Millimeter-wave dielectric tunability driven by topological polar structure switching in PbTiO_3/SrTiO_3 superlattices

Domain: physics
Published: 2026-02-13
Editorial outcome: major_revision

Abstract
Dielectric tunability induced by an external electric field in materials underpins radio frequency signal modulation devices such as phase shifters, which are critical components in wireless communication and sensing systems. However, the tunability and integrability of current devices have yet to be enhanced for emerging applications, particularly at millimeter-wave frequencies. Here, we demonstrate that topological polar structures formed in PbTiO3/SrTiO3 superlattices exhibit large tunable in-plane dielectric properties, as determined by their multiscale structural configurations and polarization switching behaviors. Under a moderate field of 30 kV cm−1, the dipole wave structure maintains a tunability exceeding 15% at 70 GHz and above 8% over the measured range up to 110 GHz, contrasting with the weakly tunable flux closure structure. Based on in situ structural characterizations and molecular dynamics simulations, we delineate the polarization switching processes and elucidate the mechanisms underlying the observed tunable millimeter-wave dielectric responses. Our results provide new insights into the high-frequency dielectric properties of topological polar phases, potentially broadening the versatility of these materials in next-generation integrated electronic applications.

Introduction
Topological polar structures have attracted considerable attention in recent years because of their unique swirling polarization configurations and topological protection mechanisms that distinguish them from conventional ferroelectric materials1. The established topological polar structures are exemplified by polar vortices, skyrmions, and merons2–7 formed in PbTiO3/SrTiO3 superlattices and multilayers, which have been extensively investigated through microscopic observations and simulations. Aside from these structures, dipole waves exhibit a continuous head-to-tail arrangement of electric dipoles along in-plane directions as opposed to separate vortex units8, forming a sinusoidal polarization distribution profile with an in-plane modulation periodicity. Flux closures, by contrast, are characterized by domains with alternating in-plane or out-of-plane polarization vectors, separated by 90° and 180° domain walls9. These domain walls act as sharp edges between areas with different polarization directions and inhomogeneous lattice strains. The formation of these different types of topological structures primarily arises from the electrostatic screening and elastic effects within the superlattices, involving a delicate balance of competing free energy components related to the polarization and strain states10,11. Therefore, their phase stabilities can be engineered by tuning the relative thickness of the component layers8,12–14.

The characteristic polarization topology of these structures leads to a distinct response to electric fields at different frequencies. Previous studies have mostly investigated their polarization evolution behaviors under the influence of low-frequency electric fields15,16, showing characteristic switching processes17,18 and anomalous dielectric responses such as negative permittivity19,20,21 linked to topological protection behaviors. Meanwhile, the dynamic response of these structures have also been explored in the higher frequency range (from terahertz to optical frequencies), revealing ultrafast collective dynamics and nonequilibrium phase transitions of polar vorties22–25. Exploring the polarization dynamics within the microwave to millimeter-wave bands (1–300 GHz) could bridge the gap between the low-frequency and subterahertz ranges and unlock novel electronic properties of topological polar structures at these frequencies. Notably, the dipole wave and flux closure structures may potentially exhibit complex in-plane polarization dynamics due to the existence of net in-plane polarization at the static states.

Meanwhile, the rapid development of next-generation wireless communications and sensing technologies has driven the demands for high-performance millimeter-wave tunable dielectrics26. These applications, including frequency-agile filters, adaptive phase shifters, and phased array antennas, hinge on efficient electric-field tuning of the dielectric permittivity of working materials. Ferroelectrics exhibit tunable dielectric properties, as observed in various material systems such as (Ba,Sr)TiO3, Pb(Zr,Ti)O3, and (K,Na)NbO327–32, which have advantages over semiconductors and other types of materials in terms of tunable high permittivity, large bandwidth and low loss33. While ferroelectric-based tunable devices were demonstrated at microwave frequencies27,34–38, few studies have extended the operating frequency to millimeter-wave bands (over 40 GHz) thus far. To address this, explorations into millimeter-wave polarization dynamics and associated dielectric tuning mechanisms becomes pivotal for extending functionalities of ferroelectrics into high-frequency regimes.

Here, we demonstrate large in-plane dielectric tunability at millimeter-wave frequencies in PbTiO3/SrTiO3 superlattices hosting dipole waves and flux closures. By systematically adjusting the thickness of PbTiO3 layers, the superlattices exhibit varying topological phase stabilities, leading to distinct polarization switching dynamics and tunable dielectric responses. Our results not only reveal the polarization evolution mechanisms of topological polar structures under electric fields extending into the millimeter-wave range, but also establish these systems as a promising material platform for tunable millimeter-wave device applications.

Results
Superlattice growth and structural characterization
We select a series of (PbTiO3)n/(SrTiO3)8 superlattices with n=12, 15, 20, and 22, and total thicknesses ranging from ~165 nm to ~183 nm (see details in Supplementary Table 1), as representative systems hosting topological polar phases. All the superlattices were epitaxially grown on (001)-cut (LaAlO3)0.3-(SrAl0.5Ta0.5O3)0.7 (LSAT) single crystal substrates buffered with a 5 nm GdScO3 buffer layer. The GdScO3 layer introduces a relaxation of the lattice mismatch strain so that the average in-plane lattice constants are determined to be ~3.93 Å from X-ray diffraction (XRD) measurements. Coupled 2θ/ω-scans across the LSAT 002 reflection (Supplementary Fig. 1a) exhibit well-defined superlattice peak fringes, indicating their excellent crystallinity and clear interfaces as a result of high-quality film growth. The out-of-plane layer periodicities of the superlattices are extracted to be ~8.0–11.8 nm from the fringes, which agrees with the designed values. The average out-of-plane lattice constant c of the superlattices can be calculated from the positions of the zeroth-order superlattice peak (SL(0), marked in Supplementary Fig. 1a). As shown in Fig. 1c, the lattice constant c increases with increasing PbTiO3 layer thickness n, suggesting a concomitant increase in the out-of-plane polarization component in the PbTiO3 layers.

X-ray reciprocal space mapping (RSM) about the LSAT 103 reflection (Fig. 1a) shows diffraction intensities of second- and even third-order satellites separated from the superlattice peaks along the Qx/Qy directions, in contrast to only the first-order diffraction satellites in previously observed polar vortices or skyrmions2,4,16,22. This indicates the existence of topological phases with strong in-plane ordering in these superlattices. As the PbTiO3 layer thickness n decreases from 22 to 12, the satellite peaks shift further from the superlattice peaks, suggesting smaller in-plane periodicities (Fig. 1b). The calculated in-plane periodicity d ranges from approximately 9.1 nm to 13.6 nm, and increases with n in a nearly linear manner (Fig. 1c). This trend fits well with the scaling equation d (nm) = 2.015 + 13.46/n + 0.497n, in agreement with previous studies39. The satellites also split into fringes along the Qz direction with the same Qz values as the superlattice peaks, evidencing an out-of-plane ordering with the same superlattice periodicities. As the diffraction satellites originate from topological phases in the PbTiO3 layers, such ordering state imply strong interlayer polarization coupling across different PbTiO3 layers. Similar interlayer coupling has been observed in vortex-hosting (PbTiO3)n/(SrTiO3)m superlattices with thicker PbTiO3 layers than SrTiO3 layers12,13, as is the case here (n=12~22, m=8). The existence of both in-plane periodicity and out-of-plane interlayer coupling renders our superlattices a model ordered system for investigating the structural evolution and dielectric properties of topological polar phases. Notably, the 002 RSM for the n=12 superlattice shows diffraction intensities apart from the superlattice peaks (see the box-marked region in Supplementary Fig. 1b), which are attributed to tilted ferroelectric a-domains similar to those of single-layer PbTiO3 films with c/a domain structures40.

