Multiband full-bandwidth anisotropic Eliashberg theory of interfacial electron-phonon coupling and high-T$_c$ superconductivity in FeSe/SrTiO$_3$
Alex Aperis Peter M. Oppeneer




alex.aperis@physics.uu.se
peter.oppeneer@physics.uu.se
Department of Physics and Astronomy, Uppsala University, P. O. Box 516, SE-75120 Uppsala, Sweden

0.4cm



We examine the impact of interfacial phonons on the superconducting state of FeSe/SrTiO_3 developing a materials' specific multiband, full bandwidth, anisotropic Eliashberg theory for this system.  
Our selfconsistent calculations highlight the importance of the interfacial electron-phonon interaction, which is hidden behind the seemingly weak coupling constant λ_m=0.4, in mediating the high-T_ c, and explain other puzzling experimental observations like the s-wave symmetry and replica bands. We discover that the formation of replica bands has a T_ c decreasing effect that is nevertheless compensated by deep Fermi-sea Cooper pairing which has a T_ c enhancing effect. We predict a strong coupling dip-hump signature in the tunneling spectra due to the interfacial coupling.



Multiband, full bandwidth anisotropic Eliashberg theory of interfacial electron-phonon coupling and high-T_ c superconductivity in FeSe/SrTiO_3
    Peter M. Oppeneer
    December 30, 2023
===============================================================================================================================================



Superconductivity in monolayer-thick FeSe on SrTiO_3 reaches amazingly high transition temperatures of typically T_ c=50–70 K <cit.> and up to 100 K <cit.>, much higher than the 8-K value of bulk FeSe <cit.>. 
A coupling between SrTiO_3 phonons and FeSe electrons occurs at the FeSe/SrTiO_3 interface, which manifests itself as electron replica bands <cit.>. The value of this coupling is estimated by experiments to be around 0.4, thus it is commonly believed to moderately enhance T_ c but not be enough to explain it <cit.>.

The superconducting state in iron-based superconductors is customarily associated  with residual spin fluctuations due to the remnant quasi-nesting between electron and hole Fermi sheets that give rise to a sign alternating gap <cit.>. 
However, for FeSe/STO the situation is markedly different. 
Charge transfer at the interface induces electron doping in FeSe <cit.>, leading to a distinct Fermi surface consisting of only two electron sheets around the corners of the tetragonal Brillouin zone (M point) <cit.>. 
The observed anisotropic superconducting gap has a more conventional form with plain s-wave symmetry <cit.> and is thus nodeless in the entire Brillouin zone.
Furthermore, angular resolved photoemission spectroscopy (ARPES) measurements <cit.> reveal an interface-induced electron-phonon interaction (EPI) between FeSe electrons and polar STO phonons that is strongly peaked at the q=0 phonon wavevector <cit.>.
There is growing experimental evidence for the pivotal role of such interfacial phonons in 
engineering high-T_ c heterostructures that involve FeSe <cit.> or even FeAs <cit.> monolayers.

Although ab initio calculations confirm the existence of small-q phonons as a strictly interfacial phenomenon in FeSe/STO <cit.> and indicate the importance of the coupling between substrate phonons and FeSe electrons <cit.>, the estimated low value of the electron-phonon coupling constant (λ≤ 0.4) has been widely considered insufficient to explain the impressive T_ c enhancement 
unless another, dominant pairing mechanism is at play <cit.>. On the other hand, Eliashberg calculations within a single band model suggest that interfacial phonons may lead to the high T_ c with a coupling of merely half of that estimated by experiments <cit.>.
However, a materials' specific  theory of superconductivity
that can account for the interplay between multiple bands, doping and small-q phonons has not yet been developed.
It remains therefore unsolved to what extend and how such phonons contribute to the peculiar superconductivity in FeSe/STO.




Here, we present the first anisotropic, full bandwidth multiband Eliashberg calculations dedicated to unveil the influence of the interfacial electron-phonon coupling in FeSe/STO.
 Our theory extends on previous single band approaches <cit.> by establishing a microscopic description of 
superconductivity in this system on a materials' specific level, thus paving the way toward more realistic calculations of higher accuracy. Our selfconsistent results provide unambiguous support for the dominant contribution of these phonons to the high T_ c and to further enigmatic experimental observations, and allows us to shed light on novel aspects of the mechanism responsible for the high-T_ c. Remarkably, bands not crossing the Fermi level provide an additional Cooper pairing channel that enhances T_ c and places the value of the gap over T_ c ratio in the strong coupling regime. 
In stark contrast to previous proposals, our here predicted deep Fermi sea Cooper pairing does not depend upon interband scattering processes mediated by bosons at large wavevectors, like e.g. antiferromagnetic spin fluctuations <cit.> or the conventional short-ranged in real space EPI <cit.>.

