Comparing the Superconductivity of Mgb2 and Sr2ruo4

Home , Cuprate superconductor, Magnesium diboride

Comparing the superconductivity of MgB2 and Sr2RuO4 Etienne Palosa) (PHYS 232: Electronic Materials) The physical properties of two superconducting materials are compared in their normal and superconducting states. Their similarities and differences are reviewed in a systematic manner, beginning with a description of the unit cell. Structural, thermodynamic, electronic and magnetic properties are compared. In the superconducting state, re compare the superconducting parameters of the materials and the effect on TC induced by physical or chemical changes to the systems. The similarities and differences between the conventional superconductor MgB2 and the unconventional superconductor Sr2RuO4 are discussed mainly within the context of experimental evidence reported in the literature.

I. INTRODUCTION

Some of the most important unresolved questions in condensed matter physics are related to superconductivity. While the phenomena was observed over one-hundred years ago and an theory that explains conventional superconductivity has been established for the better half of a century, there is no unified theory of conventional and unconventional superconductivity. Or is there? However, through continuous theoretical, experimental and collaborative efforts, much has been learned about the physics of superconducting materials. The two main categories of superconductors are (i) conventional: those explained by the Bardeen–Cooper–Schrieffer (BCS) theory, and (ii) unconventional: materials whose superconducting mechanisms do not satisfy the conditions in BCS theory. In these materials, the superconducting mechanism is not phonon-mediated. In this work, a comparison of the physical properties of one conventional and one unconventional superconductors is presented. On one hand, we have magnesium diboride, MgB2, a material whose superconductivity was discovered in 20011, decades after it’s synthesis and crystal structure was first reported. This discovery represented a milestone in superconductor physics, as at the time MgB2 had the highest TC for a (i) binary system and (ii) a non-Cuprate superconductor. A wave of studies followed, demonstrating that MgB2 is a BCS superconductor. We compare the properties of MgB2 with an unconventional superconductor, specifically a spin-triplet superconductor Sr2RuO4. With a similar history to magnesium diboride, Sr2RuO4 was synthesized decades earlier (1959) and it’s superconductivity wasn’t discovered until 19942. Strontium ruthenate also made headlines, as it too, did not contain copper. In this sense, it is the first layered perovskite superconductor without copper. We aim to review and compare the properties of these two materials in a systematic yet pedagogical manner.

II. PHYSICAL PROPERTIES IN THE NORMAL STATE

A. Structure and Symmetry

Magnesium diboride (MgB2) is hexagonal ”omega” structure structured and crystallizes in the hexagonal P 6/mmm space group. The Mg atoms alternate or are ”sandwiched” between hexagonal layers of B2. Mg is bonded to twelve equivalent B atoms to form a mixture of edge and face-sharing MgB12 cuboctahedra. All Mg–B bond lengths are '

a)Electronic mail: [email protected] 2

2.50 A.˚ B is bonded in a 9-coordinate geometry to six equivalent Mg and three equivalent B atoms. All B–B bond lengths are '1.77 A.˚ The unit cell of magnesium diboride is illustrated in Fig. 1 (a) and (b).

Strontium ruthenate, Sr2RuO4, is (La, Ba)CuO4 structured (i.e. perovskite structured) + and crystallizes in the tetragonal I4/mmm space group. Sr2 is bonded in a 9-coordinate – geometry to nine O2 atoms. There are a spread of Sr–O bond distances ranging from ∼ + – 2.45–2.77 A.˚ Ru4 is bonded to six O2 atoms to form corner-sharing RuO6 octahedra. There are four shorter (1.95 A)˚ and two longer (2.10 A)˚ Ru–O bond lengths, and two – in-equivalent O2 sites as is the case with most perovskites. The unit cell of strontium ruthenate is illustrated in Fig. 1 (c) and (d). The crystallographic properties described here can be corroborated by downloading the publicly available crystallographic information .cif ﬁles and using a visualization software such as VESTA, or any other, and comparing to the information in the reviews used as main references for this special topic paper3,4. A comparison of their lattice parameters, density and a few mechanical properties are compared in Table I. While the structures themselves may not be responsible for the superconducting properties of these two materials, they have been a motive for further investigation since, after all, the atomic arrangement of a given material is determined by its electronic structure. In the case of MgB2, for its simplicity and in the case of Sr2RuO4, because of it’s intriguing relationship between spin parity and it’s superconducting state. More on that later.