Nanobeam X-ray diffraction imaging further reveals the mesoscopic spatial distribution of different topological phases in the superlattices41,42. For the n=12 superlattice, mapping of the tilted a-domain diffraction intensities within a slice of reciprocal space around the 002 reflection (see the boxed region in Fig. 1d) reveals uniformly distributed stripes with high diffraction intensities (Fig. 1e). The observed diffraction contrast indicates the locations of tilted a-domains, occupying approximately 40% of the mapped area (see Supplementary Note 3 and Supplementary Fig. 8). Such coexistence of topological polar structures and conventional a-domains has also been reported in polar vortices15. Although similar diffraction peaks are also observed in the n=15 superlattice, their intensities are significantly weaker than those in the n=12 superlattice, along with smaller full widths at the half maximum (Supplementary Fig. 1b, c). The spatial distribution of such diffraction intensities is more concentrated along with a smaller area proportion of ~12%, which is more likely associated with dislocations in the films, as shown in Fig. 1f, g.

High-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM) reveals the real-space polarization configurations of the topological phases, as presented in Fig. 1h–j. Large field-of-view images of the superlattices confirm the microstructures of the uniform PbTiO3 and SrTiO3 layers with sharp interfaces, as well as their designed layer thicknesses (Fig. 1h). The maps of polar displacement vector (arrows in Fig. 1i, j), derived from the cation column positions in the atomically resolved HAADF images, show varying polarization configurations in the superlattices. Figure 1i, j presents typical results of a dipole wave structure for n=12 and a flux closure structure for n=22, both assigned according to the original structure models8,9. We consider that the n=15 and 20 superlattices have similar structures as n=12 and 22, respectively, and therefore divide them into two groups: dipole waves for n=12 and 15, and flux closures for n=20 and 22, as the n is closer in each group than between the two groups. Furthermore, the individual dipole wave or flux closure structures appear to be well aligned vertically across multiple PbTiO3 layers (see their cores marked in Fig. 1i, j), consistent with the X-ray RSM results. Previous studies have reported vortex-antivortex pairs in superlattices with similar PbTiO3 and SrTiO3 layer thicknesses12. In our superlattices, the difference in misfit strain and other factors may have led to the observed topological structures (see Supplementary Note 4).

Ferroelectric switching and low-frequency electric measurements
The in-plane ferroelectric switching properties of the superlattices were measured on coplanar capacitors with interdigitated electrodes (IDEs), as shown in the inset of Fig. 2a. Owing to the complex electrical attributes of IDEs, the polarization and capacitance are shown as charge or capacitance per unit length of electrode. The measured polarization–electric field (P–E) loops at 10 kHz display nonzero in-plane net polarization Px and ferroelectric-like hysteresis at the initial state in all the superlattices (Fig. 2a). This in-plane polarization matches the polar structures shown in Fig. 1i, j, where larger areas with in-plane polarization along +x direction than the negligible ones along -x direction prevent the polarization from canceling out, enabling the ferroelectric-like switching process. Capacitance–voltage (C–V) loop can be derived from the P–E loop by extracting the polarization change rate with respect to the external voltage variation43, as shown in Fig. 2b. The peaks of these butterfly-shaped loops are located at the coercive field Ec, with the highest polarization change rate. The coercive field Ec increases with increasing PbTiO3 layer thickness n in a nearly linear manner (Fig. 2a inset), similar to the trend of in-plane periodicity, suggesting a correlation between the in-plane switching dynamics and periodicities of the topological polar structures.

The observed correlation is further corroborated by molecular dynamics (MD) simulations. The simulation results (Fig. 2d, e) indicate a two concurrent in-plane switching processes: the major switching process in the PbTiO3 layers when the net polarization changes direction, in addition to a minor polarization switching in the SrTiO3 layers (contributing ~12% to total polarization at the highest field) which has little influence on the net polarization and thus is invisible in the P–E loop. With an electric field applied along the +x direction, the areas with the opposite in-plane polarization (yellow areas in Fig. 2d, e) shrink, and those with out-of-plane polarization directions (red and blue areas) expand, revealing the polarization rotation from the in-plane to out-of-plane directions44. With further increase in the electric field, the polarization in-plane flips towards the electric field direction, completing the switching process in the PbTiO3 layers. Upon removal of the electric field, the topological structure recovers into the new in-plane polarization direction (Supplementary Fig. 6), corresponding to the remanent polarization in the P–E loop. However, such switching processes result in different final states for different topological structures. In the dipole wave structure with smaller n, the superlattice turns into a single a-domain with minor out-of-plane polarization after switching, thereby losing almost all topological features. In the flux closure structure with larger n, the topological features are maintained after switching, suggesting better stability and topological protection of the flux closure. This results in a larger coercive field for the flux closure (Fig. 2c), as a higher energy barrier has to be overcome to perturb a more stable topological phase, consistent with the experimental observations. The calculated coercive fields are a few times larger than the experimental values and thus represent the intrinsic values; in the real superlattices, defects may have lowered the energy barrier for polarization switching.

Structural evolution during ferroelectric switching
To probe the structural evolution of the topological polar phases during switching, second harmonic generation (SHG) mapping under normal incidence was conducted before and after the switching process45. The mapping results for both dipole wave and flux closure superlattices (Fig. 3a and Supplementary Fig. 2a) show uniform SHG intensities in the pristine area. Post-switching SHG mapping within the IDE fingers (Fig. 3b and Supplementary Fig. 2b) reveals an nearly uniform contrast with ~3 times stronger SHG intensities, suggesting an enhancement of the in-plane polarization component. SHG polarimetry analysis (Fig. 3c and Supplementary Fig. 2c) further reveals changes in the lattice anisotropy of the topological polar phases. Both the α=0° and α=90° polar plots show a mixture of twofold intensity lobes and 45°-oriented fourfold lobes outside the IDEs. The twofold component at α=0° and the fourfold component at α=90° disappears inside the IDEs after switching. This evidences a nonvolatile symmetry change during polarization switching, which persists after field removal.

The second-order nonlinear susceptibility tensors (χ(2)) were obtained from fittings to the SHG polarimetry patterns (see details in Supplementary Note 1). According to the symmetry of the topological polar structures and the coordinate transformation methods previously described46, the twofold-fourfold mixture in the SHG originates from coexisting a/a-like domains. We use T1/T2 to denote the topological structures with different net in-plane polarization directions, thus to distinguish from the conventional a1/a2 domain nomenclature. As shown by the simulation results in Fig. 3d, these T1/T2 domains form a hatched pattern, similar to previous observations in polar vortices18,47. The changes in their populations during switching are obtained from the measured SHG patterns (see Supplementary Table 2). For the n=15 superlattice, the proportion of T1 domains (dipole wave) decreases from 62.6% to 29.3% after switching, whereas the T1 domains (flux closure) for n=22 decrease from 51.6% to 29.5%, both shifting away from 50%. This indicates a breaking of the x-/y-, two equivalent in-plane directions due to the applied electric field, corresponding to the 90° polarization switching from T1 to T2 domains, as shown in Fig. 3d.

