Bennemann K.H., Ketterson J.B. Superconductivity: Volume 1: Conventional and Unconventional Superconductors; Volume 2: Novel Superconductors

Подождите немного. Документ загружается.

22 A Spin Fluctuation Model for d-Wave Superconductivity 1411

146. H.F. Fong, P.Bourges,Y. Sidis, L.P. Regnault,A.Ivanov, G.D. Gu, N. Koshizuka, and B. Keimer, Nature 398, 588 (1999)

147. T.Holstein,Phys. Rev. 96, 535 (1954); P.B.Allen, Phys.Rev. B 3, 305 (1971)

148. P. Littlewood and C.M.Varma, Phys. Rev. B 46, 405 (1992)

149. D.J. Scalapino, J.R. Schrieffer and J.W.Wilkins, Phys. Rev. 148, 263 (1966)

150. D. Coffey, Phys. Rev. B 42, 6040 (1990)

151. W. Shaw and J.C. Swihart, Phys. Rev. Lett. 20, 1000 (1968)

152. G.D. Mahan,In: Many-Particle Physics (Plenum Press, New York, 1990)

153. A. Chainani,T.Yokoya,T. Kiss, and S.Shin, Phys.Rev. Lett. 85, 1966 (2000),and references therein.

154. H.Won and K. Maki, Phys.Rev. B 49,1397 (1994); Ye Sun and K. Maki, Phys. Rev. B 51,6059 (1995)

155. A.Abanov,A.V.Chubukov, N. Gemelke, Phys. Rev. B 61, R787 (2000)

156. A.Abanov and A. Chubukov, Phys. Rev. B 61 ,R9241 (2000)

157. A. Chubukov, G. Blumberg, and D. Morr, Solid State Comm. 112, 183 (1999)

158. J.R. Schrieffer, In: Theory of Superconductivity(W.A. Benjamin,Reading, MA, 1966)

159. S.M.Quinlan, P.J. Hirschfeld, D.J. Scalapino,Phys. Rev. B 53, 8575 (1996)

160. T.P. Devereaux, D. Einzel, B. Stadlober, R. Hackl, D.H. Leach, and J.J. Neumeier Phys. Rev. Lett. 72, 396 (1994); ibid

Phys.Rev.B54, 12 523 (1996)

161. A.Abanov,A.V. Chubukov, J. Schmalian, Phys.Rev. B 63, 180510 (2001)

162. R. Haslinger,A. Chubukov, and A. Abanov, Phys.Rev. B 63, 020503 (2001)

163. V. Barzykin and D. Pines, Phys.Rev. B 54, 7567 (1995)

164. V. Barzykin, D. Pines,A. Sokol,and D. Thelen, Phys. Rev. B 49, 1544 (1994)

165. P. Bourges,Y. Sidis, H.F. Fong, L.P. Regnault,J.Bossy,A. Ivanov, and B. Keimer, Science 288,1234 (2000)

166. P.D. Johnson,T.Valla,A. Fedorov, Z.Yusof, B.O.Wells, Q. Li,A.R. Moodenbaugh, G.D. Gu, N. Koshizuka, C. Kendziora,

Sha Jian, D.G. Hinks,Phys. Rev. Lett. 87, 177007 (2001)

167. P.V.Bogdanov,A.Lanzara,S.A.Kellar,X.J. Zhou,E.D. Lu,W. Zheng, G.Gu, K. Kishio, J.-I.Shimoyama,Z.Hussain,Z.X.

Shen, Phys. Rev. Lett. 85, 2581 (2000)

168. E. Dagotto,A.Nazarenko, and M. Boninsegni, Phys.Rev. Lett. 73, 728(1994); E. Dagotto,A.Nazarenko, andA. Moreo,

Phys. Rev. Lett. 74, 310 (1995); A. Nazarenko, K.J.E. Vos, S. Haas, E. Dagotto,and R.J. Gooding, Phys. Rev. B, 51, 8676

(1995)

169. T.Valla,A.V. Fedorov,P.D. Johnson,B.O.Wells,S.L. Hulbert,Q.Li,G.D.Gu, and N. Koshizuka,Science 285, 2210 (1999)

170. F. Bouis, M.N. Kiselev, F.Onufrieva, and P. Pfeuty, cond-mat/9906369

171. V.Yu Irkhin,A.A. Katanin and M.I. Katsnelson, cond-mat/0102381and references therein

172. S. Sachdev, In: Quantum Phase Transitions (Cambridge Univ Press., Cambridge,1999)

173. M. Lavagna, Phil.Mag.B 81, 1469 (2001)

174. More recent ARPES data [226,227] show that the magnitude of £



 is actually smaller than as thought before

175. E.Abrahams and C.Varma, Proc. Nat. Acad. Sci. 97, 5714 (2000)

176. J.J.Tu,C.C.Homes,G.D. Gu,D.N. Basov, S.M.Loureiro, R.J. Cava, M. Strongin, cond-mat/0104208

177. M.A.Quijada, D.B. Tanner, R.J. Kelley, M. Onellion, H. Berger and G. Margaritondo, Phys. Rev. B 60, 14917 (1999)

178. S. Spielman, D. Parks, J. Orenstein, D.T. Nemeth, F. Ludwig, J. Clarke, P. Merchant, and D.J. Lew, Phys. Rev. Lett. 73,

