The occurrence of superconductivity in doped SrTiO_{3} at low carrier densities points to the presence of an unusually strong pairing interaction that has eluded understanding for several decades. We report experimental results showing the pressure dependence of the superconducting transition temperature, _{c}, near to optimal doping that sheds light on the nature of this interaction. We find that _{c} increases dramatically when the energy gap of the ferroelectric critical modes is suppressed, _{c} near to a ferroelectric quantum critical point can arise due to the virtual exchange of longitudinal hybrid-polar-modes, even in the absence of a strong coupling to the transverse critical modes.

Superconductivity in doped SrTiO_{3} near to a ferroelectric quantum critical point emerges due to a strong interaction driving the formation of Cooper pairs, the nature of which has remained elusive for several decades. Here, the authors reveal that pairing is due to the exchange of longitudinal hybrid polar modes rather than transverse critical modes.

Strontium titanate is an incipient ferroelectric insulator, which can be tuned essentially continuously into the ferroelectric phase via a ‘quantum’ tuning parameter, such as chemical substitution, isotopic substitution or applied stress (see, e.g., refs. ^{1–7}). The temperature-quantum tuning parameter phase diagram of SrTiO_{3} and related materials has recently been discussed in terms of a phenomenological model involving the interaction of the local ferroelectric order-parameter field with itself, and with the strain field of the lattice^{3,S1–S13} (see Fig. _{0}, without the use of freely adjustable parameters^{3}. The phase diagram is characterized by (i) a ferroelectric transition temperature _{Curie} that terminates at a quantum critical point (QCP), (ii) a low-temperature crossover curve _{x} separating a power law (1/_{0} ~ ^{2}) and an exponential temperature dependence of 1/_{0} also terminating at the QCP, and (iii) a high-temperature crossover curve separating classical (Curie–Weiss) and quantum behaviour of the temperature dependence of 1/_{0}. An additional crossover curve _{min} that terminates at the QCP marks the position of a minimum in the temperature dependence of 1/_{0}, which arises from the coupling of the electrical polarization to the lattice strain field^{3,S7,S14–S17}. More exotic behaviour beyond that indicated above^{S8,S11,S18} is anticipated at least for sufficiently low frequencies and low temperatures below that probed experimentally thus far. Quantum phase transitions have also been observed in a diversity of different ferroelectrics including in oxides^{S19}, organics^{S14}, hydrogen-bonded crystals^{S20}, electronic ferroelectrics^{S21} and multiferroics^{S15} and have been highlighted in recent reviews^{S22}^{8}.

The vertical axis is the temperature while the horizontal axis represents a quantum tuning parameter, ^{16}O_{1−x}^{18}O_{x})_{3}, or by chemical substitution as in the cases of Sr_{1−y}Ca_{y}TiO_{3} and Sr_{1−y}Ba_{y}TiO_{3}. In strontium titanate _{Curie} and _{min} are, respectively, of order 25 K at 0 kbar and _{3}, ^{18}O substituted SrTiO_{3}, KTaO_{3}, and other materials. The curve _{min} marks the temperature of the minimum in the inverse dielectric susceptibility as observed and predicted by a model involving the electrostrictive coupling of polarization and strain and below which the system forms a quantum polar-elastic regime^{3,S7 4}. An extension of the phase diagram would include an additional orthogonal axis representing the charge carrier density,

