Bennemann K.H., Ketterson J.B. Superconductivity: Volume 1: Conventional and Unconventional Superconductors; Volume 2: Novel Superconductors
Подождите немного. Документ загружается.
18 Neutron Scattering and the Magnetic Response of Superconductors 1019
Fig. 18.33. The energy dependence of the incommensu-
rate response in YBa
2
Cu
3
O
6.7
. Peak positions are measured
along (h, h, 0) line. The horizontal bars show the widths of
the peaks. (After M.Arai et al. [137])
Fig. 18.34. The dispersion of the incommensurate peaks
in YBa
2
Cu
3
O
6.85
. Peak positions are measured along the
(1/2+h,1/2,0) line. (After P. Bourges et al. [138])
Fig. 18.35. The resonance “exciton” mode in the supercon-
ducting state of YBa
2
Cu
3
O
6.6
(T
c
=63 K, T=10K).Theplot
is
(q, !)forQ
2D
=(1/2, k). (After Hayden et al.[141])
mensurate scattering develops as the temperature is
lowered throughT
c
.Forunderdopedcompositionsat
T
c
, there is evidence for magnetic scattering at ener-
gies where the incommensurate feature develops but
a strong incommensurate structure is not seen [135].
As with La
2−x
Sr
x
CuO
4
there is a suppression of the
magnetic scattering near (1/2,1/2) at the lowest en-
ergy transfers as the temperature is lowered through
T
c
[81,83,132,142,143].This suppression is partly re-
sponsible for the formation of the incommensurate
scattering described above and the resonance mode.
We can define a spin gap (arbitrarily) as the energy
below which the response decreases on entering the
superconducting state. This spin gap energy is plot-
tedinFig.18.39andcomparedwithmeasurements
of the gap extracted from other probes.
The resonance mode is an extremely obvi-
ous feature in the magnetic response function of
YBa
2
Cu
3
O
6+x
. However when its integrated intensity
is placed on an absolute scale we find that it only
accounts for a small fraction of the total fluctuat-
ing moment per Cu atom expected for an ion with
S =1/2. Estimates for the fluctuating moment asso-
ciated with the resonance from various groups are
given in Table. 18.4. Since two of the Cu atoms in
1020 S.M. Hayden
Table 18.4.The spectral weight of the resonance obtain by integrating in wavevector and energy. The symmetry (Sym.) is
denoted by o (odd) or e (even). †Hayden et al. [141] integrate over the whole resonance mode
Composition Ref. Sym. M
2
(
2
B
f.u.
−1
)
YBa
2
Cu
3
O
6.5
Stock et al. [144] o 0.052
YBa
2
Cu
3
O
6.6
Dai et al. [80] o 0.06 ± 0.04
YBa
2
Cu
3
O
6.6
Hayden et al. [141] o 0.12 ± 0.02 †
YBa
2
Cu
3
O
6.7
Fong et al. [79] o 0.056
YBa
2
Cu
3
O
6.93
Dai et al. [80] o 0.06 ± 0.04
YBa
2
Cu
3
O
7
Fong et al. [79] o 0.043
Y
0.9
Ca
0.1
Ba
2
Cu
3
O
7
Pailh`es et al.[134] o 0.042
e 0.036
the formula unit are in the CuO
2
plane, we should
compare M
2
res
with
2 × g
2
2
B
S(S +1)=6
2
B
.
Thus, the resonance only accounts for a small frac-
tion of the total Cu
2+
moment.
As discussed in Sect. 18.6.1 at higher doping lev-
els YBa
2
Cu
3
O
6+x
is orthorhombic. The presence of
Cu-O chains makes the structure anisotropic. In-
deed this anisotropy is reflected in many physical
properties of YBa
2
Cu
3
O
6+x
, for example, the pene-
tration depth [145]and the resistivity [146].The first
measurementsof incommensurate structure forsub-
resonance energies [135,136] were made on twinned
samples. If YBa
2
Cu
3
O
6+x
is annealed without the ap-
plication of uniaxial pressure then it forms a mi-
crostructurewith two types of domains such that the
a and b axes are interchanged when moving from one
domain to the other. It is possible to prepare almost
completely “detwinned” or single domain samples.
Recent measurements on detwinned samples show
that the low energy scattering reflects the anisotropy
of the crystal lattice [147,148].Specifically, the inten-
sity is strongest near Q
2D
=(1/2±ı, 1/2)rather than
near (1/2, 1/2 ± ı).
So far we have discussedthe magnetic response for
energies less than or equal to the resonance energy.
The high-energy (! > E
res
)responseisofconsid-
erable interest for two reasons. Firstly, the cuprates
are derived from parent compounds which are an-
tiferromagnetic and have magnon excitations up to
about 320 meV. We might expect that some resid-
ual interactions remain for superconducting com-
positions. Secondly, it is likely that high-energy ex-
citations could be important for pairing. The high-
energy response has been the subject of a number of
studies [79,80,137,141,144,149,150]. Early work on
the hot source IN1 spectrometer at the ILL [149] and
on the HET and MARI chopper spectrometers at ISIS
[80,150] showed that high-energy excitations per-
sisted in underdoped YBa
2
Cu
3
O
6+x
for x ≈ 0.5–0.6
bothin theoddandevenchannels.Withtheavailabil-
ity of the MAPS spectrometer at ISIS, better high en-
ergy data has recently become available. Figure 18.36
shows data collected on underdoped YBa
2
Cu
3
O
6.6
.