1537 (1994)

179. A. Puchkov, D. Basov, and T. Timusk, J.Phys.: Cond. Matter 8, 10049 (1996)

180. L. Ioffe and A.I. Larkin, Phys.Rev. B 39,8988 (1989)

181. P.Bourges,Y.Sidis,H.F.Fong,B.Keimer,L.P.Regnault,J.Bossy,A.S.Ivanov,D.L.Milius,I.A.Aksay,In: High Temperature

Superconductivity, ed. by S.E. Barnes et al.(American Institute of Physics,Amsterdam,1999) pp. 207–212

182. S.H.Pan, E.W. Hudson,K.M. Lang, H. Eisaki, S. Uchida,J.C. Davis, Nature, 403, 746 (2000)

183. D.A.Bonn,P.Dosanjh, R. Liang, and W.N. Hardy, Phys.Rev. Lett. 68, 2390 (1992)

184. M.C. Nuss, P.M.Mankiewich, M.L. O’Malley and E.H. Westerwick, Phys.Rev. Lett. 66, 3305 (1991)

185. N. Miyakawa,J.F. Zasadzinski,L. Ozyuzer,P. Guptasarma, D.G. Hinks,C. Kendziora,and K.E. Gray, Phys.Rev. Lett. 83,

1018 (1999)

186. Y. Ando, G.S. Boebinger,A. Passner, T. Kimura, and K. Kishio,Phys. Rev. Lett. 75, 4662 (1995)

187. G.S. Boebinger, Y. Ando, A. Passner, T. Kimura, M. Okuya, J. Shimoyama, K. Kishio, K. Tamasaku, N. Ichikawa, and

S. Uchida, Phys. Rev. Lett. 77, 5417 (1996)

1412 A.V. Chubukov, D. Pines, and J. Schmalian

188. S. Ono,Y. Ando, T. Murayama, F.F.Balakirev, J.B. Betts, and G.S. Boebinger, Phys. Rev. Lett. 85, 638 (2000)

189. P.W.Anderson, Phys.Rev. Lett. 67, 2092 (1991)

190. C.M.Varma and E. Abrahams, Phys.Rev. Lett. 86, 4652 (2001)

191. J. Cerne, M. Grayson, D.C. Schmadel, G.S. Jenkins, H.D. Drew, R. Hughes, A. Dabkowski, J.S. Preston and P.-J. Kung,

Phys.Rev.Lett.84, 3418 (2000)

192. N.P. Ong, In: Physical Properties of High Temperature Superconductors, ed. by D.M. Ginsberg (World Scientific,

Singapore,1990) Vol. 2

193. B. Stojkovic and D. Pines,Phys. Rev. Lett. 70, 811 (1996); Phys. Rev. B 55, 8576 (1997)

194. H. Kontani, K. Kanki,K. Ueda, Phys. Rev. B 59, 14723 (1999)

195. G.Aeppli, T.E.Mason,S.M. Hayden, H.A. Mook, J. Kulda, Science 278, 1432 (1997)

196. Y. Zha,V. Barzykin and D. Pines, Phys. Rev. B 54, 7561 (1996)

197. J. Schmalian and P.G.Wolynes,Phys. Rev. Lett. 85, 836 (2000)

198. H.Y.Hwang, B. Batlogg, H. Takagi,H.L.Kao,J.Kwo,R.J. Cava,J.J.Krajewski, and W.F.Peck,Jr., Phys.Rev.Lett.72,2636

(1994)

199. J.L. Tallon, J.W. Loram, G.V.M. Williams, J.R. Cooper, I.R. Fisher, J.D. Johnson,M.P. Staines, C. Bernhard, Phys. Stat.