The substitution of Ti by Nb leads to an extra electron per substituted unit cell of SrTiO_{3}, which is bound to Nb only very weakly due to the high dielectric constant of the host lattice. For a doped electron density, ^{15} cm^{−3} the doped electrons are promoted to the _{2g} bands that are split by the spin–orbit interaction as well as by a structural perturbation observed below ~100 K of the starting simple cubic structure of SrTiO_{3}^{S23}. Charge carriers may also be introduced by lanthanum doping, oxygen reduction, interface engineering and electrostatic gating. The introduction of charge carriers leads to an additional axis (_{3} (conduction electrons) occupying the lowest of these _{2g} bands tend to dominate the properties of principal interest in the discussion below, but we note that additional effects due to the progressive filling of the three _{2g} bands have been reported^{S24}. Umklapp processes may be neglected under our conditions as well as inter-valley scattering processes that would require the presence of multiple Fermi surface pockets well separated in the Brillouin zone. These are not present in electron-doped SrTiO_{3} as confirmed by several quantum oscillation experiments and band structure calculations (see, for example, refs. S23 and S25 and _{3}, the first of the oxide superconductors^{9}. Despite numerous investigations, the detailed nature of the relevant effective interaction mediating superconductivity in carrier-doped SrTiO_{3}^{S24,S26-S31} ^{9–25} and related systems such as KTaO_{3}^{26}, LaAlO_{3}/SrTiO_{3} interfaces^{19,27}, ferroelectric Ca_{x}Sr_{1−x}TiO_{3−δ}^{28} and FeSe/SrTiO_{3}^{29}, continues to be debated.

We have investigated specimens with carrier densities, ^{18}–4.0 × 10^{20} cm^{−3} spanning the superconducting dome maximum of the temperature-carrier density phase diagram of SrTiO_{3} with Nb substitution (see “Methods” section and Supplementary Information Fig. _{3} and related materials.

In Fig. _{c}, as determined from resistivity measurements in a sample with nominal 0.2 at.% Nb doping (see “Methods” section), which has an intermediate carrier density near that of the dome maximum where _{c} is as high as 0.4 K. As shown in these figures we find that _{c} drops sharply with modest pressures and collapses towards absolute zero above 5 kbar. Thus, _{c} increases with decreasing 1/_{0} (Fig. ^{6,S1-S3,S32,S33}, Ω(^{19} cm^{−3} ^{S34,S35} ^{30} and its _{c} observed on approaching magnetic QCPs in nearly magnetic metals^{31}.

_{c}, as determined from resistivity data vs. pressure for a SrTi_{1-x}Nb_{x}O_{3} sample (see “Methods” section) with a carrier density depending on _{c} vs. _{0}, proportional to Ω(0)^{2}, for the undoped state^{4,6,S16,S17}. Ω(0) is only slightly changed (see Eq. (^{19} cm^{−3 S34}. _{0}, which is weakly dependent on pressure. _{c} in _{0}. For decreasing values of _{c}, _{0}. The pressures may be inferred from the inset, which shows the relative change of the ^{2} resistivity coefficient (see caption of Fig. _{c} with increasing pressure and hence 1/_{0} is seen to be extraordinarily rapid, pointing to a significant growth of the pairing strength on approaching the ferroelectric quantum critical point.

This finding is consistent with other strain and pressure measurements tuning Ω(^{32–34}, as well as recent reports of increases of _{c} observed upon calcium^{28} and oxygen-18 isotope^{35} substitutions. It is also supported by the observation of a lower value of _{c} in electron-doped KTaO_{3}^{26}, which is a quantum paraelectric further away from the QCP (having 1.5 times the value of Ω(_{3} in the undoped starting material^{3}). Note that the ‘ferroelectric’ transition in the conducting state^{36}, where the uniform static dielectric function is strictly singular for all finite _{0} is defined here as the starting uniform static dielectric function for the undoped state.

We also comment that in the normal state the resistivity varies approximately as the square of the temperature with a ^{2} coefficient that changes by ~30% in the pressure range where _{c} drops by an order of magnitude or more (Fig. ^{2} variation of the resistivity in SrTiO_{3}^{37,38}.

The above findings and our pressure measurements shown in Fig. _{c} on approaching a ferroelectric QCP, it is natural to consider first the role of the polar optical modes consisting of a multiplet of transverse and longitudinal components, the lowest transverse mode frequency Ω(_{LO}, which is ~100 meV in the starting undoped state and weakly _{LO}, varying as ^{21}, the contribution to the effective paring interaction of the intermediate TO and LO modes tend to cancel out in first approximation.

