32,33. This creates a net inward force that reduces the liquid’s surface area and alters the shape of liquid volumes. Surface tension plays an essential role in various environmental and biological processes, such as the formation of rainwater droplets34, insects walking on water35, and the ability of plants to transport water from their roots to their leaves through capillary action36. In solids, since the molecular forces at the surface differ from those in the bulk, the phenomenon of surface tension also manifests itself37,38.

In contrast to liquids, where molecules can move freely, solids resist deformation due to their atoms being arranged in a stable lattice structure. Surface tension does not significantly alter the shape of a crystal body but generates internal anisotropic stresses and deformations within the crystal lattice. The stress anisotropy depends on the geometry of a sample. In spherical particles, only isotropic hydrostatic compression is produced, which, when applied to ferroelectrics, does not affect the crystal anisotropy but merely reduces the critical temperature of the transition39. In elongated nanorods, the shrinking of the lateral surface produces different strains along and across the nanorod axis, ultimately compressing the nanorod more along its axis than in the radial direction38.

We explore the effects of surface tension in ferroelectric nanorods by evaluating the magnitude and anisotropy of lattice deformation, based on the results of analytical calculations and numerical simulations. We demonstrate that the surface-tension-induced compressive stress along the nanorod axis results in the emergence of the non-uniform easy-plane vortex states, which compete with the easy-axis uniform polar phase, having smaller gradient energy. This result opposes the conventional belief that surface tension compresses the nanorod more in the radial direction, stabilizing the uniform easy-axis polar phase—a belief based on the incomplete consideration of surface shrinkage, focusing solely on radial stress forces39,40,41. The ability to select between the easy-plane vortex state and the easy-axis uniform state by varying the radius of a nanorod and its temperature has practical significance for hands-on creation and switching of topological states.

Results

Polarization states in ferroelectric nanorods

It is intuitively expected that the spontaneous polarization P in a ferroelectric free-standing nanorod tends to align with the nanorod axis. Such a polarization distribution minimizes the depolarization energy of bound charges, which emerge at the rod surface. Specifically, in pseudocubic ferroelectric with crystallographic c-direction oriented along the rod z-axis and a, b-directions oriented in the x-y plane, perpendicular to the rod axis, this alignment forms a uniform c-phase, P = (0, 0, Pc), as shown in Fig. 1a. In this case, the depolarization field of the edge bound charges disrupts the uniform state only at the tips of the nanorod but does not significantly affect the c-phase away from the edges30. Naturally, states where P is perpendicular to the nanorod axis, such as a-phase with P = (Pa, 0, 0) and aa-phase with P = (Pa, Pa, 0), are less energetically favorable. This is because, in these configurations, bound charges are generated at the lateral surface, and the large perpendicular to the axis depolarization field extends over the entire volume of the nanorod, as shown in Fig. 1b.

Fig. 1: Polarization states in ferroelectric nanorods.
figure 1

a Uniform c-phase. Polarization P (blue arrows) is oriented along the nanorod’s axis, which coincides with the crystal c-axis. The axes orientation is shown to the side of the nanorod. The surface-bound charges shown by red symbols “+” and “−” emerge at the rod edges where P abruptly terminates. b Uniform a-/aa- phases in which P is oriented perpendicular to the nanorod’s axis. Bound charges emerging at the lateral surface induce the depolarization electric field, E, shown by the red arrows. c Charge-neutral vortex phase.

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In this work, we conduct atomic second-principle simulations and confirm them by phase-field and analytical modeling to study the polarization distribution in free-standing cylindrical PbTiO3 (PTO) nanorods, with radii R ranging from 1 to 16 nm. Surprisingly, we observe that beyond the uniform c-oriented polarization state, another steady state emerges in which the polarization is not only oriented perpendicular to the nanorod axis but also forms a non-uniform vortex configuration swirling around the axis, as illustrated in Fig. 1c. Such a situation usually occurs when compressive strain shrinks the nanorod along the z-axis, thereby making the easy-plane orientation of polarization more favorable. In this case, the vortex state, with polarization tangential to the lateral surface and corresponding absence of bound charges is stabilized to reduce the depolarization energy42. On the opposite, when the tensile strain stretches the nanorod along the z-axis, the uniform c-phase with easy-plane polarization orientation is stabilized. The corresponding strain-temperature phase diagram for different states in a loaded ferroelectric PTO nanorod is given in28. Here, to explain the origin of the topological phases in free-standing nanorods, we demonstrate that the effects of surface tension uniaxially deform the nanorods. This leads to easy-plane anisotropy and enables the switching between different polarization states.

Surface tension in solid nanorods

The surface tension arises because molecules at the surface are attracted to each other more strongly than to the molecules beneath the surface. This causes the surface to behave as if it was covered by a stretched elastic membrane, which subjects the sample to volume deformation. In elongated liquid droplets, surface tension causes them to contract along the elongation axis, transforming into a more spherical shape. While fluids can only support the isotropic component of the stress tensor, which corresponds to Laplace pressure, in solid rods, the emerging shear stresses also induce anisotropic internal compressive deformations, which, as shown below, lead to easy-plane anisotropy. In ferroelectric materials, this effect stabilizes the vortex phase.