Sol. (b)215, 531 (1999)

200. J.Loram,K.A.Mirza,J.R. Cooper,and W.Y.Liang,Phys.Rev. Lett.71,1740 (1993); J.W. Loram,K.A.Mirza,J.R. Cooper,

W.Y. Liang, J.M. Wade, J. Supercond. 7, 243 (1994)

201. C.Panagopoulos, B.D. Rainford, J.R. Cooper, and C.A. Scott, Physica C 341–348, 843 (2000)

202. R.B. Laughlin,G. Lonzarich, P.Monthoux, and D. Pines, Adv. Phys.50, 361 (2001)

203. N.J.Curro, P.C. Hammel, B.J. Suh, M. H¨ucker, B.B. ¨uchner, U. Ammerahl, and A. Revcolevschi, Phys. Rev. Lett. 85, 642

(2000)

204. J.Haase, C.P. Slichter, R. Stern, C.T. Milling, D.G. Hinks,J.Supercond. 13, 723 (2000)

205. A.Abanov, B. Altshuler, E.Yuzbashyan and A.V. Chubukov, unpublished

206. M.Vojta and S. Sachdev,Phys. Rev.Lett.83,3916 (1999); N.Read and S.Sachdev,Phys.Rev.Lett.62,1694 (1989); Phys.

Rev. B 42, 4568 (1990)

207. S. Kivelson, D.S. Rokhsar, and J. Sethna, Phys. Rev. B 35, 8865 (1987)

208. X.-G.Wen and P.A. Lee,Phys.Rev. Lett. 76 503 (1996)

209. T. Senthil and M.P.A. Fisher, Phys. Rev. B 62, 7850 (2000)

210. I.Afﬂeck and J.B. Marston, Phys. Rev. B 37, 3774 (1988)

211. G. Kotliar and J.Liu, Phys.Rev. B 38,5142 (1988)

212. Y. Suzumura,Y. Hasegawa, and H. Fukuyama, J. Phys.Soc. Jpn. 57, 2768 (1988)

213. G. Baskaran and P.W.Anderson, Phys. Rev. B 37, 580 (1988)

214. P.A. Lee and N. Nagaosa,Phys. Rev. B 46, 5621 (1992)

215. I.Afﬂeck,Z. Zou, T. Hsu,and P.W.Anderson, Phys.Rev. B. 38, 745 (1988)

216. D. Poilblanc and Y. Hasegawa, Phys.Rev. B.

41,

6989 (1990)

217. A.Kaminski,M.Randeria,J.C.Campuzano,M.R.Norman,H.Fretwell,J.Mesot,T.Sato,T. Takahashi,and K.Kadowaki,

Phys. Rev. Lett. 86, 1070 (2001); A. Lanzara, P.V. Bogdanov, X.J. Zhou, S.A. Kellar, D.L. Feng, E.D. Lu, T. Yoshida, H.

Eisaki,A. Fujimori, K. Kishio, J.-I. Shimoyama, T. Noda, S. Uchida, Z. Hussain, and Z.-X. Shen, Nature (London) 412,

510 (2001); P.D. Johnson, T.Valla,A.V. Fedorov, Z.Yusof, B.O.Wells, Q. Li,A.R.Moodenbaugh, G.D.Gu, N. Koshizuka,

C. Kendziora, Sha Jian, and D.G. Hinks, Phys.Rev. Lett. 87, 177007 (2001); F. Ronning,T. Sasagawa,Y. Kohsaka, K.M.

Shen,A.Damascelli,C.Kim,T.Yoshida,N.P.Armitage,D.H.Lu,D.L.Feng,L.L.Miller,H.TakagiandZ.-X.Shen,Phys.

Rev. B 67,165101 (2003); X.J.Zhou,T.Yoshida,A.Lanzara,P.V.Bogdanov,S.A.Kellar, K.M. Shen,W.L.Yang,F.Ronning,

T.Sasagawa,T.Kakeshita,T.Noda,H.Eisaki,S.Uchida,C.T.Lin,F.Zhou,J.W.Xiong,W.X.Ti,Z.X.Zhao,A.Fujimori,Z.

Hussain and Z.-X. Shen, Nature (London) 423, 398 (2003); T. Sato, H. Matsui, T. Takahashi, H. Ding, H.-B.Yang, S.-C.

Wang, T. Fujii, T. Watanabe,A. Matsuda, T. Terashima and K. Kadowaki, Phys. Rev. Lett. 91, 157003 (2003); T.K. Kim,

A.A.Kordyuk,S.V.Borisenko,A.Koitzsch,M.Knupfer,H.Berger,andJ.Fink,Phys.Rev.Lett.91, 167002 (2003); A.D.

Gromko,A.V.Federov,Y.-D.Chuang,J.D.Koralek,Y.Aiura,Y.Yamaguchi,K.Oka,Y.Ando,andD.S.Dessau,Phys.Rev.