In terms of the dielectric screening model^{S36 }^{16,39–42}, the above effective pairing interaction in the low _{LO}, which extends to _{0} diverges (Ω(_{c}, in agreement with observation (inset of Fig.

It has been suggested that the mediation of polar optical modes, while of key importance in reducing the Coulomb repulsion as _{0} increases, might be supplemented by an additional pairing mechanism to account for superconductivity or at least for a quantitative understanding of the magnitude of _{c}. A number of additional candidates have been proposed involving: (i) residual coupling to critical TO modes missing in the dynamical screening model^{20,43,44}; (ii) plasmons in the conduction electron system^{21}, which must be included for minimal consistency of any proposed model; (iii) multi-valley transition processes^{9,13}; (iv) non-polar acoustic phonons (see, e.g., the Appendix in ref. ^{21}; (v) non-polar soft optical phonons^{12}; (vi) currents associated with transverse polar optical modes; (vii) non-cancelling contributions of polar optical modes in between Ω(_{LO}^{21}; (viii) phonon modes localized around the dopant impurity sites^{23}; (ix) effects associated with Sr disordering at low temperatures^{45}; and other distinct models, e.g., refs. ^{46–48}. Theories involving the formation of polarons and bipolarons^{14}, as well as pre-formed pairs^{10}, have also been considered. Though of interest in their own right these mechanisms have not been shown by quantitative analyses free of adjustable parameters to play central roles in understanding superconductivity in the case of SrTiO_{3}.

Having investigated all of these proposed theories in light of realistic model parameters for SrTiO_{3}, and our new experiments, we consider here a minimal description that includes pairing due to the dipolar fluctuations of the coupled ion (polar optical modes) and charge-carrier system in a dielectric screening model of the effective long-wavelength interaction between carriers expressed in the form

where _{B} = 1. The approach as set out below using the interaction given by Eq. (^{39} and Takada^{16,49}.

By way of contrast the pairing interaction due to the virtual exchange of magnetic fluctuations on the border of a ferromagnetic QCP in the simplest case for parallel spins is given by _{m}(^{2}_{m}(_{m}(^{50}). The model defined by Eq. (

We assume that for each mode of wavevector ^{16,42,50}) as given approximately for low _{p} is the bare plasma frequency for the ions in a medium with background dielectric constant _{∞},

is the bare frequency spectrum at low _{s}, and _{el}(_{p}/^{2} and (ω_{p}/ω(^{2}, respectively, for _{p} is the bare plasma frequency for the carriers and

is the bare characteristic frequency at low _{F}. _{el}(^{21,23} similar in form to the second term on the right-hand side of Eq. (2), but in principle including effects of dissipation and a restriction to _{F}, as defined by the Lindhard function. (The role of dissipation missing in the simplest interpolation model for _{el}(

For the simplest interpolation model for _{el}(

Here _{±}(

At a low carrier density, _{p} << Ω_{p} the lower hybrid longitudinal mode of frequency _{−}(_{p} can be thought of as a carrier plasma mode as screened by the ions, so that _{−}(_{p} (and even _{F}_{F}), while the upper longitudinal hybrid mode of frequency _{+}(_{p} >> Ω_{p} the lower longitudinal hybrid mode of frequency _{−}(_{p} can be thought of as a polar optical phonon mode as screened by the charge carriers, so that _{−}(_{+}(_{±}(_{3} (see “Methods” section).

_{±}(_{F}, and coupling functions _{±}(_{3} (see “Methods” section) ℏ = _{B} = 1). _{F}_{F}/_{F}, where _{F} is the density of states of the doped carriers at the Fermi level and _{2g} band, _{c} presented in Fig. _{0} are also shown. The two steps most clearly visible for the case _{0} = 100 are associated with the two hybrid LO modes. An increase of _{0} increases the size of the upper step, but decreases the size of the lower step pointing essentially to an inversion of the trend of pairing strengths vs. _{0} as illustrated in Fig.