Synchrotron-based in situ 3D RSM measurements were performed to obtain the microscopic information during switching in the dipole wave superlattice with n=12, as presented in Fig. 3e–g and Supplementary Fig. 3. When an electric field is applied (“on” state, Supplementary Fig. 3b) along the y-direction, the satellite diffraction intensity transfers from the x- to y-direction, and meanwhile, the zeroth-order superlattice (SL(0)) peak splits, shifting toward a higher Qz, as illustrated in Fig. 3e. The observed intensity transfer corresponds to the previously described polarization switching from T1 to T2 domains. The zeroth-order peak splitting results from dipole waves switching to conventional ferroelectric a-domains, which transition reduces the out-of-plane polarization component, thus shrinking the lattice constant c from 3.960 Å to 3.926 Å (-0.86% change). Such an induced transition implies that the n=12 superlattice is close to the phase boundary of dipole waves and a-domains, consistent with the XRD results for the pristine state (Fig. 1d, e).

Upon removal of the electric field (“off” state, Fig. 3f), the induced intensity transfer and peak splitting are partially reversed, but the intensities of both the split zeroth-order peak and satellite peaks along the y direction remain higher than those in the pristine state. These remanent changes indicate a larger proportion of T2-domains along with the existence of a-domains compared with the pristine state. Figure 3g presents the field evolution of diffraction intensities for both “on” and “off” states. Overall, major intensity changes occur at approximately 40–50 kV cm−1, which agrees with the coercive field observed in the P–E loops (Fig. 2a). For the satellites along the y-axis, a maximum intensity increase of ~24%/18% for the “on/off” state, suggesting a ~20% increase in the proportion of T2 domains.

Millimeter-wave dielectric measurements
Millimeter-wave dielectric constants within the 2–110 GHz frequency range were measured via coplanar waveguides (CPWs) fabricated on the superlattices, with d.c. bias voltages applied between the ground and signal lines to measure the dielectric tunability, as illustrated in Fig. 4a (see details in Methods). The complex dielectric constants εr for different superlattices at zero d.c. voltage are shown in Fig. 4b, c, which serve as a baseline for the tunability calculation. The real part of εr (εr’) decreases monotonically while the imaginary part (εr”) increases as the measuring frequency increases. These trends agree with the Cole-Cole relaxation model which describes the complex permittivity ε*(ω) as48,49:

ε*(ω) = ε∞ + (εs - ε∞) / (1 + (iωτ)β) (1)

where εs is the static dielectric permittivity, ε∞ the “infinite frequency” permittivity, τ the relaxation time constant, and β is a parameter describing the distribution width of relaxation times. The fitted τ for all the superlattices is on the scale of ~1 ps (Fig. 4d), suggesting rapid polarization responses. The dipole waves (n=12 and 15) exhibit twice larger τ, in agreement with their stronger frequency dispersion of εr’ than the flux closures (n=20 and 22). The superlattices exhibit relatively large loss tangents (εr’/εr”) above 40 GHz, ranging from approximately 0.2 to 0.5. This high loss is likely dominated by defect-related contributions35, representing a drawback for applications. However, it could be mitigated through optimized thin-film growth processes, which hold promise for reducing defect densities.

The C–V curves were measured by sweeping a.c. frequencies as a function of d.c. bias voltages. The results at 20 GHz are illustrated in Fig. 4e for different superlattices (see Supplementary Fig. 4 for the results at other frequencies), and the dielectric tunability, calculated by |Δεr’|/εr’(0V), is presented in Fig. 4f. For the flux closures (n=20 and 22), the measured tunability is only approximately 2% in the range of 30 kV cm−1, consistent with their higher coercive fields hindering polarization switching. For the dipole waves, the in-plane polarization switching process introduces a decrease in the dielectric constant, resulting in over 5% tunability observed in all n=12–17 superlattices. Under 30 kV cm−1, the n=17 superlattice exhibits the largest dielectric tunability of ~20% at 20 GHz, which remains relatively stable in the entire measured frequency range: >15% at 70 GHz and >8% at 110 GHz. The lower overall dielectric tunability of the n=12 superlattice may be attributed to the coexistence of dipole waves and tilted a-domains (see Fig. 1d, e). Similar a-domains in single-layer PbTiO3 films grown on SrTiO3 substrates showed minimal tunability (~2%) under 700 kV cm−1, a negligible response compared with the low-field conditions in our measurements19. Considering the tilted a-domain and dipole wave structures behave as capacitors in series, the overall dielectric permittivity should be a weighted average of the two structure types, whereby an almost field-independent permittivity of the a-domains considerably reduces the overall dielectric tunability. The high overall dielectric tunability in the n=17 superlattice correlates with its large zero-field permittivity, suggesting that the dynamical polarization response is enhanced near the topological phase boundaries between dipole wave and flux closure structures.

MD simulations were performed to further correlate the dielectric tunability with the structural changes during polarization switching. The dielectric constant εr was evaluated from the simulated time-domain polarization evolution upon the excitation of a Gaussian electric field pulse (see details in Supplementary Note 2). Figure 4g presents the simulated dielectric constants under varying excitation fields, which manifest larger permittivity and tunability values in the dipole wave than the flux closure structure, well corroborating the measurement results. The structural origin of such differences is illustrated in Fig. 4h for the zero-bias case. In the dipole wave structure, the wavy area with large out-of-plane polarization components (see the blue-shadowed region in Fig. 1j) exhibits prominent local in-plane polarization response, indicating an instantaneous polarization rotation following the millimeter-wave electric field. This leads to large average polarization changes (denoted by ΔPy) in the whole superlattice, and thus higher dielectric permittivity. By contrast, the area with discernible local ΔPy in the flux closure structure concentrates along the 90°-domain walls, corresponding to the in-plane domain switching process similar to the low-frequency transitions (Fig. 2e). However, the polarization response is marginal in the domain interior, thus amounting to considerably smaller average ΔPy and lower dielectric permittivity.

Supplementary Fig. 9 (see also Supplementary Movie 1) illustrates the polarization dynamics under the d.c. bias. At the field strength, the original wavy area in the structure still displays high local ΔPy under the d.c. bias, similar to its zero-bias response behavior. This indicates that its remnant minor out-of-plane polarization can still rotate to the in-plane direction upon the millimeter-wave excitation. The ΔPy in the dipole wave under the d.c. bias is smaller than that without bias, yet still larger than that in the flux closure. The latter structure retains its topological features after the low-frequency switching, and similarly exhibits minor ΔPy concentrated along the domain walls.

Figure 4i further compares the frequency spectra of the simulated and measured dielectric tunability in the dipole wave, both of which show broad peaks around 20–30 GHz and 40–50 GHz. This possibly hints at the existence of collective modes in the dipole wave structure, with frequencies in the same order with the modes observed in polar vortices (around 80 GHz)22. Another possible contribution to the dielectric tunability is the resonant switching from T1 to T2 domains (Fig. 3d). As shown in Supplementary Fig. 10, upon application of a combination of d.c. and 20 GHz a.c. electric fields, the T1/T2 proportion further decreases, lower than that under a d.c. field with the same magnitude. Similar resonant switching processes have been reported in a previous study on polar vortices50. This process further reduces the overall dielectric permittivity under electric fields, leading to higher dielectric tunability. Taken together, the MD simulations clearly delineate the structural dynamics underlying the large dielectric tunability of the dipole wave.

Methods
Superlattice growth
(PbTiO3)n/(SrTiO3)8 superlattices were grown on (001)-cut (LaAlO3)0.3-(SrAl0.5Ta0.5O3)0.7 (LSAT) single crystal substrates via reflection high-energy electron diffraction-assisted pulsed laser deposition (PLD), with a 5-nm buffer layer of GdScO3 to relax the in-plane substrate strain. A 248 nm KrF excimer laser (COMPex 205, Coherent) was used to ablate the corresponding ceramic targets with a laser fluence and repetition rate of 1.1 J cm-2/3 Hz for GdScO3, 1.2 J cm-2/5 Hz for PbTiO3, and 1.2 J cm-2/3 Hz for SrTiO3. The GdScO3 and superlattice layers were grown at temperatures of 770 °C and 650 °C, at oxygen pressures of 5 Pa and 12 Pa, respectively. The superlattice films were then annealed for 10 min at 650 °C, followed by cooling to room temperature at a cooling rate of 60 °C min−1 under an oxygen pressure of 20 kPa.

