Are superconductivity and ferroelectricity reconcilable? Bernd T. Matthias, a pioneer in the search for superconducting materials, stated more than 50 years ago1 that “superconductivity and ferroelectricity will exclude one another”. This is known as the Matthias conjecture, which is an old guideline used in the hunt for new superconducting materials. The conflict between superconductivity and ferroelectricity motivated the study of Bednorz and Müller in which they discovered high-temperature cuprate superconductors2. Anderson and Blount proposed that some metals can be ferroelectric in the sense that they have broken inversion symmetry. However, almost all ferroelectric materials are highly insulating, and only a few of these materials have been experimentally shown to be Anderson-Blount-type ferroelectric (polar) metals3. These so-called polar metals have recently raised interest from the research community4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21. In particular, low-carrier-density polar metals have drawn attention22,23,24,25 because they show superconductivity that contradicts the Matthias conjecture. Our study focuses on one of these metals, the perovskite-type titanate SrTiO3, to reveal a missing aspect of superconductivity with ferroelectricity.

Single crystals of SrTiO3 show huge relative permittivity of approximately 42,000 at 4.2 K (ref. 26). However, the ferroelectric phase transition does not occur at temperatures above 35 mK (ref. 27). Because the ground state is predicted to be ferroelectric28,29, it is believed that strong quantum fluctuations suppress the material’s ferroelectricity27. However, this does not explain the unusual minimum observed in the inverse relative permittivity of SrTiO3 (refs. 30,31), and the origin of the paraelectric state is still a subject of intensive debate. Regardless of its origin, the paraelectric state is quite fragile; a small amount of homovalent substitution of Ca2+ (refs. 32,33) or Ba2+ (refs. 34,35) for Sr2+ turns SrTiO3 into a ferroelectric material. Exchange of the 18O isotope for 16O also results in a change into the ferroelectric phase (refs. 10,36). The domain walls of SrTiO3 in the tetragonal phase carry polar properties related to ferroelectricity37,38. Furthermore, both epitaxial strain18,19,39,40 and Sr defects41 in thin SrTiO3 films also generate ferroelectricity.

In contrast, electron doping of SrTiO3, e.g., substitution of La3+ or Sm3+ for Sr2+ (refs.4,5,6,7,42,43) or Nb5+ for Ti4+ (refs.21,44,45,46,47,48), or removal of O2− (refs.9,10,11,12,49,50,51,52,53,54), causes SrTiO3 to become metallic. In all these cases, the superconductivity appears at low temperatures. (Note that carrier doping using Nb or La is denoted by Nb:SrTiO3 or La:SrTiO3 in this paper in cases where it is not necessary to express the compositions explicitly as SrTi1−xNbxO3 or Sr1−xLaxTiO3.) A recent tunnelling study revealed that the Ti 3d t2g orbitals generate two light bands and one heavy band48. The electrons in the heavy band make the largest contribution to the superconductivity48. The most interesting property of doped SrTiO3 systems is that they are extremely low carrier density superconductors with Fermi energies that are a few orders of magnitude lower than the phonon energy. Because the Migdal-Eliashberg criterion (adiabatic condition) is violated23,24, the pairing interaction should show a significant frequency dependence that is caused by poor screening of the Coulomb repulsion24. Further doping causes the Fermi energy to increase, but the superconductivity disappears above a carrier density of 1021 cm−3, and this results in a dome-like dependence of the superconducting transition temperature on the carrier density4,49,50,51.

The physical mechanism that lies behind these phenomena has challenged the related microscopic theory for more than half a century. Although no compelling theories have been proposed that encompass the disparate existing ideas, some unconventional mechanisms have recently sparked new research interest in this field. Typical examples include the Cooper pairings driven by plasmon and plasmon-polariton coupling with longitudinal optical (LO) phonons55,56,57 and by ferroelectric fluctuations15,16,17,22,24,29, particularly those driven via the two phonons58,59,60,61 of a soft transverse optical (TO) mode, which explain the T2-dependence of the resistivity very well62,63. There were also some remarkable experimental discoveries: carrier doping of two ferroelectric matrices, (Sr,Ca)TiO3 (refs.10,11,12) and SrTi(16O,18O)3 (refs.4[,9,10), was found to raise the transition temperature Tc to be higher than that of the SrTiO3 matrix, indicating that the superconductivity appears to be enhanced in the ferroelectric SrTiO3. Several experimental investigations of SrTiO3 with epitaxial strain were also noteworthy5,6,7,43; the strain breaks the material’s spatial inversion symmetry and enhances the spin-orbit interactions5,6, and Tc then increased to 0.6 K (ref. 5). Surprisingly, although Tc is very low in these SrTiO3 systems, magnetic impurities, e.g., Sm, Eu, demonstrated no effect on Tc (ref.43).

