Unified picture of the oxygen isotope effect in cuprate superconductors
Xiao-Jia Chen, Viktor V. Struzhkin, Zhigang Wu, Hai-Qing Lin, Russell J. Hemley, and
Ho-kwang Mao
PNAS 2007;104;3732-3735; originally published online Feb 26, 2007;
doi:10.1073/pnas.0611473104
This information is current as of March 2007.
Online Information
& Services
High-resolution figures, a citation map, links to PubMed and Google Scholar,
etc., can be found at:
www.pnas.org/cgi/content/full/104/10/3732
This article has been cited by other articles:
www.pnas.org/cgi/content/full/104/10/3732#otherarticles
E-mail Alerts
Receive free email alerts when new articles cite this article - sign up in the box
at the top right corner of the article or click here.
Rights & Permissions
To reproduce this article in part (figures, tables) or in entirety, see:
www.pnas.org/misc/rightperm.shtml
Reprints
To order reprints, see:
www.pnas.org/misc/reprints.shtml
Notes:
Unified picture of the oxygen isotope effect
in cuprate superconductors
Xiao-Jia Chen*‡§, Viktor V. Struzhkin*, Zhigang Wu*, Hai-Qing Lin¶, Russell J. Hemley*, and Ho-kwang Mao*
*Geophysical Laboratory, Carnegie Institution of Washington, Washington, DC 20015; ‡Department of Applied Physics, South China University
of Technology, Guangzhou 510641, China; and ¶Department of Physics, The Chinese University of Hong Kong, Hong Kong, China
Contributed by Ho-kwang Mao, January 2, 2007 (sent for review November 1, 2006)
High-temperature superconductivity in cuprates was discovered
almost exactly 20 years ago, but a satisfactory theoretical explanation for this phenomenon is still lacking. The isotope effect has
played an important role in establishing electron–phonon interaction as the dominant interaction in conventional superconductors.
Here we present a unified picture of the oxygen isotope effect in
cuprate superconductors based on a phonon-mediated d-wave
pairing model within the Bardeen–Cooper–Schrieffer theory. We
show that this model accounts for the magnitude of the isotope
exponent as functions of the doping level as well as the variation
between different cuprate superconductors. The isotope effect on
the superconducting transition is also found to resemble the effect
of pressure on the transition. These results indicate that the role of
phonons should not be overlooked for explaining the superconductivity in cuprates.
electron–phonon interaction 兩 high-temperature superconductivity 兩
pressure effect
U
nderstanding the high-temperature superconductivity in
cuprate superconductors is at the heart of current research
in solid-state physics. The isotope effect is an important experimental probe in revealing the underlying pairing mechanism of
superconductivity. Early measurements (1) of the isotope effect
on the superconducting transition temperature, Tc, provided key
experimental evidence for phonon-mediated pairing and supported strongly the Bardeen–Cooper–Schrieffer (BCS) theory of
superconductivity in conventional materials. When hightemperature superconductivity was discovered in copper oxides,
the oxygen isotope exponents, defined by ␣ ' ⫺dlnTc/dlnM,
with M being the isotopic mass, were promptly measured. The
initial finding of the small value of ␣ in La1.85Sr0.15CuO4 (2, 3)
and near zero value in YBa2Cu3O7⫺␦ (␦ ⬃ 0) (4–6) was taken to
be convincing evidence against electron–phonon interaction as
the dominant mechanism in these materials. However, later
experiments have revealed a much richer and more complex
situation (7). There is a striking variation of ␣ with doping level
in a given cuprate material (8–11). Meanwhile, ␣ also varies
among various compounds of different cuprate families (7, 12).
Interestingly, ␣ is a remarkably monotonic function of the
number of CuO2 layers in a way opposite to Tc in optimally doped
compounds (12). Such elaborate isotope effects strongly suggest
that one should include phonons in the theory of high-Tc
superconductivity. Studies of neutron scattering (13), angleresolved photoemission spectroscopy (14, 15), and scanning
transmission electron microscopy (16) provide further evidence
for phonons being an important player in the basic physics of
high-Tc superconductivity.
