Ann. Phvsik 5 (1996) 320-332 Annalen der Physik 0 Johann Ambrosius Barth 1996 D-wave superconductivity: New paradigm? Kazumi Maki' and Hyekyung Won aDepartment of Physics and Astronomy, University of Southern California, Los Angeles, CA900890484, USA Department of Physics and Interdisciplinary Research Center in Physical Science, Hallym University, Chunchon, 200-702, South Korea Received 20 December 1995. revised version 19 February 1996, accepted 23 February 1996 Dedicated to Peter Fulde on the occasion of his 60th birthday Abstract. As the 10th anniversary of the discovery of high T, cuprate superconductors by Bednorz and Miiller is approaching, a simple picture of high T, cuprates based on d-wave superconductivity is emerging. Here we shall describe the effect of impurity scattering and compare with experiments on Zn-substituted LSCO and YBCO. We shall discuss some characteristics of the vortex state in dwave superconductor. In a magnetic field parallel to the c axis a square vortex lattice tilted by 45" from the a axis is stable, which have been seen experimentally though elongated in the b direction in monocrystals of YBCO. Keywords: d-Wave superconductivity; High T, cuprates; Vortex state. 1 Introduction We are now approaching the 10th anniversary of the epoch making discovery of high T, cuprate superconductivity by Bednorz and Miiller [l]. From the beginning to understand the origin and the nature of the superconductivity in this new ever growing class of materials has been one of the central issues of the strongly correlated electron systems. It was also clear that these materials are quite different from ordinary metallic systems like Pb, Nb, or A1 where the BCS theory [2] based on the electron-phonon interaction describes s-wave superconductivity. Already in 1987 Anderson [3] proposed one-band 2D Hubbard model should be a proper model to describe these new classes of superconductivity based on the fact that these superconductivities appear in the proximity of the antiferromagnetic phase [4]. Later the model is further refined by several people [5-91 into 2D t-J model. We believe with possible exception of NCCO [lo] 2D t-J model should be the model for high T, cuprates. If you take 2D t-J model and a certain amount of Fermi surface nesting, you will have strong antiparamagnon (i. e. antiferromagnetic spin fluctuation) [ 11-13]. Then the exchange of antiparamagnon naturally leads to d-wave superconductivity [ 14, 151. This is the antiparamagnon scenario for d-wave superconductivity and fits surprisingly K. Maki, H. Won, d-Wave superconductivity: New paradigm? 32 1 well with one of possible paths we pursue following the renormalization analysis of 2D fermion systems with repulsive interaction [16]. So there will be no possibility that the quasi-particles in the Cu-02 plane may be described by a Luttinger liquid [17] or a marginal Fermi liquid [18] as long as they are in the metallic state. It is therefore somewhat surprising that most of microscopic model analysis of the gap equation with high speed computer starts from 2D Hubbard model [19-211 rather than 2D t-J model. It is highly desirable that these analyses are redone in the framework of 2D t-J model. There is an accumulating body of experimental evidences for d-wave superconductivity though some of them may be still controversial. First AWES from Bi2212 [22], the T-linear dependence of superfluid density at low temperatures in YBCO [23], electron Raman scattering from Bi2212 [24], the T2-specific heat observed in the YBCO [25] and LSCO [26] and the dependence of the residual density of states observed in the vortex state of YBCO [25, 271 all point to the d-wave superconductivity. But more direct and clearer evidences are x-shift observed by SQUID type experiments [28-331. The most elegant among them appear to be the detection of magnetic flux in the tricrystal geometry [29, 31, 331. Finally, we should add that the absence of the Hebel-Slichter peak in the nuclear spin lattice relaxation rate of Cu, 0 and Y in YBCO [34] started the theoretical activity leading to the d-wave model [35, 361. Within RPA type analysis we have pointed out that the inelastic neutron scattering data from LSCO [37] and YBCO [4] are fully consistent with the dwave superconductivity but incompatible with the s-wave superconductivity [ 13, 381. Also a surprising paramagnetic Meissner effect (or the Wohlleben effect) in granular high T, cuprates like Bi2Sr2CaCu208 [39] is most naturally interpreted in terms of dwave superconductivity 1401. In the following section we shall give a short review of the effect of impurity scattering. Then in section 3 we describe new results on the fourfold symmetry seen in the vortex state, which are very revealing on the nature of d-wave superconductivity [41]. 