Introduction

The observation of superconductivity in “quantum paraelectrics”—materials with low temperature incipient ferroelectricity—has raised several fundamental questions on the pairing mechanism in such systems. Examples of quantum paraelectrics include strontium titanate (SrTiO3)1,2,3,4,5,6 as well as potassium tantalate (KTaO3)7,8,9,10 and lead telluride (PbTe)11. A central question involves the hierarchy of the relevant fluctuations and their consequences for superconductivity. For instance, if pairing is mediated mainly by soft critical ferroelectric fluctuations, the associated superconducting dome would be confined to low electron densities, as ferroelectricity itself is sharply defined only in an insulating phase (A “ferroelectric metal” is essentially characterized by either broken inversion, i.e. non-centrosymmetric, or spatial reflection symmetries12; dipole moments while permitted by symmetry, are strongly screened by the conduction electrons.). The fact that superconductivity, at least in niobium-doped strontium titanate, is observed only over a range in the dilute limit (~0.05% to ~0.5%)3,4,6 gives support to the notion of pairing mediated by ferroelectric fluctuations13.

The restriction of pairing to dilute carrier concentrations, however, presents several puzzling issues. First, the resulting small density of states in a 3d electron system would suggest a correspondingly small superconducting pairing strength. Second, the soft mode associated with ferroelectricity is the transverse-optical (TO) phonon, which couples less strongly to the electrons than the longitudinal optical (LO) phonons. Moreover, symmetry considerations lead to the conclusion that in the absence of orbital- or spin-dependent processes, electrons can only scatter off of pairs of TO phonons14,15,16. The resulting reduction in phase space would naturally result in reduction of the superconducting transition temperature Tc. Finally, given the dilute electron concentrations, there is the possibility that the Fermi energy may be smaller than the phonon frequency itself, resulting in an inverted “anti-adiabatic” pairing regime. Whether superconducting domes can arise in quantum paraelectrics despite these circumstances remains actively debated17,18,19,20,21. Moreover, the observation of superconductivity at interfaces of quantum paraelectrics7,8,9,10,22,23 further motivates the study of superconductivity in these systems, raising the question of the role of spatial dimensionality on all these issues.

In this Letter, we construct a self-consistent pairing scenario for quantum paraelectrics, and illustrate it in the case of SrTiO3, where recent experiments have placed significant constraints on theory. These experiments have reported a textbook BCS gap to Tc ratio4,5,6; any theory of pairing in this system must satisfy this constraint. In light of these experiments, we discuss constraints on pairing that arises from either the anti-adiabatic or the more conventional adiabatic pairing mechanisms. We construct a scenario in which pairing is mediated by TO phonons. We also present a mechanism by which electrons may couple to single TO phonons, resolving the issues of limited phase space alluded to above. We then discuss the relevance of these findings to other quantum paraelectrics, including interfacial systems.

Results

Experimental considerations

Two distinct pairing scenarios have been proposed to explain superconductivity in the dilute limit of bulk SrTiO3: a conventional one in which the phonon frequency remains smaller than the Fermi energy EF24,25,26,27, and an anti-adiabatic mechanism in which the hierarchy of energy scales are inverted28,29,30,31,32,33,34,35. In recent experiments4,6, various phonon frequencies were probed, in addition to the superconducting gap. Various experiments also show that (1) the lowest TO (TO1) phonon frequency increases with doping but remains below the Fermi energy across the superconducting dome6,36,37, (2) the LO phonon frequencies remain unchanged with doping and are either comparable to or greater than EF across the dome38, and (3) the superconducting gap to Tc ratio is close to the BCS value4,5,6.

It follows from the first observation that any pairing mechanism involving the TO1 phonons can remain conventional and adiabatic whereas LO pairing mechanisms would be in the anti-adiabatic regime in SrTiO3. We first briefly describe why the anti-adiabatic scenarios are unlikely in SrTiO3 and we then consider the adiabatic pairing scenario mediated by TO phonons.

