Ferroelectric domain walls in PbTiO 3 are effective regulators of heat flow at room temperature

. Achieving efficient spatial modulation of phonon transmission is an essential step on the path to phononic circuits using “phonon currents.” With their intrinsic and reconfigurable interfaces—domain walls (DWs)—ferroelectrics are alluring candidates to be harnessed as dynamic heat modulators. This paper reports the thermal conductivity of single-crystal PbTiO 3 thin films over a wide variety of epitaxial-strain-engineered ferroelectric domain configurations. The phonon transport is proved to be strongly affected by the density and type of DWs, achieving a 61% reduction of the room-temperature thermal conductivity compared to the single-domain scenario. The thermal resistance across the ferroelectric DWs is obtained, revealing a very high value ( » 5.0 ´ 10 -9 Km 2 W -1 ) ¾ comparable to grain boundaries in oxides ¾ , explaining the strong modulation of the thermal conductivity in PbTiO 3 . This low thermal conductance of the DWs is ascribed to the structural mismatch and polarization gradient found between the different types of domains in the PbTiO 3 films, resulting in a structural inhomogeneity that extends several unit cells around the DWs. These findings demonstrate the potential of ferroelectric DWs as efficient regulators of heat flow in one single material ¾ overcoming the complexity of multilayers systems and the uncontrolled distribution of grain boundaries ¾ , paving the way for applications in phononics.

ABSTRACT. Achieving efficient spatial modulation of phonon transmission is an essential step on the path to phononic circuits using "phonon currents." With their intrinsic and reconfigurable interfaces-domain walls (DWs)-ferroelectrics are alluring candidates to be harnessed as dynamic heat modulators. This paper reports the thermal conductivity of single-crystal PbTiO3 thin films over a wide variety of epitaxial-strain-engineered ferroelectric domain configurations.
The phonon transport is proved to be strongly affected by the density and type of DWs, achieving a 61% reduction of the room-temperature thermal conductivity compared to the single-domain scenario. The thermal resistance across the ferroelectric DWs is obtained, revealing a very high value (»5.0´10 -9 Km 2 W -1 )¾comparable to grain boundaries in oxides¾, explaining the strong modulation of the thermal conductivity in PbTiO3. This low thermal conductance of the DWs is ascribed to the structural mismatch and polarization gradient found between the different types of domains in the PbTiO3 films, resulting in a structural inhomogeneity that extends several unit cells around the DWs. These findings demonstrate the potential of ferroelectric DWs as efficient regulators of heat flow in one single material¾overcoming the complexity of multilayers systems and the uncontrolled distribution of grain boundaries¾, paving the way for applications in phononics.
Classical strategies to control the propagation of phonons are based on the design of artificial interfaces, i.e. grain boundaries 1 or interfaces in multilayer heterostructures 2,3 . Indeed, this approach has been widely used for reducing the thermal conductivity, k, in thermoelectric devices 1,4 . Unfortunately, the arrangement of these interfaces is fixed once the material is fabricated, which eliminates the possibility of dynamically tuning phonon transport. In this regard, ferroelectric DWs can be as effective as multilayer interfaces and grain boundaries in inhibiting the transmission of phonons [5][6][7] , but with the added advantage of being reconfigurable. Recent phonon-transport calculations in multidomain PbTiO3 predict a strong suppression in the transmission of transverse phonons across DWs, leading to a »30-50% calculated decrease of k 8,9 . A reduction of k with the density of DWs has been experimentally observed at low temperatures in bulk single crystals of BaTiO3, KH2PO4 and LiF [5][6][7] and more recently at room temperature in multiferroic BiFeO3 by Hopkins et al., 10 although these results were challenged by Ning et al. 11 In addition, the experiments performed by Mante and Volger 5 proved that a large increase in k can be achieved by the application of an electric field, which reduces the DW density, yet the effect is only significant at temperatures below 30 K. Ihlefeld et al. 12 achieved »11-13% electric-fieldinduced reduction of k at room temperature in polycrystalline Pb(Zr0.3Ti0.7)O3 films. In contrast, suspended membranes of the same composition show »13% increase of k when an electric field is applied. 13 The presence of grain boundaries in polycrystalline Pb(Zr0.3Ti0.7)O3 films results in a lack of homogeneity throughout the sample in terms of domain configurations-each individual grain possesses different distributions of DWs-leading to a random and nonuniform modulation of the phonon transport across the film. This severely reduces the effectiveness of the electric fieldmodulation of k when compared to theoretical values and makes it difficult to disentangle the role played by DWs from grain boundaries and vacancies. First-principles calculations revealed that k in single-domain PbTiO3 can vary (10-20%) by applying an electric field, 14 similar to what is found in polycrystalline Pb(Zr0.3Ti0.7)O3 films, 12,13 but in this case the variation is caused by the structural response (phonon-phonon scattering) without the need of changing the DW distribution. . The corresponding PFM phase images of these samples for PbTiO3/SrRuO3/SrTiO3 (c) and for PbTiO3/SrTiO3 (d). X-ray RSM around the 002 Bragg reflection of PbTiO3 for the heterostructure PbTiO3/SrRuO3/SrTiO3 heterostructure (e), and for PbTiO3/SrTiO3 (f).