TABLE I. Summary of structural and some mechanical properties of MgB2 and Sr2RuO4 in the normal state.

˚ ˚ ˚ g Compound Lattice Space Group a (A) b (A) c (A) ρ ( cm3 ) K (GPa) G (GPa) MgB2 Hexagonal P 6/mmm [191] 3.083 3.083 3.521 2.64 118 148 Sr2RuO4 Tetragonal I4/mmm [139] 7.017 7.017 7.017 5.75 108 53

FIG. 1. Illustrations of the crystal structure (conventional unit cell) is shown for both MgB2 (top panel) and Sr2RuO4 (lower panel). The unit cell is enclosed by solid black lines. We show the isostructural (a) and top (xy) in (b) of MgB2, and in (c) and (d) respectively for Sr2RuO4. Colors: Mb = orange, B = dark green; Sr = light green, Ru = grey, and O = red. Image prepared with VESTA. 3

B. Electronic Structure

The structural and electronic properties of the materials under exam provide constraints on the superconducting properties. There electronic structure and of these systems has been investigated thoroughly, by means of diﬀerent theoretical methods such as the Tight-Binding formalism, Density Functional Theory, Dynamical Mean Field Theory and others. Due to the volume of such studies, I will the reader to (in my opinion) representative references5,6. In Fig 2, we present the band structures of the materials side-to-side, along with their orbital decomposed density of states computed within the General Gradient Approximation (GGA) in Density Functional Theory. Here, we brieﬂy discuss the band structure of the materials

FIG. 2. Electronic band structure and spin-polarized density of states (DOS) of (a) MgB2 and (b) Sr2RuO4 computed by Kohn-Sham Density Functional Theory. Graphics publically avaible in the Materials Project database. in what is mostly a qualitative manner. This is to not lose generality and focus on technical details which may be specific to certain methodologies, yielding distinct quantitative results (e.g. LDA vs GGA vs DMFT). The band structure of magnesium diboride was reported well before it’s superconductivity was dsicovered, but the first ”post-enlightenment” comprehensive first-principles investigation of electronic structure and phonon coupling in MgB2 was reported by Mazin 6 and Antropov in 2001 , and refer to them for our discussion. The band structure of MgB2 resembles that of graphite, with three bonding σ bands which correspond to in-plane sp2 (spxpy) hybridization in the Boron honey-comb layer, and two bands, π, π∗ formed by hy- bridized pz orbitals. The main difference between MgB2 and graphite is that two σ bands are only partially filled. The band structure of Sr2RuO4 was first described by Tamio Oguchi shortly after the 5 discovery of it as a superconductor . He showed that Sr2RuO4 is a metal, with 12 energy bands that lye below -2 eV that correspond to the bonding states of the Ru d and O p orbitals, as well as O p non-bonding. Additionally, there are three bands that cross the Fermi energy, EF . These bands posses the antibonding character of the Ru dxz, dyz and dxy orbitals O p, π orbitals. Ultimately, the take away here is that, like in the cuprates, there exists strong dp hybridization due planar network structure. However, while these band structure calculations provide insight into the orbital nature of the material, and provides similarities and dissimilarities with related cuprates, id sheds no real insight into the superconducting mechanism of the system. Further theoretical modeling, including the proposal of effective Hamiltonians have been developed based on Oguchi’s work.

C. Vibrational Structure

Understanding the vibrational properties of materials can provide insight into it’s dynamical stability, thermal properties and the mechanism of superconductivity in the materials we are currently interested in. In Figure 3 we show the phonon dispersion (phonon band structure) in (a) and the phonon density of states in (b) for MgB2, and in the same order 4

FIG. 3. Phonon dispersion bands (a) and phonon density of states (phDOS) (b) are shown for 7–10 MgB2, and in (c), (d) for Sr2RuO4 computed. Image adapted from Refs .

