Angular dependence of the upper critical field of Sr{}_{2}RuO{}_{4}

Angular dependence of the upper critical field of SrRuO

S. Kittaka Department of Physics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan    T. Nakamura Department of Physics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan    Y. Aono Department of Physics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan    S. Yonezawa Department of Physics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan    K. Ishida Department of Physics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan    Y. Maeno Department of Physics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan
September 17, 2019

One of the remaining issues concerning the spin-triplet superconductivity of is the strong limit of the in-plane upper critical field at low temperatures. In this study, we clarified the dependence of on the angle between the magnetic field and the plane at various temperatures, by precisely and accurately controlling the magnetic field direction. We revealed that, although the temperature dependence of for is well explained by the orbital pair-breaking effect, for is clearly limited at low temperatures. We also revealed that the limit for is present not only at low temperatures, but also at temperatures close to . These features may provide additional hints for clarifying the origin of the limit. Interestingly, if the anisotropic ratio in is assumed to depend on temperature, the observed angular dependence of is reproduced better at lower temperature with an effective-mass model for an anisotropic three-dimensional superconductor. We discuss the observed behavior of based on existing theories.

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I Introduction

The layered perovskite superconductor with the transition temperature of 1.5 K has been extensively studied due to its unconventional pairing state. Maeno et al. (1994); Mackenzie and Maeno (2003) Knight shift measurements with NMR Ishida et al. (1998, 2001) and with spin-polarized neutron-scattering Duffy et al. (2000) have revealed the invariant spin susceptibility across for , which firmly indicates that the spin part of the Cooper-pair state is triplet. The orbital part is favorably interpreted as odd parity based on the measurements of the critical current through Pb//Pb proximity junctions Jin et al. (1999); Honerkamp and Sigrist (1998) and other experiments.Nelson et al. (2004); Kidwingira et al. (2006) These results establish that is an odd-parity spin-triplet superconductor. In addition, the SRLuke et al. (1998) and Kerr effect Xia et al. (2006) measurements indicate broken time-reversal symmetry in the superconducting state. The zero-field ground state consistent with all these results is expressed by the vector order parameter, the -vector, . However, recent Ru-NMR measurements under very low fields down to 20 mT revealed the invariant Knight shift for . Murakawa et al. (2004, 2007) This means  () with the following two possibilities.Murakawa et al. (2004, 2007); Kaur et al. (2005) The -vector can rotate freely in the plane,Annett et al. (2008) or the -vector pointing along the axis in zero field can flip perpendicular to the axis by a small magnetic field along the axis.Yoshioka and Miyake (2009) In either case, the spin of the Cooper pair can be polarized to any field directions at least above 20 mT.

Another unsolved issue in is the origin of the strong limit of the upper critical field , which occurs when a magnetic field is applied parallel to the plane.Deguchi et al. (2002) Similar limit is observed in another spin-triplet superconductor UPt for , van Dijk et al. (1993) as shown in the inset of Fig. 2(b). These limits are reminiscent of the Pauli effect, which results from the Zeeman energy of quasiparticles. However, in spin-triplet superconductors, the Pauli effect contributes to pair-breaking only when , because the spin of the triplet Cooper pairs can be polarized along the field direction when . As mentioned above, the -vector of is likely to be perpendicular to the magnetic field possibly except at low fields. This suggests that the Pauli effect should not affect . We note that the -vector of UPt for was revealed to be perpendicular to the magnetic field () in phase C. Tou et al. (1998) Therefore, the limit observed in UPt cannot be attributed to the Pauli effect either. The origins of these limits have not been clarified yet.

In this paper, we report the dependence of of , determined from the ac susceptibility, on the magnetic field direction between the plane and axis. Because of the large anisotropy of in , a small misalignment would lead to a large difference in the value of , especially when the field direction is nearly parallel to the plane. Therefore, in this study, we controlled the applied field direction more accurately and precisely than in the previous reports. Mao et al. (2000a); Yaguchi et al. (2002) We evaluated the curves for different field directions and revealed that the limit is clearly observed only when the angle between the magnetic field and the plane is less than 5 degrees. This limit was revealed to occur not only at low temperatures, but also at temperatures close to . We also identified the angle dependence of at several fixed temperatures. We found that is fitted better at lower temperature with an effective-mass model for an anisotropic three-dimensional superconductor. In addition, we investigated the difference between for fields in the (100) plane and for fields in the (110) plane. The difference appears only at low temperatures below roughly 1 K for small . These results allow us to reexamine the origin of the limit based on existing theories.