In the presence of a strong inhomogeneous dielectric background, the EPI develops a pronounced forward scattering peak <cit.>. Here, the interface-induced EPI is modelled by a dispersionless mode at ħΩ=81 meV <cit.> coupled to FeSe electrons via the functional form g( q)=g_0 exp(-| q|/q_c) with q_c=0.3a^-1 <cit.>, with a the FeSe lattice constant.
 This is the only mediator of superconductivity in our theory. In what follows, we do not take into account explicitly the effect of Coulomb repulsion on superconductivity, the implications of which are discussed further below. For the electron dispersions of monolayer FeSe we use a recently derived ten-band tight-binding bandstructure <cit.>. Since the  FeSe doping level is not a priori known, the electron filling is chosen such that the bottom of the electron bands around the M point in the Brillouin zone are at -50 meV as observed in experiment <cit.>. 
 We determine the value of the electron-phonon scattering strength g_0, by requiring that the ARPES replica bands are reproduced at their observed energies <cit.>. In this way, we circumvent the need for treating screening effects at the FeSe/STO interface explicitly <cit.> and our determined value for the EPI strength may considered as the overall strength of the resulting effective EPI. We obtain g_0=728 meV, which is significantly close to the ab initio calculated value for anatase TiO_2 <cit.>, but a bit lower. This discrepancy may be understood as due to an enhanced screening effect at the FeSe/STO interface <cit.>. We solve the three coupled Migdal-Eliashberg equations for  Δ( k, ω), Z( k,ω), and χ( k,ω), describing the superconductivity order parameter, electron mass and chemical potential renormalizations, respectively, selfconsistently with full bandwidth, momentum and energy dependence, while taking care to keep the electron occupancy n fixed throughout the calculations <cit.>. The latter quantity measures the electron filling (the case of half-filling corresponds to n=1). We find  n ≈0.8, indicating that the system is in the electron-doped regime. We also note that when forward-scattering processes dominate the EPI, the Migdal theorem holds even in non-adiabatic cases <cit.>.

 

Figure <ref>(a) shows the calculated electron spectral function at T=10 K for the whole energy bandwidth and momenta along the M-Γ -M high-symmetry line of the folded Brillouin zone,
which is measured in ARPES experiments. Comparison of the spectral function with the bare tight-binding band structure  <cit.> used as input in our calculations (shown by white dashed lines) reveals that the interfacial small-q phonon modifies the FeSe electrons in a manifest way.
Several shake-off effects in the band structure take place over the whole bandwidth, including the appearance of new bands near -160 meV. The opening of a superconducting gap around the Fermi level can be seen.
Most of the predicted shake-off effects in the band structure have not yet been observed by experiment <cit.>. However, the replica bands appearing near the M point (zoom-in shown in Fig. <ref>(b) at ∼110 meV distance from the main electron bands that form the Fermi surface have been experimentally resolved <cit.>. Figure <ref>(c) highlights that not only the position of the peaks but also the peak ratio, A_2/A_1=0.17, agrees well with experiment <cit.>.
Notably, within solely phononic small-q theory we obtain the superconducting T_ c=61 K, in good agreement with experiment  <cit.>, with a temperature dependence of the gap edge as shown in Fig. <ref>(d).  

The momentum dependence of the calculated superconducting gap Δ ( k_ F) is shown in Fig. <ref>(a). It has s-wave symmetry and is moderately anisotropic (∼25%) with gap values that vary from 8-11 meV over the Fermi surface with an average value of 10 meV. These values are in agreement with experiments <cit.> although the location of the gap maxima seems to somewhat deviate from those experimentally measured (e.g. <cit.>). The resulting anisotropy of the gap is a consequence of the small-q form of the interfacial EPI (cf. <cit.>). Taking the maximum required excitation energy at the gap edge to calculate the gap over T_ c ratio, we obtain the strong coupling (non-BCS) value Δ/k_ B T_ c=2.1 (in contrast to the BCS value Δ / k_ B T_ c=1.76).
The chemical potential renormalization χ( k_ F), shown in Fig. <ref>(b), has an anisotropic momentum dependence with an average Fermi surface value of ⟨χ( k_ F)⟩=5.9 meV. The fact that χ( k_ F) even changes sign at certain Fermi surface points indicates the highly non-trivial role this quantity plays in shaping the quasiparticle band structure of the monolayer. In contrast, the  mass renormalization function Z( k_ F), shown in Figs. <ref>(c), (d), is rather isotropic with an average Fermi surface value ⟨ Z( k_ F)⟩=1.37 and 1.40 for temperature below and above T_ c, respectively. This quantity is related to the electron-phonon coupling constant λ_m, by ⟨ Z ( k) ⟩ _ k_ F |_ T>T_c=1+λ_m which in our case yields λ_m= 0.4. This weak coupling value matches remarkably well to experiments <cit.>.
 Also, in the superconducting state λ^ 10 K_m≈0.37 
and satisfies A_2/A_1≈λ^ 10 K_m/2 (Fig. <ref>(c))  <cit.>.