We have studied two different polar metals, comprising Sr0.95Ba0.05TiO3 and Sr0.985Ca0.015TiO3, with substitution of Nb5+ for Ti4+ for carrier densities ranging from ~1018 to ~1021 cm−3. Higher levels of Ca or Ba substitution may extend the ferroelectric-metal region of the carrier density to be much greater. However, Ba substitution at more than 20% is accompanied by a complex sequential structural change, and Ca substitution at levels above 0.9% has not been studied in depth. Therefore, 5% for Ba and 1.5% for Ca are acceptably good settings. Sr0.95Ba0.05TiO3 exhibits Ti-site-dominated ferroelectricity, in which Ti-O hybridisation causes a strong pseudo- or second-order Jahn-Teller distortion13. The Curie temperature is ~50 K when the polarisation direction is along the [111] direction of a pseudo-cubic lattice. In contrast, Sr0.985Ca0.015TiO3 shows Sr-site-dominated ferroelectricity that originates from the off-centre position of Ca2+, which has a smaller ionic radius than Sr2+ (refs.11,12,13,14). The Curie temperature is ~30 K when the polarisation direction is [110] (ref. 32). Despite their differences in terms of ferroelectricity, we demonstrate that these two polar metals show common and simple superconducting domes with a higher Tc than that of the nonpolar matrix Nb:SrTiO3. We also show that the Tc enhancement becomes much greater if we go deeper into the polar region.

## Results

### Nonpolar matrix Nb:SrTiO3 compared to SrTiO3−δ and La:SrTiO3

The evolution of the metallic state of our Nb:SrTiO3 single crystals (see the Methods for details of our sample preparation processes) is illustrated in Fig. 1a. The results are similar to those reported by Tufte et al.64 Hall effect measurements (see Supplementary Note 3) were performed to deduce the carrier density n and the Hall mobility μ at 5 K (Fig. 1b). The values of μ for SrTiO3−δ and Nb:SrTiO3 from the literature44,64,65,66 are also plotted for comparison. In the normal metal, μ is proportional to n−1, but Behnia proposed a model67 that assumed that the mean-free path is proportional to the average distance between the dopants and the Thomas-Fermi screening length, thus giving $$\mu \propto n^{ - {\textstyle{5 \over 6}}}$$. Therefore, we tried to fit the formula $$\mu \propto n^{ - b}$$ to our experimental data (see Supplementary Fig. 2). The result that b = 0.85 ± 0.02 indicates that b = 5/6 from the literature67 falls within the error bar, whereas b = 1 does not. In fact, our data in Fig. 1b fitted the model reasonably well. In the model, the proportionality factor is dependent on the reciprocal of the effective mass and the dopant potential. Because the μ value of our Nb:SrTiO3 is greater than that of SrTiO3−δ, the dopant potential is shallower for Nb:SrTiO3. Creation of oxygen defects causes one or two electrons to be trapped at each oxygen-vacancy site and localised without contributing to the itinerant carriers67,68. However, in the case of Ti/Nb substitution, the replacement of the Ti 3d orbitals with the Nb 4d orbitals does not change the orbital characteristics, which means that the disorder is smaller in scale than that caused by the formation of oxygen defects. This is expected to increase the superconducting critical temperature because the spatial disorder generally destroys the superconducting state and suppresses Tc (ref. 69).