B 68 174520 (2003)

218. A.V.Chubukov and M.R. Norman, Phys. Rev. B, 70, 174505(2004)

219. T.P. Devereaux et al., Phys. Rev. Lett. 93, 117004 (2004); T.Cuk et al., Phys. Stat. Sol. (b) 242, 11 (2005)

22 A Spin Fluctuation Model for d-Wave Superconductivity 1413

220. H.J.A. Molegraaf, C. Presura,D. van der Marel, P.H. Kes,and M. Li, Science 295 2239 (2002)

221. A.V. Boris et al., Science 304,708 (2004)

222. A.F. Santander-Syro, R.P.S.M. Lobo, N. Bontemps, Z. Konstantinovic, Z.Z. Li, and H. Raffy, Europhys. Lett. 62, 568

(2003); N. Bontempts, private communication

223. A.Abanov and A. Chubukov, Phys. Rev. B 70,100504 (2004)

224. D. van der Marel, H.J.A.Molegraaf, J. Zaanen, Z. Nussinov, F. Carbone, A. Damascelli, H. Eisaki, M. Greven, P.H. Kes

and M. Li, Nature 425, 271 (2003)

225. M.R. Norman and A.V.Chubukov, Phys. Rev. B 73, 140501 (2006)

226. A.A. Kordyuk et al., Pys. Rev. B 67, 064504 (2003)

227. J.D. Koralek et al.,cond-mat/0508404)

228. S. Pailhes et al., cond-mat/0403609; S.M. Hayden et al., Nature (London) 429, 531 (2004)

229. I. Eremin et al.,Phys. Rev. Lett., 94, 147001 (2005)

230. Y. Tokura, H. Takagi,S. Uchida , Nature 337, 345 (1989)

231. Y.Dagan et al., Phys. Rev. Lett. 92, 167001 (2004)

232. A. Koitzsch et al.,Phys. Rev. B, 67, 184522 (2003)

233. A. Zimmers et al., Europhys. Lett. 70, 225 (2005)

234. L.Alff et al., Nature 422, 698 (2003)

235. Y. Onose et al., Phys.Rev. B 69, 024504 (2004)

236. P. Krotkov and A. Chubukov, Phys. Rev. Lett. 97, 106002 (2006)

237. H.Y.Kee, S.A.Kivelson, and G.Aeppli,Phys.Rev. Lett. 88, 257002 (2002)

238. A.Abanov et al., Phys.Rev. Lett. 89, 177002 (2002)

239. C.Timm, D. Manske,and K.H. Bennemann,Phys. Rev. B 66, 094515 (2002)

240. A.V. Chubukov and J. Schmalian, Phys.Rev. B 72, 174520 (2005)

241. T.A.Maier, M.S.Jarrell,and D.J. Scalapino,Phys. Rev. Lett. 96, 047005 (2006)

242. V. Barzykin and D. Pines, preprint cond-mat/0601396

243. D.N.Basov and T.Timusk, Rev. Mod.Phys. 77, 721 (2005)

244. A. Damascelli, Z. Hussain,and Z.-X. Shen,Rev. Mod.Phys. 75, 473 (2003)

245. M.R. Norman and C.Pepin, Rep. Prog. Phys.66, 1574 (2003)

246. M.R. Norman, D. Pines and C. Kallin, Adv. Phys.54, 715 (2005)

23 Electronic Theory for Superconductivity in High-T

Cuprates and Sr

RuO

D. Manske Max Planck Institute for Solid State Research, Stuttgart, Germany

I. Eremin Institut f¨ur Mathematische Physik,TU-Braunschweig, Germany

K. H. Bennemann Institut f¨ur Theoretische Physik, Freie Universit¨at Berlin, Germany

23.1 Introduction ............................................................................1416

23.2 Electronic Theory for Hole-Doped and Electron-Doped Cuprates .............................1428

23.2.1ElectronicStructureandHamiltonain................................................1428

23.2.2 Eliashberg Theory for the Interaction Between Quasiparticles and Antiferromagnetic

SpinFluctuations..................................................................1430

23.2.3ElementaryExcitations.............................................................1434

23.2.4DynamicalSpinSusceptibility.......................................................1437

23.2.5GapEquation:SingletandTripletCooper-Pairing.....................................1439

23.2.6 Superﬂuid Density n

...............................................................1441

23.2.7Cooper-PairPhaseFluctuations......................................................1443

23.2.8TunnelingSpectra..................................................................1446

23.2.9BilayerEffects.....................................................................1448

23.2.10Electron-DopedCuprates..........................................................1449

23.3 Electronic Theory for Ruthenates (Sr

RuO

) ................................................1451

23.3.1ElectronicStructureandtheHamiltonian.............................................1451

23.3.2 Eliashberg-Like Theory for the Coupling of Quasiparticles to Ferromagnetic and

IncommensurateAntiferromagneticSpinFluctuations.................................1453

23.3.3 Elementary Excitations in Sr

RuO

...................................................1455

23.3.4DynamicalSpinSusceptibility:MagneticAnisotropy...................................1456

23.3.5 Gap Equation: Triplet Pairing and Symmetry of the Superconducting Order Parameter . . . . 1457

23.4 Results for Hole-Doped and Electron-Doped Cuprates .......................................1460

23.4.1BareElectronicStructure ...........................................................1460

23.4.2ElementaryExcitations.............................................................1461