The surface-tension-deformed state in solids is the result of the balance of the elastic energy, E = Esurf + Ebulk, which includes the shrinking surface tension contribution,

$${E}_{{{{{\rm{surf}}}}}}=\gamma {S}_{0}+\gamma {\int}_{S}{u}_{ij}^{{{{{\rm{s}}}}}}ds,$$
(1)

and the bulk elastic contribution,

$${E}_{{{{{\rm{bulk}}}}}}=\frac{1}{2}{\int}_{V}{C}_{ijkl}{u}_{ij}{u}_{kl}\,dV,$$
(2)

which opposes the shrinking.

Here, the summation over the repetitive indices {i, j, k, l} = {x, y, z} is assumed. The coefficient γ is the surface tension energy per unit of surface, Cijkl is the stiffness tensor, \({u}_{ij}=\left({\partial }_{j}{u}_{i}+{\partial }_{i}{u}_{j}\right)/2\) is the volume strain tensor, and \({u}_{ij}^{{{{{\rm{s}}}}}}={u}_{ij}-{u}_{lk}{n}_{l}{n}_{k}{n}_{i}{n}_{j}\) is the surface strain tensor, where ui is the displacement vector, and ni is the unit vector normal to the surface. The value S0 is the surface area of a non-deformed sample; the surface integral in (1) is the surface stretching with respect to S0. The integral in (2) is given over the volume of the sample, V. For simplicity, we assume that the surface tension coefficient is isotropic in all surface directions and that deformation strains are uniform throughout the sample volume.

In cylindrical geometry, the tendency of the surface to shrink induces two types of forces at the lateral surface, produced by summing up the tangential stresses acting on each segment of the shrinking surface (Fig. 2a–c). The first, transversal ones, generate the radial compressive stress σrr (written in cylindrical coordinates (r, θ, z)), illustrated in Fig. 2a by the red arrows. The second, longitudinal forces, acting along the z-axis, produce the compressive stress σzz along the nanorod, shown by the green arrows in Fig. 2a. We find these stresses using the virtual displacement method. Considering that the work necessary to increase the cylinder radius by δR, δW = σrr(2πRh)δR contributes to increasing the surface energy of the system, δE = γ(2πh)δR, we derive the stress σrr = −γ/R, which corresponds to the Laplace pressure in liquids. Here, h is the cylinder height (h ≫ R), and 2πRh is its lateral surface area. In a similar way, considering that the work necessary to elongate the cylinder by δh along z, δW = σzz(πR2)δh contributes to increasing the surface energy of the system, δE = γ(2πR)δh, we derive the stress σzz = −2γ/R. Although both types of stress result in the compression of the crystal unit cell, their impacts on crystal anisotropy are opposite. Transversal stress tends to compress the nanorod in the x-y plane, thereby increasing the c/a ratio above unity. In contrast, longitudinal stress compresses the nanorod in the z-direction, resulting in a c/a ratio of less than 1.

Fig. 2: Surface tension in nanorods.
figure 2

a Radial Laplace (red arrows) and axial (green arrows) compressive stresses in an exemplary cylindrical section of radius R and height h of an infinitely long nanorod. The inset and its enlarged image shown in (b) explain how these stresses are caused by surface tension, which shrinks the surface, as indicated by purple arrows. The tension forces directed laterally and vertically result in radial and axial stresses, respectively. The microscopic origin of the surface tension is explained in (c). Surface molecules, shown in orange, are more strongly attracted to each other than the molecules in the bulk, shown in gray, which leads to the tendency of the nanorod’s lateral surface to shrink in all directions. d Estimation of the surface tension constant from inverse radius dependence of a and c lattice parameters in BaZrO3 (BZO) and PbTiO3 (PTO) nanorods. The dashed lines represent the fit to the simulation data, indicated by markers, using equations (3).

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The resulting deformation strain in the cylinder is obtained from the elasticity equation \({u}_{ij}={C}_{ijkl}^{-1}{\sigma }_{kl}\),

$${u}_{xx}={u}_{yy}=-\frac{1}{3}\left(\frac{4}{3K}-\frac{1}{2\mu }\right)\frac{\gamma }{R},\qquad {u}_{zz}=-\frac{1}{3}\left(\frac{4}{3K}+\frac{1}{\mu }\right)\frac{\gamma }{R}.$$
(3)

Here, K = (C11 + 2C12)/3 > 0 is the bulk compressibility, and μ = (C11 − C12)/2 > 0 is the shear modulus along the cubic directions. The Voigt notations Cij are used for the stiffness tensor Cijkl. In cubic crystal C11 = Ciiii, C12 = Ciijj. Notably, the anisotropic stiffness coefficient C44 = Cijij does not enter into the final expressions. Similar expressions were obtained in38 through energy minimization for nanorods with square cross-sections.