Compared with isotope substitution, pressure has been realized not only to be another effective way to change Tc for a
superconductor (17) but also to yield information on the interaction causing the superconductivity. So far, the record high Tc
of 164 K was achieved under high pressure in HgBa2Ca2Cu3O8⫹␦
(18). Observing how the pressure influences the individual
parameters of the lattice in the normal and superconducting
state of a single sample allows a clean quantitative test of the
3732–3735 兩 PNAS 兩 March 6, 2007 兩 vol. 104 兩 no. 10
validity of theoretical models. In early 1964, Olsen et al. (19)
showed that there is a correlation between the pressure coefficient of the effective interaction and the isotope effect in
conventional metal superconductors. For high-Tc materials
YBa2Cu3O7⫺␦, Franck and Lawrie (20) found a similar doping
dependence for both the copper isotopic exponent and pressure
coefficient of Tc, dlnTc/dP. These findings directly point to the
equivalent importance of the pressure and isotope effects on
superconductivity. Because many materials, among them 23
elements (17) and even spin-ladder cuprate (21), become superconductors only under sufficiently high pressure, measurements of the isotope effect in these materials must be performed
under high pressure. Meanwhile, Crespi and Cohen (22) proposed that cuprates under high pressure are a hopeful candidate
for attaining negative values for ␣, which would provide support
to an anharmonic model for high-Tc superconductivity. Until
now, there is little information regarding how ␣ changes with
pressure in high-Tc cuprates. Such investigations are highly
desirable for elucidating the mechanism of high-Tc superconductivity. The purpose of this paper is to offer an interpretation
for the elaborate oxygen isotope effect in cuprate superconductors based on a phonon-mediated d-wave BCS-like model. We
find that the effective next-nearest-neighbor hopping dominates
the variation of ␣ for the optimally doped cuprates. Based on the
measurements of both Tc and Raman shift of the B1g phonon
mode, we predict that the pressure dependence of ␣ for the
optimally doped YBa2Cu3O7⫺␦ is similar to that of the pressure
coefficient of Tc. A unified picture for the isotope effect in
cuprates is given at ambient condition and under high pressure,
which is supported by good agreements between theory and
experiments.
Results and Discussion
We start directly from the mean-field Hamiltonian describing
d-wave superconductivity:
H ⫽
冘
共␧k ⫺ ␮兲ck†␴ ck␴ ⫺
k␴
冘
共⌬kc†k1c†⫺k2 ⫹ h.c.兲,
[1]
k
where ␧k is the quasiparticle dispersion, ␮ is the chemical
potential, and ck†␴ is a quasiparticle creation operator. The gap
parameter ⌬k ⫽ N⫺1 兺k⬘Vkk⬘ 具c⫺k⬘2ck⬘1典, where N is the number
of sites. Considering that the interaction could be of the chargecoupled type mediated by phonons, we used d-wave pairing
potential Vkk⬘ ⫽ Vg(k)g(k⬘); ␧k ⫺ ␮ or ␧k⬘ ⫺ ␮ ⬍ ␻0, where
g(k) ⫽ cos kx ⫺ cos ky, V is the in-plane pairing interaction
strength, and ␻0 is the cutoff of the phonon frequency. For
phonon-mediated pairing, ␻0 is assumed to vary with the isotope
Author contributions: X.-J.C., V.V.S., Z.W., H.-Q.L., R.J.H., and H.-k.M. performed research.
The authors declare no conflict of interest.
Abbreviation: BCS, Bardeen–Cooper–Schrieffer.
§To
whom correspondence should be addressed. E-mail: xjchen@ciw.edu.