The resistive transitions of the superconductivity of our Nb:SrTiO3 single crystals (i.e., the same samples studied in Fig. 1a) are shown in Fig. 2a. We defined Tc as the mid-point of the resistive transition, as described in ref. 4 and as shown in Fig. 2a. The upper and lower error bars were determined from the onset and end temperatures of the superconductivity. This definition of Tc is used throughout this work. The Tc values are plotted as a function of the carrier density n at 5 K in Fig. 2b. The carrier density in the nonpolar metallic state shows little dependence on temperature, as described in the Supplementary Note 2. The Tc values of SrTiO3−δ (refs. 12,49), Nb:SrTiO3 (ref. 23) and La:SrTi(16O1−z18Oz)3 (refs. 4,42 and the three unpublished data points depicted in Supplementary Fig. 1) are also plotted for comparison. Our data clearly demonstrate that the superconducting dome of Nb:SrTiO3 is shifted toward a higher Tc and a larger n region than the superconducting dome of La:SrTiO3. The optimal Tc value for our Nb:SrTiO3 is ~0.5 K at n 1 × 1020 cm−3.

There are two noticeable differences between the superconducting domes of some of the SrTiO3−δ samples in the literature and the domes of our La:SrTiO3 and Nb:SrTiO3 samples. The first prominent difference is a shoulder peak that occurs at n 1.2 × 1018 cm−3 in SrTiO3−δ (ref. 12). If the three-fold degeneracy of the t2g band is lifted51 and each band has a different filling, it is then likely that a superconducting dome will be observed for each band. However, neither our La:SrTiO3 nor our Nb:SrTiO3 samples showed this shoulder peak, seemingly indicating that a single band of Ti 3d makes the contribution to the superconductivity48. The second difference is observed in the high n region (n 1 × 1021 cm−3), where superconductivity with Tc 0.25 K was reported for SrTiO3−δ (ref. 49). Similar to the previous case, neither our La:SrTiO3 sample nor our Nb:SrTiO3 sample showed superconductivity. One possible explanation that can account for both differences simultaneously is based on consideration of the inhomogeneity of the oxygen vacancies52. The thermal reduction procedure used to create the oxygen vacancies for the SrTiO3−δ single crystal is restricted to dislocation70 because the formation enthalpy of the oxygen vacancies near the dislocations is significantly lower than that in the stoichiometric SrTiO3 matrix71. Therefore, the inhomogeneous carrier density distribution occurs in the SrTiO3−δ sample. Hall measurements give the averaged carrier density of the bulk, whereas the observed value of Tc is the highest Tc among all the percolation paths in the doped regions where the carrier density differs from the average value.

When compared with La:SrTiO3 and SrTiO3−δ, Nb:SrTiO3 shows either comparable or higher mobility (Fig. 1b; see also Supplementary Fig. 3 of ref. 4) and a higher Tc with a simple single superconducting dome (Fig. 2b). These results mean that Nb doping of the ferroelectric derivatives of SrTiO3 represents a solid strategy for investigation of the relationship between ferroelectricity and superconductivity. Therefore, we focused on the two ferroelectric matrices: Sr0.985Ca0.015TiO3 and Sr0.95Ba0.05TiO3. Interestingly, the types of ferroelectricity exhibited by these two matrices are different.

### Polar metals Nb:(Sr,Ca)TiO3 and Nb:(Sr,Ba)TiO3

In (Sr, Ca)TiO3, the ferroelectricity is driven by dipole-dipole interactions between the off-centre Ca sites, which have smaller ionic radii than the Sr2+ ion13. Nb doping makes this material highly conductive, but its resistivity increases slowly as T decreases at low temperatures (Fig. 3a). In contrast, our nonpolar Nb:SrTiO3 single crystal does not show this resistance anomaly at all. We, therefore, defined TK as the temperature at which the resistivity reaches a minimum. The values of TK were plotted versus n at TK (n at 5 K for TK = 0) and fitted the solid line in Fig. 3b fairly well. To derive n at TK, we performed a smoothing spline interpolation72 (smoothing factor = 1) for the raw n vs. T data (see Supplementary Note 10) using Igor Pro v8.04 software (WaveMetrics, Inc., USA). Intriguingly, the ferroelectric Curie temperature $$T_{{{\mathrm{C}}}}^{{{{\mathrm{FE}}}}}$$ ~ 25 K, corresponding to the sharp peak in the relative permittivity ε (see the inset of Fig. 3b), is located almost on the same line. It was proposed that the dipole moment remains at the off-centre Ca site in the lightly carrier-doped (Sr, Ca)TiO3 because of poor screening, and the glassy dipole-dipole interaction between the Ca sites then causes the resistance anomaly12,14. The static interaction is fully screened (TK = 0) at a value of n* of 3.1 × 1019 cm−3. This experimental value of n* is almost equivalent to the n* = 3.3 × 1019 cm−3 value calculated by assuming a Thomas-Fermi screening length for the itinerant carriers that is comparable with the averaged dipole-dipole distance14.

