Anomalous change in the de Haas-van Alphen oscillations of CeCoIn 5 at ultra-low temperatures
Hiroaki Shishido, Shogo Yamada, Kaori Sugii, Masaaki Shimozawa, Youichi Yanase, Minoru Yamashita
aa r X i v : . [ c ond - m a t . s t r- e l ] M a r Anomalous change in the de Haas-van Alphen oscillations of CeCoIn at ultra-lowtemperatures Hiroaki Shishido , , ∗ Shogo Yamada , Kaori Sugii , Masaaki Shimozawa , Youichi Yanase , and Minoru Yamashita † Department of Physics and Electronics, Graduate School of Engineering,Osaka Prefecture University, Sakai, Osaka 599-8531, Japan. Institute for Nanofabrication Research, Osaka Prefecture University, Sakai, Osaka 599-8531, Japan. The Institute for Solid State Physics, The University of Tokyo, Kashiwa, 277-8581, Japan Department of Physics, Kyoto University, Kyoto 606-8502, Japan. (Dated: March 15, 2018)We have performed de Haas-van Alphen (dHvA) measurements of the heavy-fermion supercon-ductor CeCoIn down to 2 mK above the upper critical field. We find that the dHvA amplitudesshow an anomalous suppression, concomitantly with a shift of the dHvA frequency, below the transi-tion temperature T n = 20 mK. We suggest that the change is owing to magnetic breakdown causedby a field-induced antiferromagnetic (AFM) state emerging below T n , revealing the origin of thefield-induced quantum critical point (QCP) in CeCoIn . The field dependence of T n is found tobe very weak for 7–10 T, implying that an enhancement of AFM order by suppressing the criti-cal spin fluctuations near the AFM QCP competes with the field suppression effect on the AFMphase. We suggest that the appearance of a field-induced AFM phase is a generic feature of un-conventional superconductors, which emerge near an AFM QCP, including CeCoIn , CeRhIn , andhigh- T c cuprates. The competition between distinct quantum states at aquantum critical point (QCP), where the second-ordertransition temperature is suppressed to absolute zero,prevents the ground state from attaining either state andenhances quantum fluctuation in the vicinity of the QCP[1]. The fate of electronic states influenced by the en-hanced quantum fluctuations near a QCP has been acentral issue of modern physics. Many anomalous statessuch as non-Fermi liquid (NFL) behavior and unconven-tional superconductivity have been observed near QCPsin various materials including high- T c cuprates [1], iron-pnictides [2], and heavy fermions (HFs) [3]. These uncon-ventional superconductors often show a maximum tran-sition temperature near the QCP, suggesting that the en-hanced quantum fluctuation is the key to understandingtheir superconductivity.HF systems have emerged as prototypical systems forstudying QCPs, because the strong mass renormalizationthat occurs through hybridization of f -electrons withconduction electrons lowers the relevant energy scale onwhich the effects take place. Thus, the ground statecan be easily tuned at experimentally accessible pres-sures or magnetic fields. In particular, unconventionalsuperconductivity in HFs has been most extensively stud-ied in CeCoIn because a d -wave superconducting stateemerges at ambient pressure without chemical substitu-tion [4]. The d -wave superconductivity has been shown tobe located in the vicinity of an antiferromagnetic (AFM)QCP [5, 6]. Various measurements have further revealeda crossover of NFL behavior at zero field to FL behaviorat high fields, indicating the presence of a field-inducedQCP near the upper critical field, H c2 [7–10].Despite the accumulating evidence for field-inducedQCP, no AFM state corresponding to the QCP has beenobserved outside the superconducting phase. This ap- parent absence has been attributed to the AFM statebeing hidden at an inaccessible negative pressure [11] orsuperseded by the superconductivity [6]. For H//ab , aspin-density-wave order is induced inside the supercon-ducting phase near H c2 , which is recently discussed interms of a condensation of the spin resonance in the su-perconducting phase [12, 13]. This coexisting “Q-phase”[14], however, vanishes when the field is tilted from the ab plane [15] whereas the NFL behaviors are observedregardless of the field direction. Thus, it is necessary tosearch for an ordered state for H//c at lower tempera-tures to clarify the origin of the field-induced QCP andits interplay with the unconventional superconductivityof the material.In this Letter, we report de Haas-van Alphen (dHvA)measurements for
