Principales Casos de Uso del Sistema y requisitos asociados

Under sequentially decreased compressive strains, components of po-larisation can condense, first parallel to the domain walls and then parallel to the domain periodicity (Figure 5.2) forming bc^d, abc^d and finally aac^d domains through second order transitions. For each of these domains, out-of-plane stripe domains persist with the additional components superim-posed. The x-z profile of the bc^d is identical to that of the c^d (Figure 5.4).

For compressive strains between 1< η <3.75 (%) the Px component also condenses forming abc^d domains (Figure 5.6). The symmetry break-ing between the Px and Py critical temperatures (Figure 5.2b) occurs due to the requirement for dipoles to reorientate to form the closure domains parallel to the direction of domain periodicity [176, 191]. Our simulations show that the 180^◦ domain walls of the adjacent bc^d phase collapse into to a vertex creating infinitely long cylindrical chiral tubes [186] as the po-larisation parallel to the domain periodicity (Px) stabilises further reducing the symmetry. This is consistent with observations in theoretical studies on PZT films [186]. Further to previous investigations, we observe that these chiral tubes propagate towards the film surface as the epitaxial compression is reduced.

Our model shows a critical misfit strain of 1% upon which the chiral tubes reach the surfaces of the film and dissipate, equalizing the macroscopic in-plane polarisation components whilst maintaining distinct out-of-plane stripe domains, aac^d domains. The aac^d domains have a similar cross-section to the abc^d domains in Figure 5.6 with the quantitative exception Px = Py and the loss of the chiral centres such that there is no remnant of the 90^◦ domain caps (which still exist for the abc^d domains observable in Figure 5.6 above the chiral centre pointer). For compressive strains in the range 0 < η < 1 (%) the magnitude of Pz in the aac^d domains

Dielectric Response 105

Figure 5.6: abc^ddomains in PbTiO3 films under a compressive strain of η =−2.25 % at 25 K. (a) Perspective view with the chiral centre and domain structure emphasised. (b) x-z cross-section. Each unit cell is represented by a cube whose colour is proportional to the time averaged Pz component of the polarisation.

reduce continuously to a limiting vanishing point of freestanding films upon which the aa domain pattern is recovered. Second order transitions of the polarisation by means of continuous dipole rotation with the local structure transitioning through low symmetry triclinic phases, such as we observe, have been been identified in an effective Hamiltonian simulation on PZT subjected to variations in the depolarising field strength [179].

5.4 Dielectric Response

To reduce the error in our domain transition temperatures we eval-uate the characteristic dielectric response of the films as it is known that susceptibility of ferroelectrics exhibits a divergent Curie-Weiss behaviour at the phase transition temperature. The susceptibility χαβ and dielectric

_αβ tensors are calculated from fluctuations in the total polarisation using equation 4.14 [168, 197].

Figure 5.7: Temperature dependence of the static dielectric constant of a PbTiO3 film for η = −2.26% identifying transition to different ferroelectric domains. A divergent response is observed for the c^b→ bc^dtransition (yy) and bc^d → abc^d (xx) transitions. Inset: Inverse dielectric constant with linear extrapolation from the high temperature response providing a more accurate estimation of the domain phase transition temperatures.

We find that both the c^d → bc^d and bc^d → abc^d transitions are accompanied by large dielectric response. Indeed, Figure 5.7 shows asymp-totic behaviour for χxx and χyy corresponding to the condensation of Px

and Py components, respectively.

The transition temperature between the domain phases are then deter-mined from the linear high temperature dependence of the inverse dielectric constant ⁻¹ (Figure 5.7-inset). For η = −2.26% shown in Figure 5.7, the transition temperatures from the extrapolation are 0.27Tcand 0.36Tcfor the bc^d → abc^d and c^d → bc^d transitions, respectively, with an uncertainty of ±4.4%. Our studies are performed for the system under open circuit electrical boundary conditions which results in the presence of a strong depolarising field. The latter prevents the formation of an out-of-plane

Dielectric Response 107

dielectric response (zz = 0). A corresponding increase in χzz would be expected for imperfect screening approaching the Curie temperature [191].

Historically, local dielectric properties have been overlooked in ferro-electric literature owing to the experimental challenges with characterisa-tion. However, two recent reports have shown exciting prospects utilising local dielectric phenomena. In a MD study [198], it was observed that static 180^◦ domain walls can increase the total dielectric response of BaTiO3 by almost a factor of two. This is contrary to the long standing doctrine that susceptibility arises from intrinsic contributions from changes in the polar-isation of the bulk of the domain [199, 200], and secondary extrinsic con-tributions from domain wall motion induced by applied stimuli such as an electric field or mechanical stress [201–203]. Further, a joint experiment and MD study on PbTiO₃/SrTiO₃ superlattices has observed conditions upon which the capacitance becomes negative [204]. The controversal concept of negative capacitance has been proposed to overcome the power consump-tion limits of field effect transistors [205]. Consequently, it is important to report the local dielectric behaviour of the PbTiO3 film with its different domain morphologies.