Based on the tunneling measurements4,5,6, any anti-adiabatic pairing scenario that remains viable across the superconducting dome must necessarily only involve the highest LO phonon mode (LO4). Furthermore, the constraint imposed by the BCS gap to Tc ratio requires the effective attraction mediated by the LO phonon to be weak, which can occur in the anti-adiabatic regime only if the LO4 phonon frequency were significantly higher than the Fermi energy. Further restrictions from the tunneling data come from the fact that the LO frequency remains essentially unchanged with doping. Since the BCS coupling is proportional both to the density of states and the square of electron-phonon coupling, a crucial ingredient needed for Tc to decrease beyond optimal doping is the reduction of the electron-phonon coupling strength with the dopant concentration n faster than n−1/3, in order to overcome the growth of the BCS coupling with increasing density of states. It therefore seems unlikely, based in part on the tunneling measurements, that pairing in SrTi1−xNbxO3 is mediated by an anti-adiabatic LO4 phonon.

We are thus led naturally to consider an adiabatic pairing mechanism across the dome of SrTi1−xNbxO3. The only phonon mode that remains in the adiabatic regime across the dome is the TO1 mode, the softening of which leads to ferroelectricity. Furthermore, a conventional BCS framework based on the Migdal approximation should suffice to account for the BCS gap to Tc ratio within this scenario (The lower density dome that is observed in oxygen-reduced samples3 is outside the scope of the present paper as such samples have been resistant to the pairing gap measurement through the planar tunneling spectroscopy.).

Additionally, the superconducting dome from TO1 phonon exchange can be simply understood as follows. Prior experiments36,37 indicate that the TO1 phonon frequency increases with carrier concentration as \({\omega }_{T}^{2}={K}_{0}+n{K}_{1}\) with the approximate values of K0 ≈ 1 meV2 and K1 ≈ 1.8 × 10−19 meV2 cm3 > 015. Hence the BCS eigenvalue for the adiabatic pairing mediated solely by a single TO1 phonon is parametrically:

$${\lambda }_{{{{\rm{BCS}}}}}\propto \frac{{N}_{F}}{{\omega }_{T}^{2}} \sim \frac{{n}^{1/3}}{{K}_{0}+{K}_{1}n};$$
(1)

the overdoped attenuation of Tc naturally comes from the fact that the TO1 phonon hardens “faster” with Nb concentration than the increase in the density of states. Thus, in the adiabatic pairing scenario based on TO1 phonon exchange, the low density edge of the dome is dictated by the vanishing of the density of states whereas the high density edge is dictated by the hardening of the phonon frequency (in conjunction with the Coulomb pseudopotential μ*).

The only caveat in the hypothesis above is that it assumes a conventional coupling of the electrons to a single TO phonon. Symmetry considerations however, require that if the initial and final electron states in a phonon exchange process have the same symmetry with respect to reflection about the plane normal to which the TO mode displacement occurs, the process must involve a pair of TO phonons39 (for discussion on superconductivity arising from such electron-phonon coupling, see ref. 16). As we discuss below, the way around this constraint is to include multiple orbitals; a single TO phonon can scatter an electron from an orbital that is even under such a reflection to one that is odd, and vice-versa40. As we show, such processes can naturally account for a superconducting dome in this system.

Pairing from TO phonon scattering

The qualitative effect of the electronic coupling to the single TO1 phonon can be most simply obtained from a microscopic model for the SrTiO3 electronic band structure that incorporates the titanium (Ti) 3dt2g orbitals while assuming the simple cubic lattice structure. The low-energy band structure can be described well by a minimal tight-binding model whose k-space representation can be written as refs. 15,41,42,43:

$$\begin{array}{l}{H}_{0}=\mathop{\sum}\limits_{{{{\bf{k}}}},\alpha ,s}({\epsilon }_{\alpha }-\mu ){c}_{{{{\bf{k}}}},\alpha ,s}^{{\dagger} }{c}_{{{{\bf{k}}}},\alpha ,s}\\ \qquad-\frac{\xi }{2}\mathop{\sum}\limits_{{{{\bf{k}}}},\alpha ,\beta ,s,s^{\prime} }{{{{{\ell }}}}}_{\alpha \beta }\cdot {{{{\boldsymbol{\sigma }}}}}_{s,s^{\prime} }{c}_{{{{\bf{k}}}},\alpha ,s}^{{\dagger} }{c}_{{{{\bf{k}}}},\beta ,s^{\prime} },\end{array}$$
(2)

where α, β = X, Y, Z refer, respectively, to the Ti dyz, dxz, dxy orbitals, \(s,s^{\prime}\) the spin indices, ξ = 19.3 meV from the Ti atomic spin-orbit coupling with the totally antisymmetric tensor \({\ell }_{\beta \gamma }^{\alpha }\equiv -i{\varepsilon }_{\alpha \beta \gamma }\) representing the effective L = 1 orbital angular momentum of the TI t2g orbitals, and:

$${\epsilon }_{\alpha }={\epsilon }_{0}+4{t}_{1}\mathop{\sum}\limits_{\beta \ne \alpha }{\sin }^{2}\frac{{k}_{\beta }}{2}+4{t}_{2}{\sin }^{2}\frac{{k}_{\alpha }}{2},$$
(3)

is the intra-orbital hopping (with ϵ0 = 12.2 meV) whose t1 > t2 anisotropy (0.615 and 0.035 eV, respectively) can be attributed to the quantum mechanical effect of the Ti t2g orbital symmetry15,42.

The form of the electronic coupling to the TO1 phonons is determined by the interplay between the t2g orbital symmetry and the crystalline structure. As shown in Fig. 1, the tunneling between different t2g orbitals between nearest neighbors is forbidden by inversion symmetry in a static lattice, but the TO1 mode displacements break inversion symmetry and thereby induce odd-parity inter-orbital tunneling. Given its odd-parity, this tunneling at the long-wavelength limit can be described by the following electron-phonon interaction44,45:

$${H}_{e-p}=g\mathop{\sum}\limits_{{{{\bf{k}}}},{{{\bf{q}}}}}\mathop{\sum}\limits_{i,\alpha ,\beta ,s}{{{{\boldsymbol{\phi }}}}}_{{{{\bf{q}}}}}\cdot [{{{{{\ell }}}}}_{\alpha \beta }\times ({{{\bf{k}}}}+{{{\bf{q}}}}/2)]{c}_{{{{\bf{k}}}}+{{{\bf{q}}}},\alpha ,s}^{{\dagger} }{c}_{{{{\bf{k}}}},\beta ,s}$$
(4)

where ϕ is the TO1 mode displacement vector. The simplest justification for this coupling is to consider a uniform \({{{\boldsymbol{\phi }}}}\parallel \hat{{{{\bf{z}}}}}\), which would displace the Ti atom from the center of TiO6 octahedron along the z-direction by a constant amount; this will turn on the nearest-neighbor hopping between the dxy and dyz (dxz) in the x(y)-direction through the O py (px) orbital. The following two aspects of this electron-phonon coupling makes it viable as the pairing glue for superconductivity.

Fig. 1: Electron coupling to TO1 phonon.
figure 1

Two types of symmetry allowed couplings between t2g electrons and the TO phonons. In a two phonon process (a), a pair of TO phonons, each denoted by a wavy line oriented along the displacement axis, couples to the electron density. b Single-phonon processes can occur provided the tunneling occur between two distinct t2g orbitals on the nearest neighbors. Along the crystalline x axis, a TO displacement along the z-direction couples the dxy and dyz orbitals. The wavy line denotes the TO phonon displacement axis.

Full size image

First, the electron-phonon coupling of Eq. (4) is distinct from acoustic phonons coupling derivatively to the fermions. As long as there is a non-zero fermion density, the typical fermion momentum ∣k∣ ~ kF is finite, and the electron-phonon coupling in Eq. (4) survives even in the q → 0 limit. In the interest of simplicity, we consider the case where gkF is independent of density; what this implies will be discussed upon obtaining the effective BCS interaction.

