Bennemann K.H., Ketterson J.B. Superconductivity: Volume 1: Conventional and Unconventional Superconductors; Volume 2: Novel Superconductors
Подождите немного. Документ загружается.
17 Photoemission in the High-T
c
Superconductors 989
34. D.M.King,Z.-X.Shen,D.S.Dessau,B.O.Wells,W.E.Spicer,A.J.Arko,D.S.Marshall,J.DiCarlo,A.G.Loeser,C.H.Park,
E.R. Ratner, J.L. Peng, Z.Y. Li, R.L. Greene, Phys. Rev. Lett. 70, 3159 (1993); R.O. Anderson, R. Claessen, J.W. Allen,
C.G.Olson,C.Janowitz,L.Z.Liu,J.-H.Park,M.B.Maple,Y.Dalichaouch,M.C.deAndrade,R.F.Jardim,E.A.Early,S.-J.
Oh,W.P. Ellis, ibid 70, 3163 (1993); N.P.Armitage, D.H. Lu, C. Kim, A. Damascelli, K.M. Shen, F. Ronning, D.L. Feng,
P. Bogdanov, Z.-X. Shen, Y. Onose, Y. Taguchi, Y. Tokura, P.K. Mang, N. Kaneko, M. Greven, ibid 87, 147003 (2001);
N.P. Armitage, F. Ronning,D.H. Lu, C. Kim,A. Damascelli, K.M. Shen, D.L. Feng, H. Eisaki, Z.-X. Shen, P.K. Mang, N.
Kaneko, M. Greven,Y. Onose,Y. Taguchi,Y. Tokura, ibid 88, 257001 (2002)
35. A. Ino, C. Kim, T. Mizokawa, Z.-X. Shen, A. Fujimori, M. Takaba, K. Tamasaku, H. Eisaki, S. Uchida, J. Phys. Soc.
Japan 68, 1496 (1999); A. Ino,C. Kim, M. Nakamura, T. Mizokawa, Z.-X. Shen, A. Fujimori, T. Kakeshita, H. Eisaki, S.
Uchida, Phys. Rev. B 62,4137 (2000);A.Ino,C.Kim,M.Nakamura,T.Yoshida,T. Mizokawa,Z.-X.Shen,A. Fujimori,T.
Kakeshita,H.Eisaki,S. Uchida, ibid 65,094504 (2002); T.Yoshida,X.J.Zhou, M.Nakamura,S.A.Kellar, P.V. Bogdanov,
E.D.Lu,A.Lanzara,Z.Hussain,A.Ino,T.Mizokawa,A.Fujimori,H.Eisaki,C.Kim,Z.-X.Shen,T.Kakeshita,S.Uchida,
ibid, 63, 220501 (2001)
36. H. Ding, A.F. Bellman, J.C. Campuzano, M. Randeria, M.R. Norman, T.Yokoya, T. Takahashi, H. Katayama-Yoshida, T.
Mochiku,K.Kadowaki,G.Jennings,and G.P.Brivio,Phys.Rev.Lett.76, 1533 (1996)
37. M.R. Norman, M. Randeria, H. Ding, and J.C. Campuzano, Phys. Rev. B 52, 615 (1995)
38. R.L.Withers,J.G.Thompson,L.R.Wallenberg,J.D.Fitzgerald,J.S.Anderson,B.G.Hyde,J.Phys.C21, 6067 (1988)
39. P.Aebi,J.Osterwalder,P.Schwaller,L.Schlapbach,M.Shimoda,T.Mochiku andK.Kadowaki,Phys.Rev.Lett.72,2757
(1994); J. Osterwalder, P.Aebi, P. Schwaller, L. Schlapbach, M. Shimoda, T. Mochiku and K. Kadowaki, Appl. Phys.A
60, 247 (1995)
40. A. Kampf and J.R. Schrieffer, Phys.Rev. B42, 7967 (1990)
41. H. Ding, M.R. Norman, T. Yokoya, T. Takuechi, M. Randeria, J.C. Campuzano, T. Takahashi, T. Mochiku, and K.
Kadowaki,Phys. Rev. Lett. 78, 2628 (1997)
42. A.A.Kordyuk,S.V.Borisenko,M.S.Golden,S.Legner,K.A.Nenkov,M.Knupfer,J.Fink,H.Berger,L.Forro,R.Follath,
Phys.Rev.B66, 014502 (2002)
43. D.S. Dessau, Z.X. Shen, D.M. King, D.S. Marshall, L.W. Lombardo, P.H. Dickinson, A.G. Loeser, J. DiCarlo, C.H. Park,
A. Kapitulnik,and W.E. Spicer, Phys. Rev. Lett.71, 2781 (1993)
44. M.R. Norman, M. Randeria, H. Ding, J.C. Campuzano, and A.F. Bellman, Phys. Rev. B 52,15107 (1995)
45. T. Sato, T. Kamiyama, T. Taka hashi, J. Mesot, A. Kam ins ki, J. C. Campuzan o, H.M. Fret well, T. Takeuchi, H. Ding , I.
Chong , T. Terashima, M . Takan o, Phys. Rev. B 64, 054502 (2001)
46. N.L. Saini, J.Avila,A. Bianconi and A. Lanzara, M.C.Asensio, S. Tajima, G.D. Gu, and N. Koshizuka, Phys. Rev. Lett. 79,
3467 (1997)
47. J. Mesot,M.R. Norman,H. Ding, J.C.Campuzano, Phys. Rev. Lett. 82, 2618 (1999); N.L. Saini, A. Bianconi,A.Lanzara,
J.Avila,M.C.Asensio,S.Tajima,G.D.Gu,N.Koshizuka,ibid 2619 (1999)
48. Y.-D. Chuang, A. D. Gromko, D. S. Dessau,Y.Aiura, Y.Yamaguchi, K. Oka,A. J. Arko, J. Joyce, H. Eisaki, S. I. Uchida, K.