© 2007 by The National Academy of Sciences of the USA
www.pnas.org兾cgi兾doi兾10.1073兾pnas.0611473104
Hg1201
Tl2201
Tl1201
La214
Bi2201
0.6
60
α
TC (K)
80
0.4
40
0.2
20
0
0
0
0.5
nH (holes)
1
0
0.5
nH (holes)
1
Fig. 1. The theoretical results of both the superconducting transition temperature, Tc, and oxygen isotope exponent, ␣, versus the hole concentration
nH in single CuO2-layer cuprate superconductors La2⫺xSrxCuO4 (La214),
Bi2Sr2CuO6⫹␦ (Bi2201), TlBa2CuO5⫹␦ (Tl1201), Tl2Ba2CuO6⫹␦ (Tl2201), and
HgBa2CuO4⫹␦ (Hg1201).
mass, M, as M⫺1/2. The d-wave gap function has the typical form
⌬k ⫽ ⌬g(k), and the parameter ⌬ is determined by
1 ⫽
1
N
冘
Vg2共k兲␹k␪共␻0 ⫺ 兩␧k ⫺ ␮兩兲,
[2]
k
where ␹k ⫽ (2Ek)⫺1tanh(␤Ek/2), with the quasiparticle spectrum
Ek ⫽ 公(␧k ⫺ ␮)2 ⫹ ⌬k2 in the d-wave state, and the step
function ␪(x) takes care of the condition ␧k ⫺ ␮ or ␧⬘k ⫺ ␮
⬍ ␻0. The Tc equation is then obtained by solving Eq. 2 with ⌬ ⫽
0. Solving this self-consistently for a given ␮ yields Tc as a
function of hole concentration. The validity of the d-wave BCS
formalism in describing the superconducting state of cuprates
has been supported by recent measurements of angle-resolved
photoemission spectroscopy (23) and transport properties (24).
Differentiating Eq. 2 with respect to both Tc and ␻0, one can
express ␣ ⬅ (1/2)dlnTc/dln␻0 as
1
N
␣⫽
1
N
冘
冘
冉 冊
冉 冊
␻0
␦ (␻0 ⫺ 兩␧k ⫺ ␮兩)
2T c
.
兩␧k ⫺ ␮兩
2
2
g
(k)sech
␪
(
␻
⫺
兩␧
⫺
␮
兩)
k
0
k
2T c
k
T c g2(k) tanh
[3]
Once knowing the cutoff frequency ␻0, dispersion ␧k, and interaction V, one can obtain the Tc and ␣ behaviors based on Eqs. 2 and
3. We choose a typical ␻0 ⫽ 0.060 eV (13–16). For the dispersion,
we use ␧k ⫽ t⬘effcos kx cos ky ⫹ t⬙eff(cos 2kx ⫹ cos 2ky), with t⬘eff and
t⬙eff being the effective next-nearest-neighbor and next-next-nearestneighbor hoppings, respectively. Such a dispersion can reproduce
the bandwidth and Fermi-surface shape of many cuprate superconductors (25, 26). It has been shown (27) that the choice of teff ⫽
0.032 eV and V ⫽ 0.03762 eV can reproduce the experimentally
observed maximum value of the superconducting transition temperature, T max
c , ranging from 30 to 97 K of various optimally doped
monolayer cuprates when changing t⬘eff from ⫺0.032 to 0.0914 eV.
To keep the consistency of the theory, we use the same t⬙eff and V
values in the following calculations. The relative t⬘eff can be estifor an individual
mated by using the experimentally observed T max
c
material.
Fig. 1 shows the calculated Tc and ␣ as a function of hole
concentration nH for some typical monolayer cuprates. As
shown, Tc initially increases with increasing nH to a maximum
around an optimal doping level nopt
H and then decreases with
further increasing nH. This parabolic relation between Tc and nH
agrees well with experimental observations in general. The
Chen et al.
maximum in the Tc versus nH curves is shifted toward smaller nH
values among these materials, in remarkable agreement with
experiment. This behavior implies that the diluted optimal hole
concentration is in favor of the higher value of the maximum Tc
when a material is optimally doped. Because t⬘eff in dispersion is
the only other parameter used in calculation, the different Tc
behaviors of these materials should come from their electronic
structure. This character provides an explicit interpretation for
the role played by the ‘‘chemistry’’ in determining the maximum
Tc via the effect on the electronic structure, as early suggested
by Phillips (28, 29).
