Introduction

Electron-phonon coupling plays an important role in a variety of physical phenomena in solids1. In metals and doped semiconductors, low-energy electronic excitations are strongly modified by the coupling of itinerant electrons to lattice vibrations, which influences their transport and thermodynamic properties2. Furthermore, electron-phonon coupling provides an attractive electron-electron interaction, which leads to conventional (i.e., phonon-mediated) superconductivity in many metals3. Recent studies on hydrogen-rich materials show that when their electron-phonon coupling is strong enough, the transition temperature of conventional superconductors can reach as high as 260 K at 180–200 GPa4,5,6. One general way to increase the electron-phonon coupling of solids is to find a particular phonon to which itinerant electrons are strongly coupled and whose softening (i.e., the phonon frequency approaches zero) across a structural phase transition may consequently increase the total electron-phonon coupling7. However, identifying a strong coupling between a soft phonon and itinerant electrons in real materials is no easy task, which relies on material details. On the other hand, the superconductivity in doped SrTiO3 has drawn great interests from both theorists8,9,10,11,12,13,14,15 and experimentalists16,17,18,19,20,21,22,23,24. One beautiful experiment is Sr1−xCaxTiO3−δ in which Ca doping leads to a weak ferroelectric distortion in SrTiO3 and oxygen vacancies provide itinerant electrons20,25,26. Increasing the carrier concentration in Sr1−xCaxTiO3−δ induces a polar-to-centrosymmetric phase transition and a superconducting “dome” emerges around the critical concentration. The nature of the superconductivity in doped SrTiO3 is highly debatable8,9,10,11,12,13,14,15,16,17,18,19,20,21,27, because the superconductivity in doped SrTiO3 can persist to very low carrier density11,12,28, which seriously challenges the standard phonon pairing mechanism29. It is not clear why superconductivity in doped SrTiO3 vanishes above a critical concentration in spite of an increasing density of states at the Fermi level30. Attention has been paid to recent proposals on soft polar phonons, but the coupling details and strength are controversial10,11,15,27,31. Furthermore, according to Anderson and Blount’s original proposal that inversion symmetry breaking by collective polar displacements in metals relies on the weak coupling between itinerant electrons and soft phonons responsible for inversion symmetry breaking32,33,34, it is not obvious that across the polar-to-centrosymmetric phase transition the soft polar phonons can be coupled to itinerant electrons in Sr1−xCaxTiO3−δ, or more generally in doped ferroelectrics and polar metals11,15,31,35.

Motivated by the above experiments and theories, we use first-principle methods with no adjustable parameters to demonstrate a large modulation of electron-phonon coupling in doped strong ferroelectrics by utilizing soft polar phonons. We study BaTiO3 as a prototype, because (1) previous studies found that in n-doped BaTiO3, increasing the carrier density gradually reduces its polar distortions and induces a continuous polar-to-centrosymmetric phase transition36,37; and (2) the critical concentration for the phase transition is about 1021/cm3, which is high enough so that the electron-phonon coupling can be directly calculated within the Migdal’s approximation (in contrast, in doped SrTiO3, superconductivity emerges at a much lower carrier concentration 1017–1020/cm3 so that its Debye frequency is comparable to or even higher than the Fermi energy ℏωD/ϵF ~ 1 − 10238, which invalidates the Migdal’s approximation and Eliashberg equation)29. The key result from our calculation is that, contrary to Anderson/Blount’s argument for "ferroelectric-like metals”32,33,34, we find that the phonon bands associated with the soft polar optical phonons are strongly coupled to itinerant electrons across the polar-to-centrosymmetric phase transition in doped BaTiO3. As a consequence, the total electron-phonon coupling of doped BaTiO3 can be substantially modulated via carrier density and in particular is increased to about 0.6 around the critical concentration. Eliashberg equation calculations find that such an electron-phonon coupling is sufficiently large to induce phonon-mediated superconductivity of about 2K. In addition, we find that close to the critical concentration, lowering the crystal symmetry of doped BaTiO3 by imposing epitaxial strain further increases the superconducting temperature via a sizable coupling between itinerant electrons and acoustic phonon bands.

While ferroelectricity and superconductivity have little in common, our work demonstrates an experimentally viable approach to modulating electron-phonon coupling and inducing phonon-mediated superconductivity in doped strong ferroelectrics and potentially in polar metals32,39. Our results show that the weakly coupled electron mechanism in “ferroelectric-like metals” is not necessarily present in doped strong ferroelectrics and as a consequence, the soft polar phonons can be utilized to induce phonon-mediated superconductivity across a structural phase transition.