Nakamura, and Yoichi Ando, Phys. Rev. Lett. 83, 3717 (1999)
49. H.M.Fretwell, A. Kaminski, J. Mesot, J.C.Campuzano, M.R. Norman,M. Randeria, T. Sato, R. Gatt, T. Takahashi, and
K. Kadowaki,Phys. Rev. Lett. 84, 4449 (2000)
50. S.V. Borisenko, M.S. Golden, S. Legner, T. Pichler, C. Durr, M. Knupfer, J.Fink, G. Yang, S. Abell, H. Berger, Phys. Rev.
Lett. 84, 4453 (2000); S.V. Borisenko, A.A. Kordyuk, S. Legner, C. Durr, M. Knupfer, M.S. Golden, J. Fink, K. Nenkov, D.
Eckert,G.Yang,S.Abell,H.Berger,L.Forro,B.Liang,A.Mailouck,C.T.Lin,B.Keimer,Phys.Rev.B64, 094514 (2001)
51. D.L.Feng,W.J.Zheng,K.M.Shen,D.H.Lu,F.Ronning,J.-I.Shimoyama,K.Kishio,G.Gu,D.VanderMarel,Z.-X.Shen,
cond-mat/9908056
52. J.C. Campuzano, H. Ding, M.R. Norman, M. Randeria,A.F.Bellman, T.Yokoya, T. Takahashi, H. Katayama-Yoshida, T.
Mochiku, and K. Kadowaki, Phys. Rev. B
53,
R14737 (1996)
53. T.Straub,R.Claessen,P.Steiner,S.Hufner,V.Eyert,K.Friemelt andE.Bucher,Phys.Rev.B55, 13473 (1997)
54. M.C. Schabel, C.-H. Park, A. Matsuura, Z.-X. Shen, D.A. Bonn, Ruixing Liang, and W.N. Hardy, Phys. Rev. B 57, 6107
(1998)
55. K.Gofron,J.C.Campuzano,H.Ding,C.Gu,R.Liu,B.Dabrowski,B.W.Veal,W.Cramer,G.Jennings,J.Phys.Chem.Sol.
54,1193(1993); K.Gofron,J.C.Campuzano,A.A.Abrikosov,M.Lindroos,A.Bansil,H.Ding,D.Koelling,B.Dabrowski,
Phys.Rev.Lett.73, 3302 (1994)
990 J.C. Campuzano, M.R. Norman, and M. Randeria
56. S.Massida,J.Yu,andA.J.Freeman,PhysicaC 152, 251 (1988); O.K. Andersen, O. Jepsen, A.I. Liechtenstein, I.I. Mazin,
Phys. Rev. B49, 4145 (1994)
57. S. Chakravarty, A. Sudbo, P.W. Anderson and S. Strong, Science 261,337 (1994)
58. M.R. Norman, H. Ding, J.C. Campuzano, T. Takeuchi, M. Randeria, T. Yokoya, T. Takahashi, T. Mochiku, and K.
Kadowaki,Phys. Rev. Lett. 79, 3506 (1997)
59. J.C. Campuzano, H. Ding, M.R. Norman, H.M. Fretwell, M. Randeria, A. Kaminski, J. Mesot, T. Takeuchi, T. Sato, T.
Yokoya, T. Takahashi, K. Kadowaki, P. Guptasarma, D. G. Hinks, Z. Konstantinovic, Z.Z. Li, and H. Raffy, Phys. Rev.
Lett. 83, 3709 (1999)
60. J. Zasadzinski, L. Ozyuzer, Z. Yusof, J. Chen, K.E. Gray, R. Mogilevsky, D.G. Hinks, J.L. Cobb, J.T. Markert, SPIE 2696,
338 (1996); See also J. Zasadzinski et al. (unpublished) in Fig. 1 of D. Coffey, J. Phys. Chem. Solids 54, 1369 (1993)
61. P.W. Anderson, Science 256, 1526 (1992)
62. D.L. Feng, N.P.Armitage,D.H. Lu, A. Damascelli, J.P Hu, P. Bogdanov,A.Lanzara,F. Ronning, K.M. Shen, H. Eisaki, C.
Kim, and Z.-X. Shen, Phys.Rev. Lett. 86, 5550 (2001); D.L. Feng, C. Kim, H. Eisaki, D.H. Lu, A. Damascelli, K.M. Shen,
F.Ronning,N.P.Armitage,N.Kaneko,M.Greven,J.-I.Shimoyama,K.Kishio,R.Yoshizaki,G.D.Gu,Z.-X.Shen,Phys.
Rev. B 65, 220501 (2002)
63. Y.D.Chuang,A.D.Gromko,A.Fedorov,D.S.Dessau,Y.Aiura,K.Oka,Y.Ando,H.Eisaki,S.I.Uchida,Phys.Rev.Lett.87,
117002 (2001); Y.D. Chuang, A.D. Gromko, A. Fedorov, Y. Aiura, K. Oka, Y.Ando, D.S. Dessau, Phys. Rev. B69, 094515
(2004)
64. A.A. Kordyuk, S.V. Borisenko,T.K. Kim, K. Nenkov, M. Knupfer, M.S. Golden,J. Fink, H. Berger, R. Follath, Phys.Rev.
Lett. 89, 077003 (2002)
65. D.L. Feng, A. Damascelli, K.M. Shen, N. Motoyama, D.H. Lu, H. Eisaki, K. Shimizu, J.-I. Shimoyama, K. Kishio, N.
Kaneko,M.Greven,G.D.Gu,X.J.Zhou,C.Kim,F.Ronning,N.P.Armitage,and Z.-X.Shen,Phys.Rev.Lett.88, 107001
(2002); D.L. Feng, H. Eisaki, K.M. Shen, A. Damascelli, C. Kim, D.H. Lu, Z.-X. Shen, K. Shimizu, J.-I. Shimoyama, K.
