Manifestation of dipole-induced disorder in self-assembly of ferroelectric and ferromagnetic nanocubes

TL;DR

This study investigates how dipole-induced disorder affects the self-assembly of ferroelectric and ferromagnetic nanocubes, revealing that dipoles significantly influence their structural correlations during densification.

Contribution

It provides the first detailed simulation and microstructural analysis of dipolar nanocube self-assembly, highlighting the role of dipoles in structural organization.

Findings

Dipoles strongly affect positional correlations.

Dipolar interactions influence orientational order.

Structural correlations are sensitive to dipole presence.

Abstract

The colloidal processing of nearly monodisperse and highly crystalline single-domain ferroelectric or ferromagnetic nanocubes is a promising route to produce superlattice structures for integration into next-generation devices, whereas controlling the local behaviour of nanocrystals is imperative for fabricating highly-ordered assemblies. The current picture of nanoscale polarization in individual nanocrystals suggests a potential presence of a significant dipolar interaction, but its role in the condensation of nanocubes is unknown. We simulate the self-assembly of colloidal dipolar nanocubes under osmotic compression and perform the microstructural characterization of their densified ensembles. Our results indicate that the long-range positional and orientational correlations of perovskite nanocubes are highly sensitive to the presence of dipoles.

Figures

Equations

$$\beta U_{WCA}\left(r_{ij}\right)=4\left[{\left(\delta r_{ij}^{-1}\right)}^{2p}-{\left(\delta r_{ij}^{-1}\right)}^p+\frac{1}{4}\right],r_{ij}\le \delta \sqrt[p]{2}$$

$$\beta U_{dd}\left({\mathbf{\mu }}_{\mathbf{i}},{\mathbf{\mu }}_{\mathbf{j}},{\mathbf{r}}_{\mathbf{ij}}\right)=-\frac{\lambda }{r_{ij}^3}\left[3\left({\hat{\mathbf{\mu }}}_{\mathbf{i}}\cdot {\hat{\mathbf{r}}}_{\mathbf{ij}}\right)\left({\hat{\mathbf{\mu }}}_{\mathbf{j}}\cdot {\hat{\mathbf{r}}}_{\mathbf{ij}}\right)-{\hat{\mathbf{\mu }}}_{\mathbf{i}}\cdot {\hat{\mathbf{\mu }}}_{\mathbf{j}}\right]$$

$$\lambda =\frac{1}{4\pi {ϵ}_r{ϵ}_0}\frac{\beta {\mu }^2}{{\sigma }^3}$$

$$⟨n⟩=\frac{{\sum }_{n}n{g}_n}{{\sum }_n{g}_n}$$

$$\beta F_2\left(\Delta {\xi }_{12}\right)=\beta U\left(\Delta {\xi }_{12}\right)-\log J\left(\Delta {\xi }_{12}\right)+\beta {\tilde{F}}_2\left(\Delta {\xi }_{12}\right)$$

$$\beta F_2\left(\mathbf{r}\right)=-\ln g_2\left(\mathbf{r}\right)+const.$$

Full text

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††footnotetext: a* Institute of Solid State Physics, Kengaraga str. 8, LV-1063 Riga, Latvia.††footnotetext: b University of Latvia, Raina bulv. 19, LV-1586 Riga, Latvia. E-mail: dmitrijs.zablockis@lu.lv*††footnotetext: c* Max Planck Institute for Solid State Research, Heisenbergstr. 1, 70569 Stuttgart, Germany. E-mail: kotomin@fkf.mpg.de *

