In this section, we discuss the evolution of the polarization texture P(r) as a function of size, or semi-diameter R, in nanoparticles of different shapes. Fig. 6.3 aggregates representa- tive results of these studies, showing variation of P(r) in each of the four SE shapes (one row per shape) with increasing R. From the top to bottom row, OC, NO, NC and CU particle shapes are presented. Specific values of R are marked out for each nanoparticle / figure panel, as R increases from left to right. As predicted in the previous investigation [102] for spherical PbTiO3 nanoparticles in the SrTiO3 matrix, for each of the shapes considered here
the same sequence of the polarization texture transformations is observed: monodomain to vortex-like to multidomain. At small R, the nanoparticle cannot support the existence of
any domain walls and thus a monodomain polar state is found. As shown in Fig. 6.3(a), (d), (g) and (j), the polarization magnitude in such configurations is diminished roughly by 50%, compared to the RT Ps of bulk PbTiO3, but no paraelectric states are obtained for
any of the shapes. However, paraelectric states are likely to develop for weaker dielectric media [102] because of minimization of the polarization and polarizability mismatch across the particle-matrix interface.
At the increased value of R, the polarization texture transforms from the monodomain to a vortex-like state that is illustrated in Fig. 6.3(b), (e), (h) and (k) for the different shapes. The most prominent topological feature of the polarization field arrangement in all of these sketches is the presence of a cylindrically shaped ‘core’ area, completely penetrating the particle, where the polarization is either completely absent or strongly suppressed. This behavior is completely different from that of ferromagnetic vortices, where, at temperatures below TC, magnetization density at the core is constant [52, 283]. The spontaneous polar-
ization magnitude in the near particle surface areas of the vortex-like textures is closer to its RT value in bulk PbTiO3, which is especially prominent in the NC and CU shapes, as
shown in Fig. 6.3(h) and (k). The formation of the vortex-like state with polarization vectors pointing tangentially to the particle surface, leads to a minimization of the amount of the uncompensated surface charge and a decrease in the system electrostatic energy.
Finally, at even larger R, the polarization texture adopts the multidomain state that is shown in Fig. 6.3(c), (f), (i) and (l). This state is characterized by the disappearance of the paraelectric core in favor of relatively large areas of correlated P divided by domain walls
Figure 6.3: Volume P(r) field distribution for particles of different SE shapes and sizes, the latter characterized by the value of R in nm: (a-c) OC, (d-f) NO, (g-i) NC and (j-l) CU. Polarization field magnitude |P| is represented by a color map, with a scale shown on the right, while its direction is designated by the arrows. The left column assembles small particles that exhibit monodomain polarization textures. The middle column shows medium size particles that display vortex-like polarization configurations, while the right column aggregates large particles with multidomain configurations.
Figure 6.4: Normalized gradient energy Fwall as a function of particle (a) semi-diameter R
and (b) volume VSE. For most sizes, multiple minimizations of F were conducted starting
from different random initial conditions hPi ≈ 0, with the error bars shown on the plot representing variations of the converged Fwall. Data points for the spherical (SP) particles
from our previous investigation [102] are also included for comparison.
that, although closely resembling their 90◦ and 180◦ bulk variants, may still contain some localized vorticity. Individual domains also tend to orient their polarization tangentially to the inclusion surface in order to minimize the electrostatic energy.
The critical values of R (or VSE) around which the actual phase transitions take place
for each specific particle shape can be traced by following the variation of its normalized gradient energy Fwall = 1 VSE Z VSE fwalld3r, (6.7)
that constitutes a direct measure of the energy penalty associated with the presence of domain walls in the ferroelectric system. Figs. 6.4(a) and (b) display the dependence of Fwall
for the spherical particles evaluated in the prior investigation [102].
The variation of Fwall with respect to changing particle size, as depicted in Fig. 6.4(a-b),
can be separated into three different regions. At small sizes corresponding to monodomain polar states, Fwall ≈ 0, since for such configurations no polarization gradients are present in
the system. At the increased particle sizes, when the transition into the polar vortex-like state takes place, Fwall grows sharply and then goes through a maximum. This behavior
can be understood by envisioning the vortex-like state as consisting of a large number of domain walls separating small regions with sub-optimal mutual polarization arrangements (as opposed to optimal orientations of 90◦ and 180◦, in the case of PbTiO3). Finally, for
even larger particle sizes characteristic of multidomain polar configurations, Fwall gradually
diminishes and then saturates at a constant value that is greater than zero, indicating the presence of some domain walls in the system in combination with some remaining vorticity of P(r).
Turning to the analysis of the polarization texture transitions dependence on the particle shape, we observe a considerable variation of the critical sizes for the transition onset for different particle geometries. In particular, both cubic shapes exhibit a transition to the vortex-like state at very small R ' 1 nm, with a sharp peak in the Fwall(R) dependence
followed by a rapid decrease and saturation at R ' 4 nm. On the other hand, the same transition sequence occurs much later for the octahedral particles, i.e., at R ' 2 and 4 nm for the NO shape, and 5 and 10 nm for the OC shape. Furthermore, a clear tendency for an expansion of a size interval where vortex-like states are supported is observed as t is
decreased, indicating a more sharp change of polar textures for the cubic shapes and a more diffuse one for the octahedral shapes. Results obtained for the spherical particles in our previous investigation [102] are also shown in Fig. 6.4(a-b) and point to similarities in the transitional behavior of the SP and OC shapes.
It should be noted that in the previous investigation [102] it was found that the behavior of the Fwall(R) curve, and especially the critical particle size for the multidomain phase
transition, is sensitively dependent on the choice of the Gijkl coefficients parameterization.
For PbTiO3, at least three different parameterizations exist, attributed to Li et al., [98, 99],
Wang et al. [100] and Hong et al. [101] In this work, the latter set of coefficients is utilized; however, except for the changing size for the multidomain transition onset, the general nature of the phase transitions and the behavior of the polarization textures in between transitions is expected to be the same for all three parameterizations.