H//c down to 2 mK using our home-made nuclear-demagnetization cryostat. We find that thedHvA amplitudes deviate from the conventional Lifshitz-Kosevich (LK) formula [16] and show an anomalous de-crease with a shift of the dHvA frequency below a tran-sition temperature T n , suggesting an emergence of theputative AFM state.High-quality single crystals of CeCoIn were grown bythe In-flux method [17]. Measurements of the magnetictorque were performed using a capacitance cantilevertechnique up to 10 T. The lowest temperature of thecryostat was measured by a melting curve thermometercalibrated by the transition points of He [18]. To ensurethe lowest temperature of the samples, the samples wereimmersed in liquid He of which the temperature wasmonitored by a vibrating wire thermometer [19] placed inthe same liquid He (see Supplemental Material (SM) [20]for details).The dHvA oscillation for
H//c and the correspondingfast Fourier transformation (FFT) spectrum are shownin Figs. 1(a) and 1(b), respectively. The fundamentalbranches are assigned as α , α , α , β , ε , and γ as indi-cated in Fig. 1(b), which agree with those of the previousreport [21]. As shown in Fig. 1(b), the dHvA amplitudesof the all branches increased as temperature was loweredto 20 mK in accordance with the LK formula [16]. Thetemperature dependence of the dHvA amplitudes of all α branches above 20 mK can be well fitted by the LK for-mula (the solid lines in Fig. 1(c)), enabling us to estimatethe effective cyclotron mass m ∗ c for each α branch whichis also in good agreement with the previous report [21].However, the dHvA amplitudes of all α branches devi-ated from the LK formula below ∼
20 mK and decreasedas lowering the temperature (Fig. 1(c)). This anoma-lous decrease was also observed in γ branch but not in ε branch [20]. (c) α α α m * = 12.6 m m m T (mK) d H v A A m p lit ud e ( a r b . un it ) εγ αα α β T = 20 mK 2 mK 100 mK 149 mK 250 mK(b) H // cT = 2 mKCeCoIn µ H (a) FIG. 1. Quantum oscillations of CeCoIn . (a) dHvA oscilla-tion at 2 mK for 7.6–9.9 T after subtracting the backgroundsignal. (b) FFT spectra of the dHvA oscillations obtained inthe same field range as (a) at 20, 2, 100, 149, and 250 mK indescending order of the signal size. Note that the spectrumat 2 mK is smaller than that at 20 mK. See the main text andSM for details [20]. (c) The temperature dependence of thedHvA amplitudes of α (black triangles), α (blue squares),and α (red circles) obtained in the same field range of (a).The open (filled) data were taken by using nuclear demagne-tization (dilution refrigeration). The solid lines show fits forthe data above T n from the standard Lifshitz-Kosevich for-mula [16]. The cyclotron effective mass ( m ∗ c ) estimated fromthe LK fit for each α branch is indicated with the data. To analyze the field dependence in detail, the dHvAsignal of α branch, which showed the largest signal,was isolated by a steep bandpass filter. The tempera-ture dependence of the dHvA amplitude at different field strengths is shown in Fig. 2(a). The decrease of the am-plitude below T n was most clearly observed at ∼ T n slightly decreased and the decrease in the dHvAamplitudes became smaller. The decrease was not clearlyresolved at 6.0 T because of the small dHvA signal.Further, we analyzed the temperature dependence ofthe dHvA frequency by using a phase shift analysis [16].The phase shift of a dHvA oscillation, ∆ P ( T, H ) = P ( T, H ) − P ( T , H ), is proportional to the shift ofthe dHvA frequency ∆ F ( T, H ) = F ( T, H ) − F ( T , H )as, ∆ P ( T, H ) /P ( T , H ) = ∆ F ( T, H ) /F ( T , H ), where P ( T, H ) is the peak field of the dHvA oscillation and T is a reference temperature. As shown in Fig. 2(b), thetemperature dependence of ∆ F ( T, H ) is found to show akink at T ∼ T n , which was followed by