For energies above the resonance energy (34 meV
for this composition), the response exhibits a square-
shaped diffuse pattern in wavevector, possibly with
peaks near Q
2D
=(1/2 ± , 1/2 ± )and(1/2 ∓
, 1/2 ∓ ). It is interesting to compare the wavevec-
tor and energy-integrated spectral weights of the
high-energy (odd) response with the resonance and
low-energy incommensurate structure. The two val-
ues obtained for M
2
in a single experiment are
0.12±0.02
2
B
per formulaunit for the resonance and
sub-resonance structure and 0.26 ± 0.05
2
B
for the
high-energy scattering (60 < ! < 120 meV).Thus,
the higher-energy (E > E
res
) response has a signif-
icantly greater contribution to the total fluctuating
moment than the resonance structure [141].
18 Neutron Scattering and the Magnetic Response of Superconductors 1021
Fig. 18.36. The odd magnetic excitations in
YBa
2
Cu
3
O
6.6
. (After Hayden et al. [141])
1022 S.M. Hayden
The Effect of t he Bilayer Structure
Theeffectofthebilayerstructureistocreatetwo
types of excitation with odd and even symmetry.
The even excitations are gapped (see Table 18.3)
in the undoped parent compound with !
gap
≈
70 meV. For underdoped superconductors, the on-
set frequency of the even scattering drops to about
60 meV [80,149,150]. The data of Dai et al. [80] illus-
trating the gap in the even excitations are shown in
Fig. 18.31.
For higher doping levels the even excitations are
weaker and more difficult to observe. However,there
have been observations of an increase in scattering
Fig. 18.37. The odd and even resonances in Y
0.9
Ca
0.1
Ba
2
Cu
3
O
7
as shown through the difference in the response
above and below T
c
.(a) Odd resonance (l =5.2, open
circles); Even resonance (l =7,closed circles). (b)Even
resonance with a smaller temperature difference. (After S.
Pailh`es et al.[134])
on entering the superconducting state for frequen-
cies near or below the even onset energies. This“even
resonance” is weaker and at a higher energy than its
odd counterpart. Figure 18.37 shows measurements
by Pailh`es et al.[134] of the odd and even resonances
in overdoped Y
0.9
Ca
0.1
Ba
2
Cu
3
O
7
. The energy of the
even resonance drops continuously as the doping in-
creases. Figure 18.38 summarizes the doping depen-
dence of the odd and even resonance energies.
Fig. 18.38. The doping dependence of the odd (open
circles) and even (closed circles) resonance energies in
YBa
2
Cu
3
O
6+x
.Also shown is the even gap in the excitations
(closed squares). (Data collected from [79–84,115,116,134,
140,151])
18.6.5 Magnetic Excitations in Other High-T
c
Systems
There is a large body of work on YBa
2
Cu
3
O
6+x
and La
2−x
(Sr,Ba)
x
CuO
4
. In addition, a number of
other high-T
c
systems have been measured. How-
ever, less is known about the detailed (Q, !)de-
pendence of their magnetic response. Fong et al.
[152] report the resonance in optimally doped (T
c
=
91 K) Bi
2
Sr
2
CaCu
2
O
8+ı
with E
res
=43meVand
m
2
res
=0.23
2
B
. He et al. [153] report that E
res
drops
in overdoped (T
c
=83K)Bi
2
Sr
2
CaCu
2
O
8+ı
.Signif-
icantly, a resonance peak has also been observed in
the optimally doped (T
c
=92.5K)single-layercom-
pound Tl
2
Ba
2
CuO
6+ı
[154] with E
res
=47meV.
18 Neutron Scattering and the Magnetic Response of Superconductors 1023
18.7 Discussion
The theoretical description of the electronic prop-
erties of cuprates still remains controversial. Since
the magnetic response is intimately connected to the
electron dynamics, it is inevitable that its interpreta-
tion will also remain controversial. Having said this,
it should remembered that neutron scattering mea-
sures simple correlation functions which can be di-
rectly related to physical properties such as the mag-
netic susceptibility. Several excellent reviews of the
theory relating to spin dynamics (in addition to those
is this series!) have recently appeared [155–158].
The phase diagrams of the cuprates give a phe-
nological indication of how the spin fluctuations
might evolve with doping. The parent antiferromag-
nets are S =
1
2
antiferromagnetic square-lattice insu-
lators withfairly well understood magnon and multi-
magnon excitations as discussed in Sect. 18.6.2. The
overdoped cuprates show more metallic behaviour
such as AMRO oscillations [50]. Thus we might
expect that the excitations in overdoped cuprates
could be described using a Lindhard picture (See
Sect. 18.4.2). Throughout most of the phase diagram
more complicated models are required, however the
excitations may retain features from these respective
limits.
Experimentally, we find that doping causes a col-
lapse of the very-high (! ≈ 2J) frequency response
and leads to an increase in the lower energy re-
sponse and a lowering of the characteristic energy
of the magnetic excitations. This is illustrated in
La
2−x
Sr
x
CuO
4
, where ISIS measurements show that
the response for ! ≈ 15 meV increases by a fac-
tor of two when x is changed from 0 to 0.14 (see
Fig. 18.29). The doping also causes an incommensu-
rate (IC) structure to develop in the
(q, !)atlow
energies in both La
2−x
Sr
x
CuO
4
and YBa
2
Cu
3
O
6+x
.