 A plain calculation of T_ c with our obtained value of λ_m=0.4 in McMillan's formula gives T_ c=17 K. On the other hand, using our numerical results in the two T_ c formulas recently proposed for interfacial phonon-mediated superconductivity in FeSe/STO <cit.>, yields T_ c=272–283 K and 117.5 K, respectively (note that the T_ c equation in <cit.> is derived in the q_c→ 0 limit). These estimations are in stark contrast to the T_ c=61 K obtained here by our selfconsistent Eliashberg theory, which thus  resolves the controversy between a seemingly weak λ_m  and high-T_ c superconductivity in FeSe/STO.


To elucidate further the mechanism of T_ c enhancement in FeSe/STO, we carried out a series of simulations where we first solve the usual momentum-dependent Eliashberg equations for electrons only at the Fermi surface of FeSe/STO and then perform full-bandwidth calculations while sequentially adding more bands until we recover the full bandwidth multiband calculation. Our findings are summarized in Fig. <ref>(a). In the first case (blue symbol in Fig. <ref>(a) where no electronic spectral rearrangement is allowed and thus no replica bands can form, we find λ_m=0.63. We note that this value equals the one given by the standard formula, λ=⟨λ_ q⟩_ k_ F^, k^'_ F=0.63 (with λ_ q  the momentum-dependent electron-phonon coupling <cit.>). Therefore, the obtained T_ c=60.6 K is the maximum T_ c reachable by Fermi-surface Cooper pairing.
Inclusion of the complete contribution of the two bands that form the Fermi surface (purple square in Fig. <ref>(a) leads to λ_m=0.4 and T_ c=56.8 K. Compared with the previous case,  here a part of the electron-phonon coupling strength is consumed in mediating the electronic spectral weight transfer from the bands crossing the Fermi level to the replica bands. The effective interaction left available for Cooper-pair mediation is weaker and concomitantly so is the T_ c. This weak coupling picture is further witnessed by the near-BCS value of the calculated ratio Δ/k_ B T_c=1.8. Remarkably, turning on contributions from near-Fermi-level bands (red symbols in Fig. <ref>(a) gradually increases T_ c to 61.2 K (green square) but without affecting the value of λ_m. This behavior indicates that these bands contribute to superconductivity.

To quantify this remarkable finding, we projected the superconducting gap function Δ( k,E) on the different electronic bands of FeSe/STO as shown in Fig. <ref>(b). Since Δ( k,E) is a measure of the Cooper-pair binding energy,  it is positive for Cooper pairing with an s-wave gap  and negative otherwise <cit.>. Figure <ref>(b) clearly shows that superconductivity in FeSe/STO stems not only from Fermi surface regions around M, but also from regions around the Γ-X and X-M directions of the Brillouin zone. Although in the latter regions Δ( k,E) is in the μeV range (Fig. <ref>(c)), the resulting net contribution is enough to overcompensate the T_ c decreasing effect of the replica band formation. It is also enough to raise the Δ/k_ B T_c ratio to 2.1 and thus provide a strong coupling phenomenology.

We emphasize that our predicted deep Fermi sea Cooper pairing is markedly different from previous suggestions of pairing through incipient bands <cit.>. Here, the mediating interaction is not only phonon-driven but more importantly, it is local in momentum space due to its small-q shape, thus it relies explicitly on intraband processes.
 However, by the nature of our full bandwidth Eliashberg theory, different bands are coupled implicitly via the frequency sector by the interfacial EPI, due to the large characteristic energy scale of the latter, in some sense reminiscent of the incipient band scenario.
  Our findings thus generalize the usual picture where Cooper pairing relies on near Fermi surface electrons and prove that deep Fermi-sea Cooper pairing is possible in multiband systems <cit.>. The recent puzzling superconductivity observed in doped LiFeAs without a Fermi surface  <cit.> is plausibly explained within our picture.

Scanning tunneling spectroscopy (STS) measurements represent another key experimental feature reported for FeSe/STO <cit.>.  To compare to STS data we have calculated the differential conductance spectrum dI/dV which, at low temperatures, is proportional to the superconducting density of states. 
The tunneling spectrum, calculated at T=10 K, is shown in Fig. <ref>.
 Zooming-in to the low energy regime (Fig. <ref>(b)) reveals the opening of an s-wave superconducting gap in the tunneling spectra that starts to close already around ±5 meV and exhibits main coherence peaks at ±11 meV, with secondary peaks a few meV's higher.
  The calculated spectrum is in excellent agreement with STS measurements <cit.>. The position of the main coherence peaks in Fig. <ref>(b) coincides with the maximum of the gap-edge on the Fermi surface whereas the closing of the gap beginning at 5 meV is consistent with the minimum gap-edge of 8 meV (see Fig. <ref>(a)) in combination with the finite broadening of ∼3 meV used in our calculations.