2 Impurity scattering As is well known a small amount of Zn substitution of Cu in the Cu-02 plane of high T, cuprate superconductor has dramatic effects; the rapid suppression of the superconducting transition temperature T, and the concomitant rapid rise of the residual density of states as seen both by NMR in YBCO [42, 431 and in LSCO [44] and low-temperature specific heat of LSCO [26]. Now many theoretical papers on this subject are available [4547]. Within the standard treatment [45-48] the quasi particle Green's function in the presence of impurity scattering is given by G-'(L,o) = iii, - 6 ~ -3 A(L)Pi ) (1) Where pi and p3 are Pauli spin matrices operating in the Nambu space and 6 the renormalized Matsubara frequency determined by 322 Ann. Physik 5 (1996) where r = ni/zNo, A(i) = Af = Acos(2$), ni is the impurity concentration and NO is the electron density of states in the normal state per spin. Here we assumed that the impurity scattering in the unitarity limit (i.e. the Zesonance on the Fermi surface) and ( ) means average over $ and c$ is the angle k (in the a-b plane) makes from the a axis. Then following the standard procedure the superconducting transition temperature T,, the order parameter etc. are determined. In particular T, follows the standard Abrikosov-Gor'kov relation [48] where T,, is the transition temperature in the absence of impurities and Y ( z ) is the di-gamma function. T, vanishes when r = Tc = 0.88Tc,. Perhaps of particular interest is the residual density of states N (0), which is given by where COsatisfies - - %&xal-&o4 0.6 - - - - 2 - \ G 0.4 - 0.0 0.0 A I I 0.2 I I 0.4 I I N(0) / N y 1 I 0.8 I 1.o Fig. 1 T,/T,, vs. N(O)/No obtained from the specific heat data (Momono and Ido 126, 471) La2-xSr, (Cu2-,ZnY)O4.It is remarkable that the data from x =0.22 are almost right on the theoretical curve. K. Maki,H. Won, d-Wave superconductivity: New paradigm? 323 Fig. 2 The density of states for r'/A = 0 (-1, 0.01 (...), 0.05 (- - - -), 0.1 (-.-.-) and 0.2 (- - - -) is / shown as function of E/A (from Hotta [451). and K(z) is the complete elliptic integral. It is possible to eliminate r/A at T=O K and draw T n c 0 as function of N(O)/No, which is shown in Fig. 1 . In the same figure we show experimental points obtained by specific heat measurement of Zn-doped LSCO by Momono and Ido [26,49]. It is rather surprising that Zn-doped LSCO data follow SO closely the theortetical prediction especially when x =0.22. For smaller x (say x=0.16) the experimental points lie clearly inside of the theoretical curve. If we interpret this deviation as due to the strong-coupling effect, we estimate A(0) increases about 5 % due to the strong-coupling effect. In Fig. 2 we show the density of states as function of E / A [45]. Note for small T/A N(E) has a peak at E=O, while for larger r/A ( o r 6 2 0.25) N(0) is the minimum of N(E). This means first of all the electronic specific heat in d-wave superconductor changes from the even power series in temperature T in the pure limit [50]to the odd power series suddenly when Zn is added as seen by Momono and Id0 in LSCO [26]. So the specific heat is now expanded as [51] c, = N ( 0 ) (F + p (2) + -) 7' *7' - where Aoo is the superconducting order parameter at T=O K in the pure system. Then starts from a negative value for T/T, << 1 and then changes sign around r/T, = 0.25 as shown in Fig. 3. In a superconductor there are 2 supeffluid densities p,( T ) G (h(O)/h(T))' and Ps,,,,i,,( 7')in general. Then the static spin susceptibility is given by B 324 Fig. 3 The normalized coefficient rir, ~491. Ann. Physik 5 (1996) p in the T3 term in the specific heat is shown as function of Fig. 4 The superfluid density p,(T) (solid lines) and the spin superfluid density p,,,(T) dotted line are shown as function of T&, for r/& = 0,0.1,0.2,0.3 from top to bottom. Note pn,+.(T) = 1 - p,,,,,(T) is accessible by the Knight shift measurement. Here we correct an error in fig. 6 of [45]. K. Maki, H. Won, d-Wave superconductivity: New paradigm? 325 x , ( T ) = X n P n , s p j n ( T ) = ~ n ( l- P s , s p i n ( T ) ) (7) where xn is the susceptibility at T=Tc. In a pure singlet superconductor we have p,,,,,(T) z p,(T). However we have shown in the presence of impurities this simple relation is broken [47], somewhat similar to the dirty s-wave superconductor in a magnetic field [52]. In Fig. 4 we show these two superfluid densities as functions of TR,,. We corrected here an error in the corresponding figure in [47]. Then at T=O K we obtained [471: Ps,spin(O) = (8) 1 - Ps:spin(O) = N(O)/No as expected. Therefore the residual density of states is accessible by Knight shift. Indeed we can put the Knight shift data from Zn-doped YBCO by Ishida et al. [42] in Fig. 1. These points lie further inside of the