Kishio,N.Motoyama,N.Kaneko,M.Greven,G.D.Gu,Int.J.Mod.Phys.16,1691 (2002); T. Sato, H. Matsui, S. Nishina,
T.Takahashi, T.Fujii,T. Watanabe, and A. Matsuda,Phys.Rev. Lett. 89, 067005 (2002)
66. D.J.VanHarlingen,Rev.Mod.Phys67, 515 (1995); C.C. Tsuei and J.R. Kirtley, ibid 72, 969 (2000)
67. Z.X.Shen,D.S.Dessau,B.O.Wells,D.M.King,W.E.Spicer,A.J.Arko,D.Marshall,L.W.Lombardo,A.Kapitulnik,P.
Dickinson, S. Doniach, J. DiCarlo, A.G. Loeser, C.H. Park, Phys. Rev. Lett. 70, 1553 (1993)
68. H. Ding, M.R. Norman, J.C. Campuzano, M. Randeria, A. Bellman, T. Yokoya, T. Takahashi, T. Mochiku, and K.
Kadowaki, Phys. Rev. B 54, R9678 (1996)
69. J.-M.Imer,F.Patthey,B.Dardel,W.-D.Schneider,Y.Baer,Y.Petroff,A.Zettl,Phys.Rev.Lett.62, 336 (1989)
70. C.G. Olson,R. Liu,A.-B.Yang, D.W. Lynch,A.J. Arko,R.S. List, B. W.Veal,Y.C. Chang, P.Z.Jiang, A.P.Paulikus, Science
245, 731 (1989)
71. D.PinesandP.Monthoux,J.Phys.Chem.Solids56, 1651 (1995) and D.J. Scalapino,Phys. Rep. 250, 329 (1995)
72. R.J.Kelley,J.Ma,C.Quitmann,G.Margaritondo,M.Onellion,Phys.Rev.B50, 590 (1994)
73. J.M. Harris, Z.-X. Shen, P. J.White, D.S. Marshall, M.C. Schabel, J.N. Eckstein, I. Bozovic, Phys. Rev. B 54,R15665 (1996)
74. H.Ding,J.C.Campuzano,M.R.Norman,M.Randeria,T.Yokoya,T.Takahashi,T.Takeuchi,T.Mochiku,K.Kadowaki,
P. Guptasarma, and D.G. Hinks, J. Phys. Chem. of Solids 59, 1888 (1998)
75. C.Renner,B.Revaz,J.-Y.Genoud,K.Kadowaki,O.Fischer,Phys.Rev.Lett.80,149(1998); N. Miyakawa, P. Guptasarma,
J.F. Zasadzinski, D.G.Hinks, K.E. Gray,ibid 80, 157 (1998)
76. A.J.Millis,S.M.Girvin,L.B.Ioffe,andA.I.Larkin,J.Phys.Chem.Solids 59, 1742 (1998)
77.
A.C. Durst and P. A. Lee, Phys. Rev. 62, 1270 (2000)
78. M. Chiao, P. Lambert, R.W. Hill, C.Lupien, R. Gagnon, L. Taillefer, P. Fournier, Phys.Rev. 62, 3554 (2000)
79. J. Mesot, M.R. Norman, H. Ding, M. Randeria, J.C. Campuzano, A. Paramekanti, H.M. Fretwell, A. Kaminski, T.
Takeuchi,T.Yokoya,T.Sato,T.Takahashi,T.Mochiku,andK.Kadowaki,Phys.Rev.Lett.83, 840 (1999)
80. S.-F. Lee, D.C. Morgan, R.J. Ormeno, D.M. Broun, R.A. Doyle, J.R. Waldram, K. Kadowaki, Phys. Rev. Lett. 77, 735
(1996); T. Jacobs, S. Sridhar, Q. Li,G.D. Gu, N. Koshizuka, ibid 75, 4516 (1995); O.Waldmann,F. Steinmeyer, P. Muller,
J.J.Neumeier,F.X.Regi,H.Savary,J.Schneck,Phys.Rev.B53, 11825 (1996)
81. A. Paramekanti and M. Randeria, Physica C 341–348, 827 (2000)
82. W.N.Hardy,D.A.Bonn,D.C.Morgan,R.Liang,K.Zhang,Phys.Rev.Lett.70, 3999 (1993); D.A. Bonn, S. Kamal, A.
Bonakdarpour,R.Liang,W.N.Hardy,C.C.Homes,D.N.Basov,T.Timusk,Czech.J.Phys.46, 3195 (1996)
17 Photoemission in the High-T
c
Superconductors 991
83. S. Haas, A.V. Balatsky, M. Sigrist and T.M. Rice, Phys. Rev. B 56, 5108 (1997)
84. R. Fehrenbacher and M.R. Norman, Phys. Rev. B 50, 3495 (1994)
85. T.Timusk and B.W. Statt, Rep. Prog.Phys. 62, 61 (1999)
86. M.Randeria, In: Proceedings of the International School of Physics “Enrico Fermi” Course CXXXVI on High Temper-
ature Superconductors, ed. by G. Iadonisi, J.R. Schrieffer, and M.L. Chiafalo (IOS Press, 1998) pp. 53–75; [Varenna
Lectures, 1997] cond-mat/9710223
87. D.S.Marshall,D.S.Dessau,A.G.Loeser,C-H.Park,A.Y.Matsuura,J.N.Eckstein,I.Bozovic,P.Fournier,A.Kapitulnik,
W.E.Spicer,andZ.-X.Shen,Phys.Rev.Lett.76, 4841 (1996)
