Evidence for ferromagnetic spin-pairing superconductivity in UGe{}_{2}: A {}^{73}Ge-NQR study under pressure

Evidence for ferromagnetic spin-pairing superconductivity in UGe:
A Ge-NQR study under pressure

A. Harada Department of Materials Engineering Science, Osaka University, Osaka 560-8531, Japan    S. Kawasaki Department of Materials Engineering Science, Osaka University, Osaka 560-8531, Japan    H. Mukuda Department of Materials Engineering Science, Osaka University, Osaka 560-8531, Japan    Y. Kitaoka Department of Materials Engineering Science, Osaka University, Osaka 560-8531, Japan    Y. Haga Advanced Science Research Center, Japan Atomic Energy Research Institute, Tokai, Ibaraki 319-1195, Japan    E. Yamamoto Advanced Science Research Center, Japan Atomic Energy Research Institute, Tokai, Ibaraki 319-1195, Japan    Y. Ōnuki Advanced Science Research Center, Japan Atomic Energy Research Institute, Tokai, Ibaraki 319-1195, Japan Department of Physics, Osaka University, Osaka 560-0043, Japan    K. M. Itoh Department of Applied Physics and Physico-Informatics, Keio University, Yokohama 223-8522, Japan    E. E. Haller University of California at Berkeley and Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA    H. Harima Department of Physics, Faculty of Science, Kobe University, Nada, Kobe 657-8501, Japan
July 15, 2019
Abstract

We report that a novel type of superconducting order parameter has been realized in the ferromagnetic states in UGe via Ge nuclear-quadrupole-resonance (NQR) experiments performed under pressure (). Measurements of the nuclear spin-lattice relaxation rate have revealed an unconventional nature of superconductivity such that the up-spin band is gapped with line nodes, but the down-spin band remains gapless at the Fermi level. This result is consistent with that of a ferromagnetic spin-pairing model in which Cooper pairs are formed among ferromagnetically polarized electrons. The present experiment has shed new light on a possible origin of ferromagnetic superconductivity, which is mediated by ferromagnetic spin-density fluctuations relevant to the first-order transition inside the ferromagnetic states.

pacs:
PACS: 74.25.Ha, 74.62.Fj, 74.70.Tx, 76.60.Gv

The coexistence of magnetism and superconductivity (SC) has recently become an important topic in condensed-matter physics. The recent discovery of SC in ferromagnets UGe Saxena (); Huxley () and URhGe Aoki () has been a great surprise because the Cooper pairs are influenced by a non-vanishing internal field due to the onset of ferromagnetism (FM), which is believed to prevent the onset of SC. In the ferromagnet UGe with a Curie temperature  K at ambient pressure (), the emergence of -induced SC has observed in the range of 1.0 - 1.6 GPa Saxena (); Huxley (). It is noteworthy that the SC in UGe disappears above  GPa, beyond which FM is suppressed. The SC and FM in this compound have been shown to be cooperative phenomena Harada (). The superconducting transition temperature () is the highest at  GPa, where a first-order transition occurs from FM2 to FM1 as increases. Here, it should be noted that ferromagnetic moments are increased in the first-order transition from FM1 to FM2 as functions of temperature and pressure, as shown in Fig. 1(a) Pfleiderer (); Tateiwa1 (); Kotegawa (). The -induced SC in UGe coexists with FM1 and FM2 exhibiting the large magnetization of an order 1 /U even for the case of 30 K Kotegawa (). Therefore, the onset of SC is proposed to be suitable for the formation of a spin-triplet pairing state rather than a spin-singlet pairing state Huxley (). However, there are few reports that address the type of order parameter is realized in FM1 and FM2.

Figure 1: (color online). (a) Pressure versus temperature phase diagram of UGe near the superconducting phase Kotegawa (). The values of FM1 (open squares) and FM2 (open triangles), and value determined in this study are plotted. (b) Crystal structure of UGe with the ferromagnetic moment at the U site below .