In contrast, in (Sr,Ba)TiO3, the Ba2+ ion hardly moves because of its larger ionic radius and heavier mass when compared with the Sr2+ ion. Therefore, phonon softening through the pseudo- or second-order Jahn-Teller distortion that occurs because of the sizeable Ti-O hybridisation becomes the central mechanism of the ferroelectricity13. This is different in principle from the ferroelectricity mechanism in (Sr,Ca)TiO3, but is basically similar to that in SrTi(18O,16O)3, in which the Ti-O hybridisation is also crucial. Our Sr0.95Ba0.05TiO3 single crystal shows the ferroelectric transition at $$T_{{{\mathrm{C}}}}^{{{{\mathrm{FE}}}}}$$ ~ 50 K (inset of Fig. 3d), which is accompanied by a structural change from cubic Pm3m to rhombohedral R3m (ref. 34). We performed powder X-ray diffraction measurements on our metallic Sr0.95Ba0.05Ti0.998Nb0.002O3 at 10 K, and the results were examined via a Rietveld analysis (see Supplementary Note 11). The pattern was, in fact, consistent with the rhombohedral R3m symmetry that occurs with displacement of Ti along the [111] axis of a pseudo-cubic lattice. The spatial inversion symmetry is broken and we thus consider the metallic state to be polar. To our surprise, although the origin of its ferroelectricity differs from that of Nb:Sr0.985Ca0.015TiO3, a similar resistance anomaly is observed in Nb:Sr0.95Ba0.05TiO3 (Fig. 3c), and the value of TK decreases with increasing n (Fig. 3d). For samples with two local minima in their resistivity characteristics, the mid-point between the two temperatures that give these minima is defined as TK, where the lower and upper ends of the error bar correspond to each of these local minima. The carrier density dependence does not seem to be as linear as that observed in Nb:Sr0.985Ca0.015TiO3, but if a linear relationship between TK and n is assumed simply to separate the polar and nonpolar regions, the line almost reaches $$T_{{{\mathrm{C}}}}^{{{{\mathrm{FE}}}}}$$ at n = 0. We have estimated a critical carrier density of n* ~ 2.5 × 1020 cm−3 at TK = 0. This value of n* is one order of magnitude higher than that of Nb:Sr0.985Ca0.015TiO3, although the difference in $$T_{{{\mathrm{C}}}}^{{{{\mathrm{FE}}}}}$$ is only a factor of two. (Note that even if we assume that the polar/nonpolar boundary has a different shape, this does not affect the order of n*.)

Russel et al. reported that the resistance anomaly temperature TK of their strained (Sr,Sm)TiO3 film appeared at precisely the same temperature at which the second harmonic generation (SHG) signal showed a sharp increase18. Because SHG indicates breaking of the inversion symmetry of the system, it is reasonable to define the polar metal region based on TK, i.e., both Nb:Sr0.985Ca0.015TiO3 and Nb:Sr0.95Ba0.05TiO3 are polar metals in the shaded areas shown in Fig. 3b, d, respectively. Note that, at temperatures below TK, the carrier density n for Nb:Sr0.95Ba0.05TiO3 decreases with decreasing temperature (see Supplementary Note 7) in a similar manner to that reported in ref. 5. This means that some carriers are bound to the local dipole moment for screening. However, the values of n above TK are not dependent on the temperature and are similar to the value of n at room temperature. The critical carrier density n*, at which the resistance anomaly disappears, is discussed because it may represent the putative ferroelectric quantum critical point (refs.4,15,18) because breaking of the centrosymmetry occurs when n < n* (ref. 18), possibly as a result of poor screening. If so, it is quite intriguing that our two systems, i.e., Nb:Sr0.985Ca0.015TiO3 and Nb:Sr0.95Ba0.05TiO3, have different ferroelectric mechanisms and show one order of magnitude of difference in their n* values (3.1 × 1019 cm−3 for Nb:Sr0.985Ca0.015TiO3 and 2.5 × 1020 cm−3 for Nb:Sr0.95Ba0.05TiO3). It is thus essential to consider whether these significant differences will affect the appearance of the superconductivity. From this point, we investigate the superconductivity of these two systems systematically.

