Superconductivity enhancement in polar metal regions of Sr$_{0.95}$Ba$_{0.05}$TiO$_3$ and Sr$_{0.985}$Ca$_{0.015}$TiO$_3$ revealed by the systematic Nb doping

Two different ferroelectric materials, Sr$_{0.95}$Ba$_{0.05}$TiO$_3$ and Sr$_{0.985}$Ca$_{0.015}$TiO$_3$, can be turned into polar metals with broken centrosymmetry via electron doping. Systematic substitution of Nb$^{5+}$ for Ti$^{4+}$ has revealed that these polar metals both commonly show a simple superconducting dome with a single convex shape. Interestingly, the superconducting transition temperature $T_\mathrm{c}$ is enhanced more strongly in these polar metals when compared with the nonpolar matrix Sr(Ti,Nb)O$_3$. The maximum $T_\mathrm{c}$ reaches 0.75K, which is the highest reported value among the SrTiO$_3$-based families to date. However, the $T_\mathrm{c}$ enhancement is unexpectedly lower within the vicinity of the putative ferroelectric quantum critical point. The enhancement then becomes much more prominent at locations further inside the dilute carrier-density region, where the screening is less effective. These results suggest that centrosymmetry breaking, i.e., the ferroelectric nature, does not kill the superconductivity. Instead, it enhances the superconductivity directly, despite the absence of strong quantum fluctuations.

Are superconductivity and ferroelectricity reconcilable? Bernd T. Matthias, a pioneer in the search for superconducting materials, stated more than 50 years ago 1 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 superconductors 2 . Anderson and Blount proposed that some metals can be ferroelectric in the sense that they have broken inversion symmetry.
Single crystals of SrTiO3 show huge relative permittivity of approximately 42000 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 ferroelectric 28,29 , it is believed that strong quantum fluctuations suppress the material's ferroelectricity 27 . 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 Ca 2+ (refs. 32,33 ) or Ba 2+ (refs. 34,35 ) for Sr 2+ turns SrTiO3 into a ferroelectric material. Exchange of the 18 O isotope for 16 O 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 ferroelectricity 37,38 . Furthermore, both epitaxial strain 18,19,39,40 and Sr defects 41 in thin SrTiO3 films also generate ferroelectricity.
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 band 48 . The electrons in the heavy band make the largest contribution to the superconductivity 48 .
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 violated 23,24 , the pairing interaction should show a significant frequency dependence that is caused by poor screening of the Coulomb repulsion 24 . Further doping causes the Fermi energy to increase, but the superconductivity disappears above a carrier density of 10 21 cm −3 , and this results in a domelike dependence of the superconducting transition temperature on the carrier density 4,49-51 .
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  62,63 . There were also some remarkable experimental discoveries: carrier doping of two ferroelectric matrices, (Sr,Ca)TiO3 (refs. [10][11][12] ) and SrTi( 16 O, 18 O) 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 noteworthy [5][6][7]43 ; the strain breaks the material's spatial inversion symmetry and enhances the spin-orbit interactions 5,6 , 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 Nb 5+ for Ti 4+ for carrier densities ranging from ~10 18 to ~10 21 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 distortion 13 . 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 Ca 2+ , which has a smaller ionic radius than Sr 2+ (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 at 5 K (Fig. 1b). The values of for SrTiO3−δ and Nb:SrTiO3 from the literature 44,64-66 are also plotted for comparison. In the normal metal, µ is proportional to !" , but Behnia proposed a model 67 that assumed that the mean-free path is proportional to the average distance between the dopants and the Thomas-Fermi screening length, thus giving ∝ ! ! " . Therefore, we tried to fit the formula µ ∝ !# to our experimental data (see Supplementary Figure 2). The result that b = 0.85±0.02 indicates that b = 5/6 from the literature 67 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 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 carriers 67,68 . 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 Figure 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 ≃ 1×10 20 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 ≃ 1.2×10 18 cm −3 in SrTiO3−δ (ref. 12 ).
If the three-fold degeneracy of the t2g band is lifted 51 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 superconductivity 48 . The second difference is observed in the high n region (n ≃ 1×10 21 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 vacancies 52 . The thermal reduction procedure used to create the oxygen vacancies for the SrTiO3−δ single crystal is restricted to dislocation 70 because the formation enthalpy of the oxygen vacancies near the dislocations is significantly lower than that in the stoichiometric SrTiO3 matrix 71 . 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 Sr 2+ ion 13 . 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 interpolation 72 (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 $ %& ~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 anomaly 12,14 . The static interaction is fully screened (TK = 0) at a value of n * of 3.1×10 19 cm −3 . This experimental value of n * is almost equivalent to the n * = 3.3´10 19 cm −3 value calculated by assuming a Thomas-Fermi screening length for the itinerant carriers that is comparable with the averaged dipole-dipole distance 14 .
In contrast, in (Sr,Ba)TiO3, the Ba 2+ ion hardly moves because of its larger ionic radius and heavier mass when compared with the Sr 2+ 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 ferroelectricity 13 . This is different in principle from the ferroelectricity mechanism in (Sr,Ca)TiO3, but is basically similar to that in  (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 $ %& at n = 0. We have estimated a critical carrier density of n * ~ 2.5´10 20 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 $ %& 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 increase 18 . 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  for Nb:Sr0.985Ca0.015TiO3 and 2.5 × 10 () 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. 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 ≃ 1 × 10 () 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 and 3d. 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 idea 7,21,54,73 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 limit 19 . 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 superconductivity 21 . Electroforming also creates metallic filaments, and the resulting inhomogeneous structure increases the superconducting transition temperature 54 . indicating that the superconductivity is a bulk phenomenon 47 .) 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. 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.