Figure <ref>(a) shows the calculated tunneling spectra, normalized to the normal state values, at an intermediate energy range.
Remarkably, we find superconductivity related structures in the spectrum up to energies almost 30 times higher than the superconducting gap itself. Although it is well established that such non-BCS behavior is the hallmark of strong-coupling superconductivity <cit.>, this is unexpected here given the seemingly weak coupling constant of the system.
In strongly coupled superconductors, the structure of the spectral function of the mediating bosons can be visible in the tunneling spectrum <cit.>. Here, we predict that the interfacial phonon mode should manifest itself as two kinks around ±91 meV (Fig. <ref>(a)), whose location coincides with ±(Ω+Δ) where Δ is the average gap-edge value. Furthermore, at higher energies we predict a distinct dip-hump structure in the spectra with a dip at 150 meV and a hump at 220 meV (Figs. <ref>(a),(c)).
Analyzing this additional strong coupling feature we find that it originates from the competition between the real and imaginary components of the superconducting gap function at an energy scale that is larger than the characteristic boson frequencies <cit.>.
The energy location of the dip and the hump depends on the coupling strength <cit.>, and more specifically, on the Δ/k_ B T_c ratio <cit.>.
For comparison, within isotropic Eliashberg theory assuming an Einstein phonon at ħΩ=81 meV, 
we find that our predicted dip-hump spectrum in FeSe/STO can only be fitted when the obtained gap over T_ c ratio matches the one in FeSe/STO Δ/k_ B T_c=2.1 but with a strong-coupling isotropic value λ _iso=1.6 (Fig. <ref>(c)).

 The very good quantitative agreement between experiment and our selfconsistent calculations for FeSe/STO provides a consistent picture where the interfacial phonons drive the superconductivity.
 For that picture to be complete one needs to incorporate the pair-breaking effect of the Coulomb interaction on the T_ c. Inclusion of the latter effect into the full bandwidth Eliashberg calculations on an equal footing with the EPI requires knowledge of the frequency dependent renormalization of the Coulomb interaction throughout the system's bandwidth and is out of the scope of the present work. By approximating the Coulomb repulsion through the pseudopotential term μ^* <cit.>, we estimate that for μ^*= 0.1 and 0.14, T_ c=33 K and 26 K, respectively. However, the presence of an additional low-energy attractive channel due to the intrinsic EPI in FeSe monolayer <cit.>, although not sufficient to mediate the high-T_ c on its own, can balance the T_ c decrease due to Coulomb repulsion. We find that inclusion of the intrinsic EPI of freestanding monolayer FeSe <cit.> leads to T_ c=57 K and 51 K for μ^*= 0.1 and 0.14, respectively.
 
In conclusion, our first of its kind full-bandwidth multiband theory shows that the interfacial EPI in FeSe/STO with a seemingly weak λ_m=0.4, explains key experimental facts like the replica bands, superconducting gap and tunneling spectra while also producing the correct T_ c in the absence of any significant Coulomb pair-breaking. Our explicit calculations unveil the T_ c increasing effect of deep Fermi-sea Cooper pairing and the T_ c decreasing effect of replica-band formation,
and suggest new pathways to engineer high T_ c's. A definite confirmation for the former effect will be the observation of a dip-hump feature in the tunneling spectra, which will also serve as an additional fingerprint of the decisive involvement of the interfacial EPI in mediating the high-T_ c.
On a fundamental level, our findings put to the question the current standard perception of the
efficiency of EPI in mediating high-T_ c superconductivity and, whether Fermi-surface restricted theory is sufficient to capture the superconductivity of other doped high-T_ c materials.

We thank G. Varelogiannis for fruitful discussions. This work has been supported by the Swedish Research Council (VR), the Röntgen-Ångström Cluster, and the Swedish National Infrastructure for Computing (SNIC).