III. PHYSICAL PROPERTIES IN THE SUPERCONDUCTING STATE

A. Mechanisms of superconductivity

To begin the discussion of their superconducting state properties, we will make references again to their electronic structure. Let us begin with the case of MgB2. There is experimental evidence that showcases the mechanism of superconductivity in 10 MgB2, beginning with the observation of a large different in TC between Mg B2 and 11 Mg B2. This difference, ∆TC ∼ 1K is known as the ”isotope effect”. While we omit further details regarding the isotope effect coefficient (α), we highlight that in the case of MgB2 such a large isotope effect is a strong indicator that this system is a phonon- mediated superconductor. Particularly, it is the B phonon modes that are dominant for superconductivity, as there is strong coupling between the conduction electrons and the optical E2g optical phonon, where neighboring B-atoms move in opposite directions within the plane. A figure explicitly showing the isotope effect can be found in Ref4. This would classify MgB2 as a typical BCS superconductor, except for an added complexity of the boron phonon mode, which is anharmonic and the multi-band superconductivity of the material. Experimental evidence for multi-band (two-band to be precise) superconductivity is suggested from the analysis of the specific heat capacity C(T ) in the superconducting state. The data was fit by two phenomenological models, a one-band model and a two-band model which displayed optimal agreement with experiment. We showcase C(T ) for both materials in Figure 4. Additionally, let us recall that there are partially filled (partially empty) σ bands. This is important, because in the normal state, the majority of the charge carriers are hole-like, meaning it is a hole-conductor. This differentiates MgB2 from it’s relative, AlB2, with full σ bands that lead to typical electron conduction. It is well established that the Fermi surface of MgB2 has strong hole-like character, with cylindrical hole-like Fermi surfaces about the Γ symmetry point.This has been analyzed and is consistent within the BCS electron-phonon 5

picture and the theory of hole-superconductivty. The superconductivity of MgB2 (as well as other borides) has been investigated extensively, due to the fact that it possesed such a high TC , and without the complications in modeling and interpreting cuprates (due to their strongly correlated nature).

FIG. 4. Speciﬁc heat capacities of (a) MgB2 and (b) Sr2RuO4. In both cases, the symbols represent experimental data while the solid lines correspond to calculated C(T ). Image adapted from refs4 and11, respectively.

At this point, it becomes less straight-forward to compare directly MgB2 and Sr2RuO4. Let us begin by stating that unlike magnesium diboride, strontium ruthenate has a much more complex - less understood mechanism of superconductivity that is not phonon- mediated. This was evidenced by the fact that isotope-effect experiments were inconclusive for this material3. Moreover, its structure and electronic properties discussed in section II provide constraints on the pairing symmetry of this system. By pairing symmetry, we refer to the symmetry of bounded Cooper pairs in a superconductor. In principle, understanding the pairing symmetry of a superconducting material provides insight into it’s mechanism of superconductivity. In BCS superconductors, like MgB2, the pairing symmetry is of s-wave symmetry. Unconventional superconductors, like the cuprates, have d-wave symmetry. In the case of Sr2RuO4, evidence suggests that this system has a p-wave symmetry. In section II-A we described it’s crystallographic properties. The reason why we described the bond lengths and symmetry into such detail becomes apparent here. The symmetry of Sr2RuO4 is characterized by a D4g point group, meaning it has four-fold rotational and inversion symmetry. Necessarily, due to the symmetry of the bulk in this material, the pairing symmetry will be spin-triplet, with no mixed pairing allowed. This is suggested by experiments. Note the use of the word ”bulk”. This constraint on the pairing symmetry does not apply to surfaces, but let us not go down that rabbit hole. As Pauli once said, ”God made the bulk; the surface was invented by the devil”. Connecting this to it’s electronic structure, let us recall that the band structure of + Sr2RuO4 in the vecinity of EF is dominated by t2g orbitals of Ru4 , which are dxz, dyz, dxy (the label t2g corresponds to the group of three d orbitals that were stabilized upon the breaking of degeneracy due to an octahedral crystal field, with respect to the energy of an isolated Ru ion). It is also important in this material to recognize the strong electron- electron correlation, not present in MgB2. Furthermore, it hasn’t been settled that spin-triplet superconductors are understood in full. We depict an illustration for spin-triplet superconductivity in Fig 5. In this sense, Sr2RuO4 is considered a candiate of this type of materials. What is a fact, however, is that spin-triplet superfluid states are fully established in the Fermi liquid 3He, for which spin and mass supercurrents emerge in charge neutral superfluids11. The leading candidate spin-triplet superconductor among the Heavy-Fermion class, is UPt3, but there are others including inorganic and organic. 6