In a previous study, an unconventional nature of the SC has been suggested from the measurement of the Ge-NQR nuclear spin-lattice relaxation rate Kotegawa (). However, it has not been well understood whether the presence of the residual density of states (RDOS) at the Fermi level in the SC state is intrinsic or not, suggesting the occurrence of a possible extrinsic effect due to the presence of any impurity and/or imperfection in the sample Kotegawa (). In particular, it is unclear why SC emerges with the highest when the first-order transition occurs from FM1 to FM2 at  GPa. In order to gain insight into this issue, further experiments are required for understanding a -induced evolution in the FM states and a novel order-parameter symmetry emerging in the FM states in UGe.

In this letter, by performing the Ge-NQR measurements under pressure at zero field () on a newly prepared sample, we report that the SC in this compound is caused by the formation of up-spin Cooper pairs, where the gap opens only at the up-spin band in FM1 and FM2 but not at the down-spin band Ohmi (); Machida (); Fomin (). The ferromagnetic spin-pairing SC is considered to be mediated by ferromagnetic spin-density fluctuations relevant to the first-order transition inside the ferromagnetic states.

Figure 2: (color online). Comparison of the Ge-NQR spectra of the present and previous samples of UGe in (a) paramagnetic state and (b) FM1. The spectra at (a)  GPa and (b) 1.41 and 1.3 GPa are shown by solid and open circles, respectively, demonstrating that the present sample has better quality than the previous samples Kotegawa ().

A polycrystalline sample enriched by Ge was crushed into coarse powder for the NQR measurement and annealed to maintain its quality. The NQR experiments were performed by the conventional spin-echo method at in the frequency () range of 5 - 11 MHz at = 1.17, 1.2, 1.24, and 1.41 GPa. Hydrostatic pressure was applied by utilizing a NiCrAl-BeCu piston-cylinder cell filled with Daphne oil (7373) as a pressure-transmitting medium. The value of at low temperatures was determined from the of Sn measured by the resistivity measurement. The possible distribution of the pressure inside the sample was less than 3  in the present experimental setup. A He-He dilution refrigerator was used to obtain the lowest temperature of 50 mK. Figures 2(a) and 2(b) show the NQR spectra in paramagnetic (PM) phase and FM1 phase, respectively. The linewidths in these NQR spectra are narrower for the present sample than for the previous sample, demonstrating that the quality of the present sample is significantly higher than that of the previous sample. Moreover, the NQR- measurements reveal that the present sample exhibits the highest value of  K obtained thus far at  GPa, ensuring higher quality than before.

Figure 3: (color online). (a) Ge-NQR spectra at 4.2 K in the -induced paramagnetic phase. The NQR spectra in (b), (c), (d), and (e) represent in the ferromagnetic phases at 1.4 K and 1.41, 1.24, 1.2 and 1.17 GPa, respectively. The dashed lines in the figures indicate the simulated results (see text).

Figure 3(a) shows the NQR spectra for the PM phase at 4.2 K and  GPa where FM1 is completely suppressed. They reveal a structure consisting of separated peaks associated with three inequivalent Ge sites in one unit cell in the crystal structure illustrated in Fig. 1(b) Harada (); Kotegawa (). The number of Ge1 sites is twice that of Ge2 and Ge3 sites in one unit cell. The Ge1 site is closely located along the uranium (U)-zigzag chain, while the other two sites Ge2 and Ge3 are located outside this zigzag chain. From the analysis of the NQR spectra for FM1 in the previous experiment Kotegawa (), it was demonstrated that the onset of FM1 induces an internal field  T at the Ge sites that additionally causes about the Zeeman splitting in each Ge-NQR spectrum. Furthermore, the angle between the principal axis for the nuclear quadrupole Hamiltonian and a direction of was determined as . When the first-order transition occurs from FM1 to FM2, the spectral shape changes significantly from the spectra at and 1.41 GPa to those at and 1.2 GPa, as shown in Figs. 3(b)-(e). From the analysis of the spectra, it is estimated that  T for FM1 increases to  T for FM2 (see Fig. 5(b)). The sudden increase in in the narrow range should be relevant to the first-order transition at . In such a case, the spectra near are expected to reveal a mixture of both domains of FM1 and FM2 in the narrow range of close to due to an inevitable distribution of . In fact, the respective spectra at and 1.24 GPa are composed of spectra arising from FM1 ( GPa) and FM2 ( GPa) with the ratios of 1 : 9 and 7 : 3, respectively, as shown by the dashed lines in Figs. 3(b) and 3(c). By considering an inevitable distribution ( GPa) as a Gaussian function given by , we obtain  GPa. The present experimental results reveal that the first-order transition occurs at  GPa.