a slight decreaseof the frequency at lower temperatures.A similar suppression of the dHvA amplitude of α hasbeen reported below 100 mK and at 13–15 T [22], whichwas discussed in terms of a strong spin dependence ofthe effective mass. We applied the spin-dependent LKformula to our results at various fields (the blue dashedlines in Fig. 2(a)). As shown in Fig. 2(a), the spin-dependent LK formula reasonably reproduces the dataat 9.5 T, which is consistent with the previous report athigher fields [22]. However, the spin-dependent LK for-mula clearly fails to reproduce the rapid decrease of thedHvA signal below T n observed for 7.6–9.0 T, even by as-suming a very large difference between the effective massof the spin-up electrons and that of the spin-down ones(see SM [20] for details of these fittings). Moreover, thismodel cannot explain the change in the dHvA frequencybelow T n . These results indicate that a drastic changein the electronic state of the material occurs below T n ,leading us to suggest that the anomalous change in thedHvA amplitudes corresponds to a phase transition.The revised H – T phase diagram of CeCoIn with thefield-induced phase is shown in Fig. 3(a). We found thatthe field dependence of T n was very weak for 7–10 T, al-though we have to note that determining T n has a largeambiguity of ∼
20% because the dHvA measurementswere performed by sweeping fields at a constant temper-ature. We have observed similar anomalies of the dHvAamplitude in a different single crystal [20]. An anoma-lous reduction has also been observed in the magnetoresistance at ∼
20 mK and at 8 T [9, 20]. It should alsobe noted that the absence of the anomaly at low field isconsistent with the previous work done at 6–7 T [20, 23].Here, we suggest that a field-induced AFM order pro-vides the most plausible explanation for the anomalyobserved in our dHvA measurements. An AFM tran-sition in the simple tetragonal structure of CeCoIn ( P mmm ) modifies the dHvA frequencies by a foldingof the Brillouin zone. However, the paramagnetic Fermisurface can still be observed in the AFM phase with alarger attenuation by magnetic breakdown [16], as ob- ! " $ % & ’ ()* + , $ - % . / +1 )* m !$9$:06$;$:04$;$<06$;$<04$;=0>$;=0?$;=04$;>06$;>04$; " @ A" B D ! !" " $ %&’ " ( ) !" " %’’*’+’,’-’’ !" () /& (b)(a) FIG. 2. (Color online) (a) Temperature dependence ofthe dHvA amplitudes of α at different fields. The dataare shifted for clarity. The open (filled) data were takenby using nuclear demagnetization (dilution refrigeration). Aclear suppression of the dHvA amplitude from the standardLifshitz-Kosevich (LK) formula (solid lines, m ∗ c /m used forthe fits are shown right) [16] was observed. The transitiontemperature T n is determined as the onset temperature wherethe dHvA amplitude starts to deviate from the LK formula(shown by the arrows). The blue dashed lines show the fits forthe spin-dependent LK formula [22] (see the main text). Thesaturation observed at the lowest temperature may be causedby a saturation of the AFM energy gap or non-equilibriumof the sample temperature (see SM [20] for details). (b)The temperature dependence of the normalized shift of thedHvA frequency ∆ F ( T, H ) /F (150 mK , H ) of α . The dataare shifted for clarity. The dashed lines are guides to the eye. served in NdIn [24]. The magnetic breakdown proba-bility ( P MB ) is given by P MB = exp( − ǫ g / ~ ω c ǫ F ), where ǫ g is the energy gap for the magnetic breakdown, ω c isthe cyclotron frequency, and ǫ F is the Fermi energy [16].Because T n is two or three orders magnitudes smallerthan that of typical 4 f -electron AFM materials (wherethe AFM transition typically takes place at a few K orhigher e.g. ∼ [24]), the magnetic momentwould be much smaller than these AFM materials, re-sulting in a tiny energy gap at the magnetic zone bound-ary. As a result, majority of electrons undergo magneticbreakdown and the new dHvA branches from the foldedBrillouin zone in the AFM phase may not be observablewithin our experimental accuracy. This explains the sup-pression of the dHvA amplitudes below T n without newdHvA signals from the folded