FIG. 5. Illustrations of Cooper-pair spins and orbital angular momenta for spin-triplet superconductivity. The Cooper-pair spins S are depicted by small arrows in various colors; the d-vector shown by yellow arrows is deﬁned perpendicular to the spins. Taken from11

TABLE II. Summary of superconducting parameters of MgB2 and Sr2RuO4. TC is the critical temperature, λ is the penetration depth, ξ are the coherence lengths, ξc/ξab is the anisotropy ratio, ΘD is the Debye temperature and κ is the Ginzburg-Landau parameter. There are two rows for Sr2RuO4 due to it’s high anisotropy.

Material TC λ(0) (A)˚ ξ(0)ab (A)˚ ξ(0)c (A)˚ ξc/ξab ΘD (K) κ (unitless) MgB2 39-40 140 37-120 16-36 6-7 750 26 Sr2RuO4 1.5 1900(ab) 660 33 20 450±50 20 3 × 104(c) ,46

B. Magnetic and Thermal Properties

The superconducting mechanisms of the materials can be further understood through their magnetic and thermodynamic properties. In Table II, summarize some of the superconducting parameters of MgB2 and Sr2RuO4 as found in the literature. We also present the magnetic susceptibility plots for the bulk systems in Fig 6. Regarding MgB2, measurements for C(T ) enable the calculation of the Sommerfeld coeﬃcient γ and the Debye 2 temperature ΘD , with these values being γ = 3 ± 1 mJ/mol K and ΘD = 750 ± 30 K, respectively. The value for γ is relatively low in the sense that it is within the range or 7

normal metals, but ΘD is greater than that of most metals by more than a factor of 2. Mea- surements on MgB2 are somewhat difficult to perform or to consider reliable, due to the difficulty of growing large enough single-crystal samples. Thus, most early measurements were performed on poly-crystals, leading to relatively poor agreement between different studies. This mainly affected measurements of anisotropic resistivity ρ. At TC = 40K, just at the superconducting transition, the resistivity ρ(40K) ' 0.4µΩcm and at room temperature ρ(300K) ' 10µΩcm. Measured along the c and ab planes, what is observed is the ratio 4 ρc/ρab can take on values ∼ 10−100 . These values are highly temperature and orientation dependent. From this, Hall constants were also measured as seen in4. Beyond temperature dependence, what is an important fact is that they are of opposite signs for H||c and H||ab, which indicate the presence of both types of charge carriers and supporting the multi-band electronic structure of magnesium diboride.

FIG. 6. Measuremets of magnetic susceptibility χ of (a) MgB2 and (b) Sr2RuO4 are shown. Images taken from refs11.