X-ray diffraction
Laboratory-based 2θ/ω-coupled scans and reciprocal space mapping (RSM) measurements were performed with a high-resolution X-ray diffractometer (SmartLab 9kW, Rigaku) using Cu Kα radiation (λ=1.5406 Å). A 2D area detector (HyPix-3000) was used to collect the diffraction signals.

In situ electric field dependent three-dimensional reciprocal space mapping (3DRSM) was performed at the BL13XU beamline of Super Photon ring-8 GeV (SPring-8) synchrotron. A monochromated X-ray beam with a photon energy of 12.4 keV (λ=1 Å) was used. The beam was focused to a size of 12×21 μm2 on the sample surface between the interdigitated electrodes (described below). The sample was mounted on a four-circle diffractometer, connected to an external voltage generator with a tungsten probe station to apply an in-plane electric field. A 2D photo-counting detector (PILATUS3 300K) was used to record the diffraction signals. The 3DRSM results were reconstructed from raw measured data using custom Python codes.

Synchrotron X-ray nanobeam diffraction experiments were performed at the hard X-ray nanoprobe at station 26-ID-C of the Advanced Photon Source at Argonne National Laboratory. A monochromated X-ray beam with a photon energy of 10 keV and a focal spot size of 50 nm was used. The beam was focused to a footprint of ~50×160 nm2 on the sample surface. A 2D area detector positioned 1000 mm from the sample was used to record the diffraction signals.

Scanning transmission electron microscopy
Cross-sectional thin specimens of the superlattices were prepared using a focused ion beam on a dual-beam microscope (Nova Nanolab 200, Thermo Fisher Scientific) or ion milling (PIPS Model 695, Gatan) after mechanical polishing. STEM observations were performed on a double-aberration-corrected transmission electron microscope (Spectra 300, Thermo Fisher Scientific), operated at an accelerating voltage of 300 kV. High-angle annular dark-field (HAADF) STEM images were collected with a probe convergence angle of 25.0 mrad and an inner detector angle of 49 mrad. Positions for all atomic columns in the HAADF images were obtained by 2D Gaussian fits using Matlab codes55. Displacement vectors of B-site columns were calculated based on local offsets of their sublattice relative to the A-site sublattice. The direction of spontaneous polarization in each unit cell is opposite to the direction of B-site displacement.

Second-harmonic generation
SHG measurements were performed on a home-developed microscope system45. A Ti: sapphire femtosecond laser (MaiTai, Spectra-Physics) with 35 fs pulse width, 800 nm wavelength, and an average power of 25 mW was used as the 1ω excitation source. Normal incidence geometry was applied in the measurements, where the 1ω laser beam was focused on the top surface of the film using an objective lens (numerical aperture=0.55), and the backscattered 2ω signals were also collected by the lens. A zero-order half-wave plate was used to control the 1ω excitation light polarization, and a Glan-Taylor prism polarizer was used to analyze the 2ω signal polarization. A photomultiplier tube detector was used to record the 2ω light signal after band-pass filtering to exclude 1ω light signals.

Low-frequency electric measurements
Interdigitated electrodes (IDEs) of 200 nm Au/Ti were fabricated on the surface of the superlattice films via standard photolithography lift-off and sputtering techniques, with 20 μm wide electrode conductors and 10 μm gaps between these parallel conductors. The effective length of the IDEs is 1.000 mm, with 9 and 19 effective capacitor gaps in 5-pair and 10-pair IDEs, respectively. The voltage-dependent capacitance was measured via an impedance analyzer (E4990A, Keysight) with a built-in voltage source. The ferroelectric switching properties were measured with a ferroelectric tester (10-kV HVI-SC, Radiant Technology).

Millimeter-wave dielectric measurements
Coplanar waveguides (CPWs) with a cross-section comprising an 11 μm center conductor, a 15 μm gap and a 200 μm ground plane were fabricated on the superlattice films via the same processes as the IDEs. Five different effective lengths ranging from 0.260 mm to 1.740 mm were designed, together with a short-circuit reflection calibration device matching the shortest length. CPWs on all the superlattices and a bare LSAT substrate were fabricated in the same batch to ensure consistent device electrode thicknesses and shapes. The complex S-parameters of the CPW devices ranging from 2 GHz to 110 GHz were measured via a millimeter-wave vector network analyzer (N5290A, Keysight) coupled with a probe stage (EPS200 MMW, Form Factor) and a pair of ground-signal-ground (GSG) probes. D.c. voltages between the ground and signal lines were applied via a source meter (Keithley 2470).

A multiline TRL approach employing a Python package by Ziad Hatab56 was then applied to the measured S-parameters to obtain the propagation constant γ = R+iωL G+iωC, of the transmission line from the S-parameters, where R, L, C, and G represent distributed resistance, inductance, capacitance, and conductance per unit length, respectively, and ω denotes the angular frequency. R, L, C, and G of the transmission lines on LSAT substrates and films were simulated with Q3D module in Ansys Electronics Desktop Student with a 2-D cross-sectional model. The dielectric constant of LSAT and the conductivity of the metal layers can then be obtained by comparing γ calculated from the simulations and from the experimental S-parameters. The R and L on all the superlattices samples should be identical to those on the LSAT substrate, as the geometric structure and conductivity of the metal layers are identical. The C and G for the superlattices were then calculated from γ and above, simulated R and L. A functional relationship between the C, G and complex dielectric constant of the superlattice layer was established by mapping the dielectric constant and loss of the film in the 2-D model, which was later used to extract dielectric constants under different voltages and frequencies from the above calculated C and G.

Molecular dynamics simulations
Molecular dynamics (MD) simulations were conducted using a machine learning-based force field capable of modeling various topological structures57,58. To simulate the dipole wave and flux closure state, the number of layers in SrTiO3 was fixed at 8, while the layers in PbTiO3 were varied from 12 to 22. All MD simulations were carried out in the isobaric-isothermal (NPT) ensemble using LAMMPS59. The temperature and pressure was maintained at 300 K and 1.0 bar using the Nosé-Hoover thermostat and Parrinello-Rahman barostat, respectively. A timestep of 2 fs was utilized, with each simulation consisting of a 50 ps equilibrium run followed by a 50 ps production run. Initial and final configurations are given in Supplementary Data 1.