Equally significant are the coupling functions _{−}(_{+}(_{−}(_{0}. At low _{p} << Ω_{p}), where the upper longitudinal hybrid frequency tends to be far above _{F}_{F} (Fig. _{−}(_{−}(_{c} with increasing _{0}. At high _{p} >> Ω_{p}), where the upper longitudinal hybrid frequency _{+}(_{F}_{F} (Fig. _{−}(_{−}(_{+}(_{0}, i.e., on approaching the ferroelectric QCP. Note that the usual Bardeen–Pines form of the interaction often used in calculations of conventional phonon-mediated superconductivity is a special limit of Eq. (

The conditions favourable for superconductivity can also be observed by looking at the plot of the interaction versus imaginary frequency within the Matsubara formalism as shown in Fig. _{F}_{F}_{0} increases which leads to an enhancement of superconductivity. On the other hand, the step size at lower frequencies decreases as _{0} increases reducing the propensity of the system to forming a superconducting state. The relative importance of these two effects depends on the Fermi energy and correspondingly the carrier density. If the Fermi energy (shown in Fig.

In an attempt to gain more detailed insight on the conditions favourable for superconductivity for the pairing interaction defined by Eqs. (^{S38-S41} gap equation defining _{c} in the weak coupling approximation^{S39}, which is appropriate for the parameters relevant to SrTiO_{3},

It is sometimes assumed that the kernel can be approximated at least in the weak coupling limit by the interaction ^{S41}, to determine the kernel

Here _{c}. These and the remaining terms are defined more fully in the discussion of Eq. (A6) in ref. ^{50}. In the weak-coupling limit for the parameters relevant to SrTiO_{3} the frequency summation in Eq. (^{51}, originally applied to the case of SrTiO_{3} by Takada^{16}. The KMK kernel

Numerical calculations based on the full Eliashberg theory and the KMK kernel are found to be in close agreement as expected in the weak coupling limit^{52} and quite different from the predictions based on the on-shell approximation^{53}. We stress that in weak coupling, the Eliashberg theory does not reduce to the BCS gap equation with the Fröhlich or Bardeen–Pines interaction used in standard textbooks. A correction to the KMK kernel has been shown to be important where the logarithm of _{c} has a vanishing first-order term in the interaction strength^{54,55}, a special case however that we find not to be relevant to SrTiO_{3}.

It has been argued that a potentially serious challenge to the Eliashberg description can arise form vertex corrections in cases involving interactions mediated by bosons with energies well above the Fermi energy^{54,56–58}. In light of these still poorly resolved effects we limit ourselves only to consider to what extent the model interaction, Eqs. (_{c} vs. density and pressure. For this purpose we restrict ourselves to the Eliashberg approximation applied in the weak coupling limit that is more accurate than the on-shell approximation. Also, we note that at carrier densities near to and above the superconducting dome maximum, the Fermi energy is comparable to the energies of the longitudinal hybrid polar modes that mediate the pairing interaction in the model defined by Eqs. (

The gap equation, Eq. (

where Δ is a vector and _{c} is found by the condition that the highest eigenvalue, Λ_{h}, is equal to unity.

Instead of determining _{c} directly we could consider the behaviour of Λ_{h} in the low temperature limit as a function, in particular, of the carrier density and applied pressure. The region in density and pressure where superconductivity is expected to arise would be indicated by the condition Λ_{h} > 1. More generally, the maximum of Λ_{h} vs. density and pressure may be expected to indicate the regime where the contribution of the pairing interaction defined by Eqs. (_{h} is sometimes considered a kind of susceptibility of the system to forming a superconducting state.