88. A.G.Loeser,Z.-X.Shen,D.S.Dessau,D.S.Marshall,C.H.Park,P.Fournier,A.Kapitulnik,Science273, 325 (1996)
89. H. Ding, T. Yokoya, J.C. Campuzano, T. Takahashi, M. Randeria, M.R. Norman, T. Mochiku, K. Kadowaki, and J.
Giapintzakis, Nature 382, 51 (1996)
90. M.R.Norman,H.Ding,M.Randeria,J.C.Campuzano,T.Yokoya,T.Takeuchi,T.Takahashi,T.Mochiku,K.Kadowaki,
P. Guptasarma, and D.G. Hinks, Nature 392, 157 (1998)
91. M.R. Presland, J.L. Tallon,R.G. Buckley, R.S. Liu, and N.E. Flower, Physica C 176, 95 (1991)
92. D.L. Feng, D.H. Lu, K.M. Shen, C. Kim, H. Eisaki, A. Damascelli, R. Yoshizaki, J.-I. Shimoyama, K. Kishio, G.D. Gu, S.
Oh,A. Andrus, J. O’Donell, J.N. Eckstein, Z.-X. Shen, Science 289, 277 (2000)
93. H. Ding, J.R. Engelbrecht, Z.Wang, J.C. Campuzano, S.-C. Wang, H.-B.Yang, R. Rogan, T. Takahashi,K. Kadowaki, and
D. G. Hinks, Phys. Rev. Lett. 87, 227001 (2001)
94. Y.J. Uemura, G.M. Luke, B.J. Sternlieb, J.H. Brewer, J.F. Carolan, W.N. Hardy, R. Kadono, J.R. Kempton, R.F. Kiefl, S.R.
Kreitzman,P. Mulhern, T.M.Riseman, D.L.Williams, B.X.Yang, S. Uchida,H. Takagi,J.Gopalakrishnan, A.W. Sleight,
M.A. Subramanian, C.L. Chien, M.Z. Cieplak, G. Xiao, V.Y. Lee, B.W. Statt, C.E. Stronach, W.J. Kossler, X.H. Yu, Phys.
Rev. Lett. 62, 2317 (1989)
95. P.A. Lee, Normal state properties of the oxide superconductors: a review, In: High Temperature Superconductivity,
ed. by K.S. Bedell,D. Coffey, D.E. Meltzer, D. Pines, J.R. Schrieffer (Addison-Wesley, New York,1990) pp. 96-116
96. M. Randeria, N. Trivedi, A. Moreo, and R.T. Scalettar, Phys. Rev. Lett. 69, 2001 (1992); N. Trivedi and M. Randeria,
Phys.Rev.Lett.75, 312 (1995)
97. V. Emery and S. Kivelson,Nature 374, 434 (1995)
98. G. Baskaran, Z. Zou and P.W. Anderson, Solid State Commun. 63, 973 (1987); G. Kotliar and J. Liu, Phys. Rev. B 38,
5142 (1988); H. Fukuyama,Prog. Theor. Phys.Suppl. 108, 287 (1992)
99. P.A. Lee and X.G.Wen, Phys. Rev. Lett. 76, 503 (1996) and Phys. Rev. B 63, 224517 (2001)
100. J.Tallon and J. Loram, Physica C 349, 53 (2001)
101. S. Chakravarty, R.B. Laughlin,D.K. Morr and C. Nayak, Phys.Rev. B 63,094503 (2001)
102. C.M.Varma, Phys. Rev. Lett. 83, 3538 (1999)
103. A. Kaminski, S. Rosenkranz,H. Fretwell, J.C.Campuzano, Z. Li, H. Raffy, W.G.Cullen,H.You, C.G. Olson,H. Hoechst,
Nature 416,610 (2002)
104. S.H. Pan, J.P. O’Neal, R.L. Badzey, C. Chamon, H. Ding, J.R. Engelbrecht, Z. Wang, H. Eisaki, S. Uchida, A.K. Gupta,
K.-W.Ng,E.W.Hudson,K.M.Lang,J.C.Davis,Nature413, 282 (2001)
105. L.Perfetti,C.Rojas,A.Reginelli,L.Gavioli,H.Berger,G.Margaritondo,M.Grioni,R.Gaal,L.Forro,F.Rullier Albenque,
Phys.Rev.B64, 115102 (2001)
106. M.R.Norman,H. Ding, H. Fretwell, M. Randeria,and J.C. Campuzano, Phys. Rev. B 60, 7585 (1999)
107. M.R.Norman,A.Kaminski, J. Mesot, J.C.Campuzano, Phys. Rev. B 63, 140508 (2001)
108. M.R.Norman,M. Randeria, H. Ding, and J.C. Campuzano, Phys. Rev. B 57
, R11093 (1998)
109.
D.S. Dessau, B.O. Wells, Z.-X. Shen, W.E. Spicer, A.J. Arko, R.S. List, D.B. Mitzi, and A. Kapitulnik, Phys. Rev. Lett. 66,
2160 (1991); D.S. Dessau, Z.-X. Shen, B.O. Wells, D.M. King, W.E. Spicer, A.J. Arko, L.W. Lombardo,D.B. Mitzi, and A.