Discussion
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.  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.

Data availability
The  (111) interfaces. Science 371, 716-721 (2021).     In the previous paper, we reported the superconductivity of our Sr1-xLaxTiO3 single crystals 1 . La 3+ is a carrier dopant as Nb 5+ in SrTi1-xNbxO3 (denoted by Nb:SrTiO3). Therefore, Sr1-xLaxTiO3 is referred to as La:SrTiO3 in the main text. In this Supplementary Information, we follow the notation if it is not necessary to express it explicitly as SrTi1−xNbxO3 or Sr1−xLaxTiO3.

Figure Legends
After the previous work, we continued preparing the La:SrTiO3 samples, especially, in the dilute carrier-density regions (x = 0.0001 and 0.0002) and the high carrier-density region (x = 0.015). The results are shown in Supplementary Fig. 1. There is no significant upturn in the resistivity of La:SrTiO3 at low temperatures. We think this is because La:SrTiO3 is nonpolar as  The Hall effect for our Nb:SrTiO3 single crystals was almost independent of temperature.
We plotted the Hall resistivity ρH as a function of the magnetic field μ0H (μ0 is the permeability of the vacuum) at 5 K in Supplementary Fig. 2. We tried to fit the linear relationship ! = " ( ) #$ to the experimental data using the least squares method and deduced the carrier density n, where e is the elementary charge. As seen in Supplementary Fig. 2   The carrier density n is plotted as a function of temperature T for Nb:Sr0.95Ba0.05TiO3 ( Supplementary Fig. 4, Left) and Nb:Sr0.985Ca0.015TiO3 ( Supplementary Fig. 4, Right). In Nb:Sr0.95Ba0.05TiO3, n decreases at low T. The onsets of the carrier-density drop roughly correspond to the resistance anomaly temperatures TK. In contrast, in Nb:Sr0.985Ca0.015TiO3, the drop in n is not apparent. To derive n at TK, for Fig. 3b in the main text, we performed a smoothing spline interpolation 3 (smoothing factor = 1) for the n versus T data in Supplementary Fig. 4 using  Sr0.985Ca0.015Ti1-xNbxO3 Supplementary Note 8: Tc of the polar Nb:Sr0.95Ba0.05TiO3, Nb:Sr0.985Ca0.015TiO3, and nonpolar Nb:SrTiO3 plotted against n at 5K and 50K.
Since the carrier density n of Nb:Sr0.95Ba0.05TiO3 decreases below 50 K, as shown in Supplementary Fig. 4, a question arises as to which temperature n should be used to plot the relationship between Tc and n. We think it is physically more meaningful to use n at a temperature as close to Tc as possible, which is what we did in Fig. 4c in the main text. Here, however, we have plotted Tc against n at 50 K as a comparison ( Supplementary Fig. 5). Although the persistence of Tc in the low n region appears to become smaller, the onset of Tc is observed at much higher temperatures. Therefore, by connecting the onsets as indicated by the thick dashed line in Supplementary Fig. 5, it seems there is little difference between the two plots. The non-doped Sr0.95Ba0.05TiO3 and Sr0.985Ca0.015TiO3 single crystals show the ferroelectric phase transition at 50 K and 24 K, respectively, as summarised in Supplementary  Fig. 6). Supplementary Note 10: Determining the boundary separating the polar-metal and nonpolar-metal phases for the Nb:Sr0.95Ba0.05TiO3 and Nb:Sr0.985Ca0.015TiO3 single crystals.

Supplementary
We defined a specific temperature TK at which the resistivity reaches a minimum. 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 local minima. To derive n at TK, we performed a smoothing spline interpolation 3 . In Supplementary Fig. 6, the values of TK were plotted versus log(n) at TK. We    For the Nb:Sr0.95Ba0.05TiO3 single crystals with the Nb content of 0.025%, the numbers of itinerant electrons at room temperature determined by the Hall effect measurements are both smaller than that of the nominal number of 0.00025. It was 0.00008 for the sample denoted by 25α and 0.00009 for 25β. The sample 25α corresponds to x=0.00025(a) and 25β does to x=0.00025(b) in the main text. A two-step resistance anomaly was seen in the samples 25β and 25γ, while the anomalies seem to be merged into one in the sample 25α ( Supplementary Fig. 7a).
As we decrease the temperature, the carrier density decreases from around 20 K. For the twostep TK samples, the onset temperature seems to correspond to the lower TK value ( Supplementary Fig. 7b).
For the Nb:Sr0.95Ba0.05TiO3 single crystals with the Nb content of 0.2%, the variations in TK was also observed ( Supplementary Fig. 7c). In these samples, the numbers of the itinerant