However, the structure in LSCO is clearly developed
at above T
c
,whereas in YBCO the development of the
IC structure appears to be associated with the reso-
nance and pseudogap for underdoped compositions
and the superconducting transition for optimally
doped compositions. The incommensurate spin or-
der and low-energy incommensurate spin fluctua-
tionshave beeninterpreted asstripe orderand stripe
fluctuations (see e.g. Kivelson et al. [155] for a re-
view). A stripe picture can explain some features,
such as the variation of the incommensurability at
low doping (see Fig. 18.25), however, the incommen-
surability at low frequency persists to high doping
levels both in LSCO [120] and YBCO [81,148]. An-
other model is that the incommensurate structure
is due to underlying Fermi surface nesting [29–31].
The validity of this approach must be questioned
asonemovestowardstheunderdopedsideofthe
phase diagram where the cuprates are anything but
conventional metals. The nesting picture is also able
to account for the incommensurate structure (see
Fig. 18.36(b)) observed at higher frequencies (E ≈
85 meV) in YBa
2
Cu
3
O
6+x
[35,141].
A universal feature of the cuprates seems to be
that entering the superconducting state causes an in-
crease in scattering near the energy ! ≈ 5k
B
T
c
.In
LSCO this increase appears at the incommensurate
positions and in YBCO at occurs at Q
2D
=(
1
2
,
1
2
). The
effect occurs over a relatively small energy range in
YBCO and the associated peak has become known
as the “resonance” [10,132, 133]. As pointed out in
Sec. 18.4.2, the BCS coherence factors can give rise
to an increase in scattering in a d-wave supercon-
ductor for energies near 2. However, the sharpness
of the resonance in YBCO suggests that it may be a
spin-triplet exciton below 2
0
[32–38,133]. This ex-
planation also relies on the coherence factors [133].
It seems likely that the coherence factors are also re-
sponsible for the increase in the more gapless LSCO
system (see Fig. 18.27). A second explanation pro-
posedfortheresonanceisthat itis a particle–particle
antibound resonance which could only be observed
below T
c
[159]. This theory predicts an antibound
resonance at higher energies which is not observed
[156,160].
Another universal feature of the cuprates is that
the magnetic response is suppressed at low energies
on entering the superconducting state. Thus we may
(arbitrarily) define a “spin-gap” 2
spin
as the energy
below which magnetic excitations are suppressed as
we pass through T
c
.
Within a BCS model (see Sect. 18.4.2), we would
expect a suppression of the magnetic response at low
energies due to the formation of a superconduct-
1024 S.M. Hayden
Fig. 18.39.Doping dependenceof thespin gap 2
spin
and the
higher energy pseudogap 2
∗
obtained for various probes
for YBCO and Bi-2212. Doping is estimated using the em-
pirical relation T
c
/T
max
c
=1−82.6(n − n
opt
) [161] with
T
max
c
=94K.The solid line is the BCS weak coupling result
=2.14k
B
T
c
for a 2D d-wave superconductor [162]. The
spingap(YBCO)isdeterminedfromINS[81,134,141].Pen-
etration depth measurements (YBCO) [163]. In Bi-2212,
the higher energy scale pseudogap is determined from SIS
tunnelling [164] and by ARPES measurements in the su-
perconducting [165] and the normal state [166–168]
ing gap. In the case of an s-wave superconductor,
the suppression is complete at T = 0 for ! < 2.
For a d-wave superconductor the suppression is Q-
dependent and may be incomplete. Model calcula-
tions suggest that the suppression occurs below ap-
proximately 2
0
(see Sect. 18.4.2).Figure 18.39 show
a comparison of the spin gap compared with the gap
determined by other methods which yield signifi-
cantly larger values for underdoped compositions.
This suggests that the spin gap, measured by neu-
trons, is sensitive to different electronic correlations
to other probes such as ARPES and tunnelling.
18.8 Final Remarks
Although magnetic excitations in cuprates have now
been studied for some 20 years, progress contin-
ues to be made. The main reasons for this contin-
uing progress are the ongoing development of neu-
tron instrumentation and sources and the improve-
ment of samples. Inelastic neutron scattering exper-
iments are essentially flux limited because of the rel-
atively small number of neutrons produced by neu-
tron sources. The is no doubt that progress in instru-
mentation will continue unabated and many new re-
sultswillbeproduced.Itislikelythatthereisasignif-
icant part of the spectral weight that has not yet been
observed, particulary in optimally doped cuprates.
Acknowledgements
It is a pleasure to thank my collaborators in this
area G. Aeppli, S.-W. Cheong, N.B. Christensen, K.N.
Clausen,R.Coldea,P.Dai,F.Dogan,C.Frost,J.Kulda,
B. Lake, K. Lefmann, O. Lipscombe, Y. Maeno, T. E.
Mason, D. F. McMorrow, H.A. Mook, T. G. Perring, H.