1 Introduction

As a fabrication strategy self-assembly1, 2, 3, 4, 5, 6 is a promissing pathway to novel materials with structural hierarchy, multi-functionality and programmed response to mechanical stress or external field7, 8, 9, 10. It is of great interest to the nanoscale community11, 12, 13, 14 and fundamental importance for continued technological advancement. Ferroelectric perovskite-type metal oxides15, incl. BaTiO3, and their solid solutions with SrTiO3 are high-performance dielectrics widely used in a broad variety of electronic, electro-optic, photovoltaic, electro-chemical16, 17, 18, 19, microelectromechanical (MEMS) components20, 21. Likewise, efficient ambient or biomechanical energy scavenging20, 5 exploiting piezo-/pyroelectric properties22, 23 of a family of ABO3 perovskites to power low-energy (incl. wearable or medical) devices is an attractive green energy solution. In turn, further downscaling of ferroelectric devices to achieve improved properties (higher dielectric strength, lower loss), which are limited by the grain size, is in strong demand but becoming increasingly difficult in the low nano-range (sub 100 nm) via a conventional top-down paradigm20. In a different concept a key challenge to further advance the immense practical potential16 - unique spontaneous polarization and exceptional piezoelectric properties24, 20 - of these materials is fabricating highly ordered 3D functional nanoarchitectures (e.g. thin films25) over extended areas - ferroelectric supercrystals - from individual ferroelectric nanocrystals9. Recent advances in chemical synthesis26, 27, 28, 15, 4 have begun to deliver bulk quantites of nearly monodisperse highly crystalline single-domain nanocubes of some ABO3 perovskites (incl. BaTiO328, 4, SrTiO327 and a range of solid solutions), which are ideal space-filling building blocks with a 100% theoretical packing efficiency, and enabled solution processing via a simple, easily scalable and highly versatile colloidal route to produce superlattice assemblies29, 4, 9 and more complex hierarchies29, 4, 9 for integration into nanostructured ferroelectric devices30, 31, 5. Meanwhile, understanding and excercising precise control over the local behaviour of anisometric nanocrystals is absolutely imperative for fabricating highly-ordered assemblies29, e.g. for energy harvesting applications5, 32. The current picture of the nanoscale polarization structure33, 34, 35 and scaling limits of ferroelectric order in individual nanocrystals strongly suggests a potential presence of significant electrodipolar interaction between free-standing ferroelectric monodomains36. However, the understanding of the role of the dipole-dipole interaction in the condensation of nanocubes beyond the basic scenario of pair-wise attachment36 or the dilute gas-like phase37, 38, 39, 40, in the high-density limit, is an outstanding challenge. Here we report the microstructural characterization of densified ensembles produced under osmotic compression from thermalized cubic particles with embedded point dipoles as a minimal model for evaporation-driven self-assembly of ferroelectric and ferromagnetic nanocubes. The paper is organized as follows: in Section 2 we describe our model and the simulation approach: we exploit a discrete element method (DEM) coupled with particle-particle particle-mesh (P3M) approach for computing long-range dipolar interactions; ab initio quantum chemical calculations are used for parameter range estimation. In Section 3 we characterize the produced ensembles employing a variety of statistical descriptors. The experimental implications in relation to ferromagnetic and nanoperovskite nanocube assemblies are discussed in Section 4. To conclude, Section 5 provides the summary of this work.

2 Methods

2.1 Self-assembly (DEM) simulations

In practice a limited amount of self-assembly can be obtained by dropcasting a surfactant-capped nanocube solution onto a compatible substrate (TEM grid or wafer) and drying under ambient conditions 41, 42, 43, 44, 45, 46, 47, 48, 49, 50. However, dropwise deposited assemblies typically lack long-range order and the amount of defects in the produced layers is high owing to a poor control of the growth mode 51, 45, 1, 47, 52. In turn, the immersion methods (e.g. dip coating), whereby the deposition substrate is directly immersed into a nanocrystal solution evaporating under controlled conditions, are much slower (up to several days45 or weeks2) and afford continuous quasi-equilibrium crystallization into extended 3D superlattices with high degree of coherence over exceptional length scales 45, 53, 2, 49, 52, 50 by a controlled evolution of the osmotic pressure at the crystallization front. The assembly of highly crystalline superlattices grown by solution processing is achieved with high-quality nanocrystals having a tight size distribution (standard deviation < 5%), uniform shape and compatible ligand coverage 44, 53, 2, 50, 49. Hence, for self-assembly simulations of monodisperse cubes we use discrete element method (DEM)54 coupled with particle-particle-particle-mesh (P3M) approach for computing long-range dipolar interactions (Supplemental Information). Briefly, the excluded volume interactions between simulated cubes are assembled from a minimal set of lower-dimensional geometric features (i.e. vertices, edges, faces), using point-wise repulsion of all vertex-face and edge-edge pairs at their nearest points. The interaction is modeled by the WCA 12-6 potential ($p=6$), which is a combination of soft potentials

[TABLE]

and projects a uniform elastic shell $\frac{1}{2}\delta$ around the cubic shapes reflecting the soft structure of the colloids ligated by an organic coat\bibnoteThe typical interparticle spacing in colloid superlattices is $1\div 3$ nm as shown by TEM47, 52, 4, 9, which is about half as small than twice the length of the surfactant (e.g. $2$ nm for oleic acid/oleylamine) indicating interdigitation or tilt of the covalently bound molecules. Hence, we choose $\delta=0.1\sigma$ for $\sigma$=20 nm.. The dipole-dipole interaction between two assigned point dipoles $\mu_{i}$ and $\mu_{j}$

[TABLE]

where $\lambda$ is a measure of the relative strength of the dipole-dipole interaction $\propto\lambda k_{B}T$ vs thermal fluctuations $\propto k_{B}T$:

[TABLE]

The simulated consolidations of 15 625 cubes starting from low density states were done by slowly ramping the pressure within the isothermal-isobaric (NPT) ensemble using Martyna-Tobias-Klein hybrid scheme (barostat-thermostat) with periodic boundary conditions. The compression runs simulate the entire self-assembly process induced by osmotic pressure during solvent evaporation from a dilute suspension to high density states, which are subsequently measured by statistical descriptors.

