a r X i v :c o n d -m a t /0407269v 2 [c o n d -m a t .s u p r -c o n ] 15 A u g 2004Perspectives on Nodal Superconductors
Kazumi Maki,1Stephan Haas,1David Parker,1and Hyekyung Won 21Department of Physics and Astronomy,University of Southern California,Los Angeles,CA 90089-0484USA 2Department of Physics,Hallym University,Chuncheon 200-702,South Korea (Dated:February 2,2008)Abstract In the last few years the gap symmetries of many new superconductors,including Sr 2RuO 4,CeCoIn 5,κ-(ET)2Cu(NCS)2,YNi 2B 2C and PrOs 4Sb 12,have been identified via angle-dependent magnetothermal conductivity measurements.However,a controversy still persists as to the na-ture of the superconductivity in Sr 2RuO 4.For PrOs 4Sb 12,spin-triplet superconductivity has recently been propod.Here,we also propo g-wave superconductivity for UPd 2Al ,∆(k )=∆cos(2χ),χ=ck z )bad on recent thermal conductivity data.PACS numbers:
1.Introduction
After the appearance of heavy-fermion superconductors and organic superconductors in 1979the gap symmetries of the new compounds have been a central issue[1].However,un-til recently only the d x2−y2-wave symmetry of the gap function∆(k)in high-T c cuprates has been established by the elegant
Jophson interferometry[2]and the angle resolved photoe-mission spectra(ARPES)[3].Unfortunately,so far the powerful techniques are unavailable for heavy-fermion superconductors and organic superconductors with lower superconducting transition temperatures T c≤10K.
In the last few years,Izawa et al have established the gap symmetries of superconductivity in Sr2RuO4[4],CeCoIn5[5],κ-ET2Cu(NCS)2[6],YNi2B2C[7]and PrOs4Sb12[8]through the angle dependent magnetothermal conductivity.This breakthrough relies in part on the availability of high-quality single crystals of the compounds and in part on the theoretical development initiated by Volovik.[9]Last year,we have reviewed the progress in[10].
In the prent paper,we focus on3recent topics in nodal superconductors.In spite of ample evidence for f-wave superconductivity in Sr2RuO4[10]the controversy regarding this compound appears to continue.Therefore in ction2we discuss the angle dependent mag-netospecific heat data by Deguchi et al[11].Now evidence for spin-triplet superconductivity in PrOs4Sb12is mounting.In ction3,we describe p+h-wave superconductivity for the A and B phas in PrOs4Sb12[12].Recently angle-dependent thermal conductivity data in the vortex state in UPd2Al3has been reported.[13]In ction4we analyze the angle-dependent magnetothermal conduc
tivityκyy when thefield is rotated within the z-x plane,and we conclude that∆(k)in UPd2Al3is given by∆(k)=∆cos(2χ)[14].In Fig.1we show the new|∆(k)|′s so far identified.
2.F-wave Superconductivity in Sr2RuO4
Superconductivity in Sr2RuO4was discovered in1994[15].Sr2RuO4is an isocrystal to La2CuO4,but it is metallic down to low temperatures and becomes superconducting around T=1.5K.An early review on Sr2RuO4can be found in Ref.[16].From the analogy to su-perfluid3He Rice and Sigrist[17]propod2D p-wave superconductivity.Indeed spin-triplet pairing and related chiral symmetry-breaking have been established[18,19,20].As sample quality improved around1999,both the specific heat data[21]and the superfluid density [22]indicated nodal structure in the superconducting order parameter of Sr2RuO4.The findings ruled out p-wave superconductivity and its generalization[23].Therefore,a variety
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FIG.1:From top left,2D f-wave-Sr2RuO4,d x2−y2-wave-CeCoIn5andκ-(ET)2Cu(NCS)2,s+g-wave-YNi2B2C,p+h-wave-PrOs4Sb12-A pha,p+h-wave-PrOs4Sb12-B pha.
of f-wave order parameters were suggested.[24]In Fig.2and Fig.3we show the specific heat data[21]
and the superfluid density data[22]compared with a variety of models. However,the experiments cannot tell us about the nodal structure of∆(k).In a quasi-2D system such as Sr2RuO4,the line nodes in∆(k)can be either vertical or horizontal.But vertical nodes are incompatible with the angular dependent magnetothermal conductivity [4]and the ultrasonic attenuation data[25].Furthermore,Ref.4indicates that the hori-zontal nodes are far away fromχ0=0.This suggests∆(k)=d e±iφcos(χ),i.e.2D f-wave superconductivity[26].零基础学习英语
This interpretation is contested by Deguchi et al[11].They measured the magnetospe-cific heat of Sr2RuO4in a rotating magneticfield down to100mK and found cusp-like features only in the regime0.12K<T<0.31K.From our earlier analysis of s+g-wave superconductivity[27,28],we deduce that there should be a point-like minigap with
FIG.2:Specific heat for2D p-wave and f-wave models for Sr2RuO4.