Ii Experimental

We used single crystals of grown by a floating zone method.Mao et al. (2000b) In this paper, we focus on the result obtained from a single crystal with dimensions of approximately 1.0 0.5 mm in the plane and 0.08 mm along the axis. The directions of the tetragonal crystallographic axes of the sample were determined from x-ray Laue pictures. We shaped the sample so that the side surface of the sample was 10 degrees away from the (100) plane in order to avoid possible anisotropy effects due to surface superconductivity.Keller et al. (1996) The crystal was annealed in oxygen at 1 atm and 1050 °C for a week to reduce the amount of oxygen deficiencies and lattice defects. A sharp superconducting transition was observed in the ac susceptibility measurements with the midpoint at =1.503 K.

Figure 1: (Color online) Field dependence of at several for (a) and (b) . Here, denotes the azimuthal angle within the plane between the magnetic field and the [100] axis and denotes the angle between the magnetic field and the plane. is defined as the intersection of the linear extrapolations in . Thick and thin arrows represent and the anomaly due to the mosaic structure, respectively. The dip in near is attributable to the ordinary peak effect. Yaguchi et al. (2002)

We measured the ac magnetic susceptibility by a mutual-inductance technique using a lock-in amplifier with a frequency of 887 Hz. The sample was cooled down to 70 mK with a He-He dilution refrigerator. The ac magnetic field of 20 T-rms was applied nearly parallel to the axis with a small coil. The dc magnetic field was applied using the “Vector Magnet” system, Deguchi et al. (2004a) with which we can control the field direction three dimensionally and precisely. The accuracy and precision of the field alignment with respect to the plane are better than 0.1 degree and 0.01 degree, respectively. Owing to the high sensitivity of the pick-up-coil, parasitic background contributes to the signal.Kittaka et al. (2009) In order to obtain contribution only from a superconductivity , we adopt . The small deviation of the normal state values from zero indicates a good reliability of the background subtraction. We define as the intersection between the linear extrapolations of in the superconducting and normal states, as illustrated in Fig. 1 with dashed lines. The directions of the crystalline axes [100] and [001] with respect to the field direction were calibrated by making use of the anisotropy in .Yaguchi et al. (2002); Mao et al. (2000a) Our highly accurate and precise measurements revealed that the present sample has a mosaic structure dominated by two domains sharing the [100] axis; the [001] axis of one domain is tilted nearly toward the [010] axis by 0.5 degree from the [001] axis of the other part.

Iii Results

Figure 2: (Color online) (a) Field-temperature ( - ) phase diagram of at , and from top to bottom for . (b) Temperature dependence of defined as eq. (1). The inset represents the  -  phase diagram of UPt for (Ref. van Dijk et al., 1993). The dashed curve in (b) is of the boundary of the ( B + C ) phase in UPt for (the dashed curve in the inset).

Figure 2(a) is the field-temperature ( - ) phase diagram in various field directions at , where denotes the azimuthal angle within the plane between the magnetic field and the [100] axis. At , no anomaly due to the mosaic structure was seen in the raw data, as shown in Fig. 1(a). Reflecting the large anisotropy of in , becomes rapidly small when the angle between the magnetic field and the plane increases from 0. In the specific heat measurements, the second superconducting transition was observed just below at low temperatures below 0.8 K. Deguchi et al. (2002) Although such an additional transition was observed below 0.6 K in the ac susceptibility measurements, it was difficult to unambiguously identify it to be attributable to the second superconducting transition.Yaguchi et al. (2002) This is also the case for the present study. One possible reason for this difficulty is that ac susceptibility, mainly probing the vortex movements, may not be sensitive to the small change in the entropy detected by specific heat measurements. Therefore, we do not focus on the feature of the additional transition in this paper.

To characterize the limit of , we normalized by the initial slope at :


If is determined by the orbital pair-breaking effect, which originates from the kinetic energy of supercurrent around magnetic vortices, is described by the Werthamer-Helfand-Hohenberg (WHH) theory Helfand and Werthamer (1966); Werthamer et al. (1966) and its extension to -wave superconductors.Maki et al. (1999); Lebed and Hayashi (2000) In these theories, it is expected that increases linearly on cooling and is weakly suppressed at low temperatures with . In Fig. 2(b), we plot with different at . The initial slope is defined from the linear fit to in the region . For , behaves as expected from the WHH theory. In contrast, for , is strongly limited at low temperatures. This result indicates that the limit in is prominent for . To emphasize the limit in another spin-triplet superconductor UPt, we plot, in Fig. 2(b) with the dashed curve, of the boundary of the ( B + C ) phase for . Although we chose the less limited one between the two curves in UPt for , a strong limit of is clearly seen.

Figure 3: (Color online) Temperature dependence of the slope of the - phase diagram of at , and from bottom to top for . The slope of the curve for UPt (Ref. van Dijk et al., 1993), evaluated from the dashed curve in the inset of Fig. 2(b), is also plotted with crosses. The dashed curves are guides to the eye.