Figure 4: (color online). (a) Temperature dependences of for FM2 at and 1.2 GPa measured at  MHz and for FM1 at and 1.41 GPa measured at 7.09 and 7.07 MHz, respectively. (b) Temperature dependences of related to the DOS in either the SC state or the normal state at each . The solid curves represent the results calculated based on the ferromagnetic spin-pairing model (see the text).

Figure 4(a) shows the dependences of for FM1 and FM2 at pressures that are slightly lower and higher than  GPa, respectively. Clearly, for FM1 and FM2 decreases without any indication of coherence peak just below , which provides evidence for the uniform coexistence of the unconventional SC and ferromagnetism. In the previous study Kotegawa (), the line-node gap model with RDOS at the Fermi level was applied to interpret a systematic evolution in the superconducting energy gap and a fraction of RDOS /. Here is the density of state (DOS) at the Fermi level in the FM phases. It should be noted that all the data of are uniquely determined in the present sample, but not in the previous sample Kotegawa (). Therefore, we could not exclude the fact that the RDOS is present in the previous sample due to some impurity effect. Similarly for the present sample with higher quality than that of the previous sample, the application of the line-node gap model with the RDOS allows us to estimate / = 0.50, 0.48, 0.29, and 0.30 and 0.45, 0.55, 0.75, and 0.25 K ( K) at 1.17, 1.2, 1.24, and 1.41 GPa, respectively. It should be noted that decreases from 0.45 K to 0.25 K although / does decrease from 0.50 to 0.30 at  GPa in FM2 and at  GPa in FM1. These results demonstrate that the presence of the RDOS in the superconducting state is not due to the impurity effect but intrinsic in origin. Although some impurity and/or imperfection based effects, if any, are not completely ruled out, we state that the observation of the highest  K ensures that the present sample is one of the best quality samples reported thus far.
First, we address whether or not the RDOS is associated with a self-induced vortex state in the SC + FM uniformly coexisting state. By assuming the Abrikosov triangular vortex lattice, a coherence length 130 Å Tateiwa1 (), and an internal magnetic field  T MagneticField (), we consider that only 3 % of arises from the normal state inside the self-induced vortex core in the SC + FM state, which does not agree with the experimental result. Alternatively, in another promising scenario that explains the RDOS, we consider a nonunitary spin-triplet pairing model Ohmi (). In this model, the superconducting energy gap opens only in the up-spin band parallel to the magnetization of FM phases, but not in the down-spin band that remains gapless. We begin by assigning possible nuclear relaxation processes of Ge-NQR in the ferromagnetic states. One of them is caused by the transversal component of fluctuations of internal magnetic fields at the Ge sites originating from intraband and interband transitions across the Fermi level at each up-spin band and down-spin band. Another one is caused by only the interband spin-flip transition across the Fermi level between the up-spin band and down-spin band. By considering these relaxation processes, in the FM state is expressed as