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@bib@innerbibempty

[Qing-Yan et al.(2012)Qing-Yan, Zhi, Wen-Hao, Zuo-Cheng, Jin-Song, Wei, Hao, Yun-Bo, Peng, Kai,
  Jing, Can-Li, Ke,
  Jin-Feng, Shuai-Hua, Ya-Yu,
  Li-Li, Xi, Xu-Cun, and  Qi-Kun]Qing-Yan2012
  author author W. Qing-Yan, author L. Zhi,
  author Z. Wen-Hao, author Z. Zuo-Cheng, author
  Z. Jin-Song, author
  L. Wei, author D. Hao, author O. Yun-Bo, author D. Peng,
  author C. Kai, author
  W. Jing, author S. Can-Li, author H. Ke, author J. Jin-Feng, author J. Shuai-Hua,
  author W. Ya-Yu, author W. Li-Li, author
  C. Xi, author M. Xu-Cun,  and author X. Qi-Kun, http://stacks.iop.org/0256-307X/29/i=3/a=037402 journal
  journal Chin. Phys. Lett. volume 29,  pages 037402 (year 2012)NoStop
[Liu et al.(2012)Liu,
  Zhang, Mou, He, Ou, Wang, Li, Wang,
  Zhao, He, Peng, Liu, Chen, Yu, Liu,
  Dong, Zhang, Chen,
  Xu, Hu, Chen, Ma, Xue, and Zhou]Liu2012
  author author D. Liu, author W. Zhang, author D. Mou, author
  J. He, author Y.-B. Ou, author Q.-Y.  Wang, author Z. Li, author L. Wang, author L. Zhao, author
  S. He, author Y. Peng, author X. Liu, author C. Chen, author L. Yu, author
  G. Liu, author X. Dong, author J. Zhang, author C. Chen,
  author Z. Xu, author
  J. Hu, author X. Chen, author X. Ma, author Q. Xue,  and  author X. Zhou, http://dx.doi.org/10.1038/ncomms1946 journal journal Nat. Commun. volume 3, pages
  931 (year 2012)NoStop
[He et al.(2013)He,
  He, Zhang, Zhao,
  Liu, Liu, Mou, Ou, Wang, Li, Wang,
  Peng, Liu, Chen,
  Yu, Liu, Dong, Zhang, Chen, Xu, Chen,
  Ma, Xue, and Zhou]He2013
  author author S. He, author J. He, author W. Zhang, author
  L. Zhao, author D. Liu, author X. Liu, author D. Mou, author Y.-B. Ou, author
  Q.-Y. Wang, author
  Z. Li, author L. Wang, author Y. Peng, author Y. Liu, author C. Chen, author
  L. Yu, author G. Liu, author X. Dong, author J. Zhang,
  author C. Chen, author
  Z. Xu, author X. Chen, author X. Ma, author Q. Xue,  and  author X. J. Zhou, http://dx.doi.org/10.1038/nmat3648 journal journal Nat. Mater. volume 12, pages
  605 (year 2013)NoStop
[Tan et al.(2013)Tan,
  Zhang, Xia, Ye, Chen, Xie, Peng, Xu,
  Fan, Xu, Jiang, Zhang, Lai, Xiang, Hu,
  Xie, and Feng]Tan2013
  author author S. Tan, author Y. Zhang, author M. Xia, author
  Z. Ye, author F. Chen, author X. Xie, author R. Peng, author D. Xu, author
  Q. Fan, author H. Xu, author J. Jiang, author T. Zhang,
  author X. Lai, author
  T. Xiang, author J. Hu, author B. Xie,  and author D. Feng,  http://dx.doi.org/10.1038/nmat3654 journal journal Nat. Mater. volume 12, pages
  634 (year 2013)NoStop
[Peng et al.(2014)Peng,
  Xu, Tan, Cao, Xia, Shen, Huang, Wen,
  Song, Zhang, Xie,
  Gong, and Feng]Peng2014a
  author author R. Peng, author H. C. Xu,
  author S. Y. Tan, author H. Y. Cao, author
  M. Xia, author X. P. Shen, author Z. C.  Huang, author C. Wen, author Q. Song, author T. Zhang, author
  B. P. Xie, author X. G. Gong,  and author D. L. Feng, http://dx.doi.org/10.1038/ncomms6044
  journal journal Nat. Commun. volume 5, pages 5044 (year 2014)NoStop
[Lee et al.(2014)Lee,
  Schmitt, Moore, Johnston,
  Cui, Li, Yi, Liu, Hashimoto, Zhang, Lu,
  Devereaux, Lee, and Shen]Lee2014
  author author J. J. Lee, author F. T. Schmitt,
  author R. G. Moore, author S. Johnston, author
  Y.-T. Cui, author W. Li, author M. Yi, author Z. K. Liu,
  author M. Hashimoto, author Y. Zhang, author
  D. H. Lu, author T. P. Devereaux, author D.-H.  Lee,  and author Z.-X.  Shen, http://dx.doi.org/10.1038/nature13894
  journal journal Nature volume 515, pages 245 (year