The details of the evaluation of the highest eigenvalue based on Eqs. (_{3} are given under “Methods” section. Instead of presenting Λ_{h} vs. carrier density _{c} inferred from the condition Λ_{h}(_{c}, _{c}, and especially of normalized values of the highest eigenvalue Λ_{h}. The trends of Λ_{h} in particular can hence be viewed as predictions of the model, Eqs. (

We note that when calculating _{c} it is often common practice to determine parameters “_{3}, the oscillations in time of the gap function and interaction function are synchronized such that electron repulsion may be mitigated. In terms of the frequency-dependent functions, this means that the gap function as a function of frequency changes sign when the interaction function becomes positive (repulsive). This can therefore be thought of as the frequency-space analogue of the phenomenon occurring in

The predicted variations of _{c} vs. _{c} initially rises with carrier density, ^{19}–10^{20} cm^{−3}, and collapses at higher densities. This is in keeping with the trend of _{c} seen experimentally^{32} as shown in the inset of Fig. _{c} at high ^{54,58}. Moreover, the absolute magnitude of _{c}, though only estimated to logarithmic accuracy by our calculations (see “Methods” section), suggests that the pairing mechanism we have considered here as defined by Eqs. (

The relevance of this pairing mechanism (Eqs. (_{c} with increasing pressure and hence increasing 1/_{0} (Figs. ^{20} cm^{−3} where the measurements were performed, _{c} tends to increase with decreasing 1/_{0}, i.e., as the ferroelectric QCP is approached (at fixed _{c} may also be estimated via its effect on 1/_{0}. 1/_{0} can increase or decrease depending on the direction of applied stress and whether it is compressive or tensile^{6}. In cases where stresses lead to a decrease in 1/_{0} the model predicts an increase in _{c}. Experiments in which stress is varied result in anisotropies and inhomogeneities within samples, which could be the subject of future investigation. We note that the effects of isotropic or anisotropic biaxial strains in SrTiO_{3} films are of particular current interest and an important topic for future study (discussions may be found, for example, in refs. ^{59–61}).

The normalized superconducting transition temperature, _{c} calculated via the model given by Eqs. (_{3} (see “Methods” section), is shown (i) vs. pressure, ^{−3}) = 19, 20, 20.5, 21, from the upper to the lower curves, respectively, in the main figure, and (ii) vs. _{e} = 2, 3 and 4, in the inset. _{cmax} corresponds to the maximum of _{c} at _{c}/_{cmax} in contrast to _{c} itself depends only weakly on the cut-off frequency used to evaluate Eqs. (_{3}. _{cmax} is found to be in the mK range based on a realistic choice of parameters as outlined in the “Methods” section for SrTiO_{3} and as originating purely from the model described by Eqs. (

We have shown that the interaction described in terms of the dielectric function including the effects of fluctuations of the densities of both the ionic and conduction electrons can lead to a quantitative understanding of the doping and pressure dependence of the superconducting transition temperature of SrTiO_{3} and related materials. This is in manner similar to that previously proposed by Takada^{41}, but including (i) a more tractable model for the interaction, (ii) a more transparent identification of the relevant longitudinal hybrid polar modes and limitations of the model Eqs. (

For ^{31} and Fig.

The top horizontal axis is the quantum tuning parameter referred to in Fig. ^{2}) are, respectively, the coefficients of the quadratic and gradient-squared terms and the cut-off wavevector in the ϕ^{4} field theory^{3}. ^{S34} (see “Methods” section). The phase diagram is consistent with the model defined by Eqs. (^{3} for the Curie temperature, both calculated in terms of independently measured temperature-independent model parameters for SrTiO_{3}.

Some insight on this difference in behaviour can be obtained by considering the low and high _{p} << _{+} = Ω_{LO}_{p} = Ω(0)_{p}/Ω_{LO} is the lower longitudinal hybrid polar mode frequency corresponding to the plasma oscillations associated with the charge carriers as screened by the ions. On the other hand, in the low _{TF} is the Thomas–Fermi wavevector defined by _{TF}) = _{p} and _{LO}_{TF} is the lower longitudinal hybrid polar mode frequency corresponding in this case to vibrations of the ions as screened by the charge carriers. The prefactor in Eq. (_{c} may be expected to decrease with decreasing Ω(0_{c} tends to increase with decreasing Ω(0) in this regime, as observed.