2.2 Ab initio dipole moment calculations

Structurally both titanium-containing perovskites BaTiO3 and SrTiO3 exhibit a cubic (paraelectric) phase above a certain temperature, where each Ti ion is octahedrally coordinated to six oxygen ions. This structure belongs to a centrosymmetric space group, Pm$\overline{3}$m (SG:221), and therefore cannot reveal ferroelectricity, which is a specific property of non-centrosymmetric lattices. In BaTiO3 the ferroelectric order arises below the Curie temperature (TC = 120 ∘C) triggered by a spontaneous tetragonal distortion (P4mm, SG:99) of the prototype cubic symmetry - off-center shift of the Ti4+ ions along one of the cube primary axes and a corresponding displacement of ionic sublattices producing a spontaneous polarization (approx. 0.25 C/m2 bulk) in the same direction.

Ferroelectricity evolves via extended cooperative ordering of charge dipoles from local titanium off-center displacements within the Ti-O6 octahedra. Previous high-energy XRD and atomic PDF studies indicated a progressive decay of ferroelectric coherence across the free standing nanocrystals in the diminished size regime 56, 57, 58, 33, 59, whereas the tetragonal distortion itself persists down to sub-10 nm scale 56, 57, 59, 4. In turn, high fidelity mapping of ferroelectric structural distortions in discrete sub-10 nm nanocubes of BaTiO3 via aberration-corrected transmission electron microscopy confirmed a coherent ferroelectric monodomain state with a stable net polar ordering33. Likewise, in situ off-axis electron holography showed a clear room-temperature electrostatic fringing field characteristic of a linear dipole emanating from a $\langle 001\rangle$ plane 33 and stable ferroelectric polarization switching.

First-principles (ab initio) quantum-chemical estimates of the dipole moments in BaTiO3 and SrTiO3 perovskite structure (Supplemental materials) are performed using advanced hybrid functionals of the density-functional-theory (DFT), which already proved reliable for piezoelectric properties24; with Hay and Wadt effective small-core pseudopotentials (ECP), previously optimized for BaTiO3 and SrTiO3 crystals 60, for the inner core orbitals of Ba, Sr and Ti atoms and an all-electron basis set for oxygen. The outer-core sub-valence and valence electrons of Ba (5s2, 5p6, 6s2), Sr (4s2,4p, 6,5s2) and Ti (3s2, 3p6, 3d2, 4s2) are calculated self-consistently. A stoichiometric slab structure was cut from a prototype 3D cubic lattice (SG:221, BaTiO3: $a=3.993$ Å; SrTiO3: $a=3.901$ Å) along $\langle 001\rangle$ crystallographic planes. The dipole moment produced by the relaxation of ionic sublattices is used as a lowest estimate for small nanocubes.

3 Results

3.1 Ab initio parametrization

A straightforward calculation of the dipole moment for a 20 nm ferroelecric BaTiO3 nanocube based on its bulk polarization density (0.25 C/m2) yields an exceedingly large value $\mu\approx 6\cdot 10^{5}$ Debye (D) and $\lambda\approx 3.6\cdot 10^{5}$. In turn, the onset of dipole-induced self-assembly in a solution $\lambda_{crit}$ can be estimated from the second $\sim\phi^{2}$ virial coefficient of dipolar colloids $b_{2}=-\frac{e^{2\lambda}}{3\lambda^{3}}$ (calculated for dipolar spheres61): from $|b_{2}|\phi\approx 1$ for a volume fraction $\phi=5vol.\%$ (dilute state) $\lambda_{crit}\approx 4$. The estimates produced within a phenomenological Landau-Ginzburg-Devonshire theory62, attempting to account for adsorbate-induced charge screening, depolarization and potential tetragonal-cubic composite structure, yielded a dipole moment of approx. 500 D62, 36 and $\lambda<1$, however, using the bulk parameters for particles of such small size is unreliable. The primary uncontrolled factor here is the degree of adsorbate induced charge screening33, 63.