  2014)NoStop
[Ge et al.(2015)Ge,
  Liu, Liu, Gao, Qian, Xue, Liu, and Jia]Ge2015
  author author J.-F. Ge, author Z.-L. Liu,
  author C. Liu, author
  C.-L. Gao, author D. Qian, author Q.-K.  Xue, author Y. Liu,  and author J.-F. Jia, http://dx.doi.org/10.1038/nmat4153 journal
  journal Nat. Mater. volume 14,  pages 285 (year 2015)NoStop
[Hsu et al.(2008)Hsu,
  Luo, Yeh, Chen, Huang, Wu, Lee, Huang,
  Chu, Yan, and Wu]Hsu2008
  author author F.-C. Hsu, author J.-Y. Luo,
  author K.-W. Yeh, author T.-K. Chen, author
  T.-W. Huang, author
  P. M. Wu, author Y.-C. Lee, author Y.-L.  Huang, author Y.-Y. Chu, author D.-C. Yan,  and  author M.-K. Wu, 10.1073/pnas.0807325105 journal journal
  Proc. Natl. Acad. Sci. USA volume 105, pages 14262 (year 2008)NoStop
[Ding et al.(2016)Ding,
  Lv, Zhao, Wang, Wang, Song, Chen, Ma, and  Xue]Ding2016
  author author H. Ding, author Y.-F. Lv,
  author K. Zhao, author
  W.-L. Wang, author
  L. Wang, author C.-L. Song, author X. Chen, author X.-C.  Ma,  and author Q.-K.  Xue, 10.1103/PhysRevLett.117.067001
  journal journal Phys. Rev. Lett. volume 117, pages 067001 (year
  2016)NoStop
[Tian et al.(2016)Tian,
  Zhang, Li, Wu, Wu, Sun, Zhou, Wang,
  Ma, Xue, and Zhao]Tian2016
  author author Y. C. Tian, author W. H. Zhang,
  author F. S. Li, author Y. L. Wu, author
  Q. Wu, author F. Sun, author G. Y.  Zhou, author L. Wang, author X. Ma, author Q.-K. Xue,  and author J. Zhao, 10.1103/PhysRevLett.116.107001 journal journal
  Phys. Rev. Lett. volume 116, pages
  107001 (year 2016)NoStop
[Mazin et al.(2008)Mazin,
  Singh, Johannes, and Du]Mazin2008
  author author I. I. Mazin, author D. J. Singh,
  author M. D. Johannes,  and  author M. H. Du, 10.1103/PhysRevLett.101.057003 journal journal Phys. Rev. Lett. volume 101, pages 057003 (year 2008)NoStop
[Li et al.(2014)Li,
  Xing, Huang, and Xing]Li2014
  author author B. Li, author Z. W. Xing,
  author G. Q. Huang,  and  author D. Y. Xing, http://dx.doi.org/10.1063/1.4876750 journal
  journal J. Appl. Phys. volume 115,  eid 193907 (year 2014)NoStop
[Fan et al.(2015)Fan,
  Zhang, Liu, Yan,
  Ren, Peng, Xu, Xie, Hu, Zhang, and Feng]Fan2015
  author author Q. Fan, author W. H. Zhang,
  author X. Liu, author
  Y. J. Yan, author M. Q. Ren, author R. Peng, author H. C.  Xu, author B. P. Xie, author J. P. Hu,
  author T. Zhang,  and author D. L. Feng, http://dx.doi.org/10.1038/nphys3450 journal journal Nat. Phys. volume 11, pages
  946 (year 2015)NoStop
[Aperis et al.(2011)Aperis,
  Kotetes, Varelogiannis, and Oppeneer]Aperis2011
  author author A. Aperis, author P. Kotetes,
  author G. Varelogiannis,  and  author P. M. Oppeneer, 10.1103/PhysRevB.83.092505 journal journal Phys. Rev. B volume 83, pages 092505 (year 2011)NoStop
[Weger and Peter(1999)]Weger1999
  author author M. Weger and author M. Peter,  http://dx.doi.org/10.1016/S0921-4534(99)00065-9 journal journal Physica C volume
  317, pages 252  (year 1999)NoStop
[Varelogiannis(1998)]Varelogiannis1998
  author author G. Varelogiannis, 10.1103/PhysRevB.57.13743 journal journal Phys. Rev. B volume
  57, pages 13743 (year 1998)NoStop
[Kulić and Zeyher(1994)]Kulifmmodecuteclseci1994
  author author M. L. Kulić and author R. Zeyher, 10.1103/PhysRevB.49.4395 journal
  journal Phys. Rev. B volume 49,  pages 4395 (year 1994)NoStop
[Abrikosov(1994)]Abrikosov1994
  author author A. A. Abrikosov, http://dx.doi.org/10.1016/0921-4534(94)90131-7 journal
  journal Physica C volume 222,  pages 191  (year 1994)NoStop
[Rebec et al.(2017)Rebec,
  Jia, Zhang, Hashimoto,