The calculated ␣ exhibits many interesting features. The values
of ␣ are always positive over the whole doping range for all
studied monolayer materials. There exists a minimum ␣ at some
doping level on the overdoped side, away from which one can
find most large values of ␣. A significant increase in ␣ occurs
when the sample is highly underdoped or highly overdoped. In
low-Tc groups, such as La-, Bi-, and single-layer Tl-based cuprates, ␣ does not vary monotonically with doping and its value
is relatively large at optimal doping. For high-Tc two-layer Tland Hg-based materials, ␣ is found to decrease monotonically
with doping on the underdoped side, reaching a minimum on the
overdoped side after passing through the optimal doping level,
and then it is found to increase gradually when the material is
highly overdoped. Such a difference is also believed to be a result
of the electronic structure effect. Currently, La2⫺xSrxCuO4 is the
only monolayer system whose doping dependence of ␣ is well
known over the whole doping regime (2, 3, 9). Note that the
experimentally observed ␣ behavior in this system can be
reproduced by the present d-wave BCS-like model, except in the
region near x ⫽ 0.125. The large ␣ value at this doping level is
related to the appearance of the stripe order (30), which is
beyond the present theoretical treatment. Considering that the
dispersion relations used are derived from short distance antiferromagnetic correlations, our model is in reality a mixture of
both electronic and phononic contributions. Therefore, a phonon-induced attraction combines with a hole dispersion modulated by antiferromagnetic correlations to contribute to pairing.
Although the similar doping dependence of the isotope exponent
can be reproduced within van Hove scenarios (31–33), the
material-dependent dispersions considered here enable us to
investigate the variation of the isotope effect among various
cuprates.
In Fig. 2, the t⬘eff dependence of both the maximum Tc and ␣
is plotted for optimally doped materials in the range of interest
of t⬘eff. Clearly, t⬘eff dominates the maximum Tc behavior among
various compounds, confirming previous theoretical interpretations (27, 34). The most significant observation is that ␣ is not
negligible but rather varies with t⬘eff as well. For small t⬘eff values,
the material also has a small value of the maximum Tc but a large
value of ␣. As t⬘eff increases (leading to the increase of the
maximum Tc), ␣ first sharply drops and then gradually decreases,
reaching a minimum at around t⬘eff ⫽ 0.048 eV; further increase
in t⬘eff leads to slightly increasing ␣. The theoretical calculations
yield ␣ of 0.26 for the optimally doped La2⫺xSrxCuO4, which is
consistent with the experiments (2, 3). Therefore, this model
predicts that both the maximum Tc and ␣ in these optimally
doped cuprates are in reality related to the modification of the
electronic structure through the change of t⬘eff. The crucial role
of t⬘eff has been proposed (35) to be associated to the Jahn-Teller
active Q2-type mode. Our results imply that the coupling of the
electronic degrees of freedom to the Jahn-Teller Q2-type mode
is responsible for the material dependence of the isotope effect.
According to our theoretical analysis, the optimally doped
Tl2Ba2CuO6⫹␦ should have the smallest ␣ value compared with
other monolayer compounds. Thus, experiments are desirable to
answer whether the systematic evolution of ␣ among cuprates is
correctly predicted here.
PNAS 兩 March 6, 2007 兩 vol. 104 兩 no. 10 兩 3733
PHYSICS
0.8
100
120
Hg1201
100
α
20
0.04
t´eff (eV)
0.08
80
380
70
360
60
0
0.12
Fig. 2. The effective next-nearest-neighbor hopping t⬘eff dependence of both
the oxygen isotope exponent, ␣, and the maximum value of the supercon, in various optimally doped monolayer
ducting transition temperature, Tmax
c
cuprate superconductors. The corresponding t⬘eff value is marked on the top of
the frame for La2⫺xSrxCuO4 (La214), Bi2Sr2CuO6⫹␦ (Bi2201), TlBa2CuO5⫹␦
(Tl1201), Tl2Ba2CuO6⫹␦ (Tl2201), and HgBa2CuO4⫹␦ (Hg1201).