Second, the electron-phonon coupling of Eq. (4) can induce an intra-band pairing interaction due to the presence of atomic spin-orbit coupling in Eq. (2)46,47,48. This is not limited to the three C3 rotational axes of the cubic lattice, where the eigenstates of the H0 of Eq. (2) are the effective j = 1/2 and j = 3/2 states (The projections of ϕ ⋅ (ℓ × k) to the j = 1/2 and the j = 3/2 subspaces are \(\frac{4}{3}{{{\boldsymbol{\phi }}}}\cdot ({{{\bf{j}}}}\times {{{\bf{k}}}})\) and \(-\frac{2}{3}{{{\boldsymbol{\phi }}}}\cdot ({{{\bf{j}}}}\times {{{\bf{k}}}})\), respectively49,50). Even away from this band degeneracy, the intra-band Rashba coupling of the TO1 phonon27,51,52 can be obtained by treating the sum of Eq. (4) and the atomic spin-orbit coupling of Eq. (2) as perturbations48,53.

We now derive the dimensionless effective BCS pairing interaction from this electron-phonon coupling using the Dyson’s equation in the Nambu space where the electron self-energy arises entirely from the Cooper pairing. For this Dyson’s equation:

$$\begin{array}{l}{{\Sigma }}({{{\bf{k}}}},i{\nu }_{m})=2\frac{{k}_{B}T}{\hslash }\mathop{\sum}\limits_{{{{\bf{k}}}}^{\prime} ,i,j}{\chi }_{ij}({{{\bf{k}}}}-{{{\bf{k}}}}^{\prime} )\\ \qquad\qquad\quad\times {F}_{j}({{{\bf{k}}}},{{{\bf{k}}}}^{\prime} -{{{\bf{k}}}}){{{\mathcal{G}}}}({{{\bf{k}}}}^{\prime} ,i{\nu }_{m}){F}_{i}({{{\bf{k}}}},{{{\bf{k}}}}^{\prime} -{{{\bf{k}}}}),\end{array}$$
(5)

where νm ≡ (2m + 1)πkBT/ℏ (with \(m\in {\mathbb{Z}}\)) is the fermionic Matsubara frequency, \({{{\mathcal{G}}}}\) the electronic Green’s function and:

$${F}_{i}({{{\bf{k}}}},{{{\bf{k}}}}^{\prime} -{{{\bf{k}}}})\equiv g{[{{{{\ell }}}}\times ({{{\bf{k}}}}+{{{\bf{k}}}}^{\prime} )]}_{i}/4$$
(6)

is the electron-phonon interaction vertex from Eq. (4), with g ∝ n−1/3 to maintain an effective electron-phonon coupling strength independent of the carrier density n. We take advantage of the adiabaticity of the TO1 phonon to ignore the boson dynamics27 and take the static TO1 propagator:

$${\chi }^{ij}({{{\bf{q}}}})=\langle {\phi }_{-{{{\bf{q}}}}}^{i}{\phi }_{{{{\bf{q}}}}}^{j}\rangle =\frac{\hslash }{{M}_{T}}\frac{{\delta }_{ij}-{\hat{q}}_{i}.{\hat{q}}_{j}}{{\omega }_{T}^{2}+{c}_{T}^{2}{q}^{2}},$$
(7)

where MT is the TO1 phonon effective mass. Given that the self-energy for this Dyson’s equation is given as the linear combination of the pairing gap, we need to consider the form of pairing gap that would be favored by the electron-phonon coupling of Eq. (4). With regards to the Cooper pair spin states, we note that any electron-phonon coupling, even with odd-parity, favors spin-singlet pairing46,47,52,54. Therefore our pairing gap should be intra-band, even-parity, pseudospin-singlet (frequency-independence being already assumed by Eq. (5)), giving us:

$${{\Sigma }}({{{\bf{k}}}})={\tau }^{x}u({{{\bf{k}}}}){{\Delta }}({{{\bf{k}}}})(i{\sigma }^{y}\delta [{\alpha }_{{{{\bf{k}}}}}]){u}^{T}({{{\bf{-k}}}}),$$
(8)

written in orbital basis. Here, δ[αk] is a 3 × 3 matrix in band space, with unity at the (αk, αk) element and zero elsewhere for state k on band αk, and u(k) is the unitary transformation that diagonalize the normal state Hamiltonian. Hence by taking the one-loop approximation to the electronic Green’s function:

$${{{\mathcal{G}}}}({{{\bf{k}}}},i{\nu }_{m})={{{{\mathcal{G}}}}}_{0}({{{\bf{k}}}},i{\nu }_{m})+{{{{\mathcal{G}}}}}_{0}({{{\bf{k}}}},i{\nu }_{m}){{\Sigma }}({{{\bf{k}}}},i{\nu }_{m}){{{{\mathcal{G}}}}}_{0}({{{\bf{k}}}},i{\nu }_{m}),$$
(9)

where the \({{{{\mathcal{G}}}}}_{0}^{-1}({{{\bf{k}}}},i{\nu }_{m})=\frac{1}{2}(i{\nu }_{m}-{\tau }^{z}{h}_{{{{\bf{k}}}}})\) is the bare electron Green’s function (with hk being the 3 × 3 tight-binding Hamiltonian of Eq. (2) in the orbital basis), the Dyson’s equation of Eq. (5) can be readily reduced to the linearized gap equation of the form \({{\Delta }}({{{\bf{k}}}})={\sum }_{{{{\bf{k}}}}^{\prime} }V({{{\bf{k}}}},{{{\bf{k}}}}^{\prime} ){{\Delta }}({{{\bf{k}}}}^{\prime} )\) whose eigenvalues represent the dimensionless effective BCS pairing interactions for the pairing channels satisfying Eq. (8); details are given in “Methods”.

The effective BCS interaction of the above three-band model derived from this procedure plotted in Fig. 2 with comparison with the single-band estimation of Eq. (1), demonstrates that the superconducting dome arises also with the unconventional, i.e. odd-parity, electron-phonon coupling of Eq. (4). While the optimal doping (and therefore chemical potential) value may be shifted, the suppression of superconductivity on the low density dome edge by the vanishing density of states and on the high density dome edge by the TO1 phonon hardening still remains. This remains qualitatively true as long as there is any non-zero screening effect on Eq. (4) that attenuates g at sufficiently high density (For 〈ϕq=0〉 ≠ 0, Eq. (4) gives us the inversion symmetry breaking electron hopping, one of the key ingredients for the Rashba effect44,53. The reduction of the Rashba coefficient with the increasing carrier concentration found in the recent first-principle calculation for Bi2WO6, a related material55, and the experiment on the few-layer GeTe56 are possible instances of screening attenuating such electron hopping and by extension, the parameter g in Eq. (4), the simplest modeling of which is the density-independent gkF we have used for obtaining Fig. 2.

Fig. 2: BCS eigenvalue λBCS.
figure 2

Comparison between the BCS eigenvalue λBCS dome in the three-orbital model of Eq. (2) and the single-band model of Eq. (1). In the former, the weak tetragonal crystal field strength of ~2.2 meV15 has been included, which has little effect around the optimal doping. For a better comparison on the superconducting dome, both lines are rescaled with respect to their maximal value. λ/λmax is the rescaled BCS eigenvalue. Red line is the result from single-band estimation, while blue line is the result from three-band calculation.