Results

Structural phase transition induced by electron doping

In this study, electron doping in BaTiO3 is achieved by adding extra electrons to the system with the same amount of uniform positive charges in the background. For benchmarking, our calculation of the undoped tetragonal BaTiO3 gives the lattice constant a = 3.930 Å and c/a = 1.012, polarization P = 0.26 C/m2, and Ti-O and Ba-O relative displacements of 0.105 Å and 0.083 Å, respectively, consistent with the previous calculations40,41,42. We note that upon electron doping, BaTiO3 becomes metallic and its polarization is ill-defined43. Therefore, we focus on analyzing ionic polar displacements and c/a ratio to identify the critical concentration36.

We test four different crystal structures of BaTiO3 with electron doping: the rhombohedral structure (space group R3m with Ti displaced along 〈111〉 direction), the orthorhombic structure (space group Amm2 with Ti displaced along 〈011〉 direction), the tetragonal structure (space group P4mm with Ti displaced along 〈001〉 direction) and the cubic structure (space group \(Pm\bar{3}m\) with Ti at the center of oxygen octahedron). Figure 1a shows that as electron doping concentration n increases from 0 to 0.15e/f.u., BaTiO3 transitions from the rhombohedral structure to the tetragonal structure, and finally to the cubic structure. The critical concentration is such that the crystal structure of doped BaTiO3 continuously changes from tetragonal to cubic (see Supplementary Note 6). While the structural transition from tetragonal to cubic is continuous, the transition from rhombohedral to tetragonal is first-order and thus does not show phonon softening (see Supplementary Note 7). Furthermore the low electron concentration in the rhombohedral structure invalidates Migdal’s theorem and electron-phonon coupling cannot be calculated within Migdal’s approximation (see Supplementary Note 5).

Fig. 1: Structural phase transition of BaTiO3 induced by electron doping.
figure 1

a Total energies of n-doped BaTiO3 in different crystal structures: the rhombohedral structure (R, red line), the orthorhombic structure (O, green line), the tetragonal structure (T, blue line) and the cubic structure (C, setting as the zero point at each electron doping concentration n). Upon electron doping, the ground state structure of BaTiO3 changes from R to T, finally to C. b The c/a ratio and Ti-O cation displacement δ of n-doped BaTiO3. T means the tetragonal structure and C means the cubic structure. The inset shows the tetragonal structure of doped BaTiO3 where c is the long cell axis and a is the short cell axis. δ is the displacement of the Ti atom with respect to the O atom layer along the c axis.

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Figure 1b shows c/a ratio and Ti-O cation displacements δ as a function of the concentration n in the range of 0.06–0.14e/f.u. It is evident that the critical concentration nc of doped BaTiO3 is 0.10e/f.u. (about 1.6 × 1021 cm−3), at which the polar displacement δ is just completely suppressed and the c/a ratio is reduced to unity. This result is consistent with the previous theoretical studies36,37. Experimentally, in metallic oxygen-deficient BaTiO3−δ, the low-symmetry polar structure can be retained up to an electron concentration of 1.9 × 1021 cm−3 (close to the theoretical result)44,45. However, weak localization and/or phase separation may exist in oxygen-deficient BaTiO3, depending on sample quality44,46.

Electronic structure and phonon properties

Figure 2a shows the electronic structure of doped BaTiO3 in the tetragonal structure at a representative concentration (n = 0.09e/f.u., close to the critical value). Undoped BaTiO3 is a wide gap insulator. Electron doping moves the Fermi level slightly above the conduction band edge of the three Ti t2g orbitals and thus a Fermi surface is formed. We use three Wannier functions to reproduce the Ti t2g bands, upon which electron-phonon coupling is calculated. Figure 2b shows the phonon spectrum of doped BaTiO3 in the tetragonal structure at 0.09e/f.u. concentration. We are particularly interested in the zone-center (Γ-point) polar optical phonons, which are highlighted by the green dots in Fig. 2b. The vibrational modes of those polar phonons are explicitly shown in Fig. 2c. In the tetragonal structure of BaTiO3, the two polar phonons with the ion displacements along x and y directions (ωx and ωy) are degenerate, while the third polar phonon with the ion displacements along z direction (ωz) has higher frequency. Figure 2d shows that electron doping softens the zone-center polar phonons of BaTiO3 in the tetragonal structure until it reaches the critical concentration where the three polar phonon frequencies become zero. With further electron doping, the polar phonon frequencies of BaTiO3 increase in the cubic structure (see Supplementary Note 13 for a discussion about doping’s effect on polar phonon behavior).