To investigate the surface-tension-induced anisotropy, we consider the ratio of the transverse to axial deformations, uxx/uzz, expressed through the Poisson’s ratio coefficient ν,

$$\frac{{u}_{xx}}{{u}_{zz}}=\frac{1}{2}\frac{1-3\nu }{1-\nu },\qquad \nu =\frac{1}{2}\frac{3K-2\mu }{3K+\mu }.$$
(4)

Remarkably, the easy-plane anisotropy, characterized by uxx/uzz < 1 is consistently induced across the entire interval of the crystal’s elastic stability, which is defined by the condition  −1 < ν < 1/243. Furthermore, the nanorod is always contracted along the axial z-direction, and, depending on the material parameter ν, either expands, for 1/3 < ν < 1/2, or contracts, for  −1 < ν < 1/3, in the transverse x-y directions. The majority of solids possess a Poisson’s ratio coefficient ν in the range 0.2–0.344,45, leading to transversal shrinking. Such behavior should also be characteristic of auxetic materials46, displaying a negative Poisson’s ratio coefficient  −1 < ν < 0. At the same time, the transversal expansion takes place in soft materials such as rubber with ν → 1/244,45 and even liquids with ν = 1/2, tending to make droplets at free-standing conditions.

Notably, some previous publications on ferroelectrics39,40,41 claim the opposite, an easy-axis type of resulting anisotropy. The shortfall in these calculations is that only the radial compressive stress from surface forces, corresponding to the Laplace pressure, was properly accounted for, while the longitudinal stress was omitted. This incomplete consideration of surface tension effects is critical, as it leads to results opposite to those presented in this work.

Surface tension constant

We utilize atomistic second-principle simulations to explore the effects of surface tension on the deformation of solid nanorods, which arise from changes in the environment of atoms in the vicinity of the surface. As the number of interactions decreases compared to the bulk, these atoms move toward new equilibrium positions. By considering the Cauchy–Born hypothesis, which assumes that atomic positions follow macroscopic deformations47, it becomes possible to connect surface properties between the atomic model and the continuous description. The modeling details are outlined in “Methods” section.

To distinguish the deformations induced by surface tension from those caused by electrostriction in ferroelectric state, we initially focus on the non-ferroelectric dielectric material BaZrO3 (BZO). The cubic structure lattice parameter and stiffness constants in the model are a0 = 4.186 Å, C11 = 320 GPa, C12 = 99 GPa, and C44 = 95 GPa at 10 K, in good agreement with experimental values48.

The estimation of the surface tension is directly based on the strained lattice approach via Eqs. (3), where the coefficient γ is obtained by fitting the strains as a function of the inverse radius 1/R. This method is suitable for experimental validation through measuring the lattice constants a and c in surface-tension-deformed nanorods of various radii and comparing these values with the corresponding bulk crystal values, a0 and c0, which relate to strain tensor components as uxx = uyy = (a − a0)/a0 and uzz = (c − c0)/c0. The behavior of a and c as a function of the inverse radius 1/R, presented in Fig. 2b, highlights the contraction induced by the lateral surface. The changes in lattice parameters occur more strongly along the z-axis than in x-y plane, which is consistent with the Poisson’s ratio ν = 0.237 calculated from the elastic constants. As expected, surface effects increase when the radius decreases. The fitting of lattice constants with Eqs. (3) estimates γ for BZO to be in the range γ ≈ 2.94–4.01 m−1 N.

After establishing the effects of the surface on the deformation of non-ferroelectric BZO nanorods, we now consider the deformation caused by surface on ferroelectric PTO nanorods. Figure 2b also presents the calculated lattice parameters of PTO as a function of the inverse radius of the PTO nanorod at 900 K, in the cubic paraelectric phase, which enables us to distinguish the deformations caused by surface tension from those induced by spontaneous polarization, emerging below the Curie temperature Tc = 752 K.

Notably, the splitting of the lattice parameters in PTO is similar to that seen in BZO. Employing the strained lattice fit approach we obtained the surface tension coefficient in PTO to be γ ≈ 5.39–5.51 m−1 N. For the sake of self-consistency, the stiffness constants for calculations were taken from the simulations, C11 = 272.1 GPa, C12 = 125.8 GPa, and C44 = 139.5 GPa. These values are in good agreement with other ab initio calculations49 and moderately exceed the experimental values C11 = 174.6 GPa, C12 = 79.36 GPa, and C44 = 111.1 GPa, calculated from ref. 50. The corresponding Poisson’s ratio ν = 0.316, falling within the interval 0 < ν < 1/3, supports our conclusion that the surface causes compressive shrinkage of the PTO nanorods, resulting in easy-plane anisotropy.

Phase diagram of the surface-tension-tuned polarization states

Three competitive effects define the topological states in ferroelectric nanorods: (i) the electrostatic depolarization forces, which tend to align the polarization vector field tangentially to the rod’s surface; (ii) the surface-tension-induced easy-plane anisotropy, which inclines the polarization vector to lie in a plane perpendicular to the axis; and (iii) the gradient-minimizing energy, which favors uniform polarization states. As a result, the competition occurs between two distinctive states, the vortex state, see Fig. 1c, that wins the anisotropy competition but possesses the unfavorable gradient energy, and the uniform gradient-free c-phase, see Fig. 1a, which, however, may be less favorable because of the surface-tension-induced anisotropy contribution.

To uncover the conditions under which these states emerge in ferroelectric PTO nanorods and to study their stability across various system parameters, specifically. temperature, T, and nanorod radius, R, we conduct atomistic second-principles and phase-field modeling. This is complemented by analytic calculations to complete the simulations.

In the atomistic description, molecular dynamics simulations are used to compute polar and elastic system properties through the temporal evolution of atoms’ positions.