Full size image

Normal state considerations

We shall now show that the above phonon-mediated superconducting mechanism is consistent with T2 resistivity that has been observed for SrTiO3 at low doping57. This behavior has attracted attention because, given the small Fermi surface, it cannot be sufficiently explained by the electron-electron scattering process58,59,60. Our point here is that this behavior can be actually explained from the single-phonon electron-phonon scattering process, the imaginary part of the self-energy is given by the Fermi’s golden rule (see Supplementary Information for derivation):

$$\begin{array}{l}{{{\rm{Im}}}}{{\Sigma }}({{{\bf{k}}}},\epsilon )=-\pi \mathop{\sum}\limits_{{{{\bf{q}}}}}| {g}_{{{{\bf{k}}}}^{\prime} ,{{{\bf{k}}}}}{| }^{2}\left[\delta (\epsilon -{\xi }_{{{{\bf{k}}}}^{\prime} }+{\omega }_{{{{\bf{q}}}}})({n}_{F}({\xi }_{{{{\bf{k}}}}^{\prime} })+{n}_{B}({\omega }_{{{{\bf{q}}}}}))\right.\\ \qquad\qquad\qquad+\,\left.\delta (\epsilon -{\xi }_{{{{\bf{k}}}}^{\prime} }-{\omega }_{{{{\bf{q}}}}})({n}_{F}(-{\xi }_{{{{\bf{k}}}}^{\prime} })+{n}_{B}({\omega }_{{{{\bf{q}}}}}))\right],\end{array}$$
(10)

where ξk and ωq are the dispersions of electrons and phonons. \({g}_{{{{\bf{k}}}}^{\prime} ,{{{\bf{k}}}}}\) is the electron-phonon coupling strength (with \({{{\bf{k}}}}^{\prime} ={{{\bf{k}}}}+{{{\bf{q}}}}\)). nF and nB are the Fermi and Bose distributions. For the optical phonons, the Einstein model is sufficient to capture the qualitative behavior, i.e. ωq ≈ ω. Therefore, if we focus on electrons at the Fermi surface, the self-energy from scattering off a single branch of optical phonons would be:

$${{{\rm{Im}}}}{{\Sigma }}({{{\bf{k}}}},\epsilon =0)=-2\pi \left[({n}_{F}(\omega )+{n}_{B}(\omega ))\right]\mathop{\sum}\limits_{{{{\bf{q}}}}}| {g}_{{{{\bf{k}}}}^{\prime} ,{{{\bf{k}}}}}{| }^{2}\delta ({\xi }_{{{{\bf{k}}}}^{\prime} }-\omega ),$$
(11)

from which the relation between the scattering rate and the self-energy \(1/\tau =-2{{{\rm{Im}}}}{{\Sigma }}/\hslash\) gives us the scattering rate formula of the form:

$$1/\tau =A\left[({n}_{F}(\omega )+{n}_{B}(\omega ))\right].$$
(12)

The simplified scattering rate depends on the phonon energy ω, temperature T and a coefficient A, proportional to the magnitude square of the electron-phonon coupling strength. By itself, Eq. (12) cannot give rise to the T2 resistivity; the resistivity rather shows a T-linear behavior at high temperature T ≫ ω and an exponential suppression at low temperature T ≪ ω with a crossover regime for T ~ ℏω/kB.

The T2 resistivity can nevertheless arise from scattering by multiple branches of optical phonons at different frequencies. As charge carrier density increases, the energy of the TO1 phonon increases from 20 K to 100 K in the T2 resistivity measurement for Nb-doped STO57. Energies of other optical phonons are essentially doping-independent. Among them, the LO1 phonon has the lowest energy at ~200 K. Due to the strong electron-phonon coupling in the LO1 phonon channel4, its contribution to the scattering rate should not be neglected. We thus have the combined electron-phonon scattering rate:

$$\begin{array}{l}1/\tau ={A}_{T}\left[{n}_{F}({\omega }_{T},T)+{n}_{B}({\omega }_{T},T)\right]\\ \qquad+\,{A}_{L}\left[{n}_{F}({\omega }_{L},T)+{n}_{B}({\omega }_{L},T)\right]\end{array}$$
(13)

with AL ≫ AT. This leads to a broad crossover regime starting from TO1 phonon energy, up to LO1 phonon energy. This crossover regime could give an approximate T2 scattering rate, as shown in Fig. 3 for a doping at the superconducting dome. The temperature is much lower than the Fermi energy at this doping. At higher temperature, including electrons away from the Fermi surface may be needed for the computation of the scattering rate. Approximate T2 resistivity at other dopant concentrations can be found in the Supplementary Information. To summarize, we note that while other mechanisms are possible58, the measured phonon frequency values along with the calculations presented here make it impossible to rule out a scenario in which the coupling to both TO1 and LO1 phonon modes can produce the T2 resistivity at least over a range of temperatures (Fig. 3).