Fig. 2: Electronic structure and phonon properties of doped BaTiO3.
figure 2

a Electronic band structure and density of states of doped BaTiO3 in the tetragonal structure at 0.09e/f.u. concentration. In the electronic band structure, the three purple bands are generated by three maximally localized Wannier functions that exactly reproduce the original Ti t2g bands. In the electronic density of states, the blue, green and orange curves correspond to total, Ti-d projected and O-p projected partial densities of states, respectively. b Phonon band structure and phonon density of states of doped BaTiO3 in the tetragonal structure at 0.09e/f.u. concentration. In the phonon band structure, the green dots highlight the zone-center polar optical phonons. In the phonon density of states, the blue, orange, red and green curves correspond to total, Ba-projected, Ti-projected and O-projected partial densities of states, respectively. c Vibration modes of the zone-center polar optical phonons of doped BaTiO3 in the tetragonal structure at 0.09e/f.u. concentration. The left panel shows that the atoms of BaTiO3 are vibrating along the short a axis (either x-axis or y-axis, degenerate due to the tetragonal symmetry). The right panel shows that the atoms of BaTiO3 are vibrating along the long c axis (z-axis). d The frequencies of the three zone-center polar phonons of doped BaTiO3 as a function of electron concentration n. The critical concentration is at 0.1e/f.u. where the polar phonon frequencies are reduced to zero.

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Electron-phonon coupling and phonon-mediated superconductivity

The continuous polar-to-centrosymmetric phase transition in doped BaTiO3 is similar to the one in “ferroelectric-like metals” proposed by Anderson and Blount32. They first argued, later recast by Puggioni and Rondinelli33,34, that inversion symmetry breaking by collective polar displacements in a metal relies on a weak coupling between itinerant electrons and soft phonons responsible for removing inversion symmetry. According to this argument, one would expect that across the polar-to-centrosymmetric phase transition, the soft polar phonons are not strongly coupled to itinerant electrons in doped BaTiO3. In order to quantify the strength of electron-phonon coupling and make quantitative comparison, we introduce the mode-resolved electron-phonon coupling λqν and around-zone-center branch-resolved electron-phonon coupling λν:

$${\lambda }_{{\bf{q}}\nu }=\frac{1}{\pi {N}_{F}}\frac{{\rm{Im}}{{{\Pi }}}_{{\bf{q}}\nu }}{{\omega }_{{\bf{q}}\nu }^{2}}\ {\rm{and}}\ {\lambda }_{\nu }=\frac{{\int}_{| {\bf{q}}| \,{<}\,{q}_{c}}d{\bf{q}}{\lambda }_{{\bf{q}}\nu }}{{\int}_{| {\bf{q}}| \,{<}\,{q}_{c}}d{\bf{q}}}$$
(1)

where \({\rm{Im}}{{{\Pi }}}_{{\bf{q}}\nu }\) is the imaginary part of electron-phonon self-energy, ωqν is the phonon frequency, NF is the density of states at the Fermi level and qc is a small phonon momentum. The reason we define λν within ∣q∣ < qc is because: (1) exactly at the zone-center Γ point, the acoustic phonon frequency is zero and thus the contribution from the acoustic mode is ill-defined at Γ point; (2) when qc is sufficiently small, there are no phonon band crossings within ∣q∣ < qc and hence each branch ν can be assigned to a well-defined phonon mode (for a general q point, it is not trivial to distinguish which phonon band corresponds to polar modes and which to other optical modes). We choose \({q}_{c}=0.05\frac{\pi }{a}\) where a is the lattice constant (the qualitative conclusions do not depend on the choice of qc, as long as no phonon band crossings occur within ∣q∣ < qc).

Figure 3a, b show the imaginary part of electron-phonon self-energy \({\rm{Im}}{{{\Pi }}}_{{\bf{q}}\nu }\) for each phonon mode qν of doped BaTiO3 along a high-symmetry path (panel a corresponds to 0.09e/f.u. doping in a tetragonal structure and panel b corresponds to 0.11e/f.u. doping in a cubic structure). Since within the double delta approximation \({\rm{Im}}{{{\Sigma }}}_{{\bf{q}}\nu }\) is positive definite (see Supplementary Note 2), the point size in panels a and b is chosen to be proportional to the value of \({\rm{Im}}{{{\Pi }}}_{{\bf{q}}\nu }\). Our calculations find that, contrary to Anderson/Blount’s weak coupled electron mechanism32, the phonon bands associated with the zone-center polar phonons have the strongest coupling to itinerant electrons, while the couplings of other phonon bands are weaker. Specifically, in the case of 0.09e/f.u. doping:

$${\lambda }_{{\rm{acoustic}}}=\mathop{\sum}\limits_{\nu =1-3}{\lambda }_{\nu }=3.83$$
(2)
$${\lambda }_{{\rm{polar}}}=\mathop{\sum}\limits_{\nu =4-6}{\lambda }_{\nu }=10.92$$
$${\lambda }_{{\rm{others}}}=\mathop{\sum}\limits_{\nu =7-15}{\lambda }_{\nu }=0.53$$

and in the case of 0.11e/f.u. doping:

$${\lambda }_{{\rm{acoustic}}}=\mathop{\sum}\limits_{\nu =1-3}{\lambda }_{\nu }=0.27$$
(3)
$${\lambda }_{{\rm{polar}}}=\mathop{\sum}\limits_{\nu =4-6}{\lambda }_{\nu }=5.58$$
$${\lambda }_{{\rm{others}}}=\mathop{\sum}\limits_{\nu =7-15}{\lambda }_{\nu }=0.11$$

In both cases, λpolar is larger than λacoustic and λothers. An intuitive picture for the strong coupling is that in doped BaTiO3, the soft polar phonons involve the cation displacements of Ti and O atoms, and in the meantime itinerant electrons derive from Ti-d states which hybridize with O-p states (see Supplementary Note 14 for an alternative demonstration of this strong coupling, and Supplementary Note 12 for a discussion about doping’s effect on this p-d hybridization). This is in contrast to the textbook example of polar metals LiOsO3 where the soft polar phonons involve Li displacements while the metallicity derives from Os and O orbitals33. More quantitatively, we find λpolar = 0.50 for LiOsO3, which is substantially smaller than λpolar of about 5–10 for doped BaTiO3. In short, because the itinerant electrons and polar phonons are associated with the same atoms in doped BaTiO3, the coupling is strong, while in LiOsO3 the itinerant electrons and polar phonons involve different atoms and thus the coupling is weak. As a consequence of the strong interaction between the polar phonons and itinerant electrons, we expect that the total electron-phonon coupling of doped BaTiO3 can be increased by softening the polar phonons across the structural phase transition.

Fig. 3: Electron-phonon properties and phonon-mediated superconductivity in doped BaTiO3.
figure 3

a The imaginary part of the electron-phonon self-energy \({\rm{Im}}{{{\Pi }}}_{{\bf{q}}\nu }\) for each phonon mode of doped BaTiO3 at 0.09e/f.u. concentration (tetragonal structure T). The point size is proportional to \({\rm{Im}}{{{\Pi }}}_{{\bf{q}}\nu }\). The largest point corresponds to \({\rm{Im}}{{{\Pi }}}_{{\bf{q}}\nu }\) = 4.6 meV. The green dots highlight the zone-center polar optical phonons. b The imaginary part of the electron-phonon self-energy \({\rm{Im}}{{{\Pi }}}_{{\bf{q}}\nu }\) for each phonon mode of doped BaTiO3 at 0.11e/f.u. concentration (cubic structure C). The point size is proportional to \({\rm{Im}}{{{\Pi }}}_{{\bf{q}}\nu }\). The largest point corresponds to \({\rm{Im}}{{{\Pi }}}_{{\bf{q}}\nu }\) = 3.2 meV. The green dots highlight the zone-center polar optical phonons. c Electron-phonon spectral function α2F(ω) and accumulative electron-phonon coupling λ(ω) of doped BaTiO3 at 0.09e/f.u. and 0.11e/f.u. concentration. The total electron-phonon coupling λ is 0.61 for the former and 0.50 for the latter. d Total electron-phonon coupling λ of doped BaTiO3 as a function of electron concentration n. e Superconducting gap Δ of doped BaTiO3 as a function of temperature T at 0.09 e/f.u. concentration (red) and at 0.11 e/f.u. concentration (blue), calculated by the three-orbital Eliashberg equation. The Morel-Anderson pseudopotential \({\mu }_{ij}^{* }=0.1\) is used for each orbital pair. f Superconducting transition temperature Tc of doped BaTiO3 calculated by the Eliashberg equation as a function of electron concentration n. The inset shows Tc of BaTiO3 in the tetragonal structure at 0.09e/f.u. concentration as a function of Morel-Anderson pseudopotential \({\mu }_{ij}^{* }\).