### Superconductivity of Nb:(Sr,Ca)TiO3 and Nb:(Sr,Ba)TiO3

Both polar metals, i.e., Nb:Sr0.985Ca0.015TiO3 and Nb:Sr0.95Ba0.05TiO3, show superconductivity at low temperatures. The resistive transition of the former material is shown in Fig. 4a, and that of the latter material is shown in Fig. 4b. The Tc values plotted as a function of n at 5 K produce a single superconducting dome, as in the case of Nb:SrTiO3 (Fig. 4c, bottom). Moreover, the optimal Tc value of Nb:Sr0.95Ba0.05TiO3 reaches 0.75 K at n 1 × 1020 cm−3. This is the highest Tc value among the SrTiO3 families reported to date. Additionally, another interesting point can be observed in this viewgraph. Although the types of ferroelectricity in Sr0.985Ca0.015TiO3 and Sr0.95Ba0.05TiO3 are different and their n* values differ by one order of magnitude, it is common that the value of n for the maximum of the superconducting dome in both cases is almost identical to that for the nonpolar Nb:SrTiO3, whereas their superconductivity persists even in the extremely low carrier density region. For clarity, we performed the same smoothing spline interpolation procedure that was used to derive n at TK for Fig. 3b, d. The dashed lines in the bottom panel of Fig. 4c show that the spline interpolations fitted the raw data very well. Using these interpolations, we deduced the Tc difference between the polar and nonpolar systems for each value of n, as indicated by the solid lines in the top panel of Fig. 4c. We expected initially that the largest difference would be seen at around n* because we believed that the quantum fluctuation could be the main driving force of the superconductivity and that it may have become strongest at n*. However, the increase in Tc around n* was not significant (less than 0.1 K) for both Nb:Sr0.985Ca0.015TiO3 and Nb:Sr0.95Ba0.05TiO3. Surprisingly, as the system moves deeper into the polar region, in which the ferroelectric fluctuations must be suppressed, the superconductivity still persists in the two polar metals.

It is thus necessary to consider the mechanism that contributes to the considerable enhancement of the superconductivity that occurs in the dilute carrier density region away from n*. One fascinating idea7,21,54,73 is to consider the intrinsic inhomogeneity in this region. Indeed, formation of polar nanodomains is essential in most polar metals; small domains grow as the carrier density is reduced, and macroscopic ferroelectricity finally appears at the zero carrier density limit19. It has recently been revealed that application of uniaxial strain to the Nb:SrTiO3 single crystal induces both dislocations and local ferroelectricity with inhomogeneity that result in enhancement of the material’s superconductivity21. Electroforming also creates metallic filaments, and the resulting inhomogeneous structure increases the superconducting transition temperature54.

### Meissner effect of Nb:(Sr,Ba)TiO3

However, we should note that our polar Nb:Sr0.95Ba0.05TiO3 single crystal (n ~ 8.1 × 1019 cm−3), which gives the highest resistive Tc (0.75 K), is not in such a greatly inhomogeneous state that filaments would be formed. (For the nonpolar Nb:SrTiO3, filamentary superconductivity is at least ruled out at optimum doping levels46). In fact, the Tc value that is defined as the onset of the AC mutual inductance drop (0.70 K) is almost equal to the corresponding resistive value (0.75 K), as illustrated in the top panel of Fig. 5, and this suggests that the Tc distribution is homogeneous. In the bottom panel of Fig. 5, we plotted the DC volume susceptibility versus temperature. The onset of diamagnetism occurs at around 0.70 K, which nearly coincides with the zero resistivity temperature; this indicates that the superconductivity of our Nb:Sr0.95Ba0.05TiO3 cannot be regarded as being at least filamentary on the nearly static timescale. Although the volume fraction of the Meissner effect (the reversible expulsion of the magnetic flux during the field cooling measurement) appears to be small, it is almost comparable to that of La:SrTiO3 (n ~ 1.6 × 1020 cm−3) (ref. 42), where localised moments that were six orders of magnitude smaller than that of the La ions caused such a small volume fraction. Because these extremely small amounts of impurities never affect the Tc value at all, the apparently small volume fraction is not an important issue here. (Collignon et al. measured the superfluid density of nonpolar Nb:SrTiO3, which is equal to the normal state density, thus indicating that the superconductivity is a bulk phenomenon47.) Therefore, the maximum Tc is seen to be found in higher n regions, where the inhomogeneity is suppressed; in contrast, in this study, we focus on Tc enhancement in the lower n region located far away from n*, where the inhomogeneity is important.