Fig. 3: T2 scattering rate.
figure 3

(Blue) Scattering rate as a function of T2. Contributions from TO1 (Red) and LO1 (yellow) phonons are included. Parameters ωT = 6 meV, ωL = 20 meV6, AT = 1 and AL = 15 are taken. Blue line is the sum of the red (TO1 contribution) and yellow (LO1 contribution) line.

Full size image

Discussion

In this paper, we have utilized experimental data to constrain and deduce the most likely pairing mechanism underlying Nb-doped SrTiO3. Such strategies can perhaps be of broader relevance to other materials such as PbTe and KTaO3 that are close to a ferroelectric transition. It would be of considerable interest to repeat such planar tunneling measurements in these systems. For example, the ideas presented here can help shed light on the recent observations of interfacial superconductivity in KTaO3, which shows a surprisingly high Tc of ~2 K, while showing no signs of bulk superconductivity at the present time. At an interface, the presence of Rashba spin-orbit coupling allows for the coupling to a single TO1 phonon as in Eq. (4). The effective strength of the phonon coupling is enhanced by bulk spin-orbit coupling, which may account for the enhancement of superconductivity at the interface of this system. Further experimental studies in similar materials may help uncover the global phase diagram of quantum paraelectrics as a function of spin-orbit strength, dopant concentration, and spatial dimensionality.

Methods

BCS eigenvalue calculation

The linearized gap equation:

$${{\Delta }}({{{\bf{k}}}})=\mathop{\sum}\limits_{{{{\bf{k}}}}^{\prime} }V({{{\bf{k}}}},{{{\bf{k}}}}^{\prime} ){{\Delta }}({{{\bf{k}}}}^{\prime} ),$$
(14)

from the Dyson’s equation of Eq. (5) need to have:

$$\begin{array}{lll}V({\bf{k}},{\bf{k}}^{\prime})&=&C{N}_{F}\mathop{\sum}\limits_{ij}{\chi }_{ij,0}({\bf{k}}^{\prime}-{\bf{k}}){\rm{Tr}}\left\{{M}^{\dagger}({\bf{k}}){u}^{\dagger}({\bf{k}}){F}_{j}({\bf{k}},{\bf{k}}^{\prime}-{\bf{k}})\right.\\\ &&\left.u({\bf{k}}^{\prime})M({\bf{k}}^{\prime} ){u}^{T}(-{\bf{k}}^{\prime}){F}_{i}({\bf{k}},{\bf{k}}^{\prime}-{\bf{k}}){u}^*({\bf{-k}})\right\},\end{array}$$
(15)

where M(k) ≡ iσyδ[αk], while the constant C is a function of energy cutoff and critical temperature, i.e.\(C\propto \log ({\omega }_{c}/{T}_{c})\). The eigenvalues of the linearized gap equation are exactly λBCS’s that determine Tc. Numerically, the above linearized gap equation can be treated as an eigenvalue equation of matrix \(V({{{\bf{k}}}},{{{\bf{k}}}}^{\prime} )\), in a vector space spanned by N momenta. Tc is determined by the largest eigenvalue, and the corresponding eigenvector (dominant pairing channel) is s-wave. The Fermi energy EF, the carrier density n and the TO1 phonon energy ωT are taken from the tunneling experiment6. We assume \({N}_{F},{k}_{F}\propto \sqrt{{E}_{F}}\), and cT = 0.