For YBa2Cu3O7⫺␦, assuming that the ␣ behavior is affected
only negligibly by interlayer coupling, we can estimate the value
of t⬘eff from Eq. 2 when taking Tmax
⫽ 92 K at the optimal doping.
c
Thus, we are able to calculate the changes in Tc and ␣ for this
material by using Eqs. 2 and 3. Calculation results are shown in
Fig. 3 together with experimental data for comparisons. As can
be seen, all of the ␣ values are positive over the whole doping
regime. The optimally doped material has the corresponding
minimum ␣ of 3.5 ⫻ 10⫺3. An early study on YBa2Cu3O7 gives
a value of ␣ of 0 ⫾ 0.02 (4). Another careful examination of five
pairs of YBa2Cu3O7 samples (6) gives an oxygen isotope effect
of ␣ ⫽ 0.017 ⫾ 0.006. The comparison between theory and
experiments is excellent at the optimal doping level. As the
material goes away from the optimal doping level, ␣ increases
with decreasing Tc. By changing the oxygen content, Zech et al.
(8) observed that ␣ remains rather small at Tc ⯝ 61 K and then
increases for a further decrease of Tc down to ⬇53 K in
YBa2Cu3O7⫺␦ (0.47 ⬎ ␦ ⬎ 0.01). The gradual increase of ␣
with the dramatic reduction of T c was reported in
YBa2(Cu1⫺xZnx)3O7⫺␦ (7). As is clearly seen in Fig. 3, the
0.8
YBa2Cu3O7-δ
0.6
400
5
10
15
Pressure (GPa)
20
340
25
Fig. 4. Both the superconducting transition temperature, Tc, and Raman
shift of the B1g phonon mode taken at 25 K as a function of pressure up to 22.3
GPa in the optimally doped YBa2Cu3O7⫺␦.
theoretical curves cover most of these experimental data points
(4–8), except the points at ⬇50 K, where a maximum of ␣ is
believed to be associated with the chain oxygen ordering (20). It
is apparent that the present theoretical model accounts for the
observed isotope effects in this extensively studied high-Tc
material as well. Therefore, it is natural to expect a nearly zero
oxygen isotope effect in the optimally doped YBa2Cu3O7⫺␦ for
a BCS-type electron-phonon interaction pairing. It is worth
noting that the doping dependence of ␣ in YBa2Cu3O7⫺␦ resembles that of dlnTc/dP (36, 37), signaling a nice correlation
between the isotope and pressure effects.
We now show how the present d-wave BCS-like model can
predict the high-pressure behavior of ␣. Just like the case at
ambient conditions, the change of ␣ with pressure is assumed
to be through the changes of both Tc and ␻0 and is then
expressed by ␣(P) ⫽ ␣[Tc(P), ␻0(P)]. For the cutoff phonon
frequency ␻0(P), we take the pressure dependence of the
frequency ␻B1g of the B1g Cu–O bond-buckling phonon, which
has been ascribed as a bosonic mode inf luencing high-Tc
superconductivity (14). The information on ␻B1g(P) can be
obtained by high-pressure Raman scattering measurements.
We thus have ␻0(P) ⫽ [␻0(0)/␻B1g(0)]␻B1g(P). The ␣(P) behavior can be derived from Eq. 3 based on the Tc(P) and ␻B1g(P)
determinations. We choose optimally doped YBa2Cu3O7⫺␦ as
an example. Raman spectra in the y(xx)y៮ (A1g ⫹ B1g) geometry
were taken at 25 K and at different pressures up to 22.3 GPa.
The most pronounced spectral change under pressure is a
significant enhancement of the B1g phonon mode near 300
0.01
0.004
0.4
α
YBa2Cu3O7-δ
0
α
-0.01
-0.02
20
40
60
80
100
0.002
-1
0
0
TC (K)
Fig. 3. Oxygen isotope exponent, ␣, versus the superconducting transition
temperature, Tc, in YBa2Cu3O7⫺␦. The solid line represents the theory, with the
lower and upper curves representing the underdoped and overdoped sides,
respectively. The open diamond, filled circle, filled squares, and open triangles
represent the experiments from refs. 4, 6, 7, and 8, respectively. A maximum
of ␣ at Tc ⬇ 50 K is believed to be associated with the chain oxygen ordering.