theoretical curve, implying that the strong coupling correction in YBCO may be about 30%. We would like to point out also that the experimental data on N(0) rule-out completely the possibility of anisotropic s-wave models, since as long as ( A ( k ) )# 0 the energy gap will surely open up before r = Tc is reached [53-551. Also, the d-wave superconductivity with impurity has rather unusual transport properties [@I. 3 Vortex state illz. We shall first consider the vortex state when Following Luk'yanchuk and Mineev [54] the upper critical field Hc2(T ) are determined for the order parameter [57, 581 CD >= (cos(24)(1 + c(ut)4)+ w>2) I4 (9) and the interacting potential N 0 V ( 2 , 2 )= 2hcos(24') cos(24) - c1 (10) where the second term is the Coulomb repulsion in the s channel considered recently by a few people [59, 601. In JQ.(9) 10) is the Abrikosov state corresponding to swave superconductor and C and b are expansion parameters. The upper critical field (without Pauli paramagnetic term) thus determined is shown in Fig. 5 as function of the reduced temperature for a few set of parameters h and p. In drawing Fig. 5 we I T =Tc = -1.9 T / K consistent with an experimental result by Welp et assumed al. [61] from YBCO. Here we correct an error in Hc2(T) in two earlier papers [57, 621. Perhaps the most important result is the stable vortex structure, which is obtained by studying the Abrikosov parameter PA = ([A(-?) 14)/(1 A(-?) In Fig. 6 we show PA for a square lattice tilted by from the a axis and the one for usual triangular lattice which is independent of the tilt angle as function of the reduced temperature. In the vicinity of T=T, the triangular lattice is more stable as expected from Ginzburg-Landau theory. However, as temperature decreases the square lattice beCame more stable rapidly for T < 0.8Tc [58, 621. Also we note that the coefficient C in Q. (9) is crucial for the stability of the square lattice. Recently not a square lattice but an oblique vortex lattice is seen by a small angle neutron scattering by Kei- 326 Ann, Physik 5 (1996) 0 0.2 0.4 0.6 0.8 1 t=T/ Tc 1.22 1 1 I Fig. 5 H,z(T) for Ak-' + p-'=03, 10 and 7 are shown as function of Tff,,. Errors in our earlier publications [55, 601 are corrected here. 1 1.2 1.18 d 1.16 1.14 1.12 1.1 Fig. 6 The Abrikosov parameter Pa for a triangular lattice (upper set of curves) and for a square lattice tilted by 45" from the a axis are shown for A=oo (-), 10 I - - - - ) and 7 I.- - - - ),. Below around 0.8 T, the square lattice becomes more stable independent of p. ~I 0 0.2 0.4 0.6 0.8 1 t=T/ Tc mer et al. [63] and STM imaging by Maggio-Apnle et al. [64] both in YBCO. We believe that the oblique lattice is due to the anisotropy in the coherence length E,, and within the a-b plane of YBCO. In particular the observed angle of the oblicity 73"-77" appears to be consistent with E,b/E,a z 1.35 1.2. On the other hand a few calculations based on Ginzburg-Landau equation for d-wave superconductivity which includes the Coulomb term are reported recently [63, 641. Both of them find the oblique lattice the most stable configuration. Clearly experimental study of high T, cup- ch N 327 K. Maki, H. Won, d-Wave superconductivity: New paradigm? 1.4 1.2 <8 z 1 0.8 0.6 0.4 0.2 Fig. 7 The density of states for A/& = 1 for A=co (solid line), A= 10 (closed line) and 7 (dashdotted line). 0 0 0.5 1 2 1.5 2.5 E/A rates without orthorhombic distortion is highly desirable, though for us the square lattice appears to be a simple manifestation of the fourfold symmetry underlying the dwave superconductivity. Making use of the Eilenberger equation [67] we have shown elsewhere the quasiparticle spectrum around a single vortex line exhibits the fourfold symmetry [68], though this has not been seen experimentally yet. The order parameter 10) obtained for B g Hc2(T ) is used to calculate the thermodynamic properties in terms of Brandt, Pesch and Tewordt Green’s function [69]. We show in Fig. 7 the qpasi-particle density of states thus determined [58] for A/& = 1 where E = u(2eB)T. Perhaps of more interest is the residual density of states when B/Hc2(0) << 1 at low temperatures, which is given by N(0),No=((l +2(32c0s2(2+))-1)~--- I & -1.605(-); 4 A00 B Hc2 (0) (11) where use is made of at T=O K when p = 0. The residual density of states increases like d% for small B as first shown by Volovik [27] and observed recently by a specific heat measurement in YBCO in magnetic fields [25]. Also Eq. (1 1) may be used to extract A00 from the experimental result. Second we consider the case B’ in the a-b plane. In this limit we have shown the Upper critical field exhibits the fourfold symmetry [15]. The upper critical field is well approximated by 328 Ann. Physik 5 (1996) 800 600 0 0 0.2 0.4 0.6 t 0.8 1 Fig. 8 Hc2($,I ) and