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FIG.3:Superfluid density for2D p-wave and f-wave models for Sr2RuO4.
∆min∼0.1K.The simplest triplet gap function which has the minigaps is
∆(k)=d e±iφ(1+a cos(4φ)cos(χ))(1)
英语b级作文where|1−a|≤0.1.Deguchi et al have propod the Miyake-Nariyiko(MN)model[29], in order to describe the measured specific heat.However,it is easy to e that the MN model cannot give the cusp-like features in the magnetospecific heat.Also,the MN model cannot describe the obrved T2specific heat or the T-linear dependence of the superfluid density.Moreover,the angular dependent thermal conductivity data and the universal heat conduction inκxx by Suzuki et al[30]are incompatible with the MN model.Therefore further experiments on Sr2RuO4are highly desirable.We have propod that the optical conductivity[31],the Raman scattering[32]and the supercurrent[33,34]in Sr2RuO4will provide further insight on its superconductivity.
3.Triplet Superconductivity in PrOs4Sb12
Superconductivity with T c=1.8K has been discovered very recently in the skutterudite PrOs4Sb12[3
5,36,37].Angle-dependent thermal conductivity data on this system has re-vealed a multi-pha structure,characterized by a gap function with point nodes.[8]In order to account for this nodal structure s+g-wave superconductivity has been propod.[10,38] Recently there has been mounting experimental evidence for triplet superconductivity in this compound.First,fromµSR measurements Aoki et al discovered a remnant mag-netization in the B-pha of this compound,indicating triplet pairing.[39]Second,the thermal conductivity measurement in a magneticfield down to low-temperature(T>150 mk)indicatesκzz∼T and H[40],consistent with triplet pairing.Later we shall discussκzz
measured in a magneticfield rotated within the z-x plane.This data is fully consistent with triplet p+h-wave superconductivity in PrOs4Sb12.Finally,a recently reported NMR result for the Knight shift by Tou et al[41]also suggests the triplet pairing.Here we propo p+h-wave order parameters
3
japane noodle∆A(k)=
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√
Γ∆≪v
sin2(θ),A−pha(4)
8∆2
3v2eH
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and in the clean limit(v
√Γ∆)
κzz/κ00=1+3v2eH2∆
v
√
幻觉病12Γ∆
ln( Γ)sin2(θ)ln(∆eH sin(θ)),B−pha(7)
whereκn andκ00are the thermal conductivity in the normal state and the thermal conduc-tivity in the limit of universal heat conductionΓ→0,T→0.Here,Γis the quasiparticle scattering rate in the normal state,andθis the angle H makes from theˆz axis.In both Eq.(5)and Eq.(7)we have assumed that the nodes in the B-pha are parallel to the z axis. Otherwiκzz is smaller by a factor of10∼50.In Fig.4we compare the obrved angle dependent thermal conductivity with Eq.(6)and(7).The equations give an excellentfit.
FIG.4:Angular-dependent thermal conductivity in PrOs4Sb12.
From this we extract v=0.96×107cm/c andΓ=0.1K,where u is made of the weak-coupling theory gaps∆A=4.2K and∆B=3.5K for the A and B pha respectively.Note that de Haas-van Alphen measurements[44]give comparable values of v(0.7×107cm/c [α-band],0.6×107cm/c[β-band]and0.23×107cm/c[γband]).
4.G-wave superconductivity in UPd2Al3
This heavy-fermion superconductor with T c≃2K was discovered by Geibel et al[45] in1991.The reduction of the Knight shift in the superconducting state en in NMR [46]and the Pauli limiting of H c2in UPd2Al3[47]established spin-singlet pairing.Nodal superconductivity with horizontal nodes has jepn
been deduced from the thermal conductivity
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