To clarify this situation it is of paramount importance to experimentally determine the intrinsic thermal resistance of ferroelectric DWs at room temperature and disentangle it from other potential contributions, in order to evaluate the suitability and performance of ferroelectrics in the emerging field of phononics. [15][16][17] For this purpose, we have grown single-crystal PbTiO3 thin films (thickness = 50 nm) by reactive molecular-beam epitaxy (MBE) on several substrates with different lattice constants to induce a wide variety of homogenous ferroelectric domain patterns in PbTiO3 through epitaxial strain engineering. [18][19][20][21] The growth details are found in the Supp. Info.
Using this strategy, we demonstrate that k can be strongly reduced by engineering different ferroelectric DW configurations and we identify what DW patterns are most effective at achieving this reduction.
To create a single-domain reference sample free of ferroelectric DWs, we deposit an epitaxial single crystal thin-film of PbTiO3 on a (001) SrTiO3 substrate (strain = -1.36%) with a 10 nm thick buffer layer of conducting SrRuO3. 22 During the same deposition, PbTiO3 is also deposited directly onto a neighboring bare SrTiO3 substrate without the conducting layer. In Figure 1a This is extended to all the thermal conductivity measurements in this work.
The results obtained for single-domain PbTiO3, k » 3.9 ± 0.2 W m -1 K -1 is very similar to the prediction by first-principles calculations. 14,25 In contrast, the nucleation of ferroelectric 180° DWs causes a substantial suppression of the room-temperature k by »39%: k » 2.4 ± 0.1 W m -1 K -1 for the c-(up/down) multidomain structure. This remarkable reduction indicates that 180° DWs play a significant role in modulating the phonon transport-despite the fact that, structurally, the domains they separate are identical. showing the a/c domain architecture, and of PbTiO3 on GdScO3 (f), showing the coexistence of a/c and a1/a2 superdomains. g) Atomic-scale polarization maps of PbTiO3 on DyScO3 from the Ti displacements in the STEM image in (e). h) In-plane and out-of-plane lattice parameter obtained from the atomic-resolution STEM image in (f) using the substrate for reference. Sketch describing the a/c (i) and the a1/a2 (j) domain patterns.
In order to account for the intrinsic thin film thermal conductivity, the contribution from the interface thermal resistance, RInt, must be accurately determined. For that purpose, we measure the thermal conductivity for three different thicknesses of PbTiO3 on SrTiO3 with a constant 10 nm thick buffer layer of SrRuO3 in between (see Figure S4   Info.). The results show a significant modulation of k, around 34%, and an excellent correlation with the DW periodicity: the smaller the DW periodicity (ºlarger DW density), the lower the k.
