Abstract
Recent experiments on Nbdoped SrTiO_{3} have shown that the superconducting energy gap to the transition temperature ratio maintains the Bardeen–Cooper–Schrieffer (BCS) value throughout its superconducting dome. Motivated by these and related studies, we show that the Cooper pairing mediated by a single soft transverseoptical phonon is the most natural mechanism for such a superconducting dome given experimental constraints, and present the microscopic theory for this pairing mechanism. Furthermore, we show that this mechanism is consistent with the T^{2} resistivity in the normal state. Lastly, we discuss what physical insights SrTiO_{3} provides for superconductivity in other quantum paraelectrics such as KTaO_{3}.
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 (SrTiO_{3})^{1,2,3,4,5,6} as well as potassium tantalate (KTaO_{3})^{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. noncentrosymmetric, or spatial reflection symmetries^{12}; dipole moments while permitted by symmetry, are strongly screened by the conduction electrons.). The fact that superconductivity, at least in niobiumdoped 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 fluctuations^{13}.
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 transverseoptical (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 spindependent processes, electrons can only scatter off of pairs of TO phonons^{14,15,16}. The resulting reduction in phase space would naturally result in reduction of the superconducting transition temperature T_{c}. 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 “antiadiabatic” pairing regime. Whether superconducting domes can arise in quantum paraelectrics despite these circumstances remains actively debated^{17,18,19,20,21}. Moreover, the observation of superconductivity at interfaces of quantum paraelectrics^{7,8,9,10,22,23} 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 selfconsistent pairing scenario for quantum paraelectrics, and illustrate it in the case of SrTiO_{3}, where recent experiments have placed significant constraints on theory. These experiments have reported a textbook BCS gap to T_{c} ratio^{4,5,6}; 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 antiadiabatic 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 SrTiO_{3}: a conventional one in which the phonon frequency remains smaller than the Fermi energy E_{F}^{24,25,26,27}, and an antiadiabatic mechanism in which the hierarchy of energy scales are inverted^{28,29,30,31,32,33,34,35}. In recent experiments^{4,6}, 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 dome^{6,36,37}, (2) the LO phonon frequencies remain unchanged with doping and are either comparable to or greater than E_{F} across the dome^{38}, and (3) the superconducting gap to T_{c} ratio is close to the BCS value^{4,5,6}.
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 antiadiabatic regime in SrTiO_{3}. We first briefly describe why the antiadiabatic scenarios are unlikely in SrTiO_{3} and we then consider the adiabatic pairing scenario mediated by TO phonons.
Based on the tunneling measurements^{4,5,6}, any antiadiabatic 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 T_{c} ratio requires the effective attraction mediated by the LO phonon to be weak, which can occur in the antiadiabatic 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 electronphonon coupling, a crucial ingredient needed for T_{c} to decrease beyond optimal doping is the reduction of the electronphonon 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 SrTi_{1−x}Nb_{x}O_{3} is mediated by an antiadiabatic LO4 phonon.
We are thus led naturally to consider an adiabatic pairing mechanism across the dome of SrTi_{1−x}Nb_{x}O_{3}. 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 T_{c} ratio within this scenario (The lower density dome that is observed in oxygenreduced samples^{3} 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 experiments^{36,37} indicate that the TO1 phonon frequency increases with carrier concentration as \({\omega }_{T}^{2}={K}_{0}+n{K}_{1}\) with the approximate values of K_{0} ≈ 1 meV^{2} and K_{1} ≈ 1.8 × 10^{−19} meV^{2} cm^{3} > 0^{15}. Hence the BCS eigenvalue for the adiabatic pairing mediated solely by a single TO1 phonon is parametrically:
the overdoped attenuation of T_{c} 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 phonons^{39} (for discussion on superconductivity arising from such electronphonon 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 viceversa^{40}. 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 SrTiO_{3} electronic band structure that incorporates the titanium (Ti) 3dt_{2g} orbitals while assuming the simple cubic lattice structure. The lowenergy band structure can be described well by a minimal tightbinding model whose kspace representation can be written as refs. ^{15,41,42,43}:
where α, β = X, Y, Z refer, respectively, to the Ti d_{yz}, d_{xz}, d_{xy} orbitals, \(s,s^{\prime}\) the spin indices, ξ = 19.3 meV from the Ti atomic spinorbit 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 t_{2g} orbitals, and:
is the intraorbital hopping (with ϵ_{0} = 12.2 meV) whose t_{1} > t_{2} anisotropy (0.615 and 0.035 eV, respectively) can be attributed to the quantum mechanical effect of the Ti t_{2g} orbital symmetry^{15,42}.