  Lu, Moore, and Shen]Rebec2017
  author author S. N. Rebec, author T. Jia, author C. Zhang, author
  M. Hashimoto, author
  D.-H. Lu, author R. G. Moore,  and author Z.-X. Shen, 10.1103/PhysRevLett.118.067002 journal journal
  Phys. Rev. Lett. volume 118, pages
  067002 (year 2017)NoStop
[Choi et al.(2017)Choi,
  Johnston, Jang, Koepernik,
  Nakatsukasa, Ok, Lee,
  Choi, Lee, Akbari,
  Semertzidis, Bang, Kim, and  Lee]Choi2017
  author author S. Choi, author S. Johnston,
  author W.-J. Jang, author K. Koepernik, author
  K. Nakatsukasa, author
  J. M. Ok, author H.-J. Lee, author H. W.  Choi, author A. T. Lee, author A. Akbari,
  author Y. K. Semertzidis,
  author Y. Bang, author
  J. S. Kim,  and author
  J. Lee, 10.1103/PhysRevLett.119.107003 journal journal
  Phys. Rev. Lett. volume 119, pages
  107003 (year 2017)NoStop
[Xie et al.(2015)Xie,
  Cao, Zhou, Chen,
  Xiang, and Gong]Xie2015
  author author Y. Xie, author H.-Y. Cao,
  author Y. Zhou, author
  S. Chen, author H. Xiang,  and author X.-G.  Gong, http://dx.doi.org/10.1038/srep10011
  journal journal Sci. Rep. volume 5, pages 10011 (year
  2015)NoStop
[Wang et al.(2016a)Wang, Linscheid,
  Berlijn, and Johnston]Wang2016b
  author author Y. Wang, author A. Linscheid,
  author T. Berlijn,  and  author S. Johnston, 10.1103/PhysRevB.93.134513 journal journal Phys. Rev. B volume 93, pages 134513 (year 2016a)NoStop
[Zhou and Millis(2016)]Zhou2016a
  author author Y. Zhou and author A. J. Millis, 10.1103/PhysRevB.93.224506 journal journal Phys. Rev. B volume
  93, pages 224506 (year 2016)NoStop
[Huang and Pickett(2017)]Huang2017
  author author Y. N. Huang and author W. E. Pickett, 10.1103/PhysRevB.95.165107 journal journal Phys. Rev. B volume
  95, pages 165107 (year 2017)NoStop
[Rademaker et al.(2016)Rademaker, Wang, Berlijn, and  Johnston]Rademaker2016
  author author L. Rademaker, author Y. Wang,
  author T. Berlijn,  and  author S. Johnston, http://stacks.iop.org/1367-2630/18/i=2/a=022001 journal
  journal New J. Phys. volume 18,  pages 022001 (year 2016)NoStop
[Wang et al.(2016b)Wang, Nakatsukasa, Rademaker, Berlijn, and  Johnston]Wang2016c
  author author Y. Wang, author K. Nakatsukasa,
  author L. Rademaker, author T. Berlijn,  and author S. Johnston, @noop 
  journal journal Supercond. Sci. Technol.  volume 29, pages 054009 (year 2016b)NoStop
[Bang(2014)]Bang2014
  author author Y. Bang, http://stacks.iop.org/1367-2630/16/i=2/a=023029 journal journal New J. Phys. volume
  16, pages 023029 (year 2014)NoStop
[Linscheid et al.(2016)Linscheid, Maiti, Wang, Johnston, and Hirschfeld]Linscheid2016a
  author author A. Linscheid, author S. Maiti,
  author Y. Wang, author
  S. Johnston,  and author
  P. J. Hirschfeld, 10.1103/PhysRevLett.117.077003 journal journal
  Phys. Rev. Lett. volume 117, pages
  077003 (year 2016)NoStop
[Chen et al.(2015)Chen,
  Maiti, Linscheid, and Hirschfeld]Chen2015
  author author X. Chen, author S. Maiti,
  author A. Linscheid,  and  author P. J. Hirschfeld, 10.1103/PhysRevB.92.224514 journal journal Phys. Rev. B volume 92, pages 224514 (year 2015)NoStop
[Weger et al.(1996)Weger,
  Peter, and Pitaevskii]Weger1996
  author author M. Weger, author M. Peter,  and  author L. P. Pitaevskii, 10.1007/s002570050248 journal journal
  Z. Physik B volume 101, pages 573
  (year 1996)NoStop
[Hao and Hu(2014)]Hao2014
  author author N. Hao and author J. Hu,  10.1103/PhysRevX.4.031053 journal journal Phys. Rev. X volume 4, pages
  031053 (year 2014)NoStop
[SM1()]SM1
  @noop  note See Supplemental Material at [url] for
  details, which include Refs.
  