Likewise, in nanoscale perovskites the near-surface oxygen O2- ions can be slightly displaced with respect to the metal ion of the same plane - surface rumpling64 - known to be quite considerable in many oxide crystals. Here, we have calculated the atomic structure of a periodic slab of cubic BaTiO3 crystals and optimized the atomic positions in several (varied from 6 to 12, approx. 1-2 nm) $\langle 001\rangle$ surface planes using ab initio quantum chemical simulations. As a result, we obtain the relaxed slab geometry and characteristic surface dipole moments, which saturate when more than eight lattice planes are allowed to relax, indicative of a large surface polarization and the appearance of a significant electric field near the surfaces of a paraelectric crystal. For the Ti-O2 termination of the nanocubes (e.g. produced by a hydrothermal route 65, 27, 28, 4, 25) the surface rumpling of "naked" BaTiO3 would result in an overall $\langle 001\rangle$ dipole of approx. 5300 D and $\lambda=50$, whereas the bulk may still be in the cubic phase. This result is in accord with a previous estimate by some of us using a classical semi-empirical shell model 64 (2300 D and $\lambda=10$). Hence, due to a broad scatter of estimates, in the self-assembly simulations we varied $\lambda\approx 0\div 20$ within the indicated range as suggested by our quantum-chemical calculations.

3.2 Consolidation of simple cubes: reference state

Colloidal cubes assemble into a simple cubic (SC) lattice across a broad spectrum of edge lengths from approx. 5 nm to >1 $\mu$m 41, 44, 45, 53, 43, 1, 66, 67, 52, whereas slight deviations from a cubic shape - truncation or rounding (e.g. due to the adsorbed organic layer modulating excluded volume interactions) - while still maintaining a cubic symmetry, may enable alternative stable assembly pathways with an expanded phase diagram accomodating a rhombohedral (Rh) distortion68, 48, 69, 49, 70, 71, 50, 13, body-centred tetragonal (bct)52 or face-centered cubic (fcc) 2, 47, 45 packings as well as their intermediates 69, 72.

To establish a reliable reference state we start by simulating the assembly of simple cubes without dipoles (Fig. 2A, B). Under osmotic compression the cubes readily form a simple cubic (SC) crystal as the ensemble density is increased. The transition between a dilute state and a crystalline solid is first order 74 with the coexistence packing densities $\phi$ between 0.45 and 0.5-0.5211, 74, slightly modulated by the amount of delocalized vacancies in the system74. The equation-of-state (EOS) as a function of packing density (Fig. 2C) compares well to the previous Monte-Carlo simulations 11 of hard cubes. For the isotropic state the correspondence with a perturbative virial EOS73 expressed as a power series in density is reasonable as well. The long-range positional and orientational symmetries emerge robustly at high density: cubatic order parameter11 $P_{4}\in\left[0,1\right]$ (Supplemental Information) (Fig. 2C), which is sensitive to the symmetries of the cubic lattice, indicates the emergence of global rotational correlations. The virtual selected area (electron) diffraction75 (SAD) (irradiated along the cubatic director) (Fig. 2B, inset) probes the long-range order and shows a characteristic 4-fold symmetric spot pattern corresponding to the expected crystallographic orientations. Likewise, polyhedral template matching76, 77 (PTM) is used to assign the crystalline structure by graph-based matching of the convex hull formed by the local neighborhood to a set of predetermined templates (Fig. 2B). The spectrum of the root-mean-square deviation (RMSD, Fig. 2D) for a set of standard lattices reliably identifies the SC phase as the dominant motif and its structural distortions. The spectrum is sharp and well separated. The extended shoulder of the SC peak is assigned to the volume conserving finite temperature row-displacements (shearing) induced by collective thermal vibrations accommodated within the monocrystal. After manually inspecting the configurations, we have not observed a single jammed state in the assembled SC lattices, which shows that the kinetic effects are minimal and the compression protocol is adequate.

3.3 Self-assembly of dipolar nanocubes

After assigning $\langle 001\rangle$-dipoles to the initial dilute ensemble the compression protocol is repeated with varying $\lambda$-parameter to access a range of magnitudes of the dipolar interactions, after which a series of metrics are applied to the consolidated state to assess its structural properties.

We measure the positional extent of the mesoscale cubatic correlations using the orientational correlation function $G_{4}\left(r\right)\in\left[0,1\right]$ (Supplemental Information), which quantifies the degree of mutual alignment of cubes as a function of their separation distance $r$. For assemblies of simple cubes ($\lambda=0$) forming an SC lattice $G_{4}=1$ irrespective of distance indicating the presence of strong long-ranged correlations extending beyond the sampling ability of the simulation box. For all $\lambda>0$ (Fig. 3A) $G_{4}\left(r\right)\propto e^{-\kappa\left(r-\sigma\right)}$ shows an exponential loss of orientational correlation in the immediate neighborhood ($r\lesssim 10\sigma$). The characteristic length of the correlated region $l_{c}=\kappa^{-1}$ is a power law with respect to the interaction parameter $l_{c}\propto\lambda^{-a}$ (where $a\approx 2$) and quickly drops to just about a few particle sizes. For $\lambda\gg 1$ it appears that $l_{c}\rightarrow 1$ consistent with the fact that the correlation length cannot be smaller than approx. a single particle size $\sigma$.