23.4.3DynamicalSpinSusceptibility.......................................................1467

23.4.4OrderParameter:DopingDependence ...............................................1468

23.4.5ResonancePeakandMagneticCoherence.............................................1469

23.4.6PhaseDiagramforElectron-DopedandHole-DopedCuprates..........................1473

23.4.7 Doping Dependence of the Superﬂuid Density n

(x, T).................................1477

23.4.8SISandNISTunnelingSpectra.......................................................1481

23.4.9BilayerEffects.....................................................................1483

23.5 Results for Sr

RuO

......................................................................1485

23.5.1ElectronicStructure ...............................................................1485

23.5.2ElementaryExcitations:EffectofSpinFluctuations....................................1486

23.5.3DynamicalSpinSusceptibility,NMR,andINS.........................................1486

23.5.4SuperconductingOrderParameter...................................................1490

23.5.5DopingDependenceofRuthenates...................................................1494

23.6 Summary and Outlook ...................................................................1496

1416 D. Manske,I. Eremin, and K.H.Bennemann

Appendix A: Derivation of the generalized Eliashberg Equations for the Interaction

of Quasiparticles and Spin Fluctuations ............................................1497

Appendix B: Numerical Solution Method for the Generalized Eliashberg Equations .................1500

Appendix C: Derivation of the Quasiparticle Self-Energy .........................................1502

Appendix D: Cooper-Pair Phase Fluctuations ...................................................1504

References.......................................................................1511

23.1 Introduction

Superconductivity remains one of the most inter-

esting problems in solid state physics. During the

last years in particular high-transition tempera-

ture superconductivity in the cuprates with non-s-

wave pairing symmetry and unconventional super-

conductivity in the isostructural strontiumruthen-

ate (Sr

RuO

) also with a non-s-wave symmetry of

the superconducting order parameter and triplet

Cooper-pairing have been studied. Cuprates and

ruthenates havean oxidelayered structure with CuO

and RuO

-planes, respectively. These seem to play

an important role regarding superconductivity and

magnetic activity. For an illustration in Fig. 23.1 we

show the crystal structures (in Sr

RuO

the basic ele-

ment is a RuO

-plane and the superconducting tran-

sition temperature is T

=1.5 K). Undoped La

CuO

is an antiferromagnetic insulator. Superconductiv-

ity occurs only upon doping with Ba and reaches a

maximum for hole doping x =0.15 with a transi-

tion temperature T

≈ 40 K [1]. The different behav-

ior of Sr

RuO

and La

2−x

CuO

is definitely re-

lated to the electronic structure of RuO

and CuO

planes. Let us emphasize also some differences in

the crystal structures of Sr

RuO

and La

2−x

CuO

RuO

has a pure tetragonal crystal structure. The

RuO

-octahedra arenot distorted.On theother hand,

inLa

2−x

CuO

thecorresponding CuO

-octahedra

are sufficiently distorted.In general,it is important to

understand the dependence of T

based on a crystal

structure. In Fig. 23.2 we show the crystallographic

structures of Bi-based cuprates with various CuO

layers per unit cell. T

increases almost linearly from

Fig. 23.1. (a) Cuprate singlet

superconductor La

2−x

CuO

(b) Layered perovskite struc-

ture of the triplet superconduc-

tor Sr

RuO

23 Electronic Theory for Superconductivity 1417

Fig. 23.2. T

of cuprate

crystal structures with

various CuO

-layers:

(a)one-layer,

(b)two-layer,and

CuO

to the Bi

. Note that not

only the number of CuO

-layers changes, but also

other properties like the strength of the orthorhom-

bic distortions in these materials.

Cuprates

The parent compounds of the cuprate supercon-

ductors such as La

CuO

or YBa

are quasi-

two-dimensional antiferromagneticinsulators. Dop-

ing these systems with holes, for example, La

CuO

with strontium or YBa

with oxygen, causes a

transition from a Mott insulator to a metallic phase.

The metallic phase exhibits unconventional super-

conductivity with non s-wave symmetry of the or-

der parameter and the transition temperature T

in-

creases with increasing doping towards a maximum

value. Further doping decreases T

. Finally, super-

conductivitydisappears in the overdoped region.Re-

markably, despite differences between the phase di-

agrams of various hole-doped cuprates one can map

all of them into one so-called generic phase diagram

as shown in Fig. 23.3. This assumes that Cooper-

pairing and normal state dynamics already occur es-

sentially in the CuO

-plane.Then,the physical prop-

erties of each cuprate compound depend mainly on

the number of carriers within a CuO

plane and the

correlation between them.