Kapitulnik, Phys. Rev. B 45, 5095 (1992)
110. Q. Huang, J.F.Zasadzinski, K.E. Gray, J.Z. Liu, and H. Claus, Phys. Rev. B 40, 9366 (1989)
111. Y.Kuroda and C.M.Varma, Phys. Rev. B 42, 8619 (1990)
112. S. Engelsberg and J.R. Schrieffer, Phys.Rev. 131, 993 (1963)
113. D.J. Scalapino,In: Superconductivity, ed. by R.D. Parks (Marcel Dekker, New York,1969) Vol 1, p.449
114. Z.-X.Shen and J.R. Schrieffer, Phys.Rev. Lett. 78, 1771 (1997)
115. P.B. Littlewood and C.M.Varma, Phys. Rev. B 46, 405 (1992)
992 J.C. Campuzano, M.R. Norman, and M. Randeria
116. M.R.Norman and H. Ding, Phys. Rev. B 57, R11089 (1998)
117. S.M.Quinlan, P.J. Hirschfeld, and D.J. Scalapino,Phys. Rev. B. 53, 8575 (1996)
118. J.Rossat-Mignot,L.P. Regnault,C.Vettier, P.Bourges,P.Burlet, J.Bossy,J.Y.Henry,and G. Lapertot,Physica C 185-189,
86 (1991); H.A. Mook, M. Yethiraj, G. Aeppli, T. E. Mason, T. Armstrong, Phys. Rev. Lett. 70, 3490 (1993); Hung Fai
Fong,B.Keimer,P.W.Anderson,D.Reznik,F.DoganandI.A.Aksay,Phys.Rev.Lett.75, 316 (1995)
119. H.F. Fong, P.Bourges,Y. Sidis, L.P. Regnault,A.Ivanov, G.D. Gu, N. Koshizuka, and B. Keimer, Nature 398, 588 (1999)
120. In: The Gap Symmetry and Fluctuations in High T
c
S uperconductors, ed. by J. Bok, G. Deutscher, D. Pavuna, and S.A.
Wolf (Plenum, New York, 1998) p. 349
121. J.F. Zasadzinski, L. Ozyuzer, N. Miyakawa,K.E. Gray, D.G. Hinks, C. Kendziora, Phys.Rev. Lett. 87, 067005 (2001)
122. P. Dai,M.Yethiraj, H.A. Mook, T.B.Lindemer, and F. Doˇgan,Phys.Rev.Lett.77, 5425 (1996)
123. R.Preuss,W.Hanke,C.Grober,and H.G.Evertz,Phys.Rev.Lett.79,1122(1997); J.Schmalian,D.Pines,andB.Stojkovic,
ibid. 80, 3839 (1998); X.-G.Wen and P.A.Lee, ibid. 76, 503 (1996)
124. E. Demler and S.-C.Zhang, Phys. Rev. Lett. 75, 4126 (1995)
125. P. Dai, H.A. Mook, S.M. Hayden,G. Aeppli, T.G. Perring, R.D. Hunt, and F. Doˇgan, Science 284, 1344 (1999)
126. M. Hengsberger, D. Purdie, P. Segovia, M. Garnier, Y. Baer, Phys. Rev. Lett. 83, 592 (1999); T. Valla, A.V. Fedorov, P.D.
Johnson, S.L. Hulbert, ibid., 2085 (1999)
127. 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,Nature412, 510 (2001)
128. 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,
ShaJian,andD.G.Hinks,Phys.Rev.Lett.87, 177007 (2001)
129. M. Eschrig and M.R.Norman,Phys. Rev. Lett. 85, 3261 (2000); Phys. Rev. B67, 144503 (2003)
130. M. Eschrig and M.R.Norman,Phys. Rev. Lett. 89, 277005 (2002)
131. A.D. Gromko,A.V. Federov,Y.-D. Chuang, J.D. Koralek,Y.Aiura,Y.Yamaguchi, K. Oka,Y. Ando, D.S. Dessau, Phys. Rev.
B68, 174520 (2003)
132. M.R. Norman, M. Randeria, B. Janko,J.C. Campuzano, Phys. Rev. B 61,14742 (2000)
133. J.C. Campuzano, H. Ding, H. Fretwell, J. Mesot, A. Kaminski, T. Sato, T. Takahashi, T. Mochiku, and K. Kadowaki,
cond-mat/9811349
134. M.R.Norman and C. Pepin, Phys.Rev. B 66, 100506 (2002)
135. H.J.A. Molegraaf,C.Presura,D.van der Marel,P.H.Kes, M.Li, Science 295,2239 (2002); A.F. Santander-Syro,R.P.S.M.
Lobo,N.Bontemps,Z.Konstantinovic,Z.Z.Li,H.Raffy,Europhys.Lett.62, 568 (2003)
136. P.W.Anderson,Phys. Rev. B 42, 2624 (1990)
18 Neutron Scattering and the Magnetic Response of Superconductors
and Related Compounds
Stephen M.Hayden H.H. Wills Physics Laboratory, University of Bristol, Bristol, United Kingdom
18.1 Introduction .............................................................................993
18.2 The Neutron Scattering Technique ..........................................................994
18.2.1NeutronScatteringExperiments......................................................994
18.2.2ThePartialDifferentialCross-Section.................................................995
18.2.3TheGeneralizedSusceptibility........................................................995
18.2.4TheKramers–KronigRelations,Wavevector-DependentandLocalSusceptibility...........996
18.2.5TheMagneticCross-Section..........................................................998
18.2.6PolarizedNeutronScattering.........................................................998
18.2.7TheMomentSumRuleandtheFluctuation-DissipationTheorem........................999
18.3 The Static Spin Susceptibility of Superconductors ............................................999
18.4 Magnetic Excitations in Metals and Weakly Coupled Superconductors.........................1001
18.4.1ParamagneticMetals...............................................................1001
18.4.2TheSuperconductingMetal.........................................................1001
18.5 Excitations and Superconductive Pairing ...................................................1003
18.6 High Temperature Superconductivity ......................................................1004
18.6.1StructuresandPhaseDiagrams......................................................1005
18.6.2 Magnetic Excitations in the Parent Compounds of the High-T
c
Superconductors..........1008
18.6.3 Magnetic Excitations in La
2−x
Sr
x
CuO
4
................................................1015
18.6.4 Magnetic Excitations in YBa
2
Cu
3
O
6+x
................................................1017
18.6.5 Magnetic Excitations in Other High-T
c
Systems.......................................1022
18.7 Discussion ..............................................................................1023
18.8 Final Remarks ...........................................................................1024
References..............................................................................1024
18.1 Introduction
Neutron scattering has played a major role in under-
standing the properties of conventional and uncon-
ventional superconductors.Theneutron is a versatile
tool for studying condensed matter in general, and
superconductors in particular, for a number of rea-
sons. Firstly, the neutron’s interaction with matter is
weak and extremely well understood. The weak in-
teraction between matter and neutrons means that
the scattering cross-section can be interpreted using
the Born approximation.Most elements have a small
neutron absorption cross-section. Hence bulk mate-
rial, with sample dimensions up to several cm, can
be studied. Secondly, neutrons interact with matter
through the strong and electromagnetic forces. They
are able to investigate the position and motion of the
atoms through the interaction of the neutron and
994 S.M. Hayden
the nuclei via the strong force. Their magnetic in-
teraction with the electrons allows magnetic struc-
tures and excitations to be investigated. The mag-
netic interaction can be understood as an interac-
tion between the neutron and the microscopic mag-
netic field distribution. Thus magnetic field struc-
tures, such as the Abrikosov flux lattice, and their ex-
citations can be studied.A third reason which makes
neutron scattering a powerful tool is that the neutron
has a mass. This means that the energy and momen-
tum of the neutron can be matched to the collective
excitations of interest.