Once equilibrium was achieved under zero electric field conditions, an electric field was gradually increased to 200 kV cm−1 across the superlattice to obtain the P–E loop. The interaction between the electric field and each ion was determined by multiplying the electric field strength by the Born effective charge (BEC) tensor of the corresponding ion. Density functional theory (DFT) calculations were used to determine BEC values, as shown in Table 1. All DFT calculations were performed with a plane-wave basis utilizing the Vienna Ab initio Simulation Package (VASP)60,61. The exchange-correlation functional used was the Perdew-Burke-Ernzerhof functional modified for solids (PBEsol)62, with an energy cutoff of 800 eV and a k-point spacing of 0.3 Å−1.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_2": "The manuscript by Wang et al. reports the growth, structural/electrical characterization, and molecular dynamics (MD) simulation of PbTiO3/SrTiO3 superlattices with topological polar structures. By increasing the number of PbTiO3 layers from 12 to 22, the authors observed a systematic change of topological phase from dipole wave to flux closure structures, which resulted in different polarization switching behaviors and tunable dielectric responses in the millimeter-wave regime. The frequency-dependent dielectric constants of these films were studied and compared with MD simulations. The results could lead to better understandings on high-frequency dielectric properties of topological polar phases and novel microwave device applications. The paper is well written with detailed information for reproducibility of the results. The analysis of in-plane switching and dielectric response is rigorous and consistent with simulation. The demonstration of better tunability over conventional ferroelectric devices is also sound. I can recommend its publication after the authors consider the following changes.\n1. The structure of the manuscript requires some adjustments. The title, abstract, and conclusion are all emphasizing the dielectric tunability of their thin films at the millimeter-wave regime. The paper itself, however, only has the last 2 pages (out of 10) and 1 figure (out of 4) in the main text discussing the relevant high-frequency measurements. Since similar films have been grown and characterized before (Ref. 12), the authors should trim the content in the first 8 pages and substantially expand the description and discussion of millimeter-wave data. Otherwise the title and abstract do not reflect the key findings of this work.\n2. The difference between PbTiO3 layer n = 12 and 15 in Fig. 1d and 1f is rather subtle. Some line cuts like Fig. 1b would help distinguish the RSM signal for the two superlattices.\n3. According to Fig. 1f in Ref. 12, the phase-filed simulations generate a phase diagram in that n=12 and m=8 results in vortex-antivortex pair in PTO/STO superlattice, whereas n=15,20,22 and m=8 lead to vortices in the PTO layer with some antivortices in STO. The result is different from the report here, with dipole waves for n=12,15 and flux closure for n=20,22. Can the authors comment on the difference in their findings and the terminology (dipole waves and flux closures versus vortex-antivortex pairs)?", "reviewer_3": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts."}, "author_response": "Response to the reviewers’ comments\n1\n4 structure switching in PbTiO /SrTiO superlattices\n3 3\n5\n6 1. Reviewer #1 (Confidential Remarks to the Editor):\n7\n10\n11 2. Reviewer #2 (Remarks to the Author):\n13 characterization, and molecular dynamics (MD) simulation of PbTiO /SrTiO\n3 3\n14 superlattices with topological polar structures. By increasing the number of PbTiO\n3\n15 layers from 12 to 22, the authors observed a systematic change of topological phase\n16 from dipole wave to flux closure structures, which resulted in different polarization\n17 switching behaviors and tunable dielectric responses in the millimeter-wave regime.\n19 with MD simulations. The results could lead to better understandings on high-frequency\n20 dielectric properties of topological polar phases and novel microwave device\n21 applications. The paper is well written with detailed information for reproducibility of\n22 the results. The analysis of in-plane switching and dielectric response is rigorous and\n23 consistent with simulation. The demonstration of better tunability over conventional\n24 ferroelectric devices is also sound. I can recommend its publication after the authors\n25 consider the following changes.\n26\n29 the reviewer.\n30\n31 2.1 Reviewer #2’s Comment 1:\n33 conclusion are all emphasizing the dielectric tunability of their thin films at the\n34 millimeter-wave regime. The paper itself, however, only has the last 2 pages (out of 10)\n35 and 1 figure (out of 4) in the main text discussing the relevant high-frequency\n36 measurements. Since similar films have been grown and characterized before (Ref. 12),\n37 the authors should trim the content in the first 8 pages and substantially expand the\n38 description and discussion of millimeter-wave data. Otherwise the title and abstract do\n39 not reflect the key findings of this work.\n40\n43 would like to point out that the superlattice films in our study differ considerably from\n44 those in Ref. 12, as they exhibit distinct topological structures and polarization\n45 arrangements due to variations in substrate misfit strain and layer thicknesses. These\n46 structural differences influence the stability and topological protection behaviors. On\n47 the other hand, we employ a d.c. electric field to achieve dielectric tunability at\n48 millimeter-wave frequencies. The structural evolution under low-frequency electric\n49 fields determines the initial state before the excitation of millimeter-wave fields,\n50 ultimately affecting the total dielectric response. Therefore, we think it is worth\n51 discussing the structure of the films, and its evolution under low-frequency electric\n52 fields, which has also laid the basis for exploring more complex high-frequency\n53 response behaviors.\n55 insights into the origin of dielectric tunability. To further elucidate this mechanism, we\n56 have added new discussions in Pages 9–10, supported by expanded Fig. 4,\n58 model demonstrates the polarization evolution dynamics under combined d.c. electric\n59 field biasing and millimeter-wave excitation, effectively mimicking the experimentally\n60 measured dielectric responses. By deeply linking the structure and low-frequency\n61 transition behaviors, these results elucidate how the d.c. field-induced switching\n62 processes influence the millimeter-wave dielectric tunability behaviors in the dipole\n63 wave and flux closure superlattices.\n65 open to the reviewer’s suggestions regarding the title/abstract and readily adjust to\n66 enhance the presentation quality.\n67\n68\n69 2.2 Reviewer #2’s Comment 2:\n3\n71 subtle. Some line cuts like Fig. 1b would help distinguish the RSM signal for the two\n72 superlattices.\n73\nz\n76 results in Supplementary Fig. 1c to more clearly show differences in the high Q peaks\nz\n77 for the superlattices with PbTiO layer thicknesses n = 15 and 12. We also added related\n3\n78 comparison in the intensity and full width at half maximum in Page 5.\n79\n80 2.3 Reviewer #2’s Comment 3:\n82 diagram in that n=12 and m=8 results in vortex-antivortex pair in PTO/STO superlattice,\n83 whereas n=15,20,22 and m=8 lead to vortices in the PTO layer with some antivortices\n84 in STO. The result is different from the report here, with dipole waves for n=12,15 and\n85 flux closure for n=20,22. Can the authors comment on the difference in their findings\n86 and the terminology (dipole waves and flux closures versus vortex-antivortex pairs)?\n87\n90 no significant polarization is observed in SrTiO layers. Therefore, antivortices do not\n3\n91 exist in the superlattices, although similar interlayer coupling happens in the dipole\n92 wave or flux closure structures.\n94 structures is the misfit strain. As described in Ref. 12, the superlattices in that report\n95 were grown on (110)-DyScO substrates with a pseudocubic lattice constant of 3.95 Å.\n3\n97 of our superlattices grown on GdScO -buffered LSAT substrates are determined to be\n3\n98 ~3.93 Å, resulting in different lattice mismatch strains (1.1% and 0.66%, respectively,\n99 where a = 3.904 Å).\nPTO\n101 significantly affect the formation stabilities of topological polar structures. Dipole\n102 waves and flux closures form at similar or lower misfit strains compared to a /a +\n1 2\n103 vortices, but at higher strains than pure vortices. A previous study (Nano Letters 23,\n104 6602 (2023)) reported pure vortices at a misfit strain of 0.3% and a /a + vortices at\n1 2\n105 1.1%. Therefore, dipole waves and flux closures should form within a misfit strain\n106 range between these values, which matches our case here.