AHc2(t) for A=co, 10 and 7 are shown as function off. Here we took = - 1 2 T I K appropriate for YBCO. WlTzTc (13) where 8 is the angle B' makes from the a axis. We show in Fig. 8, Hc2(2, t) and Hc2($,0), and AHc2(t)and Hc2(2,0) as function of t for a few p's where we took = -12T/K for YBCO [70].The Coulomb potential tends to suppress this interesting fourfold symmetry. The theoretical value Hc2(4,0)S 750 Tesla and AHc2(0) &' 120 Tesla is much larger than the observed value Hr2(0,0) 2 350 Tesla for YBCO [70], suggesting the importance of the Pauli paramagnetism. This fourfold symmetry in Hc2(8, t) appears to be seen most readily with torque magnetometry. In the vicinity of T=T, and H,, << B << Hc2, the torque within the a-b plane is written as [71] where we assumed and where K. Maki, H. Won, d-Wave superconductivity: New paradigm? 329 0.98 30.96 E 0.94 Fig. 9 Function F(0) in the residual density of states is shown as function of 0. 0.92 a2 arises from twofold symmetry related to the orthorhombicity [72] of some of high T, cuprates like YBCO. Very recently the torque as given in Eq. (14) is seen in a YBCO monocrystal by lshida et al. [73]. The residual density of states at low temperatures and for B/Hc2(q,0)<< 1 is given by where F(8) is shown in Fig. 9. Unfortunately the anisotropy appears to be rather we& (5%). Clearly the thermal conductivity tensor in the present configuration will provide more insight into the vortex state as shown by a brilliant experiment by Yu et al. [74]. For this purpose we have to first know the quasi-particle scattering amplitude due to a single vortex line. This problem is currently under study. 4 Outlook We have Seen that d-wave superconductivity can describe a variety of experiments done in high T, cuprates in a consistent and unified way though most of experiments are still concentrated on YBCO. In particular for US the square vortex lattice when 8llc'appears to provide a convincing proof of d-wave superconductivity. In any case further experiments on the vortex state will certainly help us to understand the nature of d-wave superconductivity. In a wider context we believe that d-wave superconductivity is another paradigm the nature has chosen when the system is Coulomb interaction dominated in contrast to the electron-phonon interaction-dominated s-wave Superconductivity. For us the appearance of spin density wave or antiferromagnetism 330 Ann. Physik 5 (1996) rather than charge density wave in the proximity of superconducting state is a sure sign of the dominance of Coulomb interaction. So in quasi-low dimensional systems s-wave superconductivity is paired with charge-density wave like in NbSe3 [75] while d-wave superconductivity is paired with antiferromagnetic state. Most likely other examples are heavy fermion superconductors and organic superconductors. In the former system CeCu2Si2, UBe13, URu2Si2, UPd2A13 and UNi2A13 appear to belong to d-wave from both the presence of Knight shift and the Pauli limiting of the upper critical field [76], though the superconducting nature of most well studied UPt3 is still very controversial. Indeed a recent measurement of the thermal conductivity anisotropy [77] suggests strongly that the order parameter of UPr3 is f-wave (or hybrid 11) [78, 791. This is consistent with a recent temperature-independent Knight shift observed in UPr3 [80] suggesting the equal spin pairing (i.e. spin-triplet state), We recall also earlier a peculiar crossing of the upper critical field for BllC‘ and for B in the basal plane observed in UPt3 [81] is successfully interpreted in terms 0f.fwave superconductivity [82], though it is not exactly the same f-wave we are discussing now. We don’t know yet if UPt3 presages the appearance of a new class of superconductivity; f-wave superconductivity. In organic superconductors the nuclear spin lattice relaxation rates in (TMTSF),C104, P-(ET),I3 and K-(ET),CU[N(CN),]B~ do not exhibit the Hebel-Slichter peak [83]. Further appearance of spin density wave or antiferromagnetic phase in the proximity of the superconducting phase both in Bechaard salts and K-(BEDT),-salts suggests that there is strong parallel between organic superconductors and high T, cuprate superconductors. Indeed the low temperaappears to behave as T3 consistent with d-wave ture T;’ of K-(ET),CU[N(CN),]B~ su erconductors [84]. Also the low-temperature specific heat is found proportional to T again indicating d-wave superconductor [85]. Clearly the further study of these systems is highly desirable. We are very happy to dedicate this work to our friend Peter Fulde on the occasion of his 60th birthday. One of us (KM) recall vividly the fresh excitement generated by the appearance of the BCS theory in the early sixties. 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