N. Ronnow, H. Takagi, B.Vignolle, and Z. Fisk.
References
1. W. Marshall and S.W. Lovesey, Theory of Thermal Neutron Scattering (Oxford University Press,Oxford,1971)
2. G. E. Bacon, Neutron Diffraction, 3rd edn.(Oxford University Press, Oxford, 1975)
3. S.W. Lovesey, Theory of Neutron Scattering from Condensed Matter (Clarendon Press, Oxford, 1984)
4. G.L. Squires, Introduction to the Theory of Thermal Neutron Scattering (Dover, New York,1996)
5. P.J. Brown, In: International Tables for Crystallography (Kluwer, Dordrecht,1992) Vol. C, p. 391
6. S.Shamato,M.Sato,J.M.Tranquada,B.J.Sternlieb,andG.Shirane,Phys.Rev.B48,13817 (1993)
7. R.M.White, Quantum Theory of Magnetism (Springer, Berlin, 1983)
8. T. Izuymama,D.-J. Kim, and R. Kubo, J.Phys. Soc. Japan 18, 1025 (1963)
9. O.Steinsvoll,C.F.Majkrzak,G.Shirane,andJ.Wicksted,Phys.Rev.B30, 2377 (1984)
10. H.A.Mook,M.Yethiraj,G.Aeppli,T.E.Mason,and T.Armstrong,Phys.Rev.Lett.70, 3490 (1993)
11. H.F.Fong,B.Keimer,D.Reznik,D.L.Milius,andI.A.Aksay,Phys.Rev.B54, 6708 (1996)
18 Neutron Scattering and the Magnetic Response of Superconductors 1025
12. H.B. Callen and T.A.Welton,Phys. Rev. 83, 34 (1951)
13. K.Yosida, Phys. Rev. 110, 769–770 (1958)
14. C.G. Shull and F.A.Wedgwood, Phys. Rev. Lett. 16, 513–515 (1966)
15. J.A.Duffy,S.M.Hayden,Y.Maeno,Z.Mao,J.Kulda,andG.J.McIntyre,Phys.Rev.Lett.85, 5412 (2000)
16. J.X. Boucherle, J.Y. Henry, R.J. Papoular, J. Rossat-Mignod, J. Schweizer, F. Tasset, and G. Uimin, Physica B 192,25
(1993)
17. A.P. Mackenzie and Y. Maeno, Rev. Mod. Phys. 75, 657 (2003)
18. T.M.RiceandM.Sigrist,J.Phys.-Condens.Mat.7, L643 (1995)
19. G. Baskaran, Physica B 224, 490 (1996)
20. P.W.AndersonandP.Morel,Phys.Rev.1, 1911 (1961)
21. C. Stassis, J.Arthur, C.F. Majkrzak, J.D.Axe, B. Batlogg, J. Remeika, Z. Fisk,J.L.Smith, and A.S. Edelstein, Phys. Rev. B
34, 4382 (1986)
22. J.Lindhard, 2005, Kgl. DanskeVidenskab. Selskab, Mat.-fys. Medd. 28. No. 8 (1954)
23. E. Fawcett, H.L. Alberts,V.Y. Galkin, D.R. Noakes, and J.V.Yakhmi, Rev. Mod. Phys. 66, 25 (1994)
24. E.Fawcett,S.A.Werner,A.Goldman,andG.Shirane,Phys.Rev.Lett.61, 558 (1988)
25. S.M.Hayden,R.Doubble,G.Aeppli,T.G.Perring,and E.Fawcett,Phys.Rev.Lett.84, 999 (2000)
26. L.C. Hebel and C.P. Slichter, Phys. Rev. 113, 1504 (1959)
27. J.R. Schrieffer, Theory of Superconductivity (Addison-Wesley, New York, 1983)
28. N. Bulut and D.J. Scalapino, Phys.Rev. B 47, 3419 (1993)
29. P.B.Littlewood,J.Zaanen,G.Aeppli,andH.Monien,Phys.Rev.B48, 487 (1993)
30. Q.M.Si,Y.Y.Zha,K.Levin,andJ.P.Lu,Phys.Rev.B47, 9055 (1993)
31. T.Tanamoto,H.Kohno,andH.Fukuyama,J.Phys.Soc.Jap.62, 1455 (1993)
32. A.AbanovandA.V.Chubukov,Phys.Rev.Lett.83, 1652 (1999)
33. J.BrinckmannandP.A.Lee,Phys.Rev.Lett.82, 2915 (1999)
34. Y.J.Kao,Q.M.Si,andK.Levin,Phys.Rev.B61, 11898 (2000)
35. M.R.Norman,Phys. Rev. B 61, 14751 (2000)
36. A.V.Chubukov,B.Janko,andO.Tchernyshyov,Phys.Rev.B63, 180507 (2001)
37. D. Manske, I. Eremin, and K.H.Bennemann, Phys.Rev. B 6305, 054517 (2001)
38. F. Onufrieva and P. Pfeuty, Phys. Rev. B 65,054515 (2002)
39. I.Eremin,D.K.Morr,A.V.Chubukov,K.H.Bennemann,andM.R.Norman,Phys.Rev.Lett.94, 147001 (2005)