Additionally, we have also included in Figure 7 the measurements of the Seebeck coef- ficients of our two superconductors. Interestingly enough, the data we show for Sr2RuO4 was published in 2016 in Physical Review Letters, and the data for MgB2 is also relatively recent. This indicates the continued interest in exploring the properties of these systems. The measured thermoelectric power S(T ) shown for MgB2 was performed on different thin films. Notice the positive trend in S(T ); this corresponds and is consistent with the hole- nature of the majority current carriers, and is also consistent with the magnetic properties discussed above (and Hall effect experiments). The authors of this study8 argue that such high values of S are at odds (at least at that time) with the extremely low resistivity of the system. they commented that they could not explain this. However, as anticipated by theory, S(T ) vanishes below TC , and for T > TC there is quasi-linear behaviour in S(T ), but slowly saturating. Other studies show that S(T ) peaks at T ' 400K, with a slope of 4 − 6(×10−8) V/K2 and is agreed upon in various investigations. The plot shown forSr2RuO4 was calculated, corresponds to the in-plane Seebeck coefficient of Sr2RuO4, which was calculated within a framework combining electronic structure and dynamical mean-field theory (DMFT), a method used to model strongly-correlated systems. Again, pesky pesky spins are at it again with this mateiral. The thermopower of Sr2RuO4 at room temperature is explained by the fact that there are spin-fluctuations, while the orbital moments remain quenched up to temperatures T ∼ 500K. At this point, there is a shift and change of sign in S, clearly seen in (b) of Fig 7. By means of DFT+DMFT and the Kubo transport formalism, the authors showed that the S(T ) carries crucial physical information about decoherence temperature at which the unquenched entropy is much smaller for spin than for orbital degrees of freedom, relating Sr2RuO4 back to Hund’s rules and the definition of Hund’s metals. Compared to MgB2, it is clear once again that there are additional difficulties to accu- rately descibing the physical properties of Sr2RuO4 due to interactions not present (or not 8 important) in the diboride, which can be explained with a two-band model.

FIG. 7. Themopower and Seebeck coeﬃcient measurements of (a) MgB2 and (b) Sr2RuO4 are shown. Images taken from refs8,12.

C. Tunneling Spectra

Now, we briefly describe the Break-Junction tunneling spectra for both materials, as it gives insight into the superconducting gap. In Figure 8 (a), we show the tunneling conduc- tance from three different MgB2 break junctions. This brief section will be based on the findings of Ekino et al 13. The junction forms a SC-I-SC geometry, short for superconductor- insulator-superconductor. The innermost peak-to-peak separation is of 9 mV, and it corresponds to 4∆1/e , where 2∆1 is the smallest gap among the multiple gap values. Note that, this is consistent with our discussion that MgB2 is a two-gap superconductor. The outer-peak gap corresponds to 2(∆1 + ∆2) = 17 − 18 meV i.e. (∆2 > ∆1). can also see structures at ±13 mV. Their energy separation of ∼ 25 meV is consistent with 4∆2 . The gap values for ∆1 and ∆2 are of 2.3 and 6.5 meV, respectively, measured at T = 4.5 K, and remain up to TC . According to the authors, ∆1 is smaller than what is predicted by BCS, and the B-pz band could be responsible for this while the larger ∆2 is traced back to the B-pxy band. In the case of Sr2RuO4, we show the tunneling spectra of a Sr2RuO4 / Ru / Sr2RuO4 SC- I-SC junction that was used to study resistance oscillations of full flux quantum Φ0. This is known as Little-Parks resistance oscillation, and the interest in probing this in Sr2RuO4 is because it was recently suggested to be a topological superdoncutor, since spin-triplet superconductivity supports topological objects such as the half-flux-quantum as seen in lecture (h/4e) vortices and their associated Majorana modes, which are beyond the scope 9

FIG. 8. Tunneling spectra for (a) MgB2 and (b) a Sr2RuO4Sr2RuO4 “break junction” at various values of T as indicated. Taken from refs3,13 of this report and of course, the knowledge of the author. the interested reader is referred 3,11 to the references from 2012 and 2015 on Sr2RuO4 .