An advantage of the MD model used in this investigation is the local dielectric permittivity tensor for each unit cell j can be calculated using an analogous fluctuation formula which has been numerically validated for homogeneous and inhomogeneous structures with different dimensionlities [206, 207]:

χ^j_αβ = hv^ji

₀k_BT(hPα^jP_β^ji − hPα^jihPβ^ji) ≈ ^jαβ (5.2) where kB, T and 0 are the Boltzmann constant, simulation temperature and permittivity of free-space, respectively and hv^ji is the time-averaged volume of the unit cell. α and β correspond components of the basis vector having labels of x, y or z which define the tensor element. P^j is the local po-larisation of unit cell j andh· · · i defines a time average over the simulation.

As for the bulk permittivity, the high-frequency (optical) susceptibility χ_∞ is negligible in relation to the static component.

Each component of the local susceptibility (row) is shown in Figure 5.8 at four different temperatures (column) having domain pattern abc^d at 0.05Tc, bc^dat 0.3Tcand c^dat both 0.38Tcand 0.5Tc. An x-z cross-section of the film is shown in each instance with with the averaged local polarisation vectors superimposed. The value of the susceptibility is averaged through [010]p.

It was shown in ref. [204], that the negative capacitance arose despite the local non-shear susceptibilities always remaining positive. Our results similarly show that for each domain pattern, the diagonal components of the susceptibility remain positive at all temperatures and each unit cell. The controversial negative capacitance was identified to occur due to regions where the local susceptibility was far more responsive than the overall sys-tem [204]. For the superlattices, this occurred only in the PbTiO₃ layers and predominantly near the PbTiO3-SrTiO3 interface. In the PbTiO3 films presented in this thesis, we observe that for each domain morphology, en-hanced susceptibilities occur at domain walls and their confluences such as the chiral centre. The shear components are an order of magnitude smaller than the diagonal so are unlikely sources of negative capacitance but have been included for completeness. For the diagonal components, at each tem-perature the susceptibility of the domain is small in comparison to along the domain walls. This identifies the static domain walls have a significant, and even dominating, contribution to the overall dielectric property of the film supporting the recent claims in ref. [198].

The largest enhancements in xx and yy unsurprisingly occur at tem-peratures near the abc^d → bc^d and bc^d → c^d transitions, respectively.

This coincides with the divergences in the total susceptibilities upon the transitions (Figure 5.7). In both cases, the enhancement occurs at the 180^◦ domain wall separating the out-of-plane domains having a susceptibility

Dielectric Response 109

Figure 5.8: Local dielectric response in compressively strained PbTiO3thin films. The coloured contour map of each row identified the magnitude of the susceptibility for each component of the dielectric tensor (rows) at different temperatures (columns). The polarisation of each unit cell of the x-z cross-section are shown as unit vectors.

typically 2-4 times greater than the bulk of the domain. This identifies polarisation frustration at the domain wall and the likely nucleation site of the domain transitions.

In the bc^d and abc^d domain morphologies, zz is largest for unit cells supporting a predominantly in-plane polarisation. As the temperature is increased, the enhancement moves from the 180^◦-90^◦domain wall confluence into the 180^◦ domain wall. The susceptibility at the domain wall then increases substantially with temperature. This is a direct consequence of the overall domain pattern persisting but dipole magnitudes reducing, resulting in weakened long-range correlation and larger fluctuation amplitudes.

The identification of significant enhancements of different components of the susceptibility within the PbTiO3 identifies new possibilities for the search for negative capacitance. Further to ref. [204] observing enhanced susceptibility in PbTiO3 near the SrTiO3 interface in superlattices causing negative capacitance, we identify that regions with rapid changes in dipole direction such as domain walls and their confluences have significantly en-hanced susceptibilities relative to the rest of the film. This realisation opens possibilities of tuning the value of the capacitance by controlling the dipole frustration by shaping domain walls and, in doing so, changing the local susceptibility and therefore capacitance. Further, the polarisation frustra-tion creating the enhanced dielectric properties could provide insight into Landauer’s paradox [83], whereby calculations of single domain switching estimate an implausibly large nucleation barrier on the order of 10⁸kBT , far greater than values measured in experiment. These results show that local dipoles on surface layers of the PbTiO3 that are parallel to the surface nor-mal also have an associated anonor-malously large susceptibility, typically in xx

and yy. Therefore, when screening is incomplete, these dipoles have sub-stantial fluctuation amplitudes which will reduce the depolarisation energy, and therefore nucleation barrier due to equation 2.34, at these sites.

Novel Properties of bc^d and abc^d Domains 111

5.5 Novel Properties of bc

and abc