Thus, we anticipate a dramatic qualitative change in form of the pressure dependence of Λ_{h} (the propensity of the system to forming a superconducting state) as a function of carrier density (see ^{18} cm^{−3}, where our Nb-doped SrTiO_{3} samples are not superconducting down to at least 0.04 K, the lowest temperatures investigated. Superconductivity has been reported in oxygen-depleted specimens for carrier densities somewhat below 10^{18} cm^{−3}, but the role of inhomogeneities in these systems complicates the interpretation of the experimental results.

Along with this predicted qualitative change in behaviour with carrier density, our analyses differ, e.g., from that of refs. ^{20,44} by focusing on pairing via the virtual exchange of two longitudinal hybrid polar modes rather than the critical TO mode, and by explicitly including: (i) the Coulomb repulsion between carriers, (ii) screening due to added carriers, (iii) dynamics of the KMK versus on-shell kernel, and (iv) the retardation effect explicitly. This allows us to calculate the superconducting dome structure without the use of adjustable parameters in terms of a direct repulsion and an indirect dielectric interaction mediated via the exchange of two longitudinal hybrid polar modes in place of the critical mode alone as in, e.g., ref. ^{20}.

We would like to highlight the following features of the superconductivity as observed and modelled in Nb-doped SrTiO_{3}. Firstly, around optimal doping, the Fermi energy is not particularly small compared to the hybrid polar mode frequencies (see Fig.

The analysis presented here highlights the crucial importance of ionic polarizability in yielding an extraordinarily strong pairing interaction leading to pair formation via ‘ion mediation’, and may help our understanding of other superconductors with electron–polar phonon coupling^{62}. Indeed, this interaction may be thought of as leading to a third generation of superconductors. The first generation being traditional phonon-mediated superconductivity (involving vibrations of approximately neutral atoms), the second being magnetically mediated superconductivity (involving the exchange energy, ^{63}. There may exist circumstances under which the advantageous increase in the Fermi energy with increasing carrier density is not offset by a loss in interaction strength between the carriers, in the way implied by Eqs. (_{3} doped with potassium (see, e.g., ref. ^{62} and references therein).

^{34,64–66} and theoretical studies^{67–69} have been reported that are generally in keeping with our conclusions. A theoretical expression for the integration cut-off in the Eliashberg gap equation has been discussed^{69} and leads to numerical values of _{c} for SrTiO_{3} of the same order of magnitude as those used in our analyses in the density range near to the dome maximum (see also “Methods” section and _{3} has been given in ref. ^{47}.

Incipient ferroelectric SrTiO_{3} may be doped into a metallic state by a number of methods including oxygen reduction and niobium substitution. Metallic specimens of SrNb_{x}Ti_{1−x}O_{3} were obtained from commercial sources with niobium nominal doping of 0.02, 0.2, and 1 at.%, corresponding to nominal charge-carrier concentrations, ^{−3}) of 18.5, 19.5, and 20.3, respectively. The samples were cut into parallelepipeds with approximate dimensions 4 mm × 1 mm × 0.5 mm. Low-resistance Ohmic contacts were achieved by etching the surfaces using argon-ion plasma followed by sputtering of gold contacts over a titanium seed-layer on the top surfaces in a standard Hall bar geometry. Measurements of the Hall resistance _{xy} at liquid-helium temperatures in a field up to 9 T yield log(^{−3}) of the order of 18.0, 19.3, and 20.6 in fair agreement with the nominal values for the three samples, respectively. The value of _{c} for the sample with 0.2 at.% Nb is close to the maximum expected at log(^{−3}) of order 20 (inset of Fig. _{xx} was measured in zero-field for each sample at ambient pressure down to 50 mK using an adiabatic demagnetization refrigerator. The residual resistance ratio, defined as the resistance at room temperature divided by the resistance at 2 K, was >600 for all samples.