  <cit.>NoStop
[Gor'kov(2016)]Gorkov2016
  author author L. P. Gor'kov, 10.1103/PhysRevB.93.060507 journal journal Phys. Rev. B volume
  93, pages 060507 (year 2016)NoStop
[Verdi and Giustino(2015)]Verdi2015
  author author C. Verdi and author F. Giustino, 10.1103/PhysRevLett.115.176401 journal journal Phys. Rev. Lett. volume 115, pages 176401 (year
  2015)NoStop
[Abrikosov(2005)]Abrikosov2005
  author author A. A. Abrikosov, http://dx.doi.org/10.1016/j.physc.2005.05.006 journal
  journal Physica C volume 424,  pages 149  (year 2005)NoStop
[Varelogiannis et al.(1996)Varelogiannis, Perali, Cappelluti, and  Pietronero]Varelogiannis1996
  author author G. Varelogiannis, author A. Perali, author E. Cappelluti,
   and author L. Pietronero,  10.1103/PhysRevB.54.R6877 journal journal Phys. Rev. B volume 54, pages R6877 (year 1996)NoStop
[Kulić and Dolgov(2017)]Kulic2017
  author author M. L. Kulić and author O. V. Dolgov, http://stacks.iop.org/1367-2630/19/i=1/a=013020
  journal journal New J. Phys. volume 19, pages 013020 (year
  2017)NoStop
[Scalapino et al.(1966)Scalapino, Schrieffer, and Wilkins]Scalapino1966
  author author D. J. Scalapino, author J. R. Schrieffer,  and author J. W. Wilkins, 10.1103/PhysRev.148.263 journal
  journal Phys. Rev. volume 148,  pages 263 (year 1966)NoStop
[Miao et al.(2015)Miao,
  Qian, Shi, Richard,
  Kim, Hoesch, Xing,
  Wang, Jin, Hu, and  Ding]Miao2015
  author author H. Miao, author T. Qian, author X. Shi, author
  P. Richard, author T. K. Kim, author M. Hoesch, author L. Y.  Xing, author X.-C. Wang, author C.-Q. Jin,
  author J.-P. Hu,  and author H. Ding, http://dx.doi.org/10.1038/ncomms7056 journal journal Nat. Commun. volume 6, pages
  6056 (year 2015)NoStop
[Tang et al.(2016)Tang,
  Liu, Zhou, Li, Ding, Li, Zhang, Li,
  Song, Ji, He, Wang, Ma, and Xue]Tang2016
  author author C. Tang, author C. Liu, author G. Zhou, author
  F. Li, author H. Ding, author Z. Li, author D. Zhang, author Z. Li, author
  C. Song, author S. Ji, author K. He, author L. Wang, author X. Ma,  and author
  Q.-K. Xue, 10.1103/PhysRevB.93.020507 journal journal Phys.
  Rev. B volume 93, pages 020507
  (year 2016)NoStop
[Varelogiannis(1995)]Varelogiannis1995
  author author G. Varelogiannis, 10.1103/PhysRevB.51.1381 journal journal Phys. Rev. B volume
  51, pages 1381 (year 1995)NoStop
[Subedi et al.(2008)Subedi,
  Zhang, Singh, and Du]Subedi2008
  author author A. Subedi, author L. Zhang,
  author D. J. Singh,  and  author M. H. Du, 10.1103/PhysRevB.78.134514 journal journal Phys. Rev. B volume 78, pages 134514 (year 2008)NoStop
[Koufos et al.(2014)Koufos,
  Papaconstantopoulos, and Mehl]Koufos2014
  author author A. P. Koufos, author D. A. Papaconstantopoulos,  and author M. J.  Mehl, 10.1103/PhysRevB.89.035150
  journal journal Phys. Rev. B volume 89, pages 035150 (year
  2014)NoStop
[Coh et al.(2015)Coh,
  Cohen, and Louie]1367-2630-17-7-073027
  author author S. Coh, author M. L. Cohen,   and author S. G. Louie, http://stacks.iop.org/1367-2630/17/i=7/a=073027 journal
  journal New J. Phys. volume 17,  pages 073027 (year 2015)NoStop
[Aperis et al.(2015)Aperis,
  Maldonado, and Oppeneer]Aperis2015
  author author A. Aperis, author P. Maldonado,   and author P. M. Oppeneer,  10.1103/PhysRevB.92.054516 journal journal Phys. Rev. B volume 92, pages 054516 (year 2015)NoStop
[Bekaert et al.(2016)Bekaert, Vercauteren, Aperis, Komendová, Prozorov, Partoens, and  Milošševi ćć]Bekaert2016
  author author J. Bekaert, author S. Vercauteren, author A. Aperis,
  author L. Komendová, author R. Prozorov, author
  B. Partoens,  and author
  M. V. Milošševi ćć, 10.1103/PhysRevB.94.144506 journal journal Phys.
  Rev. B volume 94, pages 144506
  (year 2016)NoStop
[Eschrig and Koepernik(2009)]Eschrig2009
  author author H. Eschrig and author K. Koepernik, 10.1103/PhysRevB.80.104503 journal journal Phys. Rev. B volume
  80, pages 104503 (year 2009)NoStop
[Marsiglio et al.(1988)Marsiglio, Schossmann, and Carbotte]Marsiglio1988
  author author F. Marsiglio, author M. Schossmann,  and author J. P. Carbotte, 10.1103/PhysRevB.37.4965 journal journal Phys. Rev. B volume
  37, pages 4965 (year 1988)NoStop
[Degiorgi et al.(1994)Degiorgi, Nicol, Klein, Grüner, Wachter, Huang, Wiley, and Kaner]Degiorgi1994
  author author L. Degiorgi, author E. J. Nicol,
  author O. Klein, author G. Grüner, author
  P. Wachter, author S.-M. Huang, author J. Wiley,  and author R. B.  Kaner, 10.1103/PhysRevB.49.7012 journal journal Phys. Rev. B volume
  49, pages 7012 (year 1994)NoStop
[Choi et al.(2002)Choi,
  Roundy, Sun, Cohen, and  Louie]Choi2002
  author author H. J. Choi, author D. Roundy,
  author H. Sun, author
  M. L. Cohen,  and author
  S. G. Louie, @noop  journal journal Nature volume 418,  pages 758 (year 2002)NoStop
[Margine and Giustino(2013)]Margine2013
  author author E. R. Margine and author F. Giustino, @noop  journal journal
  Phys. Rev. B volume 87, pages 024505
  (year 2013)NoStop