40. R. Stedman,L.Almqvist, and G. Nilsson, Phys. Rev. 162, 549 (1967)
41. W.L. McMillan and J.M. Rowell, Phys. Rev. Lett.14, 108 (1965)
42. T. Petersen, T.E. Mason, G. Aeppli, A.P. Ramirez, E. Bucher, and R.N. Kleiman, Physica B 199,151 (1994)
43. N. Metoki,Y. Haga,Y. Koike, and Y. Onuki, Phys. Rev. Lett. 80, 5417 (1998)
44. N.Bernhoeft,N.Sato,B.Roessli,N.Aso,A.Hiess,G.H.Lander,Y.Endoh,andT.Komatsubara,Phys.Rev.Lett.81,4244
(1998)
45. M. Jourdan, M. Huth, and H. Adrian,Nature 398,
47 (1999)
46. N.K. Sato, N. Aso, K. Miyake, R. Shiina, P. Thalmeier, G. Varelogiannis, C. Geibel, F. Steglich, P. Fulde, and T. Komat-
subara, Nature 410, 340 (2001)
47. J.G. Bednorz and K.A. M¨uller, Z. Phys.B 64, 189 (1986)
48. D.Vaknin,S.K.Sinha,D.E.Moncton,D.C.Johnston,J.M.Newsam,C.R.Safinya,andJ.H.E.King,Phys.Rev.Lett.58,
2802 (1987)
49. S.Nakamae,K.Behnia,N.Mangkorntong,M.Nohara,H.Takagi,S.J.C.Yates,andN.E.Hussey,Phys.Rev.B68, 100502
(2003)
50. N. Hussey, M. Abdel-Jawad,A. Carrington,A. Mackenzie, and L. Balicas, Nature 425, 814 (2003)
51. T.Timusk and B. Statt, Rep.Prog. Phys.62, 61 (1999)
52. J.L. Tallon and J.W.Loram, Physica C 349, 53 (2001)
53. H.Alloul,T.Ohno,andP.Mendels,Phys.Rev.Lett.63, 1700 (1989)
54. P.G.Radaelli,J.D.Jorgensen,R.Kleb,B.A.Hunter,F.C.Chou,andD.C.Johnston,Phys.Rev.B49, 6239 (1994)
55. H.Takagi,R.J.Cava,M.Marezio,B.Battlog,J.J.Krajewski,andJ.W.F.Peck,Phys.Rev.Lett.68, 3777 (1992)
1026 S.M. Hayden
56. P.G.Radaelli,D.G.Hinks,A.W.Mitchell,B.A.Hunter,J.L.Wagner,B.Dabrowski,K.G.Vandervoort,H.K.Viswanathan,
andJ.D.Jorgensen,Phys.Rev.B49, 4163 (1994)
57. K. Yamada, C.H. Lee, K. Kurahashi, J. Wada, S. Wakimoto, S. Ueki, H. Kimura, Y. Endoh, S. Hosoya, G. Shirane, R.J.
Birgeneau, M. Greven, M.A. Kastner, and Y.J. Kim, Phys. Rev. B 57, 6165 (1998)
58. M.-H.Julien, Physica B 329–333, 693 (2003)
59. L.J.D. Jongh, Solid State Comm. 65, 963 (1988)
60. T. Kaplan and S. Mahanti, J. App. Phys. 69, 5382 (1991)
61. K.Yamada,E.Kudo,Y.Endoh,Y.Hidaka,M.Oda,M.Suzuki,andT.Murakami,SolidStateComm.64, 753 (1987)
62. S.Mitsuda,G.Shirane,S.Sinha,D.Johnson,M.Alvarez,D.Vaknin,andD.Moncton,Phys.Rev.B41, 1926 (1990)
63. J.Xu,T.Watson-Yang,J.Yu,andA.Freeman,Phys.Lett.12A, 489 (1987)
64. G. Luke, L. Le, B. Sternlieb, W. Wu, Y. Uemura, J. Brewer, T. Riseman, S. Ishbashi, and S. Uchida, Physica C 185–189,
1175 (1991)
65. T.Goto,S.Kazama,K.Miyagawa,and T.Fukase,J.Phys.Soc.Jpn.63, 3494 (1994)
66. P.Imbert,G.J´ehanno,P.Debray,C.Garcin,and J.Hodges,J.Phys.(Paris)I2, 1405 (1992)
67. S.M.Hayden,G.Aeppli,H.Mook,D.Rytz,M.F.Hundley,and Z.Fisk,Phys.Rev.Lett.66, 821 (1991)
68. T. Suzuki, T. Goto, K. Chiba, T. Shinoda, T. Fukase, H. Kimura, K. Yamada, M. Ohashi, and Y. Yamaguchi, Phys. Rev. B
57, R3229 (1998)
69. M. Matsuda, M. Fujita, K. Yamada, R.J. Birgeneau, M.A. Kastner, H. Hiraka, Y. Endoh, S. Wakimoto, and G. Shirane,
Phys.Rev.B62, 9148 (2000)
70. S.Katano,M.Sato,K.Yamada,T.Suzuki,and T.Fukase,Phys.Rev.B62,R14677 (2000)
71. B.Lake,H.Ronnow,N.B.Christensen,G.Aeppli,K.Lefmann,D.F.McMorrow,P.Vorderwisch,P.Smeibidl,N.Mangko-
rntong, T. Sasagawa, M. Nohara, H. Takagi, and T.E. Mason, Nature 415,299 (2002)