\n108 distinct polar structures due to kinetics stability constraints. Recent studies demonstrate\n109 that cooling from various temperatures can trigger polar topological state switching,\n110 enabling access to varied topological phases (Nature Physics 21, 464 (2025)). This\n111 suggests that distinct polar topological structures may form in superlattices with\n112 identical layer thicknesses and substrate misfit strains when grown under different\n113 temperatures and cooling rates.\n114\n115 3. Reviewer #3 (Remarks to the Author):\n117 reports. This is part of the Nature Communications initiative to facilitate training in\n118 peer review and to provide appropriate recognition for Early Career Researchers who\n119 co-review manuscripts.\n120"}, {"round": 2, "reviewer_comments": {"reviewer_2": "The authors have carefully addressed my comments and concerns in the response letter. I am particularly happy to see that they substantially expand the description and discussion of the section on field tunability of dielectric properties at millimeter-wave frequencies. However, on my Comment #3, while the authors elaborated the answers in the response letter, they made no effort to change either the main text or the SI. This does not help improve the quality of this paper. I'd like to see that they explicitly include these discussions in the next round of revised manuscript. I cannot recommend its publication in Nature Communications without that change.", "reviewer_3": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.", "reviewer_4": "The manuscript “Millimeter-wave dielectric tunability driven by topological polar structure switching in PbTiO3/SrTiO3 superlattices” by S. Wang et al. presents a study of GHz dielectric tunability of the polar flux-closure and dipole wave states commonly observed in superlattices comprising ultra-thin ferroelectric layer interleaved with dielectric spacers.\nThe authors fabricate four PbTiO (PTO)/SrTiO (STO) superlattices with PTO layer thicknesses of 12, 15, 20 and 22 unit cells. The thickness of STO layers is kept fixed to 8 unit cells. The authors then characterize polar structure of the samples using various experimental techniques. The characterization shows that 12 and 15 u.c. PTO samples feature a dipolar wave state while 20 and 22 u.c. PTO samples exhibit a dipolar wave-like flux closure domains.\nFinally, the authors perform static and dynamic dielectric measurements. The static measurement reveals the possibility of switching the in-plane polarization in all samples with coercive field increasing with increasing PTO thickness. The dynamic measurements are performed under a dc bias within ~1-100 GHz frequency range. The latter measurements show that the dc bias allows for tuning of the ac dielectric response, with a maximum variation of 10% observed in 15u.c. samples.\nOverall, the manuscript is technically sound, well written, and I do not have any concerns regarding the validity of the presented results. The results pertaining to the growth and structural characterization of the PTO/STO superlattices as well as the observed states are original and will be of interest for the specialists in the field. At the same time, the novelty of this part of the manuscript is incremental rather than transformative (unless I missed something). Similar states have been commonly observed and extensively studied during the last years in PTO/STO based materials.", "reviewer_5": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts."}, "author_response": "Response to the reviewers’ comments\n1\n4 structure switching in PbTiO /SrTiO superlattices\n3 3\n5\n6 1. Reviewer #2 (Remarks to the Author):\n8 letter. I am particularly happy to see that they substantially expand the description and\n9 discussion of the section on field tunability of dielectric properties at millimeter-wave\n10 frequencies. However, on my Comment #3, while the authors elaborated the answers\n11 in the response letter, they made no effort to change either the main text or the SI. This\n12 does not help improve the quality of this paper. I'd like to see that they explicitly include\n13 these discussions in the next round of revised manuscript. I cannot recommend its\n14 publication in Nature Communications without that change.\n15\n18 apologize for inadvertently missing the necessary revision to the main text, and thank\n19 you for pointing out the issue. We have added the related discussion in Supplementary\n21 “Previous studies have reported vortex-antivortex pairs in superlattices with\n22 similar PbTiO and SrTiO layer thicknesses12. In our superlattices, the difference in\n3 3\n23 misfit strain and other factors may have led to the observed topological structures (see\n25\n26 2. Reviewer #3 (Remarks to the Author):\n28 reports. This is part of the Nature Communications initiative to facilitate training in\n29 peer review and to provide appropriate recognition for Early Career Researchers who\n30 co-review manuscripts.\n31\n33\n34\n35 3. Reviewer #4 (Remarks to the Author):\n37 structure switching in PbTiO3/SrTiO3 superlattices” by S. Wang et al. presents a study\n38 of GHz dielectric tunability of the polar flux-closure and dipole wave states commonly\n39 observed in superlattices comprising ultra-thin ferroelectric layer interleaved with\n40 dielectric spacers.\n3 3\n42 layer thicknesses of 12, 15, 20 and 22 unit cells. The thickness of STO layers is kept\n43 fixed to 8 unit cells. The authors then characterize polar structure of the samples using\n44 various experimental techniques. The characterization shows that 12 and 15 u.c. PTO\n45 samples feature a dipolar wave state while 20 and 22 u.c. PTO samples exhibit a dipolar\n46 wave-like flux closure domains.\n48 measurement reveals the possibility of switching the in-plane polarization in all\n49 samples with coercive field increasing with increasing PTO thickness. The dynamic\n50 measurements are performed under a dc bias within ~1-100 GHz frequency range. The\n51 latter measurements show that the dc bias allows for tuning of the ac dielectric response,\n52 with a maximum variation of 10% observed in 15u.c. samples.\n54 concerns regarding the validity of the presented results. The results pertaining to the\n55 growth and structural characterization of the PTO/STO superlattices as well as the\n56 observed states are original and will be of interest for the specialists in the field. At the\n57 same time, the novelty of this part of the manuscript is incremental rather than\n58 transformative (unless I missed something). Similar states have been commonly\n59 observed and extensively studied during the last years in PTO/STO based materials.\n60\n62 agree that similar topological structures were observed in recent studies. Nevertheless,\n63 very few studies have investigated the high-frequency dielectric properties of these\n64 structures thus to link them with practical applications. In our work, we have extended\n65 beyond the growth of the superlattices and their static structural characterizations,\n66 laying the foundation for exploring electric field-driven topological structure changes\n67 that further underpin millimeter-wave tunable dielectric properties. From this viewpoint,\n68 we would trust that the novel and significance of our work is substantial.\n69 3.1 Reviewer #4’s Comment 1:\n71 measurements. Despite numerous discoveries of novel polar topologies and multiple\n72 studies characterizing such states, the full technological potential of polar states in ultra-\n73 thin ferroelectrics remains to be unlocked. Dynamic polarization response is one of the\n74 emerging areas of research where polar topological states can, potentially, make a big\n75 difference. Therefore, the authors are exploring an important area of research.\n77 bibliography search reveals reports of up to 40% tunabilities in e.g. relaxors\n82 to make a visual comparison of tunability in their system with other known materials\n83 (something similar to Fig. 1f in the above reference). This will clarify the importance\n84 and possible impact of the study.\n85\n87 related studies comparing dielectric tunability against the measurement or operating\n88 frequencies, as shown in Fig. R1 (also as Supplementary Figure 11 in the 2nd revised\n89 manuscript). We respectfully remind that the study mentioned (Adv. Funct. Mater. 2023,\n90 33, 2210709) only reports high tunability below 10 MHz, which does not fall in the\n91 range of microwave/millimeter-wave frequencies. Generally, the physical mechanisms\n92 and application scope for dielectric tunability vary significantly with the operating\n93 frequency. Therefore, it is not included in the comparison for clarity’s sake.