3734 兩 www.pnas.org兾cgi兾doi兾10.1073兾pnas.0611473104
dlnTC/dP (GPa )
0.003
0.2
-1
0
90
Raman shift (cm )
40
(K)
0.1
max
60
TC
80
0.2
0
420
YBa2Cu3O7-δ
0.3
-0.04
100
TC (K)
Tl2201
Tl1201
Bi2201
La214
0.4
-0.03
0.001
0
10
2
20
Pressure (GPa)
-0.04
30
Fig. 5. Pressure dependence of both the oxygen isotope exponent ␣ and
pressure coefficient of Tc, dlnTc/dP, in the optimally doped YBa2Cu3O7⫺␦.
Chen et al.
cm⫺1. Fig. 4 shows the measured Tc(P) and Raman shift ␻B1g(P)
of B1g phonon mode obtained by fitting the Raman data in
terms of a Green’s function approach (38). Note that the
measured Tc(P) behavior of this material is consistent with
other independent measurements (39, 40).
In Fig. 5 we present the calculated ␣ as a function of pressure
up to 25 GPa in the optimally doped YBa2Cu3O7⫺␦ together with
dlnTc/dP for comparison. As pressure is increased, ␣ slightly
decreases in a manner similar to dlnTc/dP, as expected from an
anharmonic model (22), but ␣ remains small and positive within
the pressure regime studied. Our calculations allow the prediction of isotope effect changes under pressure in cuprate superconductors. The similarity of the isotope effect and the pressure
effect both at ambient condition and under high pressure
indicates that both effects are playing an equivalent role in
shedding important light on high-Tc superconductivity in cuprates. There has been no report of isotope measurements under
high pressure in cuprates so far. The prediction made here for
␣(P) awaits further experimental studies.
We should emphasize that the present theoretical investigation is
based on a d-wave model. However, we find that the d-wave
character is not critical to the results obtained and conclusions
drawn. An s-wave model can also give the similar general trend of
␣ as functions of doping and pressure. We also would like to
mention that the strong correlation effect is not directly included in
our theoretical treatment, rather entering our model through the
hole dispersion. Although the physics in the low-doping regime
would be dominated by antiferromagnetic correlations, electron–
phonon coupling is expected to be more important in the optimal
and overdoped regime. The present theoretical model bears nicely
the experimentally observed generic phase diagram between the Tc
and doping level. Therefore, we believe that all our results and
conclusions are physically valid over a wide range of doping
regimes.
We thank O. K. Andersen, V. H. Crespi, T. Cuk, A. F. Goncharov, A.
Lanzara, J. C. Phillips, J. S. Schilling, S. Y. Zhou, and J. X. Zhu for
stimulating discussions. This work was supported by the Office of Basic
Energy Science and National Nuclear Security Administration of the U.S.
Department of Energy. H.Q.L. was supported by the Hong Kong Research
Grants Council.
1. Maxwell E (1995) Phys Rev 78:477.
2. Batlogg B, Kourouklis G, Weber W, Cava RJ, Jayaraman A, White AE, Short
KT, Rupp LW, Rietman EA (1987) Phys Rev Lett 59:912–914.
3. Faltens TA, Ham WK, Keller SW, Leary KJ, Michaels JN, Stacy AM, zur Loye
HC, Morris DE, Barbee TW, III, Bourne LC, et al. (1987) Phys Rev Lett
59:915–918.
4. Batlogg B, Cava RJ, Jayaraman, A., van Dover RB, Kourouklis GA, Sunshine
S, Murphy DW, Rupp LW, Chen HS, White A, et al. (1987) Phys Rev Lett
58:2333–2336.
5. Bourne LC, Crommie MF, Zettl A, zur Loye HC, Keller SW, Leary KL, Stacy
AM, Chang KJ, Cohen ML, Morris DE (1987) Phys Rev Lett 58:2337–2339.
6. Benitez EL, Lin JJ, Poon SJ, Farneth WE, Crawford MK, McCarron EM (1988)
Phys Rev B 38:5025–5027.
7. Franck JP (1994) in Physical Properties of High Temp Superconductors IV, ed
Ginzberg DM (World Sci, Singapore), pp 189–293.
8. Zech D, Conder K, Keller H, Kaldis E, Müller KA (1996) Physica B
219&220:136–138.
9. Crawford MK, Kunchur MN, Farneth WE, McCarron EM, III, Poon SJ (1990)
Phys Rev B 41:282–287.