Likewise, the cubatic order parameter $P_{4}$ (Fig. 3B) measured for the whole ensemble displays a progressive diminishment of the global 4-fold alignment with increasing strength of dipolar interactions. Since the asymptotic behavior of $G_{4}\left(r\right)\rightarrow P_{4}^{2}$ as $r\rightarrow\infty$ and the remaining global cubatic order as $\lambda\rightarrow\infty$ is $P_{4}\approx 0.1$, the tail of $G_{4}$ cannot decrease beyond $\propto 10^{-2}$, which is consistently observed in simulations. At the same time the decrease of the packing fraction $\phi$ in the consolidated samples constitutes a deviation from the shape-optimal packing behavior and the abundance of mechanically jammed states at high interaction strength $\lambda\gg 1$. The maximally random jammed (MRJ) packings of monodispersed cubes, defined as the jammed state with a minimal value of an order metric 78, were shown to have relatively robust densities of approx. 0.77 78, 79. Hence, the consolidated packings become MRJ states at high interaction strength.

The decay of both a global order $P_{4}$ and the correlation range $l_{c}$ is compatible with a picture of progressing policristallinity of an initially monocrystalline ensemble. To further characterize this structural transformation we perform a cluster size analysis using a proximity-based clustering criterion8 - cubes with separation $r<\sigma_{c}$, where $\sigma_{c}=1.15$ encompasses just the first coordination sphere of tightly bound (face-to-face) cubes, are considered to be a part of the same cluster - in conjunction with PTM to identify local nuclei (i.e. embedded crystallites) with roughly SC symmetry (Fig. 4B). The calculated cluster size distribution $g_{n}$ (Fig. 4A) represents the number of $n$-particle clusters per unit volume (normalized by particle volume $\sigma^{-3}$) assembled under the specific conditions of the dipolar interaction strength $\lambda$ and shows a decaying power-law dependence with respect to the cluster size $g_{n}\propto n^{-b}$. The total number of crystallites decreases ($g_{n}$ decreases in magnitude) and the size of the crystallites diminishes ($g_{n}$ shifts towards lower $n$) with stronger dipolar interactions. Fig. 4B illustrates a few ($50$) of the largest clusters. Visual inspection indicates that for $\lambda\lesssim 5$ a percolating cluster exists in the system. The ensemble is best described as a polycrystalline solid, whereas for $\lambda\gtrsim 5$ we observe a collection of well separated single-crystalline nuclei embedded in a glassy-like matrix. The characteristic size of the clusters is estimated

[TABLE]

The average cluster size is consistent with the correlation range analysis, showing a similar behavior and magnitude (cf. Fig. 3A, inset).

In turn, to examine the locality of orientational and positional correlations we produce detailed topographic maps (Fig. 5) of the local cubatic order parameter 79 and packing density fluctuations (Supplemental materials) calculated over the local neighbourhoods bounded by the first 2 coordination shells in the consolidated assemblies.

These mappings indicate progressing loss of crystallinity and fragmentation of the initial monocrystal with increasing magnitude of dipolar interactions in accord with the analysis based on the ensemble-averaged descriptors. The overlap between the locality of the cubatic order $P_{4}^{loc}$ and packing fraction $\Phi^{loc}$ marks inclusions with both a high positional and orientational order of the simple cubic lattice, separated by an interface with reduced order indicative of polycrystallinity. The dipole-induced formation of a glassy structure and porous space within the layer is an undesirable complication for the rational design of ferroelectric devices9, e.g. via strain engineering.

3.4 Mechanism of order frustration

The ordering of densified ensembles is triggered by the thermodynamic preference of crowded colloids to minimize free energy by optimizing their local packing 80, 81, 12, 82 under osmotic compression; whereby the directionality of entropic forces controlling the crystallization of hard colloids statistically emerges around the geometrical features from the collective tendency of the ensemble to increase entropy by osmotic self-depletion. We compute the effective pairwise potential of mean force and torque (PMFT) to quantify the emergent entropic valence driving the formation of the locally aligned crystalline neighborhoods and identify the enthalpic effect of added dipolar interactions, which are treated within a common context by the PMFT, derived by extracting reference coordinate pair $\Delta\xi_{12}$ (position and orientation) and treating the rest implicitly80, 81

[TABLE]

Here $U\left(\Delta\xi_{12}\right)$ is the pairwise interaction potential, i.e. steric (Eq. (1)) and dipolar (Eq. (2)), the Jacobian $J\left(\Delta\xi_{12}\right)$ encodes the entropy of the reference pair accounting for the number of ways the relative position and orientation $\Delta\xi_{12}$ can be assumed, whereas $\tilde{F}_{2}\left(\Delta\xi_{12}\right)$ is the remaining free energy of the ensemble with the reference pair fixed. The latter terms can be viewed as providing finite density correction to $U\left(\Delta\xi_{12}\right)$. We calculate the 3D topology of PMFT for both simple cubes (as reference) and dipolar cubes ($\lambda=7$) from the pair distribution function80, 81:

[TABLE]

The involved statistical integrals are computed from an additional series of NVT trajectories branching from the NPT compression runs after reaching the desired density.