Besides crystal structures,the doping dependence

(x), and the phase diagram, many important prop-

erties have been observed. For example, various ex-

periments indicate for the cuprates d

−y

-symmetry

of the superconducting order parameter and singlet

Cooper-pairing:





cos k

−cosk



. (23.1)

The superconducting transition temperatures may

be as high as 150 K. Considerable electronic corre-

lations yield a non-Fermi liquid behavior of the ele-

mentary excitations in the normal state (below T

∗

Generally, the behavior of the cuprates depends sig-

nificantly on the doping concentration x.

Note, regarding the symmetry of the supercon-

ducting order parameter in the cuprates, evidence

ford

−y

-wavepairingin hole-dopedcupratescomes

fromseveralsources,in particular fromnuclear mag-

netic resonance (NMR) [2], phase sensitive mea-

surements [3–6], penetration depth measurements

[7, 8], ARPES [9], and polarization-dependent Ra-

man scattering experiments [10–13]. In contrast to

the conventional superconductors, the behavior of

thed

−y

-wavesymmetry of the superconducting or-

der parameter in the first Brillouin zone (BZ) is very

peculiar. In particular, the presence of the line nodes

along the diagonal of the BZ and the corresponding

1418 D. Manske,I. Eremin, and K.H.Bennemann

Fig. 23.3. Schematic phase diagram of hole-doped cuprates.

High-T

superconductivity always occurs in the vicinity

of an antiferromagnetic phase transition (AF). The su-

perconducting transition temperature as a function of the

hole concentration T

(x) has a characteristic parabola-like

shape. The superconducting order parameter is of d-wave

symmetry.The normal state can be divided into two parts:

In the overdoped region the cuprates behave like a conven-

tional a Fermi-liquid, whereas in the underdoped regime

below a temperature T

∗

one finds strong antiferromag-

netic correlations and non-Fermi-liquid behavior. T

∗

is a

crossover temperature and is called the pseudogap temper-

ature,because a normal-state gap opens up in the spin and

charge degrees of freedom. T

∗

enters the T

curve roughly

at optimum doping concentration or at overdoped values

dependent on the number of CuO

-planes per unit cell

changes of sign in the gap function allow different

experimental techniques to distinguish a d

−y

-wave

symmetry from other symmetries, for example from

isotropic s-wave symmetry.

In Fig. 23.4 we show NMR/NQR results of [2] for

the temperature dependence of the spin-lattice re-

Fig. 23.4. Temperature dependence of the

Cu NQR spin-

lattice relaxation rate for various cuprate superconductors

revealing singlet d-wave Cooper-pairing (taken from [2])

laxation rate 1/T

Cu for three different high-T

cuprate superconductors belowT

.Asignatureof un-

conventional superconductivity is the absence of the

so-called Hebel–Slichter peak at T  T

and also the

temperature dependence of 1/T

.Incontrasttoalow

temperature exponential decay in the case of s-wave

pairing,therelaxationrate 1/T

has a rather different

dependence on T,namely a power law that one would

expect for d-wave symmetry of the superconducting

order parameter.Note that for a d-wave gap with line

nodes along the diagonals the density of states is not

fully gaped and thus allows the quasiparticles to oc-

cupy states for energies |!| < 2

(

is the maximal

gap at T = 0). In an NMR experiments one averages

over the first BZ and thus one takes also into account

the states at the line nodes. As a result, the tempera-

ture dependence of the spin-lattice relaxation follows

apowerlaw(∝T

).As can be seen in Fig. 23.4,this is

indeed the case for high-T

cuprate superconductors.

There is another important signature of uncon-

ventional superconductivity in the cuprates. Usually,

one observes T

∝ 

for conventional supercon-

ductors (

= (T = 0)). However, in hole-doped

cuprates this relation is not generally satisfied and is

only observed for x > x

.Forx < x

=optimal

23 Electronic Theory for Superconductivity 1419

Fig. 23.5. The superconducting transition temperature T

plotted vs low-temperature muon-spin-relaxation rate 

measured for different high-T

superconductors, see [14]

(214: La

2−x

CuO

,123:YBa

,2212: Bi

CuO

2223: Bi

). Note that  (0) is proportional to

the superﬂuid density, n

∗

doping), i.e. in underdoped cuprates, T

decreases

with decreasing doping, while the measured 

be-

comes larger. As noted by Uemura et al. [14] in un-

derdoped cuprates one finds T

∝ n

,wheren

is a

superﬂuid density. This is shown in Fig. 23.5.

In general, the normal state of the hole-doped

cuprates depends sensitively on doping, see Fig. 23.6.