\n94\n96 = 17 and materials reported in previous studies. The materials and studies included in\n3 3\n98 (BST) in Appl. Phys. Lett. 125, 223502 (2024); [3] BaTiO in Phys. Rev. B 100, 184104\n3\n99 (2019); [4] BST in Nature 560, 622 (2018); [5] (Pb,Sr)TiO (BST) in Appl. Phys. Rev.\n3\n100 2, 011302 (2015); [6] BST in Appl. Phys. Lett. 109, 112902 (2016); [7] BST in\n102 (2021); [9] BST in Journal of Advanced Ceramics, 12, 1521 (2023); [10] BST in ACS\n2\n105 97, 162909 (2010)\n106\n107 3.2 Reviewer #4’s Comment 2:\n109 would strongly encourage further characterization of the ac response that can make the\n110 study stand out more.\n112 samples is mainly related to the “softening” of the polarization response upon\n113 decreasing the PTO thickness (i.e. the coercive field diminishes with decreasing n).\n115 n=15 rather than n=12. The authors attribute this to formation of in-plane domains\n116 coexisting with dipolar waves.\n118 strongly asymmetric C-V curves at 20Ghz which appears to contribute to higher\n119 tunability. Can the authors comment on the origin of such asymmetry and whether it\n120 pertains at different frequencies?\n122 reproduced from MD simulations?\n123\n125 the raw measured S-parameters and ensured the validity of data processing routines.\n127 50 GHz or 110 GHz, as shown in Supplementary Fig. 4c,d) and in various superlattices\n128 (see Fig. R3 and R5 below), so it appears to be systematic. This is somewhat counter-\n129 intuitive, since the ground-signal-ground structure of the CPW devices are symmetric.\n132 in MD simulations. Our current speculation is that, the asymmetry observed in our\n133 experiments may originate from extrinsic factors other than the superlattice itself, e.g.\n134 contact interfaces between the superlattice and metal electrodes. We are still trying to\n135 figure out the exact origin of the asymmetry by modifying the Au/Pt metal deposition\n136 recipes (please note that this is a lengthy process for clean room facilities), but we think\n137 that such a fact does not affect the major findings and conclusions of the study.\n139 with additional experiments, further described in Part 3.6 of this response. We consider\n140 the that the “softening” is not directly due to the PbTiO thicknesses, but the different\n3\n141 types of topological polar structures. The different tunability results from the stability\n142 of varous topological structures and their proportions in the superlattices controlled by\n143 the PbTiO thicknesses.\n3\n144\n145 3.3 Reviewer #4’s Comment 3:\n147 simulations, particularly the optical phonon spectra, mainly for n=15 and n=12 cases.\n149 https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.133.096801) and identify the\n150 excitations responsible for the peaks in dielectric responses (Fig. 4b,c,i). Such analysis\n151 could allow to elaborate the existence and role of “collective modes in the dipole wave\n152 structure”.\n153\n155 clarify that the dielectric response peaks (e.g., around 20 GHz) can be attributed to the\n156 resonant switching behavior between T /T dipole wave domains. Please see details in\n1 2\n157 our response to your Comment 7 and Figure R8. We have revised the related discussion\n158 in Page 10 in the revised manuscript, and also cited the PRL paper as Ref. [50].\n159 “Another possible contribution to the dielectric tunability is the resonant switching\n160 from T to T domains (Fig. 3d). As shown in Supplementary Fig. 10, upon application\n1 2\n161 of a combination of d.c. and 20 GHz a.c. electric fields, the T /T proportion further\n1 2\n162 decreases, lower than that under a d.c. field with the same magnitude. Similar resonant\n163 switching processes have been reported in a previous study on polar vortices 50. This\n164 process further reduces the overall dielectric permittivity under electric fields, leading\n165 to higher dielectric tunability.”\n167 energy density (SED) method can provide valuable insights into the wave-like nature\n168 and characteristic frequencies associated with the T /T topological domain switching.\n1 2\n170 on the full supercell is computationally prohibitive. Instead, we consider three\n171 simplified scenarios: a single-domain structure of PbTiO (without domain features), a\n3\n1 2\n173 waves embedded in (PbTiO ) /(SrTiO ) .\n3 12 3 8\n3 1 2\n176 minimum frequency at the inverse wavelength is ~0.2 THz, closely matching the\n177 experimental value of 0.22 THz reported in Nature 592, 376 (2021). For the dipole\n178 wave structure, the lowest frequency at the inverse wavelength is found to be below 0.2\n180 comparison among the three cases suggests that the resonant switching behavior may\n181 not be solely attributed to FE T /T switching, but appears to be closely associated with\n1 2\n182 the internal topological structure of the T /T wave domains. These initial SED\n1 2\n183 calculations help corroborate the MD results in Figure R8, and we plan to further refine\n184 this approach in future work.\n1 2\n186 domains, and (c) dipole wave.\n187\n188 3.4 Reviewer #4’s Comment 4:\n190 thickness of STO layers, at least for the n=15 case, without which the study appears\n191 incomplete.\n192\n194 on two newly fabricated superlattices with n = 15, m = 5 and 11. The average in-plane\n195 lattice constants are determined to be ~3.93 Å for both superlattices from RSM\n196 measurements (Fig. R3), consistent with the previous sample superlattices. It is worth\n197 noting that the interlayer coupling and in-plane ordering is significantly weaker in the\n198 superlattice with m = 11, as the SrTiO layer is thicker.\n3\n199\n200\ny\n202 superlattices with SrTiO layer thicknesses m = 5 and 11.\n3\n205 dipole wave to polar vortex (more in-depth synchrotron characterizations is planned).\n207 topic of the current manuscript remains to be focused.\n208\n209\n211 20 GHz, 50 GHz and 100 GHz, and dielectric tunability spectra for superlattices with\n212 different SrTiO thickness m.\n3\n213\n214 3.5 Reviewer #4’s Comment 5:\n216 that are more susceptible to the applied in-plane field compared to the flux closure state,\n217 the dipolar wave exhibits a higher dielectric permittivity. As domain switching (T1/T2)\n218 occurs along the direction of the applied field, the overall permittivity decreases—since\n219 fewer regions remain available to switch in that direction. Although both configurations\n220 (dipolar wave and flux closure) undergo switching, the dipolar wave demonstrates\n221 greater tunability, owing to its larger relative change in dielectric permittivity.\n223 tunability of the dipolar wave, alluding to peaks in dielectric tunability observed in the\n224 frequency sweep (Fig. 4i). They cite [Li, Q. et al. Subterahertz collective dynamics of\n225 polar vortices, Nature 592, 376–380 (2021)], which shows that the frequency of\n226 collective modes can vary, for example, by a change in temperature. In the present study,\n227 the authors explore systems with varying thicknesses, which could similarly (due to\n228 thickness change) influence the frequency of the collective modes and thereby affect\n229 dielectric tunability. Would the authors consider examining whether the tunability of\n230 the n = 12, 20, and 22 samples improves over a broader frequency range, or whether\n232\n235 are not able to experimentally measure the dielectric tunability over 110 GHz. On the\n236 other hand, our MD simulations can indeed be extended to higher frequencies. The\n237 magnitude of high frequency components of the Gaussian pulses is smaller than that of\n238 lower frequency components, leading to smaller polarization responses. This effect\n239 affects the accuracy of the dielectric simulations, yet the overall trend should still be\n240 valid. As shown in Fig. R4, although multiple tunability peaks appear in the response\n241 spectra of both topological polar structures, there is no significant increasing trend of\n242 tunability at higher frequencies.\n243\n245 closure systems for the n = 15 and 22 superlattices. Both spectra are normalized against\n246 the zero-field values.\n247 3.6 Reviewer #4’s Comment 6:\n249 influence dielectric tunability. For instance, the n = 12 sample shows lower tunability\n250 compared to n = 15, which the authors attribute to the presence of a1/a2 domains. In\n251 that context, could the authors comment on whether there exists an optimal thickness—\n252 perhaps between n = 13 and n = 19—where tunability peaks within the dipolar wave\n253 regime? If so, does this correspond to the configuration with the largest \"wavy\" area of\n254 the dipolar wave, which the authors show is most responsive to in-plane fields? While\n255 further experiments across different thicknesses may be beyond the scope of this study,\n256 molecular dynamics simulations at additional thicknesses could offer valuable insights.