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Figure 3c shows the total electron-phonon spectral function α2F(ω) and accumulative electron-phonon coupling λ(ω) of doped BaTiO3 at 0.09e/f.u. and 0.11e/f.u. concentrations. α2F(ω) is defined as:

$${\alpha }^{2}F(\omega )=\frac{1}{2}\mathop{\sum}\limits_{\nu }\int \frac{d{\bf{q}}}{{{{\Omega }}}_{{\rm{BZ}}}}{\omega }_{{\bf{q}}\nu }{\lambda }_{{\bf{q}}\nu }\delta (\omega -{\omega }_{{\bf{q}}\nu })$$
(4)

where ΩBZ is the volume of phonon Brillouin zone. With α2F(ω), it is easy to calculate the accumulative electron-phonon coupling λ(ω):

$$\lambda (\omega )=2\mathop{\int}\nolimits_{0}^{\omega }\frac{{\alpha }^{2}F(\nu )}{\nu }d\nu$$
(5)

The total electron-phonon coupling λ is obtained by taking the upper bound ω to ∞ in Eq. (5). The green shades are α2F(ω) and the dashed lines are the corresponding cumulative electron-phonon coupling. The total electron-phonon coupling λ of doped BaTiO3 in the tetragonal structure at 0.09e/f.u. concentration is 0.61, while that in the cubic structure at 0.11e/f.u. concentration is 0.50. Both λ are sufficiently large to induce phonon-mediated superconductivity with measurable transition temperature. Figure 3d shows the total electron-phonon coupling λ of doped BaTiO3 for a range of electron concentrations (exactly at the critical concentration, we find some numerical instabilities and divergence in the electron-phonon calculations, rendering the result unreliable). An increase of λ around the critical concentration is evident, consistent with the strong coupling between the soft polar phonons and itinerant electrons in doped BaTiO3.

Based on the electron-phonon spectrum α2F(ω), we use a three-orbital Eliashberg equation (see Supplementary Note 3) to calculate the superconducting gap Δ(T) and estimate the superconducting transition temperature Tc as a function of electron concentration. Because the three Ti t2g orbitals become identical at the critical concentration, when solving the three-orbital Eliashberg equation, we set Morel-Anderson pseudopotential \({\mu }_{ij}^{* }\) to be 0.1 for each orbital pair (i.e., i, j = 1, 2, 3)47. Figure 3e shows the superconducting gap Δ(T) of doped BaTiO3 as a function of temperature T at two representative concentrations (0.09e/f.u. in the tetragonal structure and 0.11e/f.u. in the cubic structure). Since both concentrations are close to the critical value, the three Ti t2g orbitals are almost degenerate in doped BaTiO3. For clarification, we show the superconducting gap of one orbital for each concentration. From the Eliashberg equation, we find that at 0.09e/f.u. concentration, Δ(T = 0) = 0.27 meV and Tc = 1.75 K; and at 0.11e/f.u. concentration, Δ(T = 0) = 0.11 meV and Tc = 0.76 K. Thus Δ(T = 0)/(kBTc) = 1.79 at 0.09e/f.u. concentration and 1.68 at 0.11e/f.u. concentration, both close to the BCS prediction of 1.77. Figure 3f shows the estimated superconducting transition temperature Tc of doped BaTiO3 for a range of electron concentrations. Tc notably exhibits a dome-like feature as a function of electron concentration. The origin of the superconducting “dome” is that the electron-phonon coupling of doped BaTiO3 is increased by the softened polar phonons around the critical concentration. When the electron concentration is away from the critical value, the polar phonons are “hardened” (i.e., phonon frequency increases) and the electron-phonon coupling of doped BaTiO3 decreases. We note that the estimated Tc strongly depends on \({\mu }_{ij}^{* }\). Therefore in the inset of Fig. 3f, we study doped BaTiO3 at a representative concentration (0.09e/f.u.) and calculate its superconducting transition temperature Tc as a function of \({\mu }_{ij}^{* }\). As \({\mu }_{ij}^{* }\) changes from 0 to 0.3, the estimated Tc decreases from 9.3 K to 0.4 K, the lowest of which (0.4 K) is still measurable in experiment28. We make two comments here: (1) The superconducting transition temperature is only an estimation due to the uncertainty of Morel-Anderson pseudopotential \({\mu }_{ij}^{* }\) and other technical details. But the picture of an increased electron-phonon coupling around the structural phase transition in doped BaTiO3 is robust. (2) Experimentally in Sr1−xCaxTiO3, the optimal doping for superconductivity is larger than the “ferroelectric” critical concentration20, while in our calculations of doped BaTiO3, the two critical concentrations (one for optimal superconducting Tc and the other for suppressing polar displacements) just coincide due to polar phonon softening and an increased electron-phonon coupling. Comparison of these two materials implies that the microscopic mechanism for superconductivity in doped SrTiO3 is probably not purely phonon-mediated.