## Discussion

Bretz-Sullivan et al. investigated53 single-crystalline SrTiO3−δ within the dilute single band limit for 3.9 × 1016 cm−3 < n < 1.4 × 1018 cm−3. Although the system held a three-dimensional homogeneous electron gas in the normal state, the superconducting state was inhomogeneous. Interestingly, the Tc values for all the nonpolar SrTiO3−δ samples in the inhomogeneous superconducting region are not enhanced and remain almost constant at around 65 mK. This result means that the inhomogeneous superconductivity in the dilute carrier-density region may not be the dominant cause of the Tc enhancement; i.e., the ferroelectric nature of the domains would be essential to the observed enhancement. For the two-dimensional superconductivities at the LaAlO3/SrTiO3 interface, Rashba spin-orbit coupling is key to the Cooper pairing phenomenon, i.e., the superconductivity is sympathetic to inversion symmetry breaking74. In bulk SrTiO3, the spin-orbit interaction is also essential6,17,24,48,75 and the ferroelectricity arises from breaking of the centrosymmetry17. Therefore, the bulk superconductivity may involve more space symmetry breaking, which goes against the Matthias conjecture. In this sense, it is intriguing that the enigmatic 2 K superconductivity observed in the (111) surface of KTaO3 is discussed as being related to a large spin-orbit interaction76. Furthermore, based on this scenario, we may realize higher Tc values by increasing the Ba content to raise $$T_{{{\mathrm{C}}}}^{{{{\mathrm{FE}}}}}$$. Indeed, recent theoretical work on carrier-doped BaTiO3 has indicated that significant modulation of the electron-phonon coupling occurs across the polar-to-centrosymmetric phase transition, and superconductivity is then predicted to occur at 2 K (ref. 20). Another theoretical work on Dirac semimetals predicted that the superconductivity will only appear in the ferroelectric region25. The two-phonon exchange superconductivity scenario58,59,60,61 requires neither quantum criticality nor quantum fluctuations, but can predict a reasonable Tc value in the dilute carrier-density region60. Therefore, we considered whether the ferroelectric fluctuations that occur around n* may not be the main driving force for enhancement of the superconductivity. However, the dynamic movement of the polar-domain boundaries can be regarded as a type of spatiotemporal fluctuation of n*. In other words, the intrinsic inhomogeneity of the materials may still play some role in the Tc enhancement. The lattice-polarity-superconductivity dynamics must be investigated further to provide a comprehensive overview of the unique superconductivity of SrTiO3.

We have confirmed that Nb doping is an excellent carrier doping method to produce SrTiO3 with reduced disorder, superior mobility, and a higher Tc. Sr0.985Ca0.015TiO3 and Sr0.95Ba0.05TiO3 are different types of ferroelectrics, and we confirmed via X-ray structural analyses that Nb doping turns these materials into polar metals. When the carrier density n < n*, the resistance shows an anomaly at TK, which increases monotonically as the carrier density decreases and coincides with the ferroelectric Curie temperature at the zero carrier density limit. Although the values of n* for Nb:Sr0.985Ca0.015TiO3 and Nb:Sr0.95Ba0.05TiO3 are different by one order of magnitude, both materials show typical single superconducting domes with peaks commonly located around 1020 cm−3. When compared with that of nonpolar Nb:SrTiO3, the values of Tc are enhanced up to 0.75 K. However, the increase in Tc was not significant (less than 0.1 K) at around n*. This enhancement becomes much more prominent as we go deeper into the polar metal region. Space-symmetry breaking not only coexists with superconductivity, but also enhances the superconductivity. Our results call for a reconsideration of the existing microscopic models of superconductivity in SrTiO3.