The form of the electronic coupling to the TO1 phonons is determined by the interplay between the t_{2g} orbital symmetry and the crystalline structure. As shown in Fig. 1, the tunneling between different t_{2g} orbitals between nearest neighbors is forbidden by inversion symmetry in a static lattice, but the TO1 mode displacements break inversion symmetry and thereby induce oddparity interorbital tunneling. Given its oddparity, this tunneling at the longwavelength limit can be described by the following electronphonon interaction^{44,45}:
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 TiO_{6} octahedron along the zdirection by a constant amount; this will turn on the nearestneighbor hopping between the d_{xy} and d_{yz} (d_{xz}) in the x(y)direction through the O p_{y} (p_{x}) orbital. The following two aspects of this electronphonon coupling makes it viable as the pairing glue for superconductivity.
First, the electronphonon coupling of Eq. (4) is distinct from acoustic phonons coupling derivatively to the fermions. As long as there is a nonzero fermion density, the typical fermion momentum ∣k∣ ~ k_{F} is finite, and the electronphonon coupling in Eq. (4) survives even in the q → 0 limit. In the interest of simplicity, we consider the case where gk_{F} is independent of density; what this implies will be discussed upon obtaining the effective BCS interaction.
Second, the electronphonon coupling of Eq. (4) can induce an intraband pairing interaction due to the presence of atomic spinorbit coupling in Eq. (2)^{46,47,48}. This is not limited to the three C_{3} rotational axes of the cubic lattice, where the eigenstates of the H_{0} 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 \(\frac{2}{3}{{{\boldsymbol{\phi }}}}\cdot ({{{\bf{j}}}}\times {{{\bf{k}}}})\), respectively^{49,50}). Even away from this band degeneracy, the intraband Rashba coupling of the TO1 phonon^{27,51,52} can be obtained by treating the sum of Eq. (4) and the atomic spinorbit coupling of Eq. (2) as perturbations^{48,53}.
We now derive the dimensionless effective BCS pairing interaction from this electronphonon coupling using the Dyson’s equation in the Nambu space where the electron selfenergy arises entirely from the Cooper pairing. For this Dyson’s equation:
where ν_{m} ≡ (2m + 1)πk_{B}T/ℏ (with \(m\in {\mathbb{Z}}\)) is the fermionic Matsubara frequency, \({{{\mathcal{G}}}}\) the electronic Green’s function and:
is the electronphonon interaction vertex from Eq. (4), with g ∝ n^{−1/3} to maintain an effective electronphonon coupling strength independent of the carrier density n. We take advantage of the adiabaticity of the TO1 phonon to ignore the boson dynamics^{27} and take the static TO1 propagator:
where M_{T} is the TO1 phonon effective mass. Given that the selfenergy 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 electronphonon coupling of Eq. (4). With regards to the Cooper pair spin states, we note that any electronphonon coupling, even with oddparity, favors spinsinglet pairing^{46,47,52,54}. Therefore our pairing gap should be intraband, evenparity, pseudospinsinglet (frequencyindependence being already assumed by Eq. (5)), giving us:
written in orbital basis. Here, δ[α_{k}] is a 3 × 3 matrix in band space, with unity at the (α_{k}, α_{k}) element and zero elsewhere for state k on band α_{k}, and u(k) is the unitary transformation that diagonalize the normal state Hamiltonian. Hence by taking the oneloop approximation to the electronic Green’s function:
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 h_{k} being the 3 × 3 tightbinding 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 threeband model derived from this procedure plotted in Fig. 2 with comparison with the singleband estimation of Eq. (1), demonstrates that the superconducting dome arises also with the unconventional, i.e. oddparity, electronphonon 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 nonzero screening effect on Eq. (4) that attenuates g at sufficiently high density (For 〈ϕ_{q=0}〉 ≠ 0, Eq. (4) gives us the inversion symmetry breaking electron hopping, one of the key ingredients for the Rashba effect^{44,53}. The reduction of the Rashba coefficient with the increasing carrier concentration found in the recent firstprinciple calculation for Bi_{2}WO_{6}, a related material^{55}, and the experiment on the fewlayer GeTe^{56} 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 densityindependent gk_{F} we have used for obtaining Fig. 2.