Crystal symmetry and acoustic phonons

We note that in Figure 3a, b, in addition to the large \({\rm{Im}}{{{\Pi }}}_{{\bf{q}}\nu }\) in the polar optical phonon bands, there is also sizable \({\rm{Im}}{{{\Pi }}}_{{\bf{q}}\nu }\) in the acoustic phonon bands (from Γ to X) in the tetragonal structure at 0.09e/f.u. concentration. Since the mode-resolved electron-phonon coupling \({\lambda }_{{\bf{q}}\nu }\propto {\rm{Im}}{{{\Pi }}}_{{\bf{q}}\nu }/{\omega }_{{\bf{q}}\nu }^{2}\), the small frequency of acoustic phonons can lead to a substantial λqν, given a sizable \({\rm{Im}}{{{\Pi }}}_{{\bf{q}}\nu }\). However, in the cubic structure at 0.11e/f.u. concentration, \({\rm{Im}}{{{\Pi }}}_{{\bf{q}}\nu }\) in the acoustic phonon bands almost vanishes from Γ to X. To exclude that the concentration difference may have an effect, we perform a numerical experiment: we start from the cubic structure doped at 0.11e/f.u. concentration (space group \(Pm\bar{3}m\)), and then we impose a slight (001) compressive bi-axial 0.8% strain by fixing the two in-plane lattice constants (a and b) to a smaller value. This compressive strain makes the crystal structure of doped BaTiO3 tetragonal and polar (space group P4mm). Figure 4a shows the optimized crystal structures of the two doped BaTiO3. For doped BaTiO3 at 0.11e/f.u. concentration, without strain, the ground state structure is cubic and the optimized lattice constant a is 3.972 Å; under a 0.8% bi-axial (001) compressive strain, the ground state structure becomes tetragonal with the in-plane lattice constants a and b being fixed at 3.940 Å and the optimized long lattice constant c being 4.019 Å. We find that the total electron-phonon coupling λ increases from 0.50 in the \(Pm\bar{3}m\) structure to 0.57 in the P4mm structure. Figure 4b, c compare the imaginary part of the electron-phonon self-energy \({\rm{Im}}{{{\Pi }}}_{{\bf{q}}\nu }\) and the mode-resolved electron-phonon coupling λqν along the Γ → X path for the two doped BaTiO3. Similar to Fig. 3a and b, we find that there is a notable difference in \({\rm{Im}}{{{\Pi }}}_{{\bf{q}}\nu }\) from the acoustic phonon bands. The difference in \({\rm{Im}}{{{\Pi }}}_{{\bf{q}}\nu }\) is further “amplified” by the low phonon frequencies ωqν, which results in

$${\text{without}}\, {\text{epitaxial}}\, {\text{strain}}\,\ {\lambda }_{\text{acoustic}}= \, 0.27\\ {\text{under}}\, {\text{0.8\%}}\, {\text{(001)}}\, {\text{compressive}}\, {\text{strain}}\,\ {\lambda }_{\text{acoustic}}= \, 4.45$$
(6)

At the same time, we find that for polar modes,

$${\text{without}}\, {\text{epitaxial}}\, {\text{strain}}\, {\lambda}_{\text{polar}}= \, 5.58\\ {\text{under}}\, {\text{0.8\%}}\, {\text{(001)}}\, {\text{compressive}}\, {\text{strain}}\, {\lambda}_{\text{polar}}= \, 3.21$$
(7)

This shows that under 0.8% (001) compressive strain, λpolar remains substantial (albeit reduced by about 40%), but λacoustic is increased by one order of magnitude, which altogether leads to an enhancement of the total electron-phonon coupling λ. Note that in the numerical experiment, the two doped BaTiO3 have exactly the same electron concentration, indicating that the additional increase in \({\rm{Im}}{{{\Pi }}}_{{\bf{q}}\nu }\) of the acoustic phonons arises solely from the crystal structure difference. A possible explanation, which is based on our calculations, is that in the cubic structure, some electron-phonon vertices \({g}_{ij}^{\nu }({\bf{k}},{\bf{q}})\) are exactly equal to zero because some atoms are frozen in the acoustic phonons, while in the low-symmetry structure, those \({g}_{ij}^{\nu }({\bf{k}},{\bf{q}})\) become non-zero. Because \({\rm{Im}}{{{\Pi }}}_{{\bf{q}}\nu }\propto | {g}_{ij}^{\nu }({\bf{k}},{\bf{q}}){| }^{2}\)48,49, this leads to an increase in \({\rm{Im}}{{{\Pi }}}_{{\bf{q}}\nu }\). In addition, the frequencies of acoustic phonon modes ωqν are very small and \({\lambda }_{{\bf{q}}\nu }\propto {\rm{Im}}{{{\Pi }}}_{{\bf{q}}\nu }/{\omega }_{{\bf{q}}\nu }^{2}\), therefore even a slight increase in \({\rm{Im}}{{{\Pi }}}_{{\bf{q}}\nu }\) results in a substantial enhancement in λqν (see Supplementary Note 16 for the demonstration of a specific acoustic phonon). Our numerical experiment also implies that in doped BaTiO3, when the electron concentration is close to the critical value, a small (001) compressive strain that lowers the crystal symmetry may also enhance its superconducting transition temperature due to the increased electron-phonon coupling, similar to doped SrTiO318,22.