For the sake of comparison, the previously measured k of the single-domain scenario and the cup/c-down pattern (180° DWs) are also indicated in Figure 3a. As observed, a much stronger suppression of k is found upon introduction of a-domains in tensile-strained PbTiO3 thin films, reaching the remarkable value of 61% reduction with regard to the single-domain scenario, the largest reduction at room temperature ever reported in ferroelectrics. Although epitaxial strain also changes the ratio between a and c-domains, its effect on the thermal conductivity does not seem to be significant (see Figure S10 and Supp. Info.). The strong reduction of k observed suggests that DWs are very effective phonon-scatterers, probably over a wide range of wavelengths. 9 Boundary scattering due to film thickness is not critical because the thermal conductivity accumulation function shows that »70% of the room temperature k of PbTiO3 comes from phonons having a mean free path less than 10 nm (see also discussion in the Supp. Info.). 25 Given that the domain sizes-measured by PFM and TEM-are larger, the reduced k cannot result from reductions in the mean free path alone, and DW boundaries must hence offer a Kapitza thermal resistance, RDW. Our ferroelectric films can thus be considered as homogeneous domains of length d and thermal conductivity k0, separated by regularly spaced DWs with RDW. The temperature drop across the sample is divided between the interior of the domain and at the DW, which, using the model derived by Nan and Birringer, 26  . This value is similar to that calculated by Hopkins et al. 10 for BiFeO3, and is also comparable to the effect of grain boundaries in, for instance, YSZ. 27 This result further supports the strong coupling between phonon transport and DWs in ferroelectric materials. However, a word of caution should be stated regarding the Nan and Birringer model, whose validity highly depends on the isotropy of the material. As our thermal conductivity results are mainly sensitive to the cross-plane thermal transport, the measurement is significantly anisotropic. Therefore, the computed thermal resistance for the DWs in PbTiO3 should be just taken as a first approximation.
In order to shed some light into the underlying mechanisms of DWs reducing the phonon propagation, phase-field simulations were performed, in parallel with the experimental results, to calculate the evolution of the ferroelectric domain patterns with strain 19,20 (Figure 3c) and estimate the thermal conductivity. The results are in very good agreement with our experimental results ( Figure 1 and 2) as well as the effect of the DWs on thermal conductivity 28 (see Supp. Info. for details). Based on the fraction of DWs and domains for each domain pattern (Figure 3d), 19,20 the thermal conductivity of the films was then computed assuming that the k of the domains is ten times higher than that of the DWs. 28 This criterion was applied to all types of domains and DWs.
Despite the simplicity of the model, the predicted thermal conductivity qualitatively reproduces the trend observed in the experiments (Figure 3a), namely, the local minimum around 0% epitaxial strain, coincident with a local maximum in the DW density, and the presence of a local maximum at small tensile strain and small compressive strain. As the tensile strain is increased, the DW density is predicted to continuously increase and, thus the thermal conductivity decreases, precisely as observed experimentally. On the compressive side, however, phase-field simulations predict a further reduction in k as the DW density increases, which is not reflected in our experimental results on PbTiO3/SrTiO3. This is probably due to 180° DWs reducing the phonon propagation less than the 90° DWs.
In any event, both experiment and theory prove that the density of DWs is the main driving force in the modulation of k in our PbTiO3 thin films. The large thermal resistance of the 90° DWs is most probably due to the significant structural mismatch between the different types of domains ( Figure 2h and Figure S12 in Supp. Info.). The 90° DWs involve a substantial change in the interatomic distances from one domain to another when crossing the DW (Figure 2h). A certain structural mismatch is also found across 180° DWs as Ti moves off center in opposite directions, but it is definitely smaller than 90° DWs. This may explain the slightly lower impact of the 180° DWs on the phonon transport. On the other hand, a polarization gradient occurs at all types of DWs, including the exclusively ferroelectric 180° DWs. 29 This gradient extends over several unit cells as shown in the atomic-scale STEM polarization maps (Figure 2g). These changes reflect the continuous variation in the off-centering displacement of the Ti cations as the DW is approached, making the region around it structurally inhomogeneous, which severely affects phonon transmission.
In summary, we have utilized epitaxial strain in single crystal thin films of ferroelectric PbTiO3 to engineer a model system for investigating the effect of ferroelectric DWs on thermal conductivity.
Our results demonstrate that DWs are very strong phonon scattering sites, and thus their distribution and density strongly affect the thermal transport in ferroelectrics. The design of ferroelectric patterns via strain engineering in epitaxially grown ferroelectric films allows tailoring k in one single material, overcoming both the complexity involved in the fabrication of artificial multilayers systems and the random and uncontrolled distribution of grain boundaries in polycrystalline materials. In particular, a total reduction of k by 61% with respect to the singledomain case is achieved in epitaxial PbTiO3 films, the largest reduction at room temperature ever reported in ferroelectrics. This finding proves the suitability of ferroelectrics as active barriers to control thermal transport in phononic devices.

Notes
The authors declare no competing financial interest.