The overdoped cuprates behave like a conventional

Fermi-liquid, whereas the underdoped ones, in par-

ticular below the pseudogap temperature T

∗

,ex-

hibit some unusual properties. For example, a gap

is present in the elementary excitation spectrum,

strong anisotropies are observed (mainly caused by

the quasi-two-dimensional nature of the system),

and short-range antiferromagnetic ﬂuctuations ex-

ist. This different behavior of underdoped and over-

doped crystals can be seen, for example, from the

Cu spin-lattice relaxation rate and the inelastic

neutron scattering (INS) intensity in hole-doped

cuprates. In the overdoped cuprates 1/T

T increases

monotonously as T decreases down to T

,whilein

the underdoped ones 1/T

T passes through a max-

imum at the spin-gap temperature T

∗

for decreas-

ing T [16]. At the same time similar behavior is ob-

served by INS where in the underdoped cuprates the

imaginary part of the spin susceptibility Im (Q, !)

passes at fixed small frequency ! ( 10–15 meV)

through a maximum at T

∗

for decreasing T [17].

In addition, angle-resolved photoemission spec-

troscopy (ARPES) experiments on underdoped

CaCu

8+ı

show the presence of a “pseudo”-

gap with d

−y

-wave symmetry well above T

also

in the charge excitation spectrum [18, 19]. Recent

tunneling experiments also indicate a presence of

a pseudogap in the tunneling spectral density well

above T

[20]. This gap opens below the temperature

∗

. Several other experiments including heat capac-

ity [21], transport [22], Raman scattering [23], and,

in particular, scanning tunneling spectroscopy [24]

have confirmed the existence of a pseudogap in

the elementary excitations below T

∗

.Toclarifythe

origin of the pseudogap a few phenomenologi-

cal models like the marginal-Fermi-liquid (MFL)

[25], the nested-Fermi-liquid [26, 27], and nearly-

antiferromagnetic-liquid (NAFL) [28] have been de-

veloped. These describe the unusual non-Fermi-

liquid properties in the normal state of hole-doped

cuprates.

In the cuprate superconductors the CuO

-plane

seems to be most important. The surrounding en-

vironment of the CuO

-plane, usually called charge

reservoirs, somehow play a passive role supplying

charge carriers for the planes. For some cuprates

it is possible that also electrons can be doped into

the CuO

-plane. It is remarkable that these so-called

electron-doped cuprates differ drastically from the

hole-doped ones. Note the large asymmetry in their

phase diagrams, see Fig. 23.6:

1. The antiferromagnetism persists in the electron-

doped compounds up to x =0.12 or 0.14 [29],

while in hole-doped compound La

2−x

CuO

(LSCO) antiferromagnetism disappears already

at x =0.05.

2. Superconductivity in electron-doped cuprate

systems occurs in a very narrow doping range

(0.14 < x < 0.17 for Nd

2−x

CuO

(NCCO)

[30, 31] and has a smaller transition tempera-

ture T

), in the hole-doped LSCO system the

doping range for which superconductivity oc-

curs is broader (0.05 < x < 0.3). In addition,

1420 D. Manske,I. Eremin, and K.H.Bennemann

Fig. 23.6. Illustration of the phase diagram of hole (LSCO)

and electron (NCCO)-doped cuprates, temperature T ver-

sus doping concentration x. Note that the asymmetry be-

tween electron-doped and hole-doped cuprates.In the case

of hole doping the carriers mainly occupy the oxygen p-

band, while for electron doping the carriers occupy the

upper Hubbard band of the copper. The antiferromag-

netism is for rather different doping concentration de-

stroyed in electron-doped and hole-doped cuprates. Note

that it would be interesting to compare LSCO directly with

its electron-doped counterpart La

2−x

CuO

. However, a

structural phase transition at x =0.05 changes the elec-

tronic properties of the latter significantly [15]

the behavior of the electron-doped cuprates in

the normal and superconducting state is very

different in comparison to the hole-doped ones.

For example, in the normal state a pseudogap

in the excitation spectrum in hole-underdoped

cuprates is seen in many experiments [32]. In

electron-dopedones the existence of thepseudo-

gap is unclear. In optimally-doped YBa

6+y

(YBCO) and LSCO the in-plane resistivity in-

creases linearly with temperature over a wide

range [33] indicating a non-Fermi-liquid behav-

ior. In NCCO (x =0.15) the in-plane resistivity

is quadratic in temperature [34]. Photoemission

spectroscopy studies of a CuO

-plane show ﬂat

energy bands near the Fermi surface (FS) of the

high-T

hole doped cuprates [35]. However, in

NCCO one finds a ﬂat band only 300 meV below

the FS [36].

Therefore, the question arises: Are the pairing sym-

metries of the electron and hole-doped cuprates

different? It is widely believed that hole-doped

are d

−y

-wave superconductors. In electron-doped

cuprates the symmetry of the order parameter was

for a long time under debate. Early measurements

of in-plane penetration depth in NCCO have been

fitted with an exponential temperature dependence

suggesting isotropic BCS type s-wave gap [37] rather

than the power law decay expected for unconven-

tional superconductors with nodes in the gap func-

tion. Recent experiments indicate non-s-wave sym-

metry and rather d

−y

-wave symmetry of the su-

perconducting order parameter. Most recent mea-

surements of the magnetic penetration depth [38,39]

have shown that the best fit occurs for dominant the

−y

-wave plus small admixture of the s-wave con-

tribution for the underlying gapsymmetry.Themost

convincing evidence for dominant d

−y

-wave sym-

metry in electron-dopedNCCO and PCCO at optimal

doping came very recently from phase sensitive ex-

periments [31]. Note, however, that with increasing

doping the symmetry of the superconducting order

parameter seems to change towards anisotropic s-

wave type [40].