Brillouin zone.The appearance of the AFM order is also supported by the slight change of the dHvA frequency below T n (Fig.2(b)). It is known that the measured dHvA frequency f m is given by the zero-temperature extrapolation of thetrue dHvA frequency f t as f m = f t − H ( ∂f t /∂H ). Eitherthe change of the slope ( ∂f t /∂H ) and/or the Fermi sur-face size ( ∝ f t ) cause a change of f m . Even if magneticbreakdown occurs, f m can be modified by a change of theslope of f t caused by the AFM ordering. Therefore, al-thought the change in the dHvA frequency is very small,the field dependence of the dHvA frequency is consistentwith the field-induced AFM order.The absence of the anomalous suppression in ε branch[20] is also consistent with the AFM transition because ε branch corresponds to a small pocket located at the zonecenter [25] and thus is hardly affected by the band fold-ing. In addition, a kink in the temperature dependenceof the resistivity at 8 T has been observed at tempera-tures very close to T n (see the left column of Fig. 3 inref. [9]). Although the origin of the kink is not discussedin ref. [9], the kink can be consistent with a reductionof the magnetoresistance in the AFM phase. Therefore,we conclude that the anomalous change of the dHvA am-plitudes below T n is most consistent with the emergenceof an AFM phase. We note that splitting of the dHvAfrequency typically expected for an AFM phase was notobserved simply because our field range of 6–10 T wastoo narrow to allow the detection. A change of the torquesignal by AFM order was also not resolved at T n , whichwas probably hindered by the change of the dHvA sig-nal. The possibilities of multipole ordering, Lifshitz tran-sitions and nuclear spin ordering can be safely excludedas described in SM [20].The revised H – T phase diagram of CeCoIn (Fig.3(a)) reveals that the field-induced AFM phase is locatedat the boundary of the unconventional superconductiv-ity. A similar H – T phase diagram has been observedin the sister compound CeRhIn where an AFM groundstate at ambient pressure changes to a superconductingstate under pressure [11, 29]. The pressure dependenceof the H – T phase diagram of CeRhIn can be summa-rized in a schematic H – T – x phase diagram (Fig. 3(b))where x denotes pressure for CeRhIn . Given that thefield-induced phase is observed at very low temperature,the H – T phase diagram of CeCoIn at ambient pressuremay be considered to be a cross section at the vicinityof the AFM QCP in the H – T – x phase diagram. Thus,CeCoIn is a prominent superconductor where the inter-play of unconventional superconductivity, magnetic or-der, and non-Fermi liquid behaviors near the QCP canbe studied without ambiguity caused by the applicationof pressure or chemical doping. Such H – T – x phase dia-grams ( x = pressure or chemical substitution) have notonly been observed in HFs [3, 11, 25, 29] but also inhigh- T c cuprates [1, 30]. These similarities suggest thatthe H – T – x phase diagram is generic to unconventionalsuperconductors, which emerge in the vicinity of an AFM µ H ( T ) CeCoIn c FLSC1st 2nd00510 1 2 3 T (K) FLNFLNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNFNFNFNFNFFFFFFFFFFFFFFFFFFFLFLFLFLFLLLLLAFM xHT LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL SCCeCoIn QCP FI-QCP 0.60.811.2 NormalBCSFFLO + AFM FFLOAFM0 0.1 0.2 0.3 0.4 0.5 (a) (b) (c)
FIG. 3. (Color online) Phase diagrams of CeCoIn . (a) H – T phase diagram. Field-induced phase found by our measurements(pink), the Fermi liquid (FL) region (gray, taken from ref. [10]), the superconducting (SC) phase (blue), and the high-field SCphase (yellow) [26] are shown. Both T n (red circles) and T FL are multiplied by 4 for clarity. The black dashed and solid linesrepresent the first- and the second-order SC transitions, respectively. (b) A schematic H – T – x phase diagram near the AFMQCP, where x denotes pressure or chemical substitution. The cross section corresponds to the H – T phase diagram of CeCoIn ,in which a presumed field-induced QCP (FI-QCP) is also shown at the intersection with the AFM boundary. (c) A calculatedphase diagram