In the absence of dipole-induced enthalpic attraction in $U\left(\Delta\xi_{12}\right)$ the self-assembly is a competition between the osmotic self-depletion $\tilde{F}_{2}\left(\Delta\xi_{12}\right)$, produced by the ensemble on the reference pair, and the tendency of the pair to assume local dense packing configurations modulated by the particle shape (via excluded volume), in order to minimize the overall free energy $\beta F_{2}\left(\Delta\xi_{12}\right)$. The PMFT (Fig. 6 becomes anisotropic as the particle volume fraction is increased, the potential minima $\propto kT$ appear gradually and become sharper and narrower at higher $\phi$. For simple cubes the primary wells of emerging entropic valence are located at octahedral sites coordinating to the cube’s facets imposing high positional and orientational correlation at these locations to eventually produce face-to-face alignment. In turn, the logarithmic relationship (6) indicates that a relatively small change in the free energy, such as an enthalpic contribution to the pairwise interaction $U\left(\Delta\xi_{12}\right)$, may significantly influence both the local coordination environment $g_{2}(\bm{r})$ of the low density state and the crystallography of the assembled solid. For dipolar cubes the added interaction produces coordination at $\langle 0,0,\pm 1\rangle$ sites already in solution associated with the chain fluid state8, 10. Further crowding blocks these sites by potential barriers of the order of $\propto\lambda kT$, which are rather impermeable and restrict the diffusion of the cubes within the local neighborhood leading to geometric frustration during the annealing phase.

4 Discussion

4.1 Ferromagnetic nanocubes (CoxFe3-xO4)

While the polarization structure of single-domain ferroelectric nanoperovskites is highly ambiguous owing to major influence from many uncontrolled factors, the phase state of their ferromagnetic analogues is rather well understood86. Hence, we start discussing the experimental implications of dipole-mediated self-assembly using magnetic systems as a reference. In contrast to conventional magnetite Fe3O43, 14, which is a relatively soft magnetic material with $\langle 111\rangle$ easy axis, cobalt-substituted ferrite CoxFe3-xO4 ($x>0.01$) has high magnetocrystalline anisotropy and $\langle 001\rangle$ easy axis86 corresponding to the direction of a spontaneous dipole. Recently, monodisperse single-domain CoxFe3-xO4 (x=0.4-0.5) nanocubes have been accessed by solution phase thermal decomposition reaction83 of Co(acac)2 and Fe(acac)3 in high boiling point solvent in the presence of oleic acid, which may enable next generation data storage and theranostic applications, but also provide an ideal system for validating our proposed self-assembly scenario. Fig. 7B shows a scanning electron microscopy (SEM) image of approx. 35 nm nanocubes ($\lambda\gg 1$, Ref.8) self-assembled on the Si substrate by controlled evaporation of the solvent from a hexane solution in good agreement with simulated surface morphology (Fig. 7A). The spin-spin correlation function (Fig. 9) describes the dipolar structure of the simulated solid and its fast decay indicates that only the local neighborhood is correlated, whereas the solid itself is in a paraelectric (paramagnetic) state. The local neighborhood is marked by a ferroelectric motif87 reflecting the chaining of individual nanocube dipoles8, which is the most energetically advantageous ($U_{dd}=-2\lambda kT$), with noticeable antiferroelectric inclusions88 due to a side-to-side aggregation of neighboring chains. Fig. 9 shows a characteristic spin structure fragment of a crystallite embedded within a mostly disordered layer.

4.2 Nanoperovkite cubes

Experimentally, the room-temperature colloidal assembly of BaTiO3 nanocubes within an evaporative process varies from disordered assemblies with random internal structure 89, 90 to partially ordered 3D arrays with a typical domain size of about $\propto$100 nm (approx. 5 particle sizes) 89, 90. However, dense multilayered superlattice assemblies of BaTiO3 nanocubes91, 29, 84, 92, 93 were also produced in some experiments, where the kinetic effects have been systematically controlled. The surface and inner structure were highly ordered with SC lattice and a packing fraction up to 90% 29, 94 over a relatively large area of up to $\propto$ 10 $\mu m^{2}$ 91, 84, 95, 93 with thickness varying from several hundred nanometers 29, 84, 94 up to a $\propto$ 1 $\mu m$95 (several examples are shown in Fig. 8A). This leads to an important conclusion - despite the evidence discussed above, 20 nm nanocubes of BaTiO3 must have no apparent dipole or otherwise the assembly of a long-range regular lattice would not have been experimentally feasible.