D. Modifying the Critical Temperature

While MgB2 had the highest TC for a non-cuprate superconductor, it is still very far from ideal. The ideal superconductor, to be clear, is yet to be discovered. This would be a material that is in it’s superconducting state at ambient conditions (T =298.15 K, P =1atm). Hence, it is of interest to investigate the eﬀects on TC of a superconductor as a result of either a chemical (e.g. substitution or doping) or physical (e.g. strain or applied pressure) modiﬁcation to the system. The compressibility of MgB2 is anisotropic, and it

14 FIG. 9. (a) Critical temperature TC vs applied pressure. Taken from . (b) The critical tempera- 4 ture TC of MgB2 vs substituent content. Taken from Ref . has a measured bulk modulus of ∼ 150 GPa (see Table I) . Interestingly, TC decreases under pressure. It has been argues or is commonly accepted that the differences in the change of TC with respect to pressure is causd by shear-stress in non-hydrostatic media. The pressure dependence of TC in MgB2 can be traced back to strong pressure dependence of its phonon spectra. In this sense, Raman studies have shown that the anharmonic character of the E2g Raman mode and reduced TC at pressues of around ∼ 15 − 22 GPa were correlated to a phonon-assisted Lifshitz electronic topological transition. The discussion of dTC /dT P so far has been for nominally pure MgB2. If we introduce impurities, say, a few Al atoms such that we have Mg1 –xAlxB2, or C atoms, the difference in the change of TC with respect to pressure can be explained qualitatively by relating the levels and amounts of impurities in the system. The relationship dTC /dP is shown in Figure 9. In Figure 9, we show the relationship between TC and X, the substitution content for MgB2. Due to the chemistry of magneisum diboride, there is plenty of experimen- 10 tal and theoretical work in the literature reporting the effects of Al, Mn or C subsstitutions: Mg1−xAlxB2, Mg1−xMnxB2, and Mg(B1−xCx)2. In all three cases dTC /dx is negative, however the reason or explanation is not the same for all three dopants. The why ∆TC is negative is realted to the electronic structure of the dopants and thus o the new system: it is to be understood in terms of the band filling effect. For example, Al and C are both electorn dopants and therefore reduce the number of holes at the top of the σ bands, and reduce the density of states. In the case of Mn, the impurities act as magnetic impurities 4 that surpress TC via a spin-flip scattering mechanism .

FIG. 10. Critical Temperature of Sr2RuO4 vs mechanical strain in the [100] and [110] crystallographic directions. Taken from3.

The case of Sr2RuO4 is different to that of MgB2 in the sense that, for this material, the surpression of TC is necessarily related to hydrostatic pressure. Experiments have shown that the critical pressure PC for which TC is surpressed to zero is ∼ 3 GPa. An early experiment demonstrated that this rate of surpression was of 0.3 K per applied GPa. 2 Furthermore, the temperature dependence of the T coefficient A of the in-plane ρ, ρab = 2 ρ0 + AT is associated with the quasiparticle effective mass, and it was only observed to change slightly in the limited pressure range of 0 ≤ P ≤ 1.2 GPa. Other studies indicate that the T dependence of ρab at low values of T exhibits an evolution from Fermi liquid behaviour (∼ T 2) at ambient pressure to an anomalous ∼ T 4 behaviour as P → 8 GPa. This finding complements the experimental observation that ρ ∼ T 4/3, which was found to result from 2D ferromagnetic fluctuations in the neighborhood of magnetic instability. Some experiments report TC to vanish at PC ' 7 GPa. The surpression of TC by pressure is linear. Enough about TC surpression; don’t we want to make it higher? The TC of Sr2RuO4 is significantly smaller than that of MgB2, by about a factor of a little less than ∼ 20. The increase of TC and thus the enhancement of superconductivty in strontium ruthenate can be achieved by mechanical strain. We show in Figure 10 measurements of TC as a function of strain along two crystallographic directions, namely [100] and [110]. This is related to the fact that the pairing state of Sr2RuO4 features a two-fold degeneracy, which can be broken by in-plane uniaxial stress. This leads to split superconducting transitions, one raising and one lowering TC . Notice in Fig 10 that along the [100] direction, the increase of TC is strong and symmetric, a quality not observed along the [110] direction. To conclude this section, we make the remark that unlike MgB2, which was some-what resilient to mechanical perturbations, in Sr2RuO4 it has been more fruitful to investigate physical changes to the lattice rather than doping or substitution. By this we refer to vacancies and dislocations of the lattice3,11. 11