\n257\n259 and measured three new superlattices with n = 16, 17 and 19 and m = 8. The average\n260 in-plane lattice constants are determined to be ~3.93 Å for all superlattices from RSM\n261 measurements (Fig. R6), consistent with the previous sample superlattices.\n262\n263\ny\n265 superlattices with PbTiO layer thicknesses n = 16, 17 and 19.\n3\n266\n268 the highest tunability at n = 17, which is the dipole wave superlattice with highest n in\n269 the series. Combining this and the results with varying SrTiO layer thicknesses (Fig.\n3\n271 topological structures in correlation with high zero-field dielectric permittivity. We\n272 believe that the polarization response is enhanced near the topological phase boundaries,\n273 leading to the observed larger dielectric permittivity. This argument will be further\n274 substantiated by more comprehensive characterizations and simulations, as we are\n275 planning to undertake in future work. It is also worth noticing that the tunability spectra\n276 of the superlattices with n = 16, 17 and 19 also exhibit peaks at 20–30 GHz and 40–50\n279 superlattices in Fig. 4 and provided a brief discussion on the main text Page 9 in the\n280 2nd revised manuscript.\n281 “The high overall dielectric tunability in the n = 17 superlattice correlates with its\n282 large zero-field permittivity, suggesting that the dynamical polarization response is\n283 enhanced near the topological phase boundaries between dipole wave and flux closure\n284 structures.”\n285\n286\n288 20 GHz, 50 GHz and 100 GHz, and dielectric tunability spectra for superlattices with\n289 different PbTiO thickness n.\n3\n290\n291 3.7 Reviewer #4’s Comment 7:\n293 evolves after excitation of collective modes by microwave fields in the n = 15 sample.\n296 equivalent polar vortex states. The authors demonstrate that applying a DC bias alters\n297 domain proportions. If data are available, could the authors comment on whether\n298 excitation of collective modes—rather than DC bias—induces a more pronounced\n299 domain transformation? Specifically, is there evidence of a stronger shift toward the\n300 field directed domains under “resonant” microwave excitation of the collective mode\n301 and does that have an effect on the dielectric tunability?\n302\n304 identified such resonant switching behavior in both our experimental results and\n305 molecular dynamics (MD) simulations, and have accordingly cited [Phys. Rev. Lett.\n306 133, 096801 (2024)] as Ref. 50 in the revised manuscript.\n308 evidence showing that after applying an electric field to the T /T wave domains, not\n1 2\n309 only does the wave structure within each domain change, but a transition between T\n1\n310 and T domains also occurs. In this revision, we have made considerable efforts to\n2\n311 simulate such structures and investigate the evolution of them under both d.c. and\n312 d.c.+a.c. electric fields.\n3 12 3 8\n314 an in-plane size of approximately 48 nm × 48 nm, consisting of 1.44 million atoms. The\n315 simulation was performed using 48 V100 GPUs, which reaches the limit of our current\n316 computational resources. As shown in Figure R8 (Supplementary Fig. 10 and new\n1 2\n3\n319 the x-axis and y-axis are organized into T and T domains, respectively. Within each\n1 2\n320 domain, the internal structure consists of several individual wave patterns, forming a\n321 strip-like morphology. At the boundaries both inside and between T and T domains,\n1 2\n322 a/c nanostructures emerge to minimize the polarization gradient, which is consistent\n323 with experimental observations and provides additional structural details.\n1 2\n325 d.c. and combined d.c.+a.c. conditions. Our results indicate that both types of electric\n326 fields can induce transitions between T and T domains; however, the d.c.+a.c. case\n1 2\n327 requires a lower electric field amplitude. Specifically, the transition from T to T occurs\n1 2\n328 at about 100 kV/cm under a d.c. electric field, while under a combined field of 20 +\n329 20𝑠𝑖𝑛(2𝜋𝑓𝑡) kV/cm (with 𝑓 =20 GHz), the transition takes place at a maximum\n330 amplitude of only 40 kV/cm. Notably, a constant d.c. field of 40 kV/cm does not induce\n331 noticeable domain switching. Therefore, by combining the experimental and MD\n332 results, we conclude that the resonant switching behavior, which is facilitated by the T\n1\n333 to T domain transition, modulates the polarization along the y-axis and contributes to\n2\n334 the dielectric response.\n335\n336\n1 2\n338 fields.\n339\n341 dielectric tunability in Page 10 of the main manuscript. Additionally, we have replaced\n342 the original schematics in Fig. 3d with the MD simulated patterns.\n343 “Another possible contribution to the dielectric tunability is the resonant switching\n344 from T to T domains (Fig. 3d). As shown in Supplementary Fig. 10, upon application\n1 2\n345 of a combination of d.c. and 20 GHz a.c. electric fields, the T /T proportion further\n1 2\n346 decreases, lower than that under a d.c. field with the same magnitude. Similar resonant\n347 switching processes have been reported in a previous study on polar vortices 50. This\n348 process further reduces the overall dielectric permittivity under electric fields, leading\n349 to higher dielectric tunability.”\n350\n351\n352 4. Reviewer #5 (Remarks to the Author):\n354 reports. This is part of the Nature Communications initiative to facilitate training in\n355 peer review and to provide appropriate recognition for Early Career Researchers who\n356 co-review manuscripts.\n357"}, {"round": 3, "reviewer_comments": {"reviewer_2": "The authors have finally included all information in the manuscript and SI. I can recommend its publication in Nature Communications now.", "reviewer_3": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.", "reviewer_4": "I appreciate that the authors have carefully considered my comments. After having reviewed the responses, I find that my main concerns have been addressed.\nMy final suggestion would be to place the Supplementary Figure 11 comparing the tunability of known materials in the main text rather than SI (either as a panel within one of the existing figures or as a standalone figure in the conclusions section). It contains important information which would enhance the impact and help understand the importance of the study. It would also be helpful to include the discussion of the electric field magnitudes (Supplementary Figure 11 shows tunability slope with field rather than tunability in % as discussed in the manuscript), as well as the loss tangents that can be deduced from Figs.4b,c. The latter figures imply that the losses become relatively high at high frequencies (>10GHz) and it would be important to understand whether this would pose problems for applications."}, "author_response": "Response to the reviewers’ comments\n1\n4 structure switching in PbTiO /SrTiO superlattices\n3 3\n5\n6 1. Reviewer #2 (Remarks to the Author):\n8 recommend its publication in Nature Communications now.\n9\n12\n13 2. Reviewer #3 (Remarks to the Author):\n15 reports. This is part of the Nature Communications initiative to facilitate training in\n16 peer review and to provide appropriate recognition for Early Career Researchers who\n17 co-review manuscripts.\n18\n20\n21\n22 3. Reviewer #4 (Remarks to the Author):\n24 reviewed the responses, I find that my main concerns have been addressed.\n25\n27 tunability of known materials in the main text rather than SI (either as a panel within\n28 one of the existing figures or as a standalone figure in the conclusions section). It\n29 contains important information which would enhance the impact and help understand\n30 the importance of the study. It would also be helpful to include the discussion of the\n31 electric field magnitudes (Supplementary Figure 11 shows tunability slope with field\n32 rather than tunability in % as discussed in the manuscript), as well as the loss tangents\n33 that can be deduced from Figs.4b,c. The latter figures imply that the losses become\n34 relatively high at high frequencies (>10GHz) and it would be important to understand\n35 whether this would pose problems for applications.\n36\n38 original Supplementary Figure 11 to Figure 5 in the manuscript, and added a discussion\n39 about the low electric field magnitudes in the Discussion section in Page 11. We have\n40 already included a discussion about the fitted tunability in % under larger electric fields\n41 in the original manuscript, which is also moved to the Discussion section in Page 11."}]

Decision letter