Ferroelectric instability in nanotubes and spherical nanoshells

Ferroelectric nanoparticles receive a considerable research attention [1–13] and novel fabrication methods are being developed [14,15]. The case of ferroelectric nanotubes and nanoshells is particularly interesting, as their specific topology can be exploited for engineering additional functionalities relevant for technological applications [1,2,16]. However, one of the limiting factors of these systems is the ferroelectric instability itself, as the corresponding transition temperature T_c can drop drastically due to finite-size effects [7,17–24]. In this paper we address this crucial point within the Ginzburg-Landau-like formalism, with which we describe analytically the ferroelectric transition in nanotube and nanoshell geometries. Thus, we extend the considerations reported in [4,5,7,8,12,25,26] to novel geometries of experimental relevance. Specifically, in the case of ferroelectric nanotubes, we will consider the electric polarization perpendicular to their axis. In addition to the overall size effect, we analyse the impact of the thickness, relative permittivities, and boundary conditions on the possible competition between different type of polarization distributions.

The emergence of ferroelectricity in a finite-size system is ultimately determined by two fundamental factors [21,27]. On the one hand, there is the tendency towards ferroelectricity itself, which can satisfactorily be modelled within the Ginzburg-Landau formalism [21]. This provides the constitutive equation in the ferroelectric that, to our purposes, can be linearised and taken as $(a - g{\nabla^2})\mathbf{P}= - \nabla \Phi$, where P is the ferroelectric polarization and Φ is the electric potential. Here ${a = a'(T - {T_{c0}})}$ is the inverse susceptibility, with T_c0 being the nominal transition temperature $({a'}= \text{const})$, and g is associated to the gradient term in free energy. For the sake of simplicity, the response of the ferroelectric is assumed to be isotropic —either completely (nanoshell) or within the ferroelectric plane (nanotube). As in [7,12], this approximation is expected to capture the key qualitative features of the ferroelectric instability¹ . On the other hand, there is a purely electrostatic aspect described by Gauss's law: $\nabla \cdot (\varepsilon \nabla \Phi - {\mathbf{P}}) = 0$, where ε is the so-called background permittivity [28], and the corresponding boundary conditions [21,27]. Thus, whenever $\nabla \cdot{\mathbf P} \not =0$, the spontaneous polarization will be penalised by the accompanying electric potential and the corresponding increase of electrostatic energy.

Following [7], the task is to find the nontrivial solution of the above equations that can appear at the highest T (i.e., for the maximum value of the coefficient a). This search can be restricted to the family of divergenceless distributions of polarization $(\nabla \cdot \mathbf P = 0)$ that automatically minimize (most of) the electrostatic energy in the ferroelectric. Furthermore, two subfamilies can be identified among the targeted distributions: i) irrotational distributions $(\nabla \times \mathbf{P} =0)$ and ii) vortex-like states $(\nabla \times \mathbf{P} \not =0)$. In the first case the gradient energy is minimised at the expense of some electrostatic energy generated by interfacial bound charges (depolarizing field). In the second case the situation is reversed, and the electrostatic energy is minimised at the expense of some gradient energy in the ferroelectric. These cases will be analysed separately for the different geometries of interest, and the results will be illustrated by considering the material parameters of BaTiO₃.

In the case of a cylinder or a sphere, the only possible irrotational distribution of polarization corresponds to the $\mathbf{P} = \text{const}$ (homogeneous polarization). The presence of the internal boundary in the nanotube or the nanoshell, however, introduces more complex patterns. In this case, since $\nabla^2{\mathbf P} = \nabla(\nabla\cdot{\mathbf P}) - \nabla \times (\nabla \times {\mathbf P}) = 0$, the above equations reduce to the Laplace equation ${\nabla^2}\Phi = 0\ (\mathbf P = -a^{-1} \nabla \Phi)$. We thus adopt cylindrical $(r, \theta, z)$ and spherical $(r, \theta, \phi)$ coordinates for the nanotube and the nanoshell, respectively, and consider the solutions

$$\begin{equation} \Phi_{2D} (r, \theta) =({A_n}{r^n} + {B_n}{r^{- n}})\cos (n\theta ), \end{equation} \tag{ 1 }$$

$$\begin{equation} \Phi_{3D} (r, \theta) =({A_n}{r^n} + {B_n}{r^{- n - 1}}){P_n}(\cos \theta), \end{equation} \tag{ 2 }$$

for the electrostatic potential, where $P_n(x)$ represent the Legendre polynomials. Hereafter $R_{1(2)}$ represents the internal (external) radius. The irrotational distributions of polarization are illustrated in fig. 1. n = 1 corresponds to the homogeneous polarization for ${R_1 =0}$ (see fig. 1(a)). Whenever ${R_1 \ne 0}$, however, the resulting polarization is inhomogeneous (fig. 1(b)), and this inhomogeneity increases with the corresponding order n (fig. 1(c)).