In phase-field modeling we are based on the relaxation of the time-dependent variational Ginzburg–Landau–Khalatnikov equation:

$$-\tau \frac{\partial {P}_{i}}{\partial t}=\frac{\delta }{\delta {P}_{i}}\int\,{{{{{\mathcal{F}}}}}}^{{{{{\rm{u}}}}}}({P}_{i},{u}_{i},{\partial }_{i}\varphi )dV.$$
(5)

Here, τ represents the relaxation time scale parameter, and \({{{{{\mathcal{F}}}}}}^{{{{{\rm{u}}}}}}({P}_{i},{u}_{i},{\partial }_{i}\varphi )\) is the Ginzburg–Landau free energy functional specified in Methods. This function depends on the polarization order parameter Pi, displacements ui, and the electric field Ei  = − ∂iφ, where φ is the electrostatic potential.

In analytical calculations, we compare the energies of the arising polar phases employing reasonable analytical approximations for the Ginzburg-Landau free energy \({{{{{\mathcal{F}}}}}}^{{{{{\rm{u}}}}}}({P}_{i},{u}_{i},{\partial }_{i}\varphi )\). For the detailed description of all approaches, see Methods.

The resulting R-T phase diagram of emerging topological states in PTO nanorods is displayed in Fig. 3a. The colored regions indicate the stability zones for the emerging states, as obtained by the phase-field simulation method: the paraelectric phase (p), the vortex phase (v, \({v}^{{\prime} }\)), and the c-phase (c) are represented by green, orange, and blue, respectively. There are two types of vortex states, the axisymmetric ones (v), and the deformed ones (\({v}^{{\prime} }\)). Their stability regions are distinguished by the light-orange and dark-orange shades. The transition temperatures between the paraelectric phase and vortex phase, and between the vortex phase and the c-phase, obtained by atomistic simulations, are depicted by square and circle markers respectively. The solid red and solid blue lines, which correspond to these transitions, are calculated analytically. For comparison, the surface-tension-induced transitions between the paraelectric and uniform ferroelectric phases, namely the c-phase (see Fig. 1a) and the aa-phase (see Fig. 1b), which are stabilized in the absence of depolarization charges, are presented by dotted lines, see “Methods”. The bottom panel demonstrates polarization distribution in the emergent topological states, calculated by the phase-field method (upper row) and atomistic methods (lower row).

Fig. 3: Phase diagram of topological states in a PbTiO3 nanorod.
figure 3

a Regions of stability for the paraelectric phase (p), vortex phase (v, \({v}^{{\prime} }\)), and uniform c-phase (c), as obtained from phase-field simulations, are represented by green, orange, and blue colors, respectively. The light and dark orange shades indicate axisymmetric vortices (v) and deformed vortices (\({v}^{{\prime} }\)) regions. Transition temperatures, marked by red squares and blue circles, along with solid red and blue lines, are determined through atomistic and analytical methods, respectively. Dotted lines depict the transitions between the paraelectric phase and the emerging without depolarization charges uniform aa-phase (in red), and between aa-phase and c-phase (in blue). The lower panel displays the polarization distribution in the emerging topological states, obtained from phase-field simulations (top row), and atomistic simulations (bottom row). Temperature dependence of average lattice constants, a and c (b), and average square polarization  <P2> (c) for the nanotube with R = 10 nm along the line A-B in the phase diagram, obtained by different methods. Markers correspond to atomistic simulations, solid lines represent the phase-field results, and thin dashed lines show the analytical estimations in the uniform phase model. Note that atomistic simulations qualitatively reproduce but underestimate the magnitude of P(T), see “Methods”.

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Discussion

The coherence of the results obtained by atomistic, phase-field, and analytical calculations confirms the emergence of the polar phases with vanishing bound charge density ρ, as defined by the fundamental topological condition7,51,52

$$\rho =\,-{{{{\rm{div}}}}}\,{{{{\bf{P}}}}}\,=\,0,$$
(6)

and stabilized by the competition between the surface-tension-induced anisotropy energy and gradient energy contributions. The axisymmetric vortex phase (v) appears on the R-T phase diagram (Fig. 3a) in regions where surface-tension-induced anisotropy dominates the gradient forces. This particularly occurs at small radial sizes of nanorods and temperatures close to the transition to the paraelectric phase. Atomistic simulations reveal a more anisotropic vortex pattern, characterized by a quadratic flux-closure shape. Notably, as the radius of the rod increases, the vortex core becomes deformed, taking on the shape of a domain wall (phase \({v}^{{\prime} }\)). This deformation, clearly evident in atomistic simulations, is also observed in phase-field simulations, and it occurs to minimize the elastic energy of the core region30. For a more detailed comparison of the vortex textures obtained through various types of simulations see Fig. 4. At large radii and low temperatures, the significance of surface-tension-induced anisotropy diminishes, stabilizing the uniform c-phase.

We discuss now the temperature dependence of the quantitative parameters of the nanorod, shown in Fig. 3. As follows from the calculated phase diagram, see Fig. 3a, all the transition temperatures scale as R−1 when the nanorod radius decreases. Interestingly, a similar reduction of the Curie temperature, scaling with the radius as R−1, was experimentally observed in barium titanate BaTiO3 (BTO) nanowires53, and was attributed to the chemical properties of the surface. However, the nanowires studied in this work are not free-standing but substrate-deposited, which restricts shrinking along the nanorod’s axis. In contrast, another research focusing on free-standing BTO nanorods at zero temperature revealed a large suppression of the axial lattice parameter at the nanoscale, with c/a becoming less than 1, as opposed to the bulk BTO54, similar to the behavior shown in Fig. 2d for PTO and BZO nanorods.

