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The gap structure of Sr_{2}RuO_{4}, which is a longstanding candidate for a chiral *p*-wave superconductor, has been investigated from the perspective of the dependence of its specific heat on magnetic field angles at temperatures as low as 0.06 K (∼ 0.04*T*_{c}). Except near *H*_{c2}, its fourfold specific-heat oscillation under an in-plane rotating magnetic field is unlikely to change its sign down to the lowest temperature of 0.06 K. This feature is qualitatively different from nodal quasiparticle excitations of a quasi-two-dimensional superconductor possessing vertical lines of gap minima. The overall specific-heat behavior of Sr_{2}RuO_{4} can be explained by Doppler-shifted quasiparticles around horizontal line nodes on the Fermi surface, whose in-plane Fermi velocity is highly anisotropic, along with the occurrence of the Pauli-paramagnetic effect. These findings, in particular, the presence of horizontal line nodes in the gap, call for a reconsideration of the order parameter of Sr_{2}RuO_{4}.

Sr_{2}RuO_{4}, a layered-perovskite superconductor with ^{1}^{)} has attracted enormous attention ever since Knight-shift experiments provided favorable evidence that it exhibits spin-triplet pairing.^{2}^{–}^{5}^{)} Numerous experiments have demonstrated that Sr_{2}RuO_{4} has non *s*-wave properties,^{6}^{,}^{7}^{)} and some of the experimental reports indicate a degenerate order parameter.^{8}^{,}^{9}^{)} The simple Fermi-surface topology of Sr_{2}RuO_{4} comprising of three cylindrical sheets (*α*, *β*, and *γ*)^{10}^{,}^{11}^{)} together with its well-characterized Fermi-liquid behavior has led to the construction of several theoretical models to describe superconductivity.^{6}^{)} Among these models, a spin-triplet chiral *p*-wave pairing characterized by

However, several experimental facts exist that cannot be explained in the framework of this spin-triplet scenario.^{12}^{)} A serious controversy is the mechanism of the first-order superconducting transition along with the ^{13}^{–}^{16}^{)} It is reminiscent of the Pauli-paramagnetic effect that is not allowed in the spin-triplet scenario. The superconducting gap structure of Sr_{2}RuO_{4} is also a contentious issue. In general, a chiral *p*-wave gap opening on the cylindrical Fermi-surface sheets has no symmetry-protected node. Nevertheless, the gap amplitude of Sr_{2}RuO_{4} has been widely accepted to be modulated, and lines of deep minima (or nodes) are suggested to be present somewhere in the gap because of the power-law temperature dependence of various physical quantities.^{17}^{–}^{20}^{)} Furthermore, universal heat transport has raised the possibility of a nodal gap.^{18}^{)} Various gap structures including vertical and horizontal line node gaps have been proposed so far;^{21}^{–}^{25}^{)} however, the location of gap minima has not yet been established.

During this decade, field-angle-dependent measurements that probe quasiparticle density of states, ^{26}^{–}^{29}^{)} This technique takes advantage of the Doppler energy shift,

In 2004, Deguchi et al. reported the in-plane field-angle *ϕ* dependence of the specific heat, _{2}RuO_{4} in the temperature range ^{23}^{,}^{24}^{)} They proposed that the gap has fourfold anisotropy within the *ab* plane, i.e., four vertical lines of gap minima, based on the assumption that ^{28}^{,}^{29}^{)} have suggested that the Doppler-predominant condition [*ϕ* dependence of ^{28}^{,}^{29}^{)} Such a sign change was indeed observed in nodal superconductors CeCoIn_{5}^{30}^{)} and KFe_{2}As_{2},^{31}^{)} though it has not yet been detected in Sr_{2}RuO_{4}, at least above 0.12 K (^{23}^{,}^{24}^{)}

Here, we report the results of high-precision *T* and *H* without a sign change. By comparing our results with microscopic calculations, we find that the observed features in

High-quality single crystals of Sr_{2}RuO_{4} were grown by a floating-zone method,^{32}^{)} and a single 11.2-mg piece was used. The crystal was oriented using the backscattering X-ray Laue method. The angle-resolved specific heat *ϕ* (*θ*) denotes an azimuthal (polar) field angle relative to the [100] ([001]) axis, as represented in Fig. 1(b). The addenda of our calorimeter mainly consisted of a stage (silver foil), thermometer, and heater. A ruthenium-oxide chip resistor (Panasonic, ERJ-XGNJ, 8.2 kΩ) was used as a thermometer; it was cut into thirds (resulting in reducing the resistance to 3.3 kΩ) and whose substrate was polished to reduce its heat capacity. A ruthenium-oxide chip resistor (ROHM, MCR004, 240 Ω) was used as a heater, whose substrate was also polished. In this study, we have measured the addenda heat capacity carefully and subtracted its contribution, as depicted in the lower inset of Fig. 1(b). The magnetic field was generated using a vector magnet; the field was up to 3 T along the *z*-axis and 5 T along the *x*-axis. By using a stepper motor mounting on top of the Dewar, the refrigerator could be rotated around the *z*-axis. Thus, the orientation of the magnetic field was controlled three-dimensionally with high accuracy (better than 0.05°).

Figure 1. (Color online) (a) Temperature dependence of the specific heat divided by temperature _{2}RuO_{4}; at 0.06 K in *ϕ* at 0.1 K.