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We consider first the (2D) case of a nanotube. The electric potential Φ has to be continuous at R₁ and R₂, while its gradient has to be such that $\varepsilon\,n \cdot \nabla \Phi \big| {_{R_i^-}^{R_i^+}} = \sigma_{R_i}$. Here n is the normal unit vector to the interface while $\sigma_{R_i}$ represents the interfacial charge density. In order to ensure charge neutrality, the interfacial charge densities can be taken as $\sigma_{R_1}={{- \bigl({\frac{{{R_1}}}{{{R_2}}}}\bigr)}}P_0\cos (n\theta )$ and $\sigma_{R_2}=P_0\cos (n\theta )$, with P₀ being a constant. Thus, the solutions (1) become compatible with the boundary conditions whenever the condition

$$\begin{eqnarray} &&({{\varepsilon _{FE}} + {\varepsilon _2}}) ({{\varepsilon _{FE}} + {\varepsilon _1}}) =({{\varepsilon _2} - {\varepsilon _{FE}}}) ({{\varepsilon _1} - {\varepsilon _{FE}}}){\left( {\frac{{{R_1}}}{{{R_2}}}} \right)^{2n}} \end{eqnarray} \tag{ 3 }$$

is satisfied. Here $\varepsilon _{FE} = \varepsilon + a^{-1}$ is the permittivity of the ferroelectric, while ${\varepsilon _1}$ and ${\varepsilon _2}$ are those of the inner and outer medium, respectively. Equation (3) determines the hypothetical T_c associated to the irrotational solutions (1) as a function of ${R_1/R_2}$ and the corresponding order n, which is illustrated in fig. 2. As we can see, while all orders tend to be degenerate in the limits $R_1 = 0$ and $R_1 = R_2$, the highest T_c corresponds to the n = 1 solution and this hierarchy is maintained for all the radii $R_1/R_2$.

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In the (3D) case of the nanoshell, the interfacial charge densities can be taken as $\sigma_{R_1}={{- \big({\frac{{{R_1}}}{{{R_2}}}} \big)}^2}P_0 P_n(\cos \theta)$ and $\sigma_{R_2}=P_0 P_n(\cos \theta)$. Thus, the compatibility between the solutions (2) and the electrostatic boundary conditions implies

$$\begin{eqnarray} & \left[ {n{\varepsilon_{FE}} + (n + 1){\varepsilon_2}} \right]\left[ {(n + 1){\varepsilon_{FE}} + n{\varepsilon_1}} \right]= \nonumber \\ & \,\quad n\left( {n + 1} \right)\left( {{\varepsilon_2} - {\varepsilon_{FE}}} \right)\left( {{\varepsilon_1} - {\varepsilon_{FE}}} \right){\left( {\frac{{{R_1}}}{{{R_2}}}} \right)^{2n + 1}}. \end{eqnarray} \tag{ 4 }$$

We now have two different situations depending on the relative permittivities $\varepsilon _1$ and $\varepsilon _2$. If ${\varepsilon _1 <\varepsilon _2}$ (see fig. 3(a)) the degeneracy at $R_1=0$ is lifted, although the n = 1 solution has always the highest T_c as in the previous (2D) case. If ${\varepsilon _1 >\varepsilon _2}$ (see fig. 3(b)), however, this hierarchy is reversed for small R₁ and, interestingly, a crossover is obtained as the $R_1/R_2$ ratio increases.

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Interestingly, in both 2D and 3D cases, the strong suppression of the T_c of the irrotational polarization can be moderated in the limit $R_1/R_2 \to 1$. However, the question of whether they can be realised experimentally eventually depends on the competition with other families of solutions. In the following we consider the vortex-like patterns, as they are the most serious contenders.