If the orbital pair-breaking effect is mainly responsible for determining , the slope of the  -  phase diagram should be constant down to well below . To identify the limit of near , we evaluate the slope at temperature as , where and are temperatures of adjacent data points. The results are shown in Fig. 3. For , the slope is constant down to approximately 1 K and approaches zero at low temperatures, which is well explained by the orbital pair-breaking effect. However, for , the slope near is not temperature-independent any more. This result suggests that the limit observed for is present not only at low temperatures, but also at temperatures close to . The slope of the curve of UPt for (the dashed curve in Fig. 2(b)) also continues to vary up to , as plotted in Fig. 3.

Figure 4: (Color online) (a) Field-angle dependence of the upper critical field of at various temperatures for . (b) Angle dependence of normalized by . Solid curves are the fitting results using eq. (3) in the range . Insets are enlarged views near .

In order to characterize the non-linear temperature dependence of in , we fitted for by with fitting parameters and . In any fitting range, is obviously larger than , which is expected for the two-dimensional (2D) superconductivity;Abrikosov (1988) the fitting in the range yields . In addition, the coherence length along the axis is estimated to be 3.2 nm using the GL equation


with  T and  T. Here, is the flux quantum. This value of is five times larger than the spacing of the conductive RuO layers (0.62 nm).Mackenzie and Maeno (2003) Even if the WHH value  T is used for ,  nm is obtained. These facts indicate that the superconductivity of cannot be classified as a 2D superconductivity.

Figure 4(a) represents the dependence of at various temperatures for . The dependence of normalized by is also plotted in Fig. 4(b). We found that, although for is nearly independent of temperature, it decreases on cooling for .

Figure 5: (Color online) Error of the fitting of eq. (3) to the data for in the range for several . is likely to give the most appropriate fitting range.
Experiment Fitting parameters
(K) Γ
0.1 1.517 T 21.4 1.574 T 22.1
0.5 1.399 T 23.7 1.504 T 25.5
0.9 1.130 T 30.5 1.243 T 33.2
1.3 0.496 T 41.3 0.568 T 46.1
Table 1: The upper critical field and its anisotropy. Fitting parameters are obtained by the fit of for using eq. (3) with at each temperature.

We found that the observed is well explained by the Ginzburg-Landau (GL) theory for anisotropic three-dimensional (3D) superconductors,Morris et al. (1972) if we allow the anisotropic ratio to depend on temperature. The angular dependence of is expressed as


We fit eq. (3) to the observed at temperature with two fitting parameters and . We chose the fitting range as so that the range is as wide as possible while the fitting yields a good result in the whole chosen range. Figure 5 represents the error of the fitting for different at each temperature. When , is well fitted by eq. (3) in the chosen fitting range. By contrast, for , exhibits systematic deviation from eq. (3) around . Thus, we conclude that is the most appropriate. The fitting results are plotted in Fig. 4 with the solid curves and the obtained fitting parameters are listed in Table 1. Interestingly, the observed is fitted by eq. (3) better at lower temperatures, as being clear in the inset of Fig. 4(b). This tendency is also clear when the fit ratio Γ is compared with the experimental ratio . We should mention that the thin-film model Tinkham (1996) applied to a 2D superconductor, Zuo et al. (2000) in which exhibits a cusp at , cannot account for our data. While we carefully examined the dependence of , a kink in around revealed by the specific heat measurements at 0.1 K (Ref. Deguchi et al., 2002) was not detected in the present study. We note that a kink in was not detected in the thermal conductivity measurements at 0.32 K, either (Fig. 4(a) in Ref. Deguchi et al., 2002). On the basis of the presently available results, we cannot clarify why the kink in was observed only in the specific heat measurement at 0.1 K.

Figure 6: (Color online) Comparison between at (circles) and (crosses).
Figure 7: (Color online) Temperature dependence of the in-plane anisotropy between and at . is defined as .

In Fig. 6, we compare the dependence of at angles between and . As indicated by Fig. 1(b), two onset features appear in the field dependence of at , reflecting the fact that the present sample consists mainly of two domains. Since it is possible to separate the contribution from each of these domains, we plot in Fig 6 for the major domain. From Fig. 6, we found that at is both qualitatively and quantitatively similar to at . Small difference in at angles between and is observed only when the magnetic field is applied nearly parallel to the plane. Figure 7 represents the temperature dependence of the in-plane anisotropy between and . Here, we define as . Although the temperature at which starts to increase on cooling depends on samples (0.9 K  K), of 40 mT at low temperatures is nearly the same among different samples with best .Mao et al. (2000a)

On the basis of the phenomenological theory proposed by Gorkov,Gor’kov (1987) superconductivity with a two-component order parameter, , should be accompanied by a substantial four-fold anisotropy in the in-plane . However, as presented in Fig. 7, no in-plane anisotropy is observable above about 1 K; the anisotropy grows on cooling, but reaches at most 3% at low temperatures. This lack of the large in-plane anisotropy is attributable to the multiband effect.Agterberg (2001); Kusunose (2004); Mineev (2008) Because the directions of the gap minima are 45 degrees different between the active () and passive ( and ) bands, Nomura (2005) the anisotropy reflecting the gap structure on different Fermi surface sheets can be cancelled. Agterberg (2001); Kusunose (2004)

Iv Discussion

Let us discuss the origin of the limit in . For both 2D and 3D superconductors in which the main pair-breaking effect is due to the ordinary orbital effect, such a limit is not expected. Thus, in order to explain the limit, we need an additional pair-breaking mechanism.