Figure 1(a) shows the temperature dependence of _{2}RuO_{4}.^{33}^{)} At 2 T (^{99}Ru, ^{101}Ru, and ^{87}Sr nuclei.^{34}^{,}^{35}^{)} The resulting

In Fig. 1(b), *H* applied parallel to the [100] and [110] axes. The increase in ^{2}), which is comparable to results of a previous study,^{23}^{)} although the absolute value is enhanced due to the difference in the definition of ^{36}^{)} This fact indicates that the superconducting volume fraction of the present sample is above 90%, and that an offset of ∼6 mJ/(mol K^{2}) is given to ^{14}^{,}^{37}^{,}^{38}^{)} At low fields, ^{39}^{)}

In previous reports,^{17}^{,}^{23}^{)} a shoulder anomaly was detected in the low-temperature _{2}^{40}^{)} and KFe_{2}As_{2}.^{31}^{,}^{41}^{)} However, no such anomaly is observed in our data after precise subtraction of the addenda contribution. It is noted that the total heat capacity shows a shoulder anomaly which can be attributed to the addenda contribution [the lower inset of Fig. 1(b)]. The lack of the multigap feature indicates relatively strong coupling between the three gaps on the *α*, *β*, and *γ* bands; all three gaps survive up to relatively high fields.

To examine the in-plane gap anisotropy, *ab* plane. The results of ^{42}^{)} Here,

Figure 2. (Color online) Field-angle *ϕ* dependence of *ab* plane. Here,

In order to explore the out-of-plane gap anisotropy, we investigated the polar-angle *θ* dependence of the specific heat at 0.06 K under a rotating field within the (010) plane (^{43}^{)} (I), however, little information on the out-of-plane gap anisotropy can be extracted from ^{44}^{)} because ^{15}^{)} This fact, however, suggests that the previously reported steep suppression of ^{24}^{)} is not due to the compensation of antiphase gap anisotropies between active and passive bands but due to this scaling by

Let us discuss the origin of the _{2}RuO_{4}. By combining the present ^{23}^{)} a contour plot of ^{29}^{)} In particular, Fig. 3(b) shows a sign change of ^{45}^{)} the line is shifted toward a lower (higher) field and a lower (higher) temperature if vertical line nodes are present on large (small) curvature parts of the warped Fermi surface.^{46}^{)} For Sr_{2}RuO_{4}, the quasi-two-dimensional *γ* band is nearly cylindrical, which yields the sign-changing line around approximately *α* and *β* bands have flat and high-curvature parts in the *α* and/or *β* bands and dominantly contribute to the

Figure 3. (Color online) (a) Contour plot of ^{23}^{)} experimental data. (b) ^{29}^{)} ^{46}^{,}^{54}^{)} (see text). Circles in (c) are the experimental data at 0.1 K.

An alternative, more promising scenario is to assume a horizontal line node gap. According to the recent band-structure calculation,^{47}^{)} the in-plane ^{48}^{)}) for the *γ* band (*α* and *β* bands). If the in-plane anisotropy of the Fermi velocity, ^{49}^{)} To examine this possibility, microscopic calculations were performed by assuming a horizontal line node gap ^{46}^{)} on the rippled cylindrical Fermi surface; this method was similar to that used in previous reports.^{29}^{,}^{50}^{,}^{51}^{)} Here, we use the anisotropic parameter *γ* band) and the Maki parameter

The calculated result of ^{52}^{,}^{53}^{)} Slight mismatch at low fields can be attributed to thermal excitations of quasiparticles in ^{46}^{,}^{54}^{)} is compared with the experimental result at 0.1 K (circles). The absence of a sign change in ^{43}^{)} (II)] and the low-temperature ^{55}^{)} This phase might be the origin of the strange ^{37}^{,}^{56}^{)} which is inverted to the anisotropy expected from *γ* band. Figure 3(d) shows a contour map of the calculated *γ* band^{57}^{)} and the Pauli-paramagnetic effect.

The horizontal line node gap is, however, incompatible with the *p*- and *f*-wave scenarios suggested in the previous reports.^{23}^{,}^{25}^{)} In addition, the absence of multigap feature in ^{58}^{)} In the tetragonal *μ*SR and Kerr-rotation experiments.^{8}^{,}^{9}^{)} Non-zero residual thermal conductivity along the *c*-axis^{25}^{)} might support the ^{59}^{)} is also incompatible with the vertical-line-node scenario.

In summary, we have investigated the gap anisotropy of Sr_{2}RuO_{4} from field-angle-dependent specific-heat measurements. In *α*, *β*, and *γ* bands equivalently survive and possess lines of deep minima in some directions. We have revealed that the fourfold oscillation in the low-field _{2}RuO_{4}.

## Acknowledgments

We thank Y. Yoshida and H. Yaguchi for their support with the experiments. We also thank Y. Maeno and S. Yonezawa for the fruitful discussion. A part of the numerical calculations was performed by using the HOKUSAI supercomputer system in RIKEN. This work was supported by a Grant-in-Aid for Scientific Research on Innovative Areas “J-Physics” (15H05883,18H04306) from MEXT and KAKENHI (18H01161, 15K05158, 15H03682, 26400360, 15K17715, 15J05698, 17K05553, 18K04715) from JSPS.

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