In our systems, a vortex-like distribution of polarization implies ${\nabla \cdot \bm{P} = 0}$ everywhere, and hence $\Phi =0$. Thus, the emergence of this type of polarization is simply governed by the equation $(a - g{\nabla^2})\bm{P} = 0$ under the corresponding boundary conditions. The solutions of interest can be written as $\bm{P} = {P_\varphi (r)} \hat{\mathbf{e}}_\varphi$, where

$$\begin{equation} {P^{2D}_\varphi (r)} = {C_1}{J_1}(r/r_c) + {C_2}{Y_1}(r/r_c), \end{equation} \tag{ 5 }$$

$$\begin{equation} {P^{3D}_\varphi (r)} = {C_1}{j_1}(r/r_c) + {C_2}{y_1}(r/r_c), \end{equation} \tag{ 6 }$$

for the (2D) nanotube and (3D) nanoshell geometries, respectively. Here $r_c = (g / a)^{1/2}$ is the correlation length, J₁ and Y₁ are Bessel functions of first and second kind, while j₁ and y₁ are spherical Bessel functions of first and second kind, respectively.

The T_c associated to these vortex-like distributions of polarization depends on the boundary conditions. We consider the general boundary conditions $\left.({1 \pm \lambda {\partial _r}})P\right|_{R_i} = 0$, where λ is the so-called extrapolation length [21]. Thus, in the (2D) case of a nanotube T_c is determined by

$$\begin{eqnarray} & \Big[J_1\Big({\frac{{{R_1}}}{{{r_c}}}}\Big) - {\frac{{{\lambda}}}{{{r_c}}}} J_1'\Big({\frac{{{R_1}}}{{{r_c}}}}\Big)\Big] \Big[Y_1\Big({\frac{{{R_2}}}{{{r_c}}}}\Big) +{\frac{{{\lambda}}}{{{r_c}}}} Y_1'\Big({\frac{{{R_2}}}{{{r_c}}}}\Big) \Big] =\nonumber\\ & \Big[J_1\Big({\frac{{{R_2}}}{{{r_c}}}}\Big) + {\frac{{{\lambda}}}{{{r_c}}}} J_1'\Big({\frac{{{R_2}}}{{{r_c}}}}\Big)\Big]\Big[Y_1\Big({\frac{{{R_1}}}{{{r_c}}}}\Big) -{\frac{{{\lambda}}}{{{r_c}}}} Y_1'\Big({\frac{{{R_1}}}{{{r_c}}}}\Big) \Big]. \end{eqnarray} \tag{ 7 }$$

A similar equation is obtained for the (3D) nanoshell case with $J_1\ (Y_1)\ \to\ j_1\ (y_1)$. For the sake of simplicity, we consider that the two interfaces are described by the same λ (see footnote ² ).

We find that the T_c as a function of R₁ and R₂ can show rather different behaviors when these parameters are varied separately. This is eventually determined by the extrapolation length λ, as illustrated in fig. 4 for the case of a ferroelectric nanotube. Specifically, the "topography" of the $T_c(R_1,R_2)$ map changes in such a way that its maximum gradient rotates by 45° as λ goes from 0 to ∞. Thus, for ${\lambda=0}$, T_c decreases by decreasing the thickness of the shell. That is, by either increasing R₁ or decreasing R₂ (A-O and B-O paths, respectively, in fig. 4(a), which correspond to blue and orange lines in the bottom plot). For a finite λ (fig. 4(b), (c)), however, T_c initially increases by increasing R₁ and then decreases after reaching a maximum. By decreasing R₂, in contrast, the behavior is monotonous and T_c always decreases. For $\lambda = \infty$, which corresponds to the so-called natural boundary conditions $\partial_r P = 0$, the dependency on the nanotube thickness is different for different paths (fig. 4(d)). While T_c increases by increasing R₁, it decreases by decreasing R₂. This unequivalence in the finite-size effect is related to the specific topology of the systems under consideration. In fact, in the case of the nanoshell, the T_c associated to the vortex-like distribution of polarization behaves qualitatively in the same way within the approximations of our model.

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We note that, compared to the irrotational states, the T_c associated to vortex-like distributions of polarization is generally much closer to its nominal value T_c0 (irrespective of the properties of the surrounding media). However, when ${R_1/R_2} \to 1$, the T_c for the vortices can drop significantly while that of the irrotational distributions approaches T_c0. Thus, we find that the specific topology of these systems enables the competition between different type of polarization distributions in the ultra-thin limit.

In summary, we have studied theoretically the ferroelectric instability in nanotubes and spherical nanoshells. Specifically, we have considered semi-analytically different families of polarization distributions and examined how their emergence is affected by the thickness of the nanoparticle, the dielectric properties of the surrounding media, and the interfacial boundary conditions. We have found an intriguing topological finite-size effect that can promote the competition between different types of ferroelectricity in the ultra-thin limit. These results illustrate new routes to control the ferroelectric instability and engineer ferroelectric properties at the nanoscale. This possibility is expected to motivate both extended theoretical analyses and future experimental work.