72. B. Lake, K. Lefmann, N. B. Christensen, G. Aeppli, D.F. Mcmorrow, H.M. Ronnow, P. Vorderwisch, P. Smeibidl, N.
Mangkorntong,T.Sasagawa,M.Nohara,andH.Takagi,Nat.Mater.4, 658 (2005)
73. J. Tranquada, B. Sternlieb, J.Axe,Y. Nakamura, and S. Uchida, Nature 375, 561 (1995)
74. A.Damascelli,Z.Hussain,andZ.-X.Shen,Rev.Mod.Phys.75,473 (2003)
75. S.Sanna,G.Allodi,G.Concas,A.D.Hillier,,and R.D.Renzi,Phys.Rev.B93, 207001 (2004)
76. M. v. Zimmermann, J.R. Schneider, T. Frello, N.H. Andersen, J. Madsen, M. K¨all, H.F. Poulsen, R. Liang, P. Dosanjh, ,
and W.N. Hardy, Phys. Rev. B 68,104515 (2003)
77. R.J.Cava,A.W.Hewat,E.A.Hewat,B.Batlogg,M.Marezio,K.M.Rabe,J.J.Krajewski,W.F.Peck,Jr.,andL.W.Rupp,
Phys.Rev.B165, 419 (1990)
78. H. Casalta, P. Schleger, P. Harris, B. Lebech, N. H. Andersen, R. Liang, P. Dosanjh, and W.N. Hardy, Phys. Rev. B 258,
321 (1996)
79. H.F.Fong,P.Bourges,Y.Sidis,L.P.Regnault,J.Bossy,A.Ivanov,D.L.Milius,I.A.Aksay,andB.Keimer,Phys.Rev.B61,
14773 (2000)
80. P.C.Dai,H.A.Mook,S.M.Hayden,G.Aeppli,T.G.Perring,R.D.Hunt,and F.Dogan,Science284, 1344 (1999)
81. P.C.Dai,H.A.Mook,R.D.Hunt,andF.Dogan,Phys.Rev.B6305, 054525 (2001)
82. H.A.Mook,P.C.Dai,andF.Dogan,Phys.Rev.Lett.88, 097004 (2002)
83. C.Stock,W.J.L.Buyers,R.A.Cowley,P.S.Clegg,R.Coldea,C.D.Frost,R.Liang,D.Peets,D.Bonn,W.N.Hardy,
and R.J. Birgeneau, Phys. Rev. B 71, 024522 (2005)
84. C.Stock,W.J.L.Buyers,Z.Yamani,C.L.Broholm,J.-H.Chung,Z.Tun,R.Liang,D.Bonn,W.N.Hardy,andR.J.Birgeneau,
Phys.Rev.B73, 100504 (2006)
85. P. Anderson, Phys.Rev. 86,
694 (1952)
86. R. Kubo,Phys. Rev. 87, 568 (1952)
87. T.Oguchi, Phys. Rev. 117, 117 (1960)
88. R. Singh, Phys. Rev. B 39, 9760 (1989)
89. Y.Endoh,K.Yamada,R.Birgeneau,D.Gabbe,H.Jenssen,M.Kastner,C.Peters,P.Picone,T.Thurston,J.Tranquada,G.
Shirane, Y. Hidaka, M. Oda, Y. Enomoto, M. Suzuki, and T. Murakami, Phys. Rev. B 37, 7443 (1988)
90. G.Shirane,Y.Endoh,R.J.Birgeneau,M.A.Kastner,Y.Hidaka,M.Oda,M.Suzuki,and T.Murakami,Phys.Rev.Lett.59,
1613–1616 (1987)
18 Neutron Scattering and the Magnetic Response of Superconductors 1027
91. G.Aeppli,S.M.Hayden,H.A.Mook,Z.Fisk,S.W.Cheong,D.Rytz,J.P.Remeika,G.P.Espinosa,andA.S.Cooper,Phys.
Rev. Lett. 62, 2052 (1989)
92. S.M.Hayden,G.Aeppli,H.A.Mook,S.W.Cheong,and Z.Fisk,Phys.Rev.B42, 10220 (1990)
93. S. Chakravarty, B. Halperin, and D. Nelson, Phys. Rev. B 38,2344 (1989)
94. S.M.Hayden,G.Aeppli,R.Osborn,A.D.Taylor,T.G.Perring,S.W.Cheong,andZ.Fisk,Phys.Rev.Lett.67, 3622(1991)
95. R. Coldea, S.M. Hayden, G. Aeppli, T.G. Perring, C.D. Frost, T.E. Mason, S.W. Cheong, and Z. Fisk, Phys. Rev. Lett. 86,
5377 (2001)
96. H.M. Ronnow, D.F. McMorrow, R. Coldea, A. Harrison, I.D. Youngson, T.G. Perring, G. Aeppli, O.S. sen, K. Lefmann,
andC.Rischel,Phys.Rev.Lett.87, 037202 (2001)
97. R.R. Singh and M.P. Gelfand, Phys.Rev. B 52,R15695 (1995)
98. O.F. Sylju˚asen and H.M. Rønnow, J. Phys.-Condens. Mat. 12, L405(2000)
99. A.W. Sandvik and R.R.P. Singh, Phys.Rev. Lett. 86, 528–531 (2001)
100. J. ichi Igarashi and T. Nagao,Phys. Rev. B 72, 014403 (2005)
101. H. Eskes and J.H.Jefferson,Phys. Rev. B 48, 9788–9798 (1993)
102. P.W.Anderson,Phys. Rev. 79, 350–356 (1950)
103. P. Bourges, H. Casalta, A.S. Ivanov, and D. Petitgrand, Phys. Rev. Lett. 79, 4906 (1997)