near H c2 by a mean field approximation (solid symbols and solid lines) [27] and a FLEX approximation (opensymbols and dashed line) [28]. The temperature and the magnetic field are normalized by the transition temperature T c andthe Pauli limiting field H p = 1 . T c , respectively. The lines are guides to the eye (see SM [20] for details). QCP, including CeCoIn .Remarkably, the transition temperature of the field-induced AFM phase depended on field only weakly for7–10 T even though the Zeeman energy at 10 T is aboutthree orders of magnitude larger than k B T n . This weakfield dependence implies that an enhancement effect onAFM order by the magnetic field competes with the sup-pression effect. If the critical spin fluctuation near theQCP [5–10] suppresses AFM order, the magnetic fieldsuppresses the critical spin fluctuation, giving rise to anenhancement effect on AFM order. To take into accountthe spin fluctuation effect near an AFM QCP, we ex-amined the AFM transition line in the normal state ofCeCoIn by adopting the fluctuation exchange (FLEX)approximation [28]. We also calculated the H – T phasediagram in the superconducting phase of CeCoIn witha neighboring AFM phase by the mean field approxima-tion [27] and plot them in Fig. 3 (c). Whereas the meanfield method can reproduce the first order superconduct-ing transition near H c2 , the critical spin fluctuation canbe included only in the FLEX method. As shown in Fig.3 (c), the AFM transition temperature calculated by theFLEX approximation (dashed line) is found to increaseas field increases, in contrast to the transition line witha negative slope by the mean field approximation (solidline above H c2 ). This difference demonstrates the fieldenhancement effect on the AFM transition temperatureby suppressing the critical spin fluctuation as consistentwith our results. In fact, such a transition line has beenreported in CeIrSi near an AFM QCP under pressure of ∼ . H//c , which has been discussed as a spatiallyinhomogeneous superconducting state [26] termed as aFulde–Ferrell–Larkin–Ovchinnikov (FFLO) state [31, 32].In the case of d -wave superconductivity located in thevicinity of an AFM QCP, such as that found in CeCoIn ,calculations by the FLEX approximation have indicatedthat an FFLO state is stabilized [28], which is also con-sistent with NMR measurements [26]. For H//ab , AFMorder coexists with the superconductivity in the Q-phase[14]. This Q-phase has also been discussed in terms ofan FFLO state because an FFLO state enhances AFMorder through the appearance of an Andreev bound statelocalized around the gap nodes in real space and by cou-pling between AFM order and pair-density-wave [27, 33].As shown in Fig. 3 (c), the transition temperature of theAFM phase increases in the FFLO phase. Therefore, wespeculate that, if a field-induced AFM phase is also hid-den at ultra-low temperatures for
H//ab , the AFM phaseis enhanced in the FFLO state, as shown in Fig. 3(c),and is observed as the Q-phase. Confirming AFM orderfor both
H//c and
H//ab and identifying the q vector bylocal probe measurements such as nuclear magnetic res-onance or muon spin resonance will be important futureissues to clarify the relation between these phases.In summary, we have observed anomalous changes inthe dHvA oscillations in the normal state of CeCoIn be-low T n = 20 mK. We attribute these anomalies to theemergence of an AFM state which is the origin of theAFM QCP in CeCoIn . We suggest that CeCoIn sharesa phase diagram with other materials near an AFM QCP.We also suggest that AFM order is enhanced by sup-pressing the critical spin fluctuations in the vicinity ofthe AFM QCP in high field, which gives rise to the weakfield dependence of T n as supported by the FLEX cal-culation. We believe that the development of the dHvAmeasurements under ultra-low temperature has extensivepotential to shed a new light on unexplored phenomenaat ultra-low temperatures.We thank D. Aoki, S. Kambe, Y. Matsuda, Y. Mat-sumoto, Y. ¯Onuki, H. Sakai, T. Shibauchi, Y. Tada, andH. Tokunaga for discussions. This work was performedunder the Visiting Researcher’s Program of the Institutefor Solid State Physics, University of Tokyo, and was sup-ported by the Toray Science Foundation, and KAKENHI(Grants-in-Aid for Scientific Research) Grant Numbers15K05164, 15H05745, 15H05884, 15K17691, 16H00991,16K05456, 16K17742, 16K17743, and 17K18747. ∗ [email protected] † [email protected][1] S. Sachdev, Phys. Status Solidi B , 537 (2010).