Scattering rate derivation

The formula for the scattering rate of electrons due to a single branch of phonons, as given in Eq. (12), can be derived from the one-loop electron self-energy. The generalized form of the electron-phonon coupling:

$${H}_{{{{\rm{e-ph}}}}}=\mathop{\sum}\limits_{{{{\bf{q}}}},\lambda }\mathop{\sum}\limits_{{{{\bf{k}}}}}{g}_{\lambda ;ss^{\prime} }({{{\bf{q}}}},{{{\bf{k}}}})({a}_{{{{\bf{q}}}},\lambda }+{a}_{-{{{\bf{q}}}},\lambda }^{{\dagger} }){c}_{{{{\bf{k}}}}+{{{\bf{q}}}},s}^{{\dagger} }{c}_{{{{\bf{k}}}},s^{\prime} },$$
(16)

where a(a†) is the phonon annihilation (creation) operator and λ the phonon polarization, can be taken as the starting point to obtain the one-loop electron self-energy:

$$\begin{array}{l}{{\Sigma }}({{{\bf{k}}}},i{\omega }_{n})=\frac{{k}_{B}T}{\hslash }\frac{1}{\hslash }\mathop{\sum}\limits_{{{{\bf{q}}}}}| {g}_{\lambda }({{{\bf{q}}}},{{{\bf{k}}}}){| }^{2}\mathop{\sum}\limits_{i{\nu }_{m}}\frac{1}{i{\omega }_{n}+i{\nu }_{m}-{\xi }_{{{{\bf{k}}}}+{{{\bf{q}}}}}}\frac{2{\omega }_{{{{\bf{q}}}},\lambda }}{{\nu }_{m}^{2}+{\omega }_{\lambda ;{{{\bf{q}}}}}^{2}}\\ \qquad\quad\quad\,\,=\,\frac{1}{\hslash }\mathop{\sum}\limits_{{{{\bf{q}}}}}| {\tilde{g}}_{\lambda }({{{\bf{q}}}},{{{\bf{k}}}}){| }^{2}\left[\frac{{n}_{B}({\omega }_{\lambda ;{{{\bf{q}}}}},T)+{n}_{F}({\xi }_{{{{\bf{k}}}}+{{{\bf{q}}}}},T)}{i{\omega }_{n}+{\omega }_{\lambda ;{{{\bf{q}}}}}-{\xi }_{{{{\bf{k}}}}+{{{\bf{q}}}}}}+\frac{{n}_{B}({\omega }_{\lambda ;{{{\bf{q}}}}},T)+{n}_{F}(-{\xi }_{{{{\bf{k}}}}+{{{\bf{q}}}}},T)}{i{\omega }_{n}-{\omega }_{\lambda ;{{{\bf{q}}}}}-{\xi }_{{{{\bf{k}}}}+{{{\bf{q}}}}}}\right],\end{array}$$
(17)

where ωq,λ is the phonon eigenfrequency, in the Matsubara frequency. One can see see how Eq. (10) can be obtained by taking the imaginary part of Eq. (17). Applying the Einstein model for phonons with ωq = ω0 for all q, the scattering rate of Eq. (12) is obtained the imaginary part of the electron self-energy:

$$\frac{1}{\tau }=-\frac{2}{\hslash }{{{\rm{Im}}}}{{\Sigma }}(\omega +i\delta \to 0),$$
(18)

with:

$$A=\frac{2\pi }{{\hslash }^{2}}\mathop{\sum}\limits_{{{{\bf{k}}}}^{\prime} }| \tilde{g}({{{\bf{k}}}}^{\prime} -{{{\bf{k}}}},{{{\bf{k}}}}){| }^{2}[\delta ({\xi }_{{{{\bf{k}}}}^{\prime} }/\hslash -{\omega }_{0})+\delta ({\xi }_{{{{\bf{k}}}}^{\prime} }/\hslash +{\omega }_{0})].$$
(19)

Note added to proof

While drafting this manuscript, we have learnt of the recent preprint by Gastiasoro et al.40 which which has some overlap with the ideas presented here. However, our motivation here is distinct in the use of recent tunneling experiments to constrain pairing mechanisms.