The 0.2 at.% Nb sample with the highest superconducting transition temperature _{c} = 0.4 K was selected for the high-pressure experiments. To check for repeatability, low-temperature resistivity measurements under hydrostatic pressures were carried out on two different adiabatic demagnetization refrigerators, one in Cambridge and one in Rio de Janeiro, using two different piston-cylinder clamp cells. In these experiments, hydrostatic pressure was applied to the sample at room temperature using fluorinert (1:1, FC84–FC87) as the pressure-transmitting fluid. For each fixed pressure, four-terminal resistance of the sample was measured using a Cambridge Cryogenics mK measurement system with a lock-in amplifier and constant current source as a function of temperature down to 50 mK. The pressure was determined in the low-temperature range by measuring the superconducting transition temperature of a high-purity tin manometer. For all observations of superconductivity in our SrNb_{x}Ti_{1−x}O_{3} samples, we defined _{c} to be the temperature at which the resistivity dropped by 10% from its value on entering the superconducting state. Measurements were collected during cooling and heating runs at a rate of ~1 K/h. Estimates of the uncertainties in our determinations of pressure, _{c} and the ^{2} coefficient of the normal state resistivity,

_{±}(_{±}(_{±}(_{±}(

^{40}. Note that in the KMK approximation the kernel falls off with increasing values of the single-particle energies individually and not their difference as in the incompletely physical case of the on-shell approximation. The KMK kernel is derived from the Eliashberg equation referred to in the main text (Eq. (_{3}.

_{∞} = 5, _{0} = 2.5 × 10^{4 3,S42}, _{e}^{S25,S35} (relevant near to optimal doping), Ω_{LO} = 100 meV ^{S34} (see caption of Fig. _{s} = 5 meVÅ ^{S33} and^{19} cm^{−3} and Δ_{LO}. The cut-off affects the magnitude of _{c} or more generally the highest eigenvalue Λ_{h} in Eq. (_{h} on the carrier density, ^{69} by taking into account the effects of quasiparticle damping is in the range of cut-offs considered in _{c} itself, Λ_{h}, and the position and width in

We thank K. Apostolidou, K. Behnia, P. Chandra, P. Coleman, A.V. Chubukov, J.R. Cooper, F. Dinola-Neto, C. Durmaz, R.M. Fernandes, M. Fontes, D.E. Khmelnitskii, D.L. Maslov, A. Mello, R. Ospina, J.F. Scott, and C.M. Varma for useful help and discussions. We thank A. Edelman and P.B. Littlewood for discussions of the electronic properties and superconductivity in SrTiO_{3} based on the Eliashberg formalism. C.E. thanks ICAM for financial support. E.B.S. acknowledges support from FAPERJ and CNPq grants, and a CNPq BP and Emeritus FAPERJ fellowship. S.S.S. acknowledges support from the ‘Increase Competitiveness Programme’ of the Ministry of Education of the Russian Federation under grant number NUST MISiS K2-2017-024. G.G.L. acknowledges support from the Engineering and Physical Sciences Research Council (EPSRC) Grant No. EP/K012894/1 and the CNPq/Science without Borders Programme. S.E.R. acknowledges support from a CONFAP Newton grant, the Engineering and Physical Sciences Research Council of the United Kingdom and the Royal Society. We would like to dedicate this article to Prof. James F. Scott FRS, a key supporter of this research, who passed away in April 2020.

This work resulted from a collaborative effort of all of the authors, C.E., J.F.d.O., D.A.T., E.B.S., S.S.S., G.G.L., and S.E.R., each of whom played an essential role.

The data supporting the findings of this study are available within the paper. Any additional data connected to the study are available from the corresponding author upon reasonable request.

The authors declare no competing interests.

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Supplementary Information