[2] T. Shibauchi, A. Carrington, and Y. Matsuda,Annual Review of Condensed Matter Physics , 113 (2014).[3] P. Gegenwart, Q. Si, and F. Steglich,Nature Physics , 186 (2008).[4] K. Izawa, H. Yamaguchi, Y. Matsuda,H. Shishido, R. Settai, and Y. ¯Onuki,Phys. Rev. Lett. , 057002 (2001).[5] Y. Kawasaki, S. Kawasaki, M. Yashima,T. Mito, G. qing Zheng, Y. Kitaoka,H. Shishido, R. Settai, Y. Haga, and Y. ¯Onuki,Journal of the Physical Society of Japan , 2308 (2003).[6] Y. Tokiwa, E. D. Bauer, and P. Gegenwart,Phys. Rev. Lett. , 107003 (2013).[7] A. Bianchi, R. Movshovich, I. Vekhter, P. G. Pagliuso,and J. L. Sarrao, Phys. Rev. Lett. , 257001 (2003).[8] J. Paglione, M. A. Tanatar, D. G. Hawthorn,E. Boaknin, R. W. Hill, F. Ronning, M. Suther-land, L. Taillefer, C. Petrovic, and P. C. Canfield,Phys. Rev. Lett. , 246405 (2003).[9] L. Howald, G. Seyfarth, G. Knebel,G. Lapertot, D. Aoki, and J.-P. Brison,Journal of the Physical Society of Japan , 024710 (2011).[10] S. Zaum, K. Grube, R. Sch¨afer, E. D. Bauer,J. D. Thompson, and H. v. L¨ohneysen,Phys. Rev. Lett. , 087003 (2011).[11] J. L. Sarrao and J. D. Thompson,Journal of the Physical Society of Japan , 051013 (2007).[12] C. Stock, C. Broholm, Y. Zhao, F. Demmel,H. J. Kang, K. C. Rule, and C. Petrovic,Phys. Rev. Lett. , 167207 (2012). [13] S. Raymond and G. Lapertot,Phys. Rev. Lett. , 037001 (2015).[14] M. Kenzelmann, T. Str¨assle, C. Niedermayer, M. Sigrist,B. Padmanabhan, M. Zolliker, A. D. Bianchi,R. Movshovich, E. D. Bauer, J. L. Sarrao, and J. D.Thompson, Science , 1652 (2008).[15] V. F. Correa, T. P. Murphy, C. Martin, K. M. Purcell,E. C. Palm, G. M. Schmiedeshoff, J. C. Cooley, andS. W. Tozer, Phys. Rev. Lett. , 087001 (2007).[16] D. Shoenberg, Magnetic oscillations in metals (Cam-bridge University press, Cambridge, 1984).[17] H. Shishido, R. Settai, D. Aoki, S. Ikeda, H. Nakawaki,N. Nakamura, T. Iizuka, Y. Inada, K. Sugiyama,T. Takeuchi, K. Kindo, T. C. Kobayashi, Y. Haga,H. Harima, Y. Aoki, T. Namiki, H. Sato, and Y. ¯Onuki,Journal of the Physical Society of Japan , 162 (2002).[18] D. S. Greywall, Phys. Rev. B , 2675 (1985).[19] D. C. Carless, H. E. Hall, and J. R. Hook,Journal of Low Temperature Physics , 583 (1983).[20] See Supplemental Material at http://yamashita.issp.u-tokyo.ac.jp/arXiv/Shishido.pdf for more information.[21] R. Settai, H. Shishido, S. Ikeda,Y. Murakawa, M. Nakashima, D. Aoki,Y. Haga, H. Harima, and Y. ¯Onuki,Journal of Physics: Condensed Matter , L627 (2001).[22] A. McCollam, S. R. Julian, P. M. C. Rourke, D. Aoki,and J. Flouquet, Phys. Rev. Lett. , 186401 (2005).[23] A. McCollam, J.-S. Xia, J. Flouquet, D. Aoki, and S. Ju-lian, Physica B: Condensed Matter , 717 (2008).[24] R. Settai, T. Ebihara, H. Sugawara, N. Kimura,M. Takashita, H. Ikezawa, K. Ichihashi, and Y. ¯Onuki,Physica B: Condensed Matter , 102 (1994).[25] R. Settai, K. Katayama, D. Aoki, I. Sheikin,G. Knebel, J. Flouquet, and Y. ¯Onuki,Journal of the Physical Society of Japan , 094703 (2011).[26] K. Kumagai, M. Saitoh, T. Oyaizu, Y. Furukawa,S. Takashima, M. Nohara, H. Takagi, and Y. Matsuda,Phys. Rev. Lett. , 227002 (2006).[27] Y. Yanase and M. Sigrist,Journal of the Physical Society of Japan , 114715 (2009).[28] Y. Yanase, Journal of the Physical Society of Japan , 063705 (2008).[29] T. Park, F. Ronning, H. Q. Yuan, M. B. Salamon,R. Movshovich, J. L. Sarrao, and J. D. Thompson,Nature , 65 (2006).[30] B. Lake, H. M. Ronnow, N. B. Christensen, G. Aep-pli, K. Lefmann, D. F. McMorrow, P. Vorderwisch,P. Smeibidl, N. Mangkorntong, T. Sasagawa, M. Nohara,H. Takagi, and T. E. Mason, Nature , 299 (2002).[31] P. Fulde and R. A. Ferrell, Phys. Rev. , A550 (1964).[32] A. I. Larkin and Y. N. Ovchinnikov, Sov. Phys. JETP , 762 (1965).[33] D. F. Agterberg, M. Sigrist, and H. Tsunetsugu,Phys. Rev. Lett.102