IV. REFLECTION AND CLOSING REMARKS

Up to this point, we have reviewed and compared side-by-side the physical properties of MgB2 and Sr2RuO4 in the normal and superconducting states. We have emphasized on understanding their electronic structure and mechanisms of superconductivity, highlight- ing the relationship between the two. Now, let us hypothesize that the mechanism for superconductivity is be the same for both materials. Due to time and length limitations, this comparison was focused primarily on experimentally measured properties of the conventional superconductor MgB2 and the spin-triplet superconductor Sr2RuO4. This focus enables a side-by-side comparison, limiting our discussion to what is known vs what is predicted or expected. Clearly, the case of Sr2RuO4 is more complex than that of MgB2, with a mechanism not yet completely understood. However, the fact that it is more complex and maybe ”interesting” to some from a theory standpoint, does not make it more interesting of a material. The simplicity yet record-high TC of MgB2 make it an exemplary material. In both materials, it is seen that hole-like behaviour is what drives the superconducting properties, and chemically analogous systems that are electron-typical conductors, do not display superconductivity. It is difficult to believe that there is a unique explanation for the superconductivity in both mateirals, but the fingerprints for there to exist such a theory are clearly evident by comparing stripping their superconductivity to a few properties. What I believe, at this point, is that there are fundamental properties of matter - physical and chemical - that are yet to be discovered to solve the mystery of a unified theory of superconductivity.

1J. Nagamatsu, N. Nakagawa, T. Muranaka, Y. Zenitani, and J. Akimitsu, “Superconductivity at 39 k in magnesium diboride,” nature 410, 63–64 (2001). 2Y. Maeno, H. Hashimoto, K. Yoshida, S. Nishizaki, T. Fujita, J. Bednorz, and F. Lichtenberg, “Super- conductivity in a layered perovskite without copper,” nature 372, 532–534 (1994). 3Y. Liu and y. . Zhi-Qiang Mao, . 4S. Bud’ko and y. . Paul C. Canfield, . 5T. Oguchi, “Electronic band structure of the superconductor sr 2 ruo 4,” Physical Review B 51, 1385 (1995). 6I. Mazin and V. Antropov, “Electronic structure, electron–phonon coupling, and multiband effects in mgb2,” Physica C: Superconductivity 385, 49–65 (2003). 7K.-P. Bohnen, R. Heid, and B. Renker, “Phonon dispersion and electron-phonon coupling in mgb 2 and alb 2,” Physical review letters 86, 5771 (2001). 8V. Drozd, A. Gabovich, P. Gierlowski, M. Peka la, and H. Szymczak, “Transport properties of bulk and thin-film mgb2 superconductors: effects of preparation conditions,” Physica C: Superconductivity 402, 325–334 (2004). 9T. Bauer and C. Falter, “The phonon dynamics of sr2ruo4: microscopic calculation and comparison with that of la2cuo4,” Journal of Physics: Condensed Matter 21, 395701 (2009). 10P. K. Jha and S. P. Sanyal, “Phonon density of states and lattice specific heat in sr2ruo4,” Journal of Physics and Chemistry of Solids 61, 943–946 (2000). 11Y. Maeno, S. Kittaka, T. Nomura, S. Yonezawa, and K. Ishida, “Evaluation of spin-triplet superconductivity in sr2ruo4,” Journal of the Physical Society of Japan 81, 011009 (2012), https://doi.org/10.1143/JPSJ.81.011009. 12 J. Mravlje and A. Georges, “Thermopower and entropy: Lessons from sr2ruo4,” Phys. Rev. Lett. 117, 036401 (2016). 13T. Ekino, T. Takasaki, T. Muranaka, H. Fujii, J. Akimitsu, and S. Yamanaka, “Tunneling spectroscopy of mgb2 and li0. 5 (thf) yhfncl,” Physica B: Condensed Matter 328, 23–25 (2003). 14C. Buzea and T. Yamashita, “Review of the superconducting properties of mgb2,” Superconductor Science and Technology 14, R115 (2001).