It appears that the polar state of nanoperovskites is extremely sensitive to the electrochemical state of the surface 96, 97. For one, the retention of the nanoscale ferroelectric order crucially relies on the ability of the molecular adsorbates to screen the diverging depolarizing field due to surface charges associated with a spontaneous polarization by inducing an electric field in the opposite direction98, 63, 33. Specifically, surface hydroxyls (OH) typically adsorbed on metal oxide nanoparticles during the synthetic process and carboxylates (R-COO, e.g. oleate) extensively used to passivate VdW interactions in colloidal processing of perovskites 90, 31, 31, 4 can produce an effective charge compensation as previously shown by quantum chemical calculations98. Hence, dipole-dipole interaction is stronger with weaker presence of adsorbates and incomplete charge screening, whereas the observation of electrostatic fringing fields emanating from 20 nm oleic acid capped BaTiO3 nanocubes in electron holography experiments 33 suggests that the magnitude of the dipole moment is an unknown variable with a strong dependence on the delicate nuance of the synthetic route 33, 63 with a potentially destructive effect in self-assembly experiments.

Recent systematic work done by Mimura and Kato89, 91, 90, 29, 84, 85, 92, 93, 94 provides strong evidence that the outcome of self-assembly can in fact be significantly affected by the phase state of nanoperovskites. The 20 nm strontium titanate SrTiO3 nanocubes27 could not be assembled and produced a mostly random glassy structure with some embedded SC crystallites (Fig. 8B, left), whereas a solution of BaTiO3-SrTiO3 nanocubes mixed in equal proportions yielded under an identical procedure a long-ranged regular cubic lattice where both types of nanocubes are homogeneously distributed 91, 29 (Fig. 8B, right). Dynamic light scattering (DLS) of SrTiO3 nanocube solutions has shown that they were slightly aggregated29, which is compatible with the chain-fluid state8, 10. Similar equilibrium structures have been imaged by cryo-TEM in vitrified solutions of PbSe and piezoelectric CdSe nanoparticles99. The analysis based on statistical thermodynamics places the interaction strength at 8$\div$10 $kT$ ($\lambda=8\div 10$). Previous simulation and experimental studies suggest that dense equilibrated structures of cubes well tolerate particle size100, 2 and shape69, 72, 2 imperfections robustly producing cubatic order $P_{4}$ during self-assembly, whereas a strong enough interaction is sufficient to block the development of long-range orientational order during consolidation (cf. Fig. 3).

While bulk SrTiO3 is an "incipient ferroelectric", which never achieves ferroelectric coherence101, however, the surface of pristine SrTiO3 probed by graphene was reported to be in a ferroelectric-like polar state35 having a non-zero static dipole moment, possibly due to an outward displacement of oxygen atoms (surface rumpling 102). We did quantum chemical calculations for this system, which indicate that the surface rumpling of "naked" SrTiO3 could result in an overall $\langle 001\rangle$ dipole of approx. 3560 D and $\lambda=22$ (1500 D and $\lambda=4$ within a shell model64) for a 20 nm nanocube. The examination of the pre-edge spectra of x-ray absorption reported that the single phase monodispersed oleic acid-capped SrTiO3 nanoparticles acquire a significant off-centering of Ti characteristic of a tetragonal distortion in the size range somewhere between 10 and 82 nm entering a room-temperature polar (and possibly ferroelectric) structural state 34, which could explain the disordered/polycrystalline assembly of SrTiO3 nanocubes. Similarly, it is well known that the dipolar interaction can be screened by coordination with non-dipolar particles in a mixed system.