Figure 3b, c presents the temperature dependence of the quantitative parameters of the nanorod when sweeping along the line A-B in Fig. 3a. The temperature evolution of the average lattice parameters is shown in Fig. 3b. In the paraelectric phase, above the critical temperature Tc ≃ 720 K, the nanorod exhibits a distinct easy-plane anisotropy with a lattice anisotropy ratio c/a < 1. This anisotropy promotes the transition into the vortex phase at T < Tc, with the ratio c/a decreasing as the temperature decreases. At temperature Tcv ≃ 450 K the transition to the uniform c-phase occurs. It is accompanied by an inversion of the ratio c/a, which becomes more than one. The temperature dependence of the averaged square polarization  <P2> , as shown in Fig. 3c, becomes nonzero below Tc and exhibits a jump to higher values at Tcv. Remarkably, the temperature dependencies of the lattice parameters and polarization magnitude, obtained through analytical calculations and numerical simulations, agree well with each other. Note here that although the observed transition from the vortex to the c-phase occurs abruptly, there exists another continuous scenario28. In this case, the vortex phase gradually unwinds towards the uniform c-phase, accompanied by the emergence of the z-component of the polarization and the formation of an intermediate helical phase. Such a scenario may be realized in ferroelectric materials under different elastic conditions.

Calculated from the atomistic simulations value of surface tension constant γ ≃ 5–5.5 m−1 N is suitable to effectively tune the transition between different topological states in ferroelectric nanorods with radii R = 1–16 nm. Notably, these values exceed those found in liquids, even surpassing the highest surface tension constant observed in mercury, which is γ = 0.48 m−1 N55. Considering the current lack of developed experiments on surface tension determination in solids and the inconsistencies in theoretical estimations, which vary from 1 to 50 m−1 N39,56,57,58, the discovered relationship between surface tension and stabilized topological states in ferroelectric nanotubes could offer a useful tool for experimentally uncovering surface tension in solids.

The surface tension effect offers an alternative to the currently employed method of driving the transition between different polarization phases through substrate-induced misfit strain59. Specifically, it can contribute to shape engineering for creating and tuning complex heterogeneous dipole configurations, such as packets of 90∘ domains60, or various sequences of a- and c-domains in elastically anisotropic thin films61. Furthermore, the surface tension effect can be combined with external mechanical loads28,62. This combination can serve as an efficient tool for establishing the desired ferroelectric states in nanorods, nanoparticles, thin films, and heterostructures. The discovery of the surface-tension-driven superposition of v- and c-phases, in which the polarization forms a helical texture28, would be of particular interest, as such states possess a definite chirality that is useful for bio-medical and optoelectronic applications63,64.

Methods

Atomic-level simulations

The atomic-level simulations are based on the shell model description. The model approach describes each atom as two charged and coupled particles: a massive core and a massless shell, while the potential energy of the system is expressed as a function of the interatomic (core and shell) distances rij:

$$E=\sum \frac{{q}_{i}{q}_{j}}{{r}_{ij}}+\sum \frac{1}{2}{k}_{2}{r}_{ij}^{2}+\frac{1}{4!}{k}_{4}{r}_{ij}^{4}+\sum \left(A+B{r}_{ij}\right){e}^{(-{r}_{ij}/\rho )}-\frac{C}{{r}_{ij}^{6}},$$
(7)

where the first term corresponds to Coulomb interactions between cores and shells of different atoms, the next two terms represent the contribution of the core-shell coupling in each atom and the last two account for the contributions of short-range interactions between shells. The coefficient values for describing BZO and PTO with energy (7) are those reported in ref. 65 and ref. 66, respectively. The nanorods are constructed from elementary perovskite ABO3 unit cells centered on B-type atoms and with axes along pseudocubic directions (x, y and z axes lie along the [100], [010], and [001] directions, respectively). Furthermore, we consider that all surface faces consist of AO planes.

For all atomistic calculations, we use the LAMMPS package67 to perform molecular dynamics simulations. A constant temperature algorithm was used to determine the structure of the nanocylinders. The cylindrical sections of a nanorod are surrounded by a vacuum with periodic boundary conditions along z-axis. We analyzed the effects of the size of the exemplary sections of different radii (5–40 unit cells) by varying their height between 10 and 30 unit cells and observed no changes in the structure’s topology or transition temperatures. The time step was 0.1 fs, which provided enough accuracy for the integration of the shell coordinates. The total time of each simulation, after 5 ps of thermalization, was at least 40 ps. Since the PTO model overestimates the Curie temperature of the bulk, temperatures were rescaled so that Tc in the simulations matches the experimental bulk value. Such temperature adjustment is a common practice in different atomistic model approaches, showing good agreement of temperature-dependent characteristics with their experimental values after rescaling68,69,70,71,72,73. The elastic constants of both compounds were calculated from their relationship with the stress fluctuations according to the Born matrix method74. In this case, simulations were performed for constant volume and temperature ensembles in system sizes of 12 × 12 × 12 unit cells with periodic boundary conditions.