Normal state considerations
We shall now show that the above phononmediated superconducting mechanism is consistent with T^{2} resistivity that has been observed for SrTiO_{3} at low doping^{57}. This behavior has attracted attention because, given the small Fermi surface, it cannot be sufficiently explained by the electronelectron scattering process^{58,59,60}. Our point here is that this behavior can be actually explained from the singlephonon electronphonon scattering process, the imaginary part of the selfenergy is given by the Fermi’s golden rule (see Supplementary Information for derivation):
where ξ_{k} and ω_{q} are the dispersions of electrons and phonons. \({g}_{{{{\bf{k}}}}^{\prime} ,{{{\bf{k}}}}}\) is the electronphonon coupling strength (with \({{{\bf{k}}}}^{\prime} ={{{\bf{k}}}}+{{{\bf{q}}}}\)). n_{F} and n_{B} are the Fermi and Bose distributions. For the optical phonons, the Einstein model is sufficient to capture the qualitative behavior, i.e. ω_{q} ≈ ω. Therefore, if we focus on electrons at the Fermi surface, the selfenergy from scattering off a single branch of optical phonons would be:
from which the relation between the scattering rate and the selfenergy \(1/\tau =2{{{\rm{Im}}}}{{\Sigma }}/\hslash\) gives us the scattering rate formula of the form:
The simplified scattering rate depends on the phonon energy ω, temperature T and a coefficient A, proportional to the magnitude square of the electronphonon coupling strength. By itself, Eq. (12) cannot give rise to the T^{2} resistivity; the resistivity rather shows a Tlinear behavior at high temperature T ≫ ω and an exponential suppression at low temperature T ≪ ω with a crossover regime for T ~ ℏω/k_{B}.
The T^{2} 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 T^{2} resistivity measurement for Nbdoped STO^{57}. Energies of other optical phonons are essentially dopingindependent. Among them, the LO1 phonon has the lowest energy at ~200 K. Due to the strong electronphonon coupling in the LO1 phonon channel^{4}, its contribution to the scattering rate should not be neglected. We thus have the combined electronphonon scattering rate:
with A_{L} ≫ A_{T}. This leads to a broad crossover regime starting from TO1 phonon energy, up to LO1 phonon energy. This crossover regime could give an approximate T^{2} 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 T^{2} resistivity at other dopant concentrations can be found in the Supplementary Information. To summarize, we note that while other mechanisms are possible^{58}, 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 T^{2} resistivity at least over a range of temperatures (Fig. 3).
Discussion
In this paper, we have utilized experimental data to constrain and deduce the most likely pairing mechanism underlying Nbdoped SrTiO_{3}. Such strategies can perhaps be of broader relevance to other materials such as PbTe and KTaO_{3} 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 KTaO_{3}, which shows a surprisingly high T_{c} of ~2 K, while showing no signs of bulk superconductivity at the present time. At an interface, the presence of Rashba spinorbit 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 spinorbit 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 spinorbit strength, dopant concentration, and spatial dimensionality.
Methods
BCS eigenvalue calculation
The linearized gap equation:
from the Dyson’s equation of Eq. (5) need to have:
where M(k) ≡ iσ^{y}δ[α_{k}], 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 λ_{BCS}’s that determine T_{c}. 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. T_{c} is determined by the largest eigenvalue, and the corresponding eigenvector (dominant pairing channel) is swave. The Fermi energy E_{F}, the carrier density n and the TO1 phonon energy ω_{T} are taken from the tunneling experiment^{6}. We assume \({N}_{F},{k}_{F}\propto \sqrt{{E}_{F}}\), and c_{T} = 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 oneloop electron selfenergy. The generalized form of the electronphonon coupling:
where a(a^{†}) is the phonon annihilation (creation) operator and λ the phonon polarization, can be taken as the starting point to obtain the oneloop electron selfenergy:
where ω_{q,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 ω_{q} = ω_{0} for all q, the scattering rate of Eq. (12) is obtained the imaginary part of the electron selfenergy:
with:
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.
Data availability
Relevant data in this paper are available upon reasonable request.
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Acknowledgements
We thank Piers Coleman, Rafael Fernandes, Changyoung Kim, Dmitrii Maslov, and Hyeok Yoon for useful discussions. Y.Y., H.Y.H., and S.R. were supported by the Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering, under contract No. DEAC0276SF00515. S.B.C. was supported by the National Research Foundation of Korea (NRF) grants funded by the Korea government (MSIT) (2020R1A2C1007554) and the Ministry of Education (2018R1A6A1A06024977).
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Yu, Y., Hwang, H.Y., Raghu, S. et al. Theory of superconductivity in doped quantum paraelectrics. npj Quantum Mater. 7, 63 (2022). https://doi.org/10.1038/s41535022004662
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DOI: https://doi.org/10.1038/s41535022004662
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