104. I.U.Heilmann, J.K. Kjems,Y. Endoh, G.F. Reiter, G. Shirane,and R.J. Birgeneau, Phys.Rev. B 24, 3939 (1981)
105. R. Coldea, D.A. Tennant, and Z. Tylczynski, Phys.Rev. B 68,134424 (2003)
106. S. Chakravarty, B. Halperin, and D. Nelson, Phys.Rev. Lett. 60, 1057 (1988)
107. E.Manousakis,Rev. Mod.Phys. 63, 1 (1991)
108. A. Chubukov, S. Sachdev, and J.Ye, Phys.Rev. 49, 11919 (1994)
109. L. de Jongh and A. Miedema,Adv. Phys. 23, 1 (1974)
110. D. Grempel,Phys. Rev. Lett. 61, 1041 (1988)
111. O.K.Andersen,A.I. Liechtenstein, O. Jepsen, and F. Paulsen,J.Phys.Chem. Solids 56, 1573 (1995)
112. A.I. Liechtenstein, O. Gunnarsson, O.K.Andersen, and R.M. Martin, Phys. Rev. B 54, 12505–12508 (1996)
113. J.Rossat-Mignod,L.P. Regnault,C.Vettier, P. Burlet, J.Y. Henry, and G. Lapertot, Physica B 169,58 (1991)
114. S. Shamoto, M. Sato, J.M. Tranquada, B.J. Sternlieb, and G. Shirane, Phys. Rev. B 48, 13817 (1993)
115. D. Reznik, P.Bourges, H.F. Fong, L.P. Regnault, J. Bossy,C.Vettier, D.L. Milius, I.A. Aksay, and B. Keimer, Phys. Rev. B
53, R14741 (1996)
116. S.M.Hayden, G. Aeppli, T.G. Perring, H.A. Mook,and F. Dogan,Phys. Rev. B 54, R6905 (1996)
117. A.T.Boothroyd,N.H.Andersen,B.H.Larsen,A.A. Zhokhov, C.D. Frost, and D.T.Adroja, Phys. Rev. B 71,094514 (2005)
118. T.R. Thurston, R.J. Birgeneau, M.A. Kastner, N.W. Preyer, G. Shirane, Y. Fujii, K. Yamada, Y. Endoh, K. Kakurai, M.
Matsuda, and Y.H.T. Murakami, Phys. Rev. B 40, 4585 (1989)
119. S.W. Cheong, G. Aeppli, T.E. Mason, H. Mook, S.M. Hayden, P.C. Canfield, Z. Fisk, K.N. Clausen, and J.L. Martinez,
Phys.Rev.Lett.67, 1791 (1991)
120. S.Wakimoto,H. Zhang, K.Yamada, I. Swainson, H. Kim, and R.J. Birgeneau, Phys. Rev. Lett. 92, 217004 (2004)
121. G.Aeppli, T.E. Mason, S.M. Hayden, H.A. Mook, and J. Kulda, Science 278
, 1432 (1997)
122.
T.E. Mason,G.Aeppli, and H.A. Mook,Phys. Rev. Lett.68
, 1414 (1992)
123. J.M. Tranquada, H. Woo, T.G. Perring, H. Goka, G.D. Gu, G. Xu, M. Fujita, and K.Yamada, Nature 429, 534 (2004)
124. N.B. Christensen, D.F. McMorrow,H.M. Ronnow, B. Lake, S.M.Hayden,G.Aeppli,T.G.Perring, M. Mangkorntong, M.
Nohara, and H. Takagi, Phys.Rev. Lett. 93, 147002 (2004)
125. S.M.Hayden, G.Aeppli,H.A.Mook,T.G.Perring, T.E.Mason,S.W.Cheong, and Z.Fisk,Phys.Rev.Lett. 76,1344(1996)
126. T.R. Thurston, P.M. Gehring, G. Shirane, R.J. Birgeneau, M.A. Kastner,Y. Endoh, M. Matsuda, K. Yamada, H. Kojima,
and I. Tanaka, Phys. Rev. B 46, 9128 (1992)
127. T.E. Mason,G.Aeppli, S.M. Hayden,A.P. Ramirez, and H.A.Mook, Phys. Rev. Lett. 71, 919 (1993)
128. M. Matsuda, K. Yamada, Y. Endoh, T.R. Thurston, G. Shirane, R.J. Birgeneau, M.A. Kastner, I. Tanaka, and H. Kojima,
Phys.Rev.B49, 6958 (1994)
129. T.E. Mason,A.Schroder, G.Aeppli, H.A. Mook,and S.M. Hayden, Phys. Rev. Lett. 77, 1604 (1996)
130. B. Lake, G. Aeppli, T.E. Mason,A. Schroder, D.F. McMorrow, K. Lefmann, M. Isshiki, M. Nohara, H. Takagi, and S.M.
Hayden, Nature 400, 43 (1999)
1028 S.M. Hayden
131. P. Bourges, Y. Sidis, H. Fong, B. Keimer, L. Regnault, J. Bossy, A. Ivanov, D. Milius, and I. Aksay, 1999, In: High
Temperature Superconductivity, ed. by S.E.Barnes et al.(CP483 American Instituteof Physics,Amsterdam,1999) pp.