Note that the polarization is not an explicit variable in the atomistic description, and local values are calculated using the charges and positions of cores and shells as:

$${{{{\bf{p}}}}}=\frac{1}{v}\sum \frac{{z}_{i}}{{\omega }_{i}}\left({{{{{\bf{r}}}}}}_{i}-{{{{{\bf{r}}}}}}_{{{{{\rm{B}}}}}}\right)$$
(8)

where v is the volume of the cell, zi and ri denote the charge and the position of the i-th particle, respectively, rB denotes the B-type atom position, and ωi is a weight factor equal to the number of cells to which the particle belongs. The model generally underestimates the magnitude of bulk polarization66, so the values of P in nanorods and nanoparticles are likely to exhibit similar behavior. Nevertheless, the qualitative results obtained in the atomic-level simulations are in complete agreement with those from the phase-field method. The consistency of polarization behavior, when explored by different approaches, can be seen from the comparison of radial polarization distribution in vortex phases (Fig. 4).

Fig. 4: Vortices in PbTiO3 nanorods with surface tension: consistency of simulation methods.
figure 4

a Atomic-level simulations of the polar vortex phase v. b Atomic-level simulations of the deformed vortex (phase \({v}^{{\prime} }\)). c Phase-field modeling of the polar vortex phase v. d Phase-field modeling of the deformed vortex (phase \({v}^{{\prime} }\)). Upper colorbar corresponds to atomic-level results (a, b), lower colorbar to phase-field results (c, d). Note that atomic-level simulations underestimate the magnitude of the polarization yet match the features of polarization pattern.

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Free energy functional of a strain-coupled ferroelectric system

Following the general approach to the description of ferroelectrics based on the Ginzburg–Landau–Devonshire functional \(F={\int}_{V}{{{{\mathcal{F}}}}}dV\), we model the free energy density of the pseudo-cubic perovskite crystal in terms of the polarization components Pi, elastic stress tensor σij, and electrostatic potential φ as9,75,76,77:

$${{{{\mathcal{F}}}}}=\, {\left[{a}_{i}(T){P}_{i}^{2}+{a}_{ij}^{\sigma }{P}_{i}^{2}{P}_{j}^{2}+{a}_{ijk}{P}_{i}^{2}{P}_{j}^{2}{P}_{k}^{2}\right]}_{i\le j\le k}+\frac{1}{2}{G}_{ijkl}({\partial }_{i}{P}_{j})({\partial }_{k}{P}_{l})\\ -{Q}_{ijkl}{\sigma }_{ij}{P}_{k}{P}_{l}-\frac{1}{2}{s}_{ijkl}{\sigma }_{ij}{\sigma }_{kl}+({\partial }_{i}\varphi ){P}_{i}-\frac{1}{2}{\varepsilon }_{0}{\varepsilon }_{{{{{\rm{b}}}}}}{(\nabla \varphi )}^{2}.$$
(9)

Here, indices {i, j, k, l} = {1, 2, 3} (or {x, y, z}), and the summation over repeated indices is assumed. The coefficients for PTO in free energy density (9) take standard values50,76,78,79: a1 = A (T − Tc) with A = 3.8 × 105 C−2 m2 N K−1, Tc = 752 K (Curie temperature), \({a}_{11}^{\sigma }=-0.73\times 1{0}^{8}\) C−4 m6 N, \({a}_{12}^{\sigma }=7.5\times 1{0}^{8}\) C−4 m6 N, a111 = 2.6 × 108 C−6m10 N, a112 = 6.1 × 108 C−6m10 N, and a123 = −37 × 108 C−6 m10 N. The stress-polarization coupling tensor components are Q1111 = 0.089 C−2 m4, Q1122 = −0.026 C−2 m4, Q1212 = 0.03375 C−2 m4, and the compliance tensor components are s1111 = 8 × 10−12 m−2 N−1, s1122 = −2.5 × 10−12 m−2 N−1, and s1212 = 9 × 10−12 m−2 N−1. The gradient energy coefficients76 are G1111 = 2.77 × 10−10 C−2 m4 N, G1122 = 0, and G1212 = 1.38 × 10−10 C−2 m4 N. The constant ε0 is the vacuum permittivity, and εb ≃ 10 is the background permittivity of PTO due to nonpolar ions80.

The stress-coupled free energy density (9) is commonly used to describe the bulk free-standing systems. However, for the description of confined nanosystems, it is more practical to use the deformations corresponding to the confined degrees of freedom as driving parameters rather than stress variables. Therefore, we perform the Legendre transformation, \({{{{{\mathcal{F}}}}}}^{{{{{\rm{u}}}}}}={{{{\mathcal{F}}}}}+{\sigma }_{ij}{u}_{ij}\), and switch to the strains, as to variables conjugated to the stresses, \({u}_{ij}=-\partial {{{{\mathcal{F}}}}}({P}_{i},{\sigma }_{ij},{\partial }_{i}\varphi )/\partial {\sigma }_{ij}\), to partially substitute stresses by corresponding strains. Thus, we arrive at a mixed description in terms of both stress and strain variables, which allows to adjust to the particular confined geometry depending on the studied properties. Finally, we substitute the remaining stress components for our particular system (a cylindrical nanorod of radius R) by taking into account the surface-tension-induced mechanical relations σxx + σyy = σzz = −2γ/R, as derived in the main text. The surface tension constant γ = 5.45 m−1 N is taken from the atomistic simulation results. The resulting strain-coupled functional is:

$${{{{{\mathcal{F}}}}}}^{{{{{\rm{u}}}}}} = \, {\left[{a}_{i}(T){P}_{i}^{2}+{a}_{ij}^{{{{{\rm{u}}}}}}{P}_{i}^{2}{P}_{j}^{2}+{a}_{ijk}{P}_{i}^{2}{P}_{j}^{2}{P}_{k}^{2}\right]}_{i\le j\le k}+\frac{1}{2}{G}_{ijkl}({\partial }_{i}{P}_{j})({\partial }_{k}{P}_{l})\\ -{q}_{ijkl}{u}_{ij}{P}_{k}{P}_{l}-\frac{1}{2}{C}_{ijkl}{u}_{ij}{u}_{kl}+\frac{\gamma }{R}({u}_{ii}+{u}_{zz})\\ +({\partial }_{i}\varphi ){P}_{i}-\frac{1}{2}{\varepsilon }_{0}{\varepsilon }_{{{{{\rm{b}}}}}}{(\nabla \varphi )}^{2}.$$
(10)

Here, the stiffness tensor components are calculated as the inverse of the compliance tensor, \({C}_{{ijkl}}={s}_{{ijkl}}^{-1}\), the electrostrictive coefficients are derived as \({q}_{{ijkl}}={s}_{{ijkl}}^{-1}{Q}_{{ijkl}}\), and the coefficients \({a}_{ij}^{{{{{\rm{u}}}}}}\) are recalculated for the strain-coupled functional following the procedure in ref. 81.

To gain an initial understanding of the polar states that can be stabilized in the presence of surface tension, we explore the uniform phases with constant polarization within a nanorod’s volume, in the absence of depolarization charges. At this stage, we minimize energy functional (10) without electrostatic terms and compare the energies of three distinct ferroelectric phases: (i) c-phase, P = (0, 0, Pc); (ii) aa-phase, P = (Pa, Pa, 0); (iii) r-phase, P = (Pa, Pa, Pc). The latter configuration appears to be unfavorable for free-standing PTO nanorods; however, we observe that surface tension induces the occurrence of in-plane aa-phase, which is a precursor to the topological vortex phase. The region of in-plane phase stability in the radius-temperature diagram expands with higher values of surface tension, as illustrated in Fig. 5. The uniform phase transitions corresponding to γ = 5.45 m−1 N are shown in the diagram in Fig. 3a by dotted lines.

Fig. 5: Uniform states diagram in the presence of surface tension.
figure 5

Solid blue lines correspond to the transition lines between the uniform aa-phase and c-phase in PbTiO3 for different values of surface tension γ. The transition from the paraelectric phase to aa-phase, shown by the red dashed line, corresponds to γ = 1 m−1 N and is only slightly affected by surface tension.

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The next step is to analytically describe the features of the polarization distribution in a strain-coupled axisymmetric vortex state. To do this, we use a simplified isotropic form of the free energy functional (10):

$${{{{{\mathcal{F}}}}}}_{{{{{\rm{iso}}}}}}^{{{{{\rm{u}}}}}}= \, {a}_{1}(T){P}^{2}+{\bar{a}}_{11}^{{{{{\rm{u}}}}}}{P}^{4}+\bar{G}\,{{{{{\rm{rot}}}}}}^{2}\,{{{{\bf{P}}}}}-{\bar{q}}_{12}{u}_{ii}{P}_{j}{P}_{j}-{\bar{q}}_{44}{u}_{ij}{P}_{i}{P}_{j}\\ +\frac{1}{2}{\bar{C}}_{12}{u}_{ii}^{2}+{\bar{C}}_{44}{u}_{ij}^{2}+\frac{\gamma }{R}({u}_{ii}+{u}_{zz}),$$
(11)

in which the isotropic coefficients (denoted with an overbar) follow the relations \({\bar{a}}_{11}^{{{{{\rm{u}}}}}}=2{\bar{a}}_{12}^{{{{{\rm{u}}}}}}\), \({\bar{q}}_{44}={\bar{q}}_{11}-{\bar{q}}_{12}\), and \(2{\bar{C}}_{44}={\bar{C}}_{11}-{\bar{C}}_{12}\) (using the Voigt notations). We take a1(T) as in (9) to have the same bulk critical temperature, Tc = 752 K. The 6th-order polarization terms are neglected in the Ginzburg–Landau part of the functional, and the isotropic coefficients are expressed through the PTO coefficients in a way to match the energy of the uniform c-phase derived from (10). The electrostatic terms vanish as we consider only the depolarization charge-free configurations with \({{{{\rm{div}}}}}\,{{{{\bf{P}}}}}=0\), which also simplifies the gradient term. The gradient energy in PTO has already an isotropic form, therefore we take \(\bar{G}={G}_{1212}\).