In view of the above, one might expect that neu-
tron scattering would have made a significantimpact
on our understanding of superconductors.In this re-
view we will concentrate on the study on spin cor-
relations in superconductors and their related com-
pounds. It is therefore pertinent to mention some
areas not covered in detail by this review. These in-
clude the determination of the atomic structure of
superconducting compounds, the study of lattice ex-
citations (phonons) and the study of the flux line
lattice by small angle neutron scattering (SANS).
18.2 The Neutron Scattering Technique
The neutron scattering technique and its interpre-
tation has been described in detail in a number of
classic textbooks [1–4]. Neutrons interact with con-
densed matter through the strong nuclear force and
the electromagnetic force. Nuclear scattering is used
to determine the atomic positions and motion in
condensed matter systems. It is a powerful tool for
determining phonon dispersion relations in solids.
Magnetic neutron scattering arises because of the
neutron’s magnet moment. It is interesting to note
that there are coherent cross terms between the nu-
clear andmagnetic cross-sectionswhich are also use-
ful experimentally (see Sect. 18.3). In this section
we summarize the basic features of magnetic neu-
tron scattering.The resultsof magneticneutronscat-
tering experiments are most succinctly discussed in
terms of the generalized magnetic susceptibilityand
the partial differential cross-section. We will intro-
duce these quantitiesbelow.
18.2.1 Neutron Scattering Experiments
Neutrons for condensed matter research are pro-
duced either by a nuclear reactor or by a spallation
source. It is well known that reactors produce neu-
trons through the fission of U-235 in a chain reac-
tion. A spallation source produces neutrons when
a high-energy (e.g. 800 MeV) proton beam hits a
heavy metal (Ta, Hg, U) target. A spallation source
can be easily pulsed, this allows the neutron’s energy
to be measured by time-of-flight methods. In this
review, we are mainly concerned with the measure-
ment of excitations using inelastic neutron scatter-
ing. Pulsed-spallation and reactor sources and their
associated instrumentation each have their own ad-
vantages and disadvantages for such measurements.
Reactors currently have more neutron flux and have
better developed instrumentation for the use of po-
larized neutrons. Spectrometers with a low intrinsic
background can be built at pulsed neutron sources.
Pulsed spallation sources also lend themselves to ex-
periments with epithermal neutrons. Figure 18.1(b)
shows a schematic diagram of a triple-axis spec-
Fig. 18.1. (a) Illustration of the geometry of a neutron scat-
tering experiment. Neutrons with wavevector k
i
are inci-
dent on a sample (S). Scattered neutrons are detected in
asolidangled§ over an energy range dE.(b)Schematic
diagram of a triple-axis spectrometer,which is widely used
at reactor sources for measuring excitations. The incident
wavevector of the neutrons is selected by a monochroma-
tor (M) crystal (typically graphite). Neutrons scatter at the
sample (S). The final wavevector (energy) of the scattered
neutrons is selected by an analyzer (A) crystal. Neutrons
are detected in a He-3 ionization detector
18 Neutron Scattering and the Magnetic Response of Superconductors 995
Fig. 18.2. The MAPS spectrometer at the ISIS pulsed
spallation source of the Rutherford Appleton Labora-
tory. MAPS is an example of a direct-geometry time-
of-time chopper spectrometer. Monochromated pulsed
neutrons are produced using a Fermi chopper. Neu-
trons are detected by an array of position sensitive de-
tectors (Figure: RAL.)
trometer,the workhorse for inelastic experiments on
single crystals at reactor sources. The energies of the
monochromated and scattered neutrons are deter-
mined by Bragg reflection from crystals of graphite,
silicon, copper, or, in the case of polarized neutrons,
a Heusler alloy. Neutrons are detected by a He-3 ion-
ization chamber. Figure 18.2 shows a chopper spec-
trometer at a pulsed spallation source with position
sensitive detectors (PSD) and time-of-flight energy
analysis. This type of spectrometer is a relatively re-
cent development which complements in many re-
spects the three-axis spectrometer and is used for
measuring excitations in single crystals.
18.2.2 The Partial Differential Cross-Section
Figure 18.1(a) shows the geometry of a typical neu-
tron scattering experiment. Neutrons with (initial)
wavevector k
i
and energy E
i
are incident on the sam-
ple. Some neutrons will be scattered such that they
have final wavevector k
f
and final energy E
f
.Wecan
determine the momentum and energy transfer be-
tween the neutron and an excitation,
Q = k
i
− k
f
,
! = E
i
− E
f
.