207–212, cond-mat/9902067
132. J.Rossat-Mignod, L.P.Regnault, C.Vettier, P. Bourges, P. Burlet, J.Bossy, J.Y. Henry, andG. Lapertot, Physica C 185,86
(1991)
133. H.F. Fong, B. Keimer, P.W. Anderson, D. Reznik,F. Dogan, and I.A.Aksay, Phys.Rev. Lett. 75, 316 (1995)
134. S. Pailh`es,Y.Sidis,P.Bourges,C.Ulrich,V.Hinkov,L.P.Regnault,A.Ivanov,B.Liang,C.T.Lin,C.Bernhard,andB.
Keimer, Phys. Rev. Lett. 91, 237002 (2003)
135. H.A. Mook, P.C. Dai,S.M. Hayden,G.Aeppli, T.G. Perring, and F. Dogan, Nature 395, 580 (1998)
136. P. Dai, H.A. Mook,and F. Doˇgan,Phys.Rev.Lett.80, 1738 (1998)
137. M. Arai, T. Nishijima, Y. Endoh, T. Egami, S. Tajima, K. Tomimoto,Y. Shiohara, M. Takahashi, A. Garrett, and S.M.
Bennington, Phys.Rev. Lett. 83, 608 (1999)
138. P. Bourges,Y. Sidis, H.F. Fong, L.P. Regnault,J.Bossy,A. Ivanov, and B. Keimer, Science 288,1234 (2000)
139. D. Reznik,P.Bourges,L. Pintschovius,Y. Endoh,Y. Sidis, T. Masui,and S. Tajima,Phys. Rev. Lett. 93, 207003 (2004)
140. S. Pailh`es, Y. Sidis, P. Bourges, V. Hinkov,A. Ivanov, C. Ulrich, L.P. Regnault, and B. Keimer, Phys. Rev. Lett. 93, 167001
(2004)
141. S.M.Hayden,H.A. Mook,P.C. Dai,T.G. Perring, and F. Dogan, Nature 429, 531 (2004)
142. P.M. Gehring, J.M. Tranquada, G. Shirane, J.R.D. Copley, R.W. Erwin, M. Sato, and S. Shamoto, Phys. Rev. B 44, 2811
(1991)
143. P. Dai,M.Yethiraj, H.A. Mook, T.B. Lindemer, and F. Dogan, Phys. Rev. Lett. 77, 5425 (1996)
144. C. Stock, W.J.L. Buyers, R. Liang, D. Peets, Z. Tun, D. Bonn, W.N. Hardy, and R.J. Birgeneau, Phys. Rev. B 69, 014502
(2004)
145. K.Zhang,D.A.Bonn,S.Kamal,R.Liang, D.J. Baar,W.N. Hardy,D.Basov,andT. Timusk, Phys.Rev. Lett.73,2484 (1994)
146. R. Gagnon,C. Lupien,and L. Taillefer, Phys. Rev. B 50, 3458 (1994)
147. H.A. Mook,P.Dai, F. Doˇgan, and R.D. Hunt,Nature 404, 729 (2000)
148. V. Hinkov, S. Pailhes,P. Bourges,Y. Sidis,A. Ivanov,A.Kulakov,C.T.Lin,D.P. Chen,C.Bernhard,and B.Keimer,Nature
430, 650 (2004)
149. P. Bourges, H.F.Fong, L.P. Regnault, J.Bossy, C.Vettier, D.L. Milius, I.A.Aksay, and B.Keimer, Phys. Rev. B 56,R11439
(1997)
150. S.M. Hayden, G. Aeppli, P. Dai, H.A. Mook, T.G. Perring, S.W. Cheong, Z. Fisk, F. Dogan, and T.E. Mason, Physica B
241, 765 (1998)
151. S.Pailh`es,C.Ulrich,B.Fauqu´e,V.Hinkov,Y.Sidis,A.Ivanov,C.Lin,B.Keimer,and P.Bourges,2005,cond-mat/0512634.
152. H.F. Fong, P. Bourges,Y. Sidis, L.P. Regnault,A. Ivanov, G. D. Gu,N. Koshizuka, and B. Keimer, Nature 398,588 (1999)
153. H. He, Y. Sidis, P. Bourges, G.D. Gu, A. Ivanov, N. Koshizuka, B. Liang, C.T. Lin, L.P. Regnault, E. Schoenherr, and B.
Keimer, Phys. Rev. Lett. 86, 1610 (2001)
154. H. He, P. Bourges, Y. Sidis, C. Ulrich, L.P. Regnault, S. Pailhes, N.S. Berzigiarova, N.N. Kolesnikov, and B. Keimer,
Science 295,1045 (2002)
155. S.A. Kivelson, I. Bindloss, E. Fradkin,V. Oganesyan, J. Tranquada, A. Kapitulnik, and C. Howard, Rev. Mod. Phys. 75,
1201 (2003)
156. M.R. Norman and C.Pepin, Rep. Prog. Phys.66, 1547 (2003)
157. S. Sachdev, Rev. Mod.Phys.75, 913 (2003)
158. P.A. Lee, N. Nagaosa,and X.-G.Wen, Rev. Mod. Phys.
78, 17 (2006)
159. E. Demler and S.C. Zhang, Phys.Rev. Lett.
75, 4126 (1995)
160. O. Tchernyshyov, M. Norman, and A.V.Chubukov, Phys. Rev. B 63, 144507 (2001)
161. J.L. Tallon, C. Bernhard, H. Shaked,R.L. Hitterman,and J.D. Jorgensen, Phys.Rev. B 51, 12911 (1995)
162. H.Won and K. Maki, Phys.Rev. B 49,1397 (1994)
163. C.Panagopoulos and T. Xiang, Phys. Rev. Lett. 81, 2336 (1998)
164. J.F. Zasadzinski, L. Ozyuzer,N. Miyakawa, K.E. Gray, D.G. Hinks,and C.Kendziora, Phys.Rev. Lett.87, 067005 (2001)
165. 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, T. Mochiku, K. Kadowaki, P. Guptasarma, D.G. Hinks, Z. Konstantinovic, Z.Z. Li, and H. Raffy,
Phys.Rev.Lett.83, 3709 (1999)