where , , and indicate the former contributions and represents the latter contribution which is only possible for . When the energy dependence of the DOS is neglected near the Fermi level, all contributions of , , and are independent of temperature. Here, and are the DOS at the up-spin and the down-spin bands at the Fermi level in the normal FM state, respectively. is an angle between the quantization axis of the Ge-nuclear-quadrupole Hamiltonian and that of at the Ge site in the FM state, which is estimated as from the analysis of the NQR spectra in the FM state, and is a Fermi distribution function. In the ferromagnetic spin-pairing model, the line-node gap of is assumed only for the density of states at the up-spin band, but not for at the down-spin band. It should be noted that if , should behave as well below . In the present case, because of , the gapless term gives rise to the RDOS at the Fermi level in the superconducting state, as shown in Fig. 4(b). In fact the experimental results are actually in good agreement with this theoretical model, as indicated by the solid lines in Figs. 4(a) and 4(b). Therefore, the SC energy gap and / are estimated as , 3.8, 4.0, and 3.7 with / = 0.57, 0.57, 0.82, and 0.80 at = 1.17, 1.2, 1.24, and 1.41 GPa, respectively. Here, .
In order to gain further insight into the novel SC state, Fig. 4(b) shows the dependence of related to the DOS at the Fermi level in either the SC or the normal FM state. As shown in Fig. 5(c), the most interesting finding is that dramatically increases as increases slightly from to 1.24 GPa across  GPa, accompanying the sudden reduction of shown in Fig. 5(b). By contrast, remains almost constant and the -linear coefficient of the specific heat gradually increases with across Tateiwa3 (). These results reveal that the Fermi level in FM2 is located just above a sharp peak in the majority up-spin band, and as increases across , it shifts down toward the peak when it enters the FM1 phase. In the ferromagnetic spin-pairing SC state, the large DOS in the up-spin band in FM1 enhances , leading to the highest value of  K, whereas its reduction in FM2 decreases , as shown in Figs. 5(a) and 5(c). However, it should be noted that as increases further up to  GPa, even though for FM1 remains rather larger than that for FM2,  K for FM1 at  GPa becomes lower than  K for FM2 at  GPa . In this context, the large increase in is not always a main factor that increases . Rather, the first-order transition from FM2 to FM1 at is responsible for the mediation of the up-spin Cooper pairing. The longitudinal FM spin fluctuations along the -axis Huxley2 () softens in energy at the critical end point for the first-order transition at . Therefore, we suggest that this longitudinal FM fluctuations along the -axis would be a mediator of the ferromagnetic spin-pairing SC where the majority up-spin band in the FM phases is gapped, while the minority down-spin band is not.

Figure 5: (color online). Pressure dependence of (a) ; (b) internal magnetic field, , at the Ge1 site; and (c) relative dependence of (solid circles) and (solid squares) estimated from the ferromagnetic spin-pairing model on a scale of and the -linear coefficient of the specific heat Tateiwa3 () (open squares) across (see text). It should be noted that dramatically increases when the first-order transition from FM2 at  GPa to FM1 at 1.24 GPa occurs across  GPa.

In another context, it is predicted that could be identified with the formation of a simultaneous charge- and spin-density wave (CSDW) induced by the imperfect nesting of the Fermi surface for the up-spin band and hence the superconducting pairing is mediated by CSDW fluctuations around Watanabe (). Although the NQR spectrum does not directly evidence an onset of static CSDW states, the remarkable increase in () across is relevant to the nesting at the Fermi surface for the up-spin band below .
In conclusion, the Ge-NQR measurements under pressure on well characterized UGe have revealed that the superconducting energy gap opens only with the line-node at the Fermi level in the majority up-spin band, but the down-band remains gapless. It is therefore concluded that ferromagnetic spin-pairing SC occurs in UGe. We have also shown that the first-order transition from FM1 to FM2 at  GPa occurs because the Fermi level is located just on the peak in the DOS of the up-spin band. The ferromagnetic spin-density fluctuations emerging in the vicinity of the critical end point for this first-order transition are considered to be the mediator of the onset of novel SC realized in ferromagnet UGe.
We thank H. Kotegawa, N. Tateiwa, S. Watanabe, and K. Miyake for fruitful discussions and comments. This study was supported by Grant-in-Aid for Creative Scientific Researchi15GS0213); the Ministry of Education, Culture, Sports, Science and Technology (MEXT); and the 21st Century COE Program (G18) supported by the Japan Society for the Promotion of Science (JSPS). A. H. was financially supported by a Grant-in-Aid for Exploratory Research of MEXT (No. 17654066).

References

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