Likewise, the phase state of ABO3 perovskites can be rationally modified by homovalent substitution into the A or B sites. It was found that B -site doping by Zr4+ in BaZrxTi1-xO3 perovskite reduces the magnitude and disrupts the coherence of cooperative dipolar order formed by the off-center Ti4+ displacements in their Ti-O6 octahedra103, 20. At low Zr content ($x<0.1$) a normal ferroelectric behavior was shows, whereas further substitution proceeds via a multiphase point at $x\approx 0.15$, in which a superposition of rhombohedral, orthorhombic, tetragonal and cubic perovskite phases coexist near room temperature, to a predominantly cubic phase ($x\approx 0.2$). Here the compositional dependence of the room temperature local crystal structure is not affected by size effects down to sub-20 nm BaZrxTi1-xO3 nanocrystals103. Indeed, the assemblies of BaZrxTi1-xO3 recently produced by dip-coating85 (Fig. 10) did show a strong dependence on Zrx content under identical experimental procedure - forming distinct crystalline phases embedded within a largely polycrystalline morphology at $x=0.1$ and a superior long range ordering at a higher Zr substitution ($x=0.2$). In turn, upon A-site substitution of Ba2+ by Sr2+ the critical levels necessary to induce a tetragonal-to-cubic phase transition are higher $x\approx 0.30\div 0.50$103, 20 and monodisperse oleic acid/oleylamine-capped 10 nm Ba0.8Sr0.2TiO3 nanocubes were assembled by the spin coating procedure into a mostly disordered 300 nm thick film (Fig. 10, right), where unusual embedded periodical stripe patterns were observed composed of 10 nm nanocubes oriented along the same directions25, which is strongly indicative of an anisotropic pairwise interaction.

5 Conclusions

We have studied for the first time the morphology of densified ensembles of dipolar nanocubes obtained under their osmotic compression. Emerging materials technologies demand superior film deposition strategies that can be absorbed into an entirely solution-based processing compatible with a current array of coating, spraying, (ink-jet) printing or patterning methods using solubilized nanoperovskites. Deterministic assembly via a colloidal route is a core platform for exploiting the potential for variable property design and deployment of mesoscale architectures for energy-harvesting devices and their integration into a successful energy-management concept. Based on their ability to tile space free-standing monodisperse nanocubes are premier candidates for this design paradigm, however, achieving long-range order in self-assembled supercrystals is still challenging.

Our study indicates that while simple cubes order reliably into regular lattices, nanocrystal cubes interacting via a long-range anisotropic potential go through a different pathway leading to unavoidable polycrystallinity and glassiness, whereas the strength of the dipolar interaction affects the final size of the crystallites embedded in a disordered glassy matrix. This behavior is also very different from the assembly of spherical nanocrystals where ferroelectric and ferromagnetic dipolar coupling should stabilize exotic hcp (hexagonal close-packed) and sh (simple hexagonal) superlattices88. Our results reproduce the morphologies observed in the experiments and are compatible with a variety of practical osmotic pressure-based techniques starting from a dilute solution.

Exploring the features of mesoscale disorder induced by competing anisotropies we find characteristically similar behavior in a wide array of systems of different physical nature - both ferromagnetic and ferroelectric. The basic understanding of the spontaneous polarization in soluble ferroelectric monodomains is incomplete and it is difficult to ascertain its presence by direct measurement, whereas the self-assembled phases reflect the structure of pair-wise interactions. We show that in fact for solubilized BaTiO3 nanocubes the polarization screening must be complete without appreciable dipole moment (below a few $\propto kT$), whereas stray interactions from incompletely screened surface charges are likely major sources of disorder in self-assembled architectures of some nanoscale perovskites, most specifically SrTiO3. Similarly, the distortion of the central symmetry in zincblende piezoelectric ZnSe104 by the surface electronic states105 produces a large size-dependent permanent dipole. The polar nature of nanocrystals (e.g. rocksalt PbSe 99, zincblende CdTe 106, wurtzite piezoelectric CdSe nanocubes 99, transition metal oxides 107, 108) driving their anisotropic assembly into a variety of dipolar mesostructures109, 99, 106 ($\lambda\gg 1$99) may be a universal feature of nanodielectrics 104, 105. Hence, operating the capping ligands to control the surface chemistry110, 111, 112, 96 might be central to produce long-range order and will require thorough understanding of the adsorbate induced polarization screening113 in future studies.

Understanding the origins of order frustration82 in perovskite superlattices is highly desirable for the rational design of ferroelectric devices from elementary building block. Colloidal shape engineering provides access to a class of entropically assembled morphologies encoded solely by the shape anisotropy inducing angularly specific interactions. Harnessing competing or synergistic entropic-enthalpic anisotropies for purposeful valence engineering may lead to a broader class of design strategies in nonclassical crystallization, which could be tuned to target specific morphologies 80, 87.

6 Acknowledgments

The authors thank Marjeta Maček Kržmanc for many useful discussions. The financial support of M-ERA.NET Project HarvEnPiez (Innovative nano-materials and architectures for integrated piezoelectric energy harvesting applications) is gratefully acknowledged. D.Z. acknowledges the support of the postdoctoral research program at the University of Latvia (Project No. 1.1.1.2/VIAA/1/16/072). The computing time of the LASC cluster was provided by the Institute of Solid State Physics (ISSP).