Neutron scattering experiments are best described
in terms of a partial differential cross-section. This
is a measure of the probabilitythat a neutron scatters
from an initialwavevector k
i
to a final wavevector k
f
.
Consider a beam of neutrons with the same energy
E
i
incident on a small sample. Let us set up a neutron
counter and measure the number of neutrons scat-
tered in a given directionas a functionof their (final)
energy E
f
.Let the countersubtend a solid angle d§ at
the sample (see Fig. 18.1(a)). The partial differential
cross-section is defined by the equation
d
2
d§ dE
=
(number of neutrons scattered per sec-
ond into d§ with final energy between
E
f
and E
f
+dE)/¥ d§ dE ,
where ¥ is the flux of incident neutrons, that is the
number per unit area per second incidenton the sam-
ple.
18.2.3 The Generalized Susceptibility
To most physicists, the magnetic susceptibility is the
ratio of the average magnetisation M to the uniform
field H which induces it. This susceptibility is typi-
cally measured using a SQUID or vibrating sample
magnetometer.More generally,a system will respond
to a magnetic field which varies in space and/or
time,H
Q!
exp[i(Q ·r − !t)]. If the spatial frequency
of the magnetic field is comparable to the atomic
spacing or the temporal frequency of the magnetic
field is comparable to the magnetic excitations of
the system, then the response will contain impor-
tantinformationabout the microscopic magnetic dy-
996 S.M. Hayden
namics of the system. Such information about the
magnetic dynamics can be obtained using a num-
ber of probes including nuclear magnetic resonance
(NMR),muon spin resonance (SR), Raman scatter-
ing and, of course, neutron scattering. The informa-
tion obtained from these probes about the micro-
scopic magnetic response is best described in terms
of the generalized magnetic susceptibility, which we
will now introduce.
In general, a susceptibility is used to character-
ize the response of a system to a probing perturba-
tion or field. For example, the perturbation might be
an electric field, a magnetic field or a force experi-
enced by a particle. If the response is proportional
to the perturbation, then we can define a suscepti-
bility or response function as the ratio of a “gen-
eralized force” (the perturbation) to a “generalized
displacement” (the response). In this review, we will
be concerned with the magnetic susceptibility,thus,
the relevant response is the magnetization M(r, t)
and the probing field is the magnetic field H(r, t).
Neutron scattering is usually most easily understood
in the frequency domain. We therefore introduce
the corresponding Fourier components M(Q, !)and
H(Q, !).Thegeneralized wavevector and frequency-
dependent magnetic susceptibility(Q, !)cannow
be defined through the equation
M
(Q, !)=
(Q, !) H
(Q, !) , (18.1)
wherewehaveonlyconsideredtheresponsein-
duced at the same Q as the perturbing field (i.e.
(Q, !) ≡
(Q, Q, !)) and the sum is over the
different spatial directions , = x,y and z.
The Reduced Susceptibility
The susceptibility defined in Eq. 18.1 is that associ-
ated with the total magnetization of the system. In
model calculations (e.g. a calculation of spin waves
in the Heisenberg model), we usually assume that
the magnetic moment is either localized at atomic
sites or completely itinerant. Such calculations yield
a reduced susceptibility (Q, !)
reduced
[7]. The total
susceptibility (Q, !)
total
of Eq. 18.1 includes the ef-
fect of spreading out the magnetization associated
Fig. 18.3. The spin form factor of the Cu
2+
d
x
2
−y
2 orbital
[5,6]. The solid and dashed lines correspond to wavevec-
tors parallel to the x and z directions, respectively and
demonstrate the anisotropy of the form factor in this case
with each atom. The electronic states (and hence the
magnetisation distribution in space) are described
by atomic orbitals orWannier functions [7,8].The to-
tal and reduced susceptibilitiesare related through a
form factor|F(Q)|
2
whichis the Fourier transformof
the normalized magnetization associated with each
site or orbital:
(Q, !)
total
= |F(Q)|
2
(Q, !)
reduced
. (18.2)
The results of magnetic neutron scattering experi-
ments are usually expressed in terms of the reduced
susceptibility so that they can be directly compared
with theory. Throughout the rest of this article we
will use the symbol (Q, !)todenotethereduced
susceptibility unless noted explicitly. The measured
response needs to be corrected for the form factor
|F(Q)|
2
and this is done using an equation such as
Eq. 18.9. The form factor is usually evaluated from
published parameterizations based on atomic calcu-
lations [5]. Fig. 18.3 shows the form factor for the
d
x
2
−y
2
orbital of a Cu
2+
ion. Note that the form factor
is anisotropic in this case.
18.2.4 The Kramers–Kronig Relations,
Wavevector-Dependent and Local
Susceptibility
Information obtained from inelastic neutron scat-
tering is most easily expressed in terms of the real
and imaginary parts of the generalized susceptibility.
18 Neutron Scattering and the Magnetic Response of Superconductors 997
Fig. 18.4. The real and imaginary parts
of the response function for a damped
harmonic oscillator. The peak in the
imaginary part of the response is used
to estimate the frequency of the oscilla-
tor
Thus a few words about the physical interpretation of
these qualities seems in order. The generalized sus-
ceptibility can be written in terms of its real part
(Q, !) and imaginary part
(Q, !),
(Q, !)=
(Q, !)+i
(Q, !) . (18.3)
To illustrate the interpretation of
(Q, !), we con-
sider the case of a harmonic oscillator with a single
pole (see Fig.18.4).The imaginary part of the suscep-
tibility is peaked near the frequency of the oscillator,
while the real part changes sign at this frequency.