In general, for increasing doping itinerancy in-

creases and antiferromagnetic activity disappears.

Thestronginterplay ofantiferromagneticexcitations

and superconductivity is in particular seen by in-

elastic neutron scattering (INS) and elementary ex-

citations measured by angle-resolved photoemission

spectroscopy (ARPES) experiments. The feedback of

superconductivity on the spin susceptibility (q, !)

and the recently observed “kink” in the dispersions

is illustrated in Figs. 23.7 and 23.8, respectively.

In Fig. 23.7 we show the INS results for optimally-

doped YBa

6.92

[41].Employing linear response

theory, the intensity in INS is proportional to the

magnetic structure factor that yields the imaginary

23 Electronic Theory for Superconductivity 1421

Fig. 23.7. Spin excitation spectrum I

(q = Q, !) vs fre-

quency for optimally-doped YBCO as observed by INS

experiments in cuprates [41]. In the normal state an

Ornstein–Zernike behavior is observed. Note the feedback

of superconductivity (T =5K< T

) on the spin suscepti-

bility and the formation of the so-called “resonance” peak

at !

res

=41meV

part of the dynamical spin susceptibility. Thus, INS

provides a direct probe of the spin susceptibility,

(q, !). The normal state spin dynamics can be

described approximately at the important antiferro-

magnetic wave vector Q =(, )bytheOrnstein–

Zernicke response function

Im ∝

+ !

. (23.2)

Note, the rapid increase of Im and the maximum at

! = !

. In the superconducting state the behavior

of Im is very different. The spin excitation spec-

trum renormalizes strongly and becomes gaped at

lower frequenciesand then exhibits a sharp so-called

“resonance”peak at an energy !

res

.Inthecaseofthe

optimally-doped YBCO one gets !

res

=41meV.

The appearance of the resonance peak firstly re-

ported in [42] suggests a strong renormalization of

the spin ﬂuctuation spectrum in the superconduct-

ing state that cannot be accounted for by the sim-

ple phenomenological form of the spin susceptibil-

ity, (q, !) (see (23.2)). Note that phonon-mediated

superconductivity is difficult to reconcile with the

appearance of a resonance peak. Thus,the important

resonance peak is a fingerprint of Cooper-pairing

dueto spinexcitations.In a recent INSmeasurements

[43]aresonantspinexcitationintheoverdopedand

optimally-doped Bi

CaCu

8+y

(BSCCO) is also

observed with mode energies E

res

=38meVand

43 meV,respectively.

This observation together with

the previous measurements in underdoped phase of

YBCO [41, 44] indicates a constant ratio (for two

CuO

-planesperunitcell) forthe whole dopingrange

for the odd mode

res

=5.4 . (23.3)

The resonance peak emerged as a key property of

the copper oxide superconductors. The observation

of a resonance like feature in the superconducting

state of single CuO

-layer cuprate superconductor

CuO

6+x

near optimal doping seems very im-

portant [45].Note thatno resonance peak is observed

in electron-doped cuprates.

In view of this one may conclude that the analysis

of the spin ﬂuctuation spectrum above and below T

and its doping dependence is significant for a micro-

scopic theory of cuprate superconductivity.

Similarly important is the behavior of the elemen-

tary excitations above and below T

. Regarding the

“kink” structure shown in Fig. 23.8 we note the fol-

lowing. With certain assumptions ARPES measures

the spectral density, A(k, !), of the electrons in the

CuO

-planes. Due to recent developments in ARPES

the spectral density can be studied with high ac-

curacy versus frequency at fixed momentum (so-

called energy distributioncurves (EDC)) as well as a

function of momentum at fixed frequency (so-called

momentum distribution curves (MDC)). One of the

most important information that one gets by analyz-

ing MDC and EDC is the renormalized energy dis-

persion !

for the elementary excitations. Note that

MDC curves are better suited for determining the

energy dispersion due to a weaker momentum than

frequency dependence of the self-energy, £(k, !).

In Fig. 23.8 we show the results of Lanzara et

al. [46] for the (0, 0) → (, ) direction in the first

Due to the bilayer structure of YBCO and BSCCO a resonating mode is expected in both the even and odd channels of

the spin excitation spectrum,respectively.Here,we refer only to the odd channel which exhibits a much larger intensity.