## Methods

### Single crystal growth

For Nb:SrTiO3, we mixed powders of SrCO3, TiO2, and Nb2O5 in a ratio of 1 : 1−x : x/2. For Nb:Sr0.985Ca0.015TiO3, we mixed powders of SrCO3, CaCO3, TiO2. and Nb2O5 in a ratio of 0.985 : 0.015 : 1−x : x/2. For Nb:Sr0.95Ba0.05TiO3, we mixed powders of SrCO3, BaCO3, TiO2, and Nb2O5 in a ratio of 0.95 : 0.05 : 1−x : x/2. In all these cases, the values of x multiplied by 100 correspond to the atomic % of the nominal Nb content in the sample. The powders were calcined at 700 °C in the air for a few hours. The calcined powders were sintered at 1000 °C in the air for five hours. Then, the powders were pulverised and formed into a rod, about 4 mm in diameter and about 50 mm in length. Each rod was fired at 1250 °C–1350 °C for five hours in an argon gas flow. The crystal growth was conducted in a floating zone furnace with double hemi-ellipsoidal mirrors coated with gold. Two halogen lamps were used as the heat source. The crystals were grown in a stream of argon gas, and the growth rate was settled at 10–15 mm/h.

### Structural analyses

Powder X-ray diffraction (XRD) patterns were collected on the Cu radiation diffractometer (SmartLab, Rigaku Co., Ltd) using the θ-2θ step scanning method in the range of 15° ≤ 2θ ≤ 110°. The pattern indicated that the samples were of a single phase. For some crystals, Rietveld refinements for the lattice parameters and crystal symmetry were performed at various temperatures from 300 K to 10 K. Crystal alignments were done using back-reflection Laue diffraction. Results are summarised in Supplementary Note 11.

### Transport properties

We cut the samples into rectangular shapes of 0.5 × 0.3 × 7 mm3 with the longest edge parallel to the [100] direction of the cubic indices. Electrodes for the measurements were made by ultrasonic indium soldering, and the current was injected parallel to the [100] direction. The resistivity and the Hall voltage for 5 K ≤ T ≤ 300 K were measured in the Physical Property Measurement System (PPMS, Quantum Design Inc.). The resistivity below 1 K was measured using an AC resistance bridge (LR700, Linear Research Inc.) in a cryostat using a 3He/4He dilution refrigerator (μ dilution, Taiyo-Toyo Sanso Inc.).

### Relative permittivity

Same as the transport measurements, the samples were cut along the [100] direction with the typical dimensions of 1.5 × 0.4 × 3 mm3. The (110) plane is the widest surface on which the electrodes were formed by painting and drying the silver paste. The relative permittivity for 5 K ≤ T ≤ 300 K were measured in PPMS.

### Mutual inductance

The measurements were performed using the LR700 resistance bridge in a cryostat using a 3He/4He dilution refrigerator (μ dilution, Taiyo-Toyo Sanso Inc.). Induction (detection) coils with the diameter/length of 3 mm/11 mm (2 mm/6 mm) were set directly on the single crystals. We used a superconducting NbTi wire to avoid a possible temperature rise due to the Joule heating of the coils. The excitation current was 2 mA. The amplitude of the AC magnetic field μ0H (μ0 is the permeability of the vacuum) was estimated to be approximately 0.02 mT. The frequency was set at 15.9 Hz. Mutual inductance is proportional to the voltage of the detection coil, and the transition temperature is defined as the onset of the mutual inductance drop.

### Diamagnetism

The DC magnetisation measurement for the Nb:Sr0.95Ba0.05TiO3 single crystal (n ~ 8.1 × 1019 cm−3) of 1.1 × 1.1 × 7.0 mm3 was performed using a superconducting quantum interference device (SQUID) magnetometer (MPMS Quantum Design Inc.) equipped with a 3He refrigerator (iHelium 3, IQUANTUM Inc.). The DC magnetic field was applied along the longest edge of the crystal parallel to the [100] direction of the pseudo-cubic lattice. The demagnetising factor along this direction is estimated to be less than 0.04677; thus, we can ignore the demagnetising effect. In the zero-field cooling (ZFC) protocol, the sample temperature was lowered to 0.4 K in zero field. The DC magnetisation was measured in the presence of a static DC magnetic field μ0H of 0.02, 0.05, and 0.1 mT while warming the sample up to above Tc. In the field cooling (FC) protocol, the sample temperature was lowered to 0.4 K in the presence of μ0H, and the DC magnetisation was measured in the static μ0H while warming. There is a tiny field-independent background in magnetisation. Since the contributions from the sample to the SQUID signals are so small, the raw data are far from the ideal dipole response because of the background. We therefore subtracted a SQUID curve above Tc from the curves below Tc before performing the fitting.