Fig. 4: Comparison of doped BaTiO3 between the cubic structure and the strain-induced tetragonal structure.
figure 4

a Doped BaTiO3 at 0.11e/f.u. concentration. Left is the cubic crystal structure of BaTiO3 with no strain (space group \(Pm\bar{3}m\)) and right is the polar tetragonal crystal structure of BaTiO3 under 0.8% bi-axial strain (space group P4mm). b The imaginary part of the electron-phonon self-energy \({\rm{Im}}{{{\Pi }}}_{{\bf{q}}\nu }\) for each phonon mode of doped BaTiO3, at 0.11e/f.u. in the P4mm structure (red, left) and in the \(Pm\bar{3}m\) structure (blue, right). The point size is proportional to \({\rm{Im}}{{{\Pi }}}_{{\bf{q}}\nu }\). The largest point corresponds to \({\rm{Im}}{{{\Pi }}}_{{\bf{q}}\nu }\) = 3.8 meV. c Mode-resolved electron-phonon coupling λqν for each phonon mode of doped BaTiO3, at 0.11e/f.u. in the P4mm structure (red, left) and in the \(Pm\bar{3}m\) structure (blue, right). The point size is proportional to λqν. The largest point corresponds to λqν = 5.1.

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Discussion

Finally we discuss possible experimental verification. Chemical doping44,45,50,51,52,53,54 and epitaxial strain55,56 have been applied to ferroelectric materials such as BaTiO3. La-doped BaTiO3 has been experimentally synthesized. High-temperature transport measurements show that Ba1−xLaxTiO3 exhibits polar metallic behavior but ultra-low-temperature transport measurements are yet to be performed50,51,52,53,54. We note that La doping in BaTiO3 may result in some chemical disorder. While the randomness of La distribution in LaxBa1−xTiO3 may affect the transport properties in the normal state, Anderson’s theorem asserts that superconductivity in a conventional superconductor is robust with respect to non-magnetic disorder in the host material57. As a consequence, the superconducting transition temperature Tc of a conventional superconductor barely depends on the randomness of defects. In our case, the superconductivity in doped BaTiO3 is phonon-mediated (i.e., conventional) and La is a non-magnetic dopant. Therefore Anderson’s theorem applies and we expect that even if chemical disorder may arise in actual experiments, it does not affect the superconducting properties of doped BaTiO3. In addition, we perform supercell calculations which include real La dopants. We find that even in the presence of real La atoms, the conduction electrons on Ti atoms are almost uniformly distributed in LaxBa1−xTiO3 (see Supplementary Note 8 and Supplementary Note 9 for details). Since our simulation does not consider dopants explicitly, a more desirable doping method is to use electrostatic carrier doping58,59,60, which does not involve chemical dopants and has been successfully used to induce superconductivity in KTaO361. We clarify two points concerning the electrostatic doping method. (1) The electrostatic gating by ionic liquid can achieve a two-dimensional carrier density as high as 8 × 1014 cm−262. The induced electrons are usually confined in a narrow region that is a few nanometers from the surface/interface, which leads to an effective three-dimensional carrier density of about 1 × 1021 ~ 5 × 1021 cm−361,63. In our current study, the critical concentration of doped BaTiO3 is about 1.6 × 1021 cm−3, which is feasible by this approach. (2) While the electrostatic doping method induces the carriers in the surface/interface area, we show that our results on bulk doped BaTiO3 can still be used as a guidance to search for superconductivity in the surface area of BaTiO3. We perform calculations of Pt/BaTiO3 interface (see Supplementary Note 10) and find that just in the second unit cell of BaTiO3 from the interface, the Ti-O displacement saturates and a bulk-like region emerges with almost uniform cation displacements. In addition, we calculate the electron-phonon properties of bulk KTaO3 at 0.14e/f.u. doping (based on the experiment61) (see Supplementary Note 15). We find that the total electron-phonon coupling of KTaO3 at 0.14e/f.u. doping is 0.36. Using McMillian equation (take μ* = 0.1) as a rough estimation of superconducting transition temperature Tc, we obtain a Tc of about 68 mK, which is in reasonable agreement with the experimental value of 50 mK. While there is definitely room for improvement, our results demonstrate that for a given target material, its desirable bulk electron-phonon property can point to the right direction in which superconductivity is found in surface/interface regions.