Such a response function would apply to a well de-
fined excitation such as a spin wave in a ferromagnet.
Magnetic excitations can also be strongly damped,
this is particularly the case in metals. The response
function in this limit might be better described by
an over-dampedharmonic oscillator(Lorentzian) re-
sponse. The real part of the susceptibility at zero fre-
quency (Q)=
(Q, ! = 0) hasa particular physical
significance: it measures the response to a spatially
varying static magnetic field.
It is well known that the real and imaginary parts
of causal response functions are connected through
the Kramers–Kroning relations [7],
(Q, !)−
(Q, ∞)=
1
∞
−∞
(Q, !
)
!
− !
d!
,
(18.4)
(Q, !)=−
1
∞
−∞
(Q, !
)−
(Q, ∞)
!
− !
d!
.
(18.5)
The Kramers–Kroning relations are useful because
the absorption or scattering cross section is propor-
tional to
(Q, !). By integrating over frequency we
canobtaintherealpart
(Q, !).
Wavevector-Dependent Susceptibility
The wavevector-dependent susceptibility (Q)=
(Q, ! = 0) is the response to a static spatially vary-
ing magnetic field
(Q)= M
(Q)/H
(Q). This may
be obtained fromthe firstKramers–Kroning relation
(Eq. 18.4),
(Q)=
1
∞
−∞
(Q, ! =0)
!
d! . (18.6)
Local Susceptibility
Another useful quantity which can be derived from
(Q, !)isthelocal (imaginary)susceptibility
(!).
The local susceptibility is a “weighted density of
states” obtained by integrating over the Brillouin
zone for a given frequency,
(!)=
BZ
(Q, !)d
3
Q
BZ
d
3
Q
. (18.7)
We shall see in Sect. 18.2.7 that
9
M
2
(Q, !)
:
∝
(Q, !). Hence for a system with a single magnetic
site in each unit cell, the moment fluctuating with a
given frequency is proportional to
(!).
998 S.M. Hayden
18.2.5 The Magnetic Cross-section
Neutron scattering directly measures the imagi-
nary part of the generalized magnetic susceptibility
(Q, !).Wewill start by considering a systemwitha
diagonal susceptibility tensor with two independent
components i.e.
xx
(Q, !)=
yy
(Q, !)=
⊥
(Q, !)
and
zz
(Q, !). The axial direction might represent
the direction of an applied field, i.e. the quantiza-
tion direction or the tetragonal axis of a crystal. The
scattering cross-section for unpolarized neutrons is,
d
2
d§ dE
f
=( r
e
)
2
k
f
k
i
|F(Q)|
2
×
1
1−exp(−!/kT)
1
g
2
2
B
×
(1 +
ˆ
Q
2
)
⊥
(Q, !)
+(1−
ˆ
Q
2
)
zz
(Q, !)
, (18.8)
where ( r
e
)
2
=0.2905 barn sr
−1
−2
B
, k
i
and k
f
are the
incident and final neutron wavevectors and |F(Q)|
2
is the magnetic form factor.
If thesystem is isotropic i.e.
⊥
(Q, !)=
zz
(Q, !)
then Eq. 18.8 becomes
d
2
d§ dE
f
=( r
e
)
2
k
f
k
i
|F(Q)|
2
×
1
1−exp(−!/kT)
×
2
g
2
2
B
(Q, !) . (18.9)
It is relatively easy to place neutron scattering mea-
surements on an absolute scale unit either by mea-
suring the elastic incoherent scattering from a vana-
dium standard or by measuring the low-frequency
coherent phonon scattering from the sample [9].
Throughout this review, we label momentum trans-
fers (Q
x
, Q
y
, Q
z
)inunitsofÅ
−1
by their reciprocal
space positions (h, k, l), where Q= ha
∗
+ kb
∗
+ lc
∗
.
18.2.6 Polarized Neutron Scattering
Magnetic neutron scattering has a number of unique
features. Implicit in Eq. 18.8 is the fact that neutrons
are only sensitiveto momentor spin fluctuationsper-
pendicular to the momentum change Q.Iftheneu-
tron spin isquantized along the z-axis,e.g.by a small
magnetic field applied along the z-axis, then mag-
netic scattering can either involve a change of the
neutronspin(S = 1), or no change in the neutron
spin (S = 0). These are known as “spin-flip” and
“non-spin-flip”scattering respectively.Spin-flip scat-
tering is caused by moment fluctuationsperpendicu-
lar to the quantizationaxis and non-spin-flipscatter-
ing by fluctuations parallel to the quantization axis.
Polarized neutronscattering allows the separation of
magnetic scattering from nuclear scattering and the
separation of
zz
from
⊥
. In systems without spin
anisotropy (e.g. paramagnets and conventional su-
perconductors), there will be no distinction between
zz
and
⊥
and scattering will, in general, be a mix-
ture of spin-flip and non-spin-flip. However, even in
this case, polarized neutron scattering can be a pow-
erful tool in isolating the magnetic response.A num-
ber of geometries have been used for neutron scat-
tering measurements (with polarization analysis) on
high-T
c
superconductors [10,11].The cross sections
for an isotropic system are summarized below,
d
2
d§ dE
↑→↑,HQ
=0,
d
2
d§ dE
↑→↓,HQ
=( r
e
)
2
k
f
k
i
|F(Q)|
2
×
2/g
2
2
B
1−exp(−!/kT)
×
(Q, !)
d
2
d§ dE
↑→↓,H⊥Q
=( r
e
)
2
k
f
k
i
|F(Q)|
2
×
1/g
2
2
B
1−exp(−!/kT)
×
(Q, !) , (18.10)
where the arrows ↑ and ↓ represent the spin polar-
ization along the quantizing magnetic field H applied
to the sample. In the first two configurations H is
applied parallel to Q and all magnetic scattering is
spin-flip.