The procedure of the minimization of functional (11) and derivation of the vortex structure in frame of the isotropic model is described in30. The resulting long-range asymptotic behavior of polarization with account of surface tension is:

$${P}^{2}\simeq \Lambda \left({a}_{1}+\frac{8{\bar{C}}_{44}{\bar{q}}_{12}+\left(2{\bar{C}}_{44}-{\bar{C}}_{12}\right){\bar{q}}_{44}}{2{\bar{C}}_{44}\left(3{\bar{C}}_{12}+2{\bar{C}}_{44}\right)}\frac{\gamma }{R}\right){\left(\frac{r}{R}\right)}^{\lambda -1}.$$
(12)

Here, coefficients λ and Λ depend on material parameters \({\bar{a}}_{11}^{{{{{\rm{u}}}}}}\), \({\bar{C}}_{ij}\), and \({\bar{q}}_{ij}\) as given in30. Note that the account of surface tension in (12) shifts the coefficient a1 and defines the temperature of the transition from the paraelectric phase to the vortex state (solid red line in Fig. 3a). Ultimately, comparing the minimized energies of the vortex v-state and the previously obtained c-phase allows us to estimate the corresponding transition temperature Tcv, shown in Fig. 3a by the solid blue line.

Phase-field approach

The phase-field approach was used to numerically solve the time-dependent Ginzburg-Landau relaxation equation:

$$-\tau \frac{\partial {P}_{i}}{\partial t}=\frac{\delta }{\delta {P}_{i}}\int\,{{{{{\mathcal{F}}}}}}^{{{{{\rm{u}}}}}}({P}_{i},{u}_{i},{\partial }_{i}\varphi )dV.$$
(13)

Here, parameter τ describes the time scale of the energy dissipation and was equal to unity for convenience, and the free energy density functional \({{{{{\mathcal{F}}}}}}^{{{{{\rm{u}}}}}}({P}_{i},{u}_{i},{\partial }_{i}\varphi )\) is given in (10), with \({u}_{ij}=\left({\partial }_{j}{u}_{i}+{\partial }_{i}{u}_{j}\right)/2\).

To obtain displacements ui and electrostatic potential φ the following linear equations were solved:

$${C}_{ijkl}{\partial }_{j}{u}_{kl}-{q}_{ijkl}{\partial }_{j}{P}_{k}{P}_{l}=0.$$
(14)
$${\varepsilon }_{0}{\varepsilon }_{{{{{\rm{b}}}}}}{\nabla }^{2}\varphi ={\partial }_{i}{P}_{i}.$$
(15)

To check the stability of the calculations with respect to extending the Ginzburg–Landau–Devonshire functional (10) to higher-order terms, we systematically included these terms in the minimization procedure and confirmed that they have a minor influence on the obtained results. In particular, we tested the effect of the widely discussed flexoelectric effect82,83, described by the additional term \(\frac{1}{2}{f}_{ijkl}\left({P}_{k}{\partial }_{l}{u}_{ij}-{u}_{ij}{\partial }_{l}{P}_{k}\right)\) in (10), with typical values of flexoelectric coefficients84∣ f1111 − f1122∣ = 3 V, f1212 = 1 V. Figure 6 illustrates the minor modification in the spatial (radial) dependence of the polarization magnitude P and the elastic tensor component urr in a vortex state when accounting for the flexoelectric effect. We observe that flexoelectric terms do not substantially impact the sequence of the emerging topological structures, and thus these terms can be neglected in the considered system.

Fig. 6: Account for flexoelectric effects in phase-field modeling.
figure 6

Radial distribution of polarization magnitude P (a) and radial strain tensor component urr (b) along x-axis in a polar vortex phase in the nanorod with R = 8 nm and T = 533 K. Dashed lines in both panels present the phase-field results without flexoelectric terms while solid lines correspond to distributions obtained with an account of flexoelectricity.

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Phase-field simulations were performed within the temperature range of 10–800 K with the help of FEniCS software85. To construct the radius–temperature phase diagram in Fig. 3 by phase-field modeling, a set of cylindrical finite element meshes of unstructured tetrahedrons was created with 3D mesh generator gmsh86. To solve both linear (14), (15), and nonlinear (13) equations, we employed the generalized minimal residual method with restarts87. Additionally, we utilized the BDF2 time stepper with adaptive step size88 to approximate the left-hand side of the equation (13). The variables Pi, φ, and ui were approximated by a function from the space of piecewise linear polynomials.

For simulations, we selected an exemplary cylindrical section of the very long (infinite) nanorod with heights h from 1 to 5 nm and radii R ranging from 1 to 16 nm. We applied periodic boundary conditions along z for Pi, checking that the resulting polarization distribution is z-independent and does not depend on h. Constant (zero) boundary conditions were imposed on φ at the top and bottom surfaces of the exemplary cylinders, which reflects the absence of the in-build electric field along the nanorod axis. Additionally, the sample was allowed to elastically relax. With regard to the initial conditions, three different distributions for Pi were employed: (i) a paraelectric phase in which the values of Pi were set randomly with amplitudes not exceeding 10−6 m−2 C, (ii) an axisymmetric vortex phase, and (iii) a c-phase. After setting the initial conditions and temperature, the linear equations were solved to obtain φ and ui, which were then used to obtain Pi on the subsequent time step. This process was repeated until the value of the functional reached a minimum. Thereafter, for each point on the radius-temperature diagram, the configuration with the lowest energy was selected.