One of the possible additional pair-breaking effects in is an unusual orbital pair-breaking effect. For example, in a nearly 2D superconductor (TMET-STF)BF,Uji et al. (2001) it is proposed that for is limited due to the limit of the coherence length by the layer spacing, which leads to the decrease of the anisotropy ratio of on cooling. For , estimated using eq. (2), for which the ordinary orbital pair-breaking effect is assumed, is limited to be approximately 3.2 nm below 1 K. In fact, if we strictly apply eq. (2), takes a minimum at 0.8 K and even increases by about 3% at low temperatures. However, we cannot find a clear answer to this limit because the limited value of , 3.2 nm, is five times larger than the layer spacing. Therefore, the origin of the apparent limit of the coherence length in seems different from that in (TMET-STF)BF.

Recently, Machida and Ichioka proposed the Pauli effect as an additional pair-breaking effect leading to the limit in .Machida and Ichioka (2008) Using a model with a single-band spherical Fermi surface and by assuming the Pauli effect, they reproduced the observed at 0.1 K (Ref. Deguchi et al., 2002) as well as field dependences of the specific heat Deguchi et al. (2004b) and magnetization.Tenya et al. (2006) Interestingly, we found that the Machida-Ichioka model well reproduces our results of , too. Nevertheless, this would not lead to the conclusion that the limit in is attributable to the Pauli effect because the Machida-Ichioka model overlooks some key experimental as well as theoretical facts. First, Machida-Ichioka model does not include the multiband effect. Their single-band model explains the field dependence of the specific heat at low temperatures. However, has three cylindrical Fermi surfaces, , , and .Mackenzie and Maeno (2003); Bergemann et al. (2003) Although the active band is dominant in the superconductivity in high fields, the passive bands and also contribute to the superconductivity in low fields.Deguchi et al. (2004c) The contribution from the and bands is essential to explain the plateau-like dependence quantitatively.Deguchi et al. (2004c) In fact, inclusion of the multiband effect is needed to explain the dependence of the specific heat at low temperatures in zero field.Agterberg et al. (1997); Zhitomirsky and Rice (2001); Nomura and Yamada (2002) Secondly, as mentioned in Sec. I, the Pauli effect contradicts the results of the Knight shift experiments.Ishida et al. (1998, 2001); Duffy et al. (2000) Although they proposed the possibility that the spin part of the Knight shift was too small to be detected in the NMR experiments, the spin part at the Ru site is in reality as large as 4%.Ishida et al. (2001) In addition, the superconductivity was distinctly observed in through Ru NMR in the identical setup.Murakawa et al. (2007) These facts exclude the possibility of the Pauli mechanism. Therefore, an alternative mechanism needs to be introduced to explain both the Knight shift behavior and the limit.

V summary

We have clarified the temperature and field-angle dependence of of . Our experiments were performed with an accurate and precise control of the applied magnetic field to avoid errors due to the misalignment. We revealed that the limit is clearly observed for and it occurs not only at low temperatures but also at temperatures close to . We also found that, by assuming a temperature-dependent anisotropic ratio, the GL theory for an anisotropic 3D superconductor can explain the angular dependence of well, particularly at lower temperatures. The observed behavior of is qualitatively the same between and . Only a small in-plane anisotropy was observed at low temperatures, which disappears rapidly as the magnetic-field direction leaves from the plane. Until now, the origin of the effective pair-breaking effect, which is compatible with both the invariance to the Knight shift and the limiting behavior of , remains unclear.

We thank K. Machida, M. Ichioka, R. Ikeda, H. Ikeda, K. Deguchi, H. Yaguchi, Y. Nakai, H. Takatsu and M. Kriener for useful discussions and supports. This work is supported by a Grant-in-Aid for Global COE program “The Next Generation of Physics, Spun from Universality and Emergence” from the Ministry of Education, Culture, Sports, Science, and Technology (MEXT) of Japan. It is also supported by Grants-in-Aid for Scientific Research from MEXT and from the Japan Society for the Promotion of Science (JSPS). One of the authors (S. K.) is financially supported by JSPS.


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