In summary, we use first-principles calculations to demonstrate a large modulation of electron-phonon coupling and an emergent superconducting “dome” in n-doped BaTiO3. Contrary to Anderson/Blount’s weak electron coupling mechanism for “ferroelectric-like metals”32,33,34, our calculations find that the soft polar phonons are strongly coupled to itinerant electrons across the polar-to-centrosymmetric phase transition in doped BaTiO3 and as a consequence, the total electron-phonon coupling increases around the critical concentration. In addition, we find that lowering the crystal symmetry of doped BaTiO3 by imposing epitaxial strain can also increase the electron-phonon coupling via a sizable coupling between acoustic phonons and itinerant electrons. Our work provides an experimentally viable method to modulating electron-phonon coupling and inducing phonon-mediated superconductivity in doped strong ferroelectrics. Our results indicate that the weak electron coupling mechanism for “ferroelectric-like metals”32,33,34 is not necessarily present in doped strong ferroelectrics. We hope that our predictions will stimulate experiments on doped ferroelectrics and search for the phonon-mediated superconductivity that is predicted in our calculations.

Methods

We perform first-principles calculations by using density functional theory64,65,66,67 as implemented in the Quantum ESPRESSO package68. We use norm-conserving pseudo-potentials69 with local density approximation as the exchange-correlation functional. For electronic structure calculations, we use an energy cutoff of 100 Ry. We optimize both cell parameters and internal coordinates in atomic relaxation, We find that the optimized crystal structures are in good agreement with experiments (see Supplementary Note 1). The detailed structural information is reported in Supplementary Note 7. In the strain calculations, the in-plane lattice constants are fixed while the out-of-plane lattice constant and internal coordinates are fully optimized. The electron Brillouin zone integration is performed with a Gaussian smearing of 0.005 Ry over a Γ-centered k mesh of 12 × 12 × 12. The threshold of total energy convergence is 10−7 Ry; self-consistency convergence is 10−12 Ry; force convergence is 10−6 Ry/Bohr and pressure convergence for variable cell is 0.5 kbar. For phonon calculations, we use density functional perturbation theory66 as implemented in the Quantum ESPRESSO package68 (see Supplementary Note 11 for the validation of this method on a prototypical oxide SrTiO3). The phonon Brillouin zone integration is performed over a q mesh of 6 × 6 × 6. For the calculations of electron-phonon coupling and superconducting gap (see Supplementary Note 2), we use maximally localized Wannier functions and Migdal-Eliashberg theory, as implemented in the Wannier9070 and EPW code71. The Fermi surface of electron-doped BaTiO3 is composed of three Ti t2g orbitals. We use three maximally localized Wannier functions to reproduce the Fermi surface. The electron-phonon matrix elements \({g}_{ij}^{\nu }({\bf{k}},{\bf{q}})\) are first calculated on a coarse 12 × 12 × 12 k-grid in the electron Brillouin zone and a coarse 6 × 6 × 6 q-grid in the phonon Brillouin zone, and then are interpolated onto fine grids via maximally localized Wannier functions. The fine electron and phonon grids are both 50 × 50 × 50. We check the convergence on the electron k-mesh, phonon q-mesh and Wannier interpolation and no significant difference is found by using a denser mesh. Details can be found in Supplementary Note 4. We solve a three-orbital Eliashberg equation to estimate the superconducting transition temperature Tc (see Supplementary Note 3).

We only use Eliashberg equation when electron doping concentration is high enough so that λTD/TF < 0.1 and Migdal’s theorem is valid29 (λ is electron-phonon coupling, TD is Debye temperature and TF is Fermi temperature). Validation test of Migdal’s theorem is shown in Supplementary Note 5.

We solve a three-orbital Eliashberg equation to estimate the superconducting transition temperature Tc. This method is compared to McMillan Equation. Details of Eliashberg Equation and McMillan Equation can be found in Supplementary Note 3.

Reporting summary

Further information on research design is available in the Nature Research Reporting Summary linked to this article.