10. Bornemann HJ, Morris DE, Liu HB, Narwankar PK (1992) Physica C
191:211–218.
11. Pringle DJ, Williams GVM, Tallon JL (2000) Phys Rev B 62:12527–12533.
12. Zhao GM, Kirtikar V, Morris DE (2001) Phys Rev B 63:220506.
13. McQueeney RJ, Sarrao JL, Pagliuso PG, Stephens PW, Osborn R (2001) Phys
Rev Lett 87:077001.
14. Cuk T, Baumberger F, Lu DH, Ingle N, Zhou XJ, Eisaki H, Kaneko N, Hussain
Z, Devereaux TP, Nagaosa N, et al. (2004) Phys Rev Lett 93:117003.
15. Gweon GH, Sasagawa T, Zhou SY, Graf J, Takagi H, Lee DH, Lanzara A
(2004) Nature 430:187–190.
16. Lee J, Fujita K, McElroy K, Slezak JA, Wang M, Aiura Y, Bando H, Ishikado
M, Masui T, Zhu JX, et al. (2006) Nature 442:546–550.
17. Schilling JS (2006) in High-Temperature Superconductivity: A Treatise on Theory
and Applications, ed Schrieffer JR (Springer, Berlin).
18. Gao L, Xue YY, Chen F, Xiong Q, Meng RL, Ramirez D, Chu CW, Eggert
JH, Mao HK (1994) Phys Rev B 50:4260–4263.
19. Olsen JL, Bucher E, Levy M, Muller J, Corenzwit E, Geballe T (1964) Rev Mod
Phys 36:168–170.
20. Franck JP, Lawrie DD (1996) J Low Temp Phys 105:801–806.
21. Uehara M, Nagata T, Akimitsu J, Takahashi, H., Môri N, Kinoshita K (1996)
J Phys Soc Jpn 65:2764–2767.
22. Crespi VH, Cohen ML (1993) Phys Rev B 48:398–406.
23. Matsui H, Sato T, Takahashi T, Wang SC, Yang HB, Ding H, Fujii T, Watanabe
T, Matsuda A (2003) Phys Rev Lett 90:217002.
24. Latyshev, Yu I, Yamashita T, Bulaevskii LN, Graf MJ, Balatsky AV, Maley MP
(1999) Phys Rev Lett 82:5345–5348.
25. Dagotto E, Nazarenko A, Moreo A (1995) Phys Rev Lett 74:310–313.
26. Belinicher VI, Chernyshev AL, Shubin VA (1996) Phys Rev B 53:335–342;
54:14914–14917.
27. Chen XJ, Lin HQ (2004) Phys Rev B 69:104518.
28. Phillips JC (1987) Phys Rev B 36:861–863.
29. Phillips JC (1994) Phys Rev Lett 72:3863–3866.
30. Emery VJ, Kivelson SA, Lin HQ (1990) Phys Rev Lett 64:475–478.
31. Tsuei CC, Newns DM, Chi CC, Pattnaik PC (1990) Phys Rev Lett 65:
2724 –2727.
32. Nazarenko A, Dagotto E (1996) Phys Rev B 53:R2987–R2990.
33. Xing DY, Liu M, Wang YG, Dong JM (1999) Phys Rev B 60:9775–9781.
34. Pavarini E, Dasgupta I, Saha-Dasgupta T, Jepsen O, Andersen OK (2001) Phys
Rev Lett 87:047003.
35. Bussmann-Holder A, Keller H (2005) Eur Phys J B 44:487–490.
36. Almasan CC, Han SH, Lee BW, Paulius LM, Maple MB, Veal BW, Downey
JW, Paulikas AP, Fisk Z, Schirber JE (1992) Phys Rev Lett 69:680–683.
37. Chen XJ, Lin HQ, Gong CD (2000) Phys Rev Lett 85:2180–2183.
38. Goncharov AF, and Struzhkin VV (2003) J Raman Spectrosc 34:
532–548.
39. Tissen VG, and Nefedova MV (1990) Sverkhprovodn Fiz Khim Tekh 3:999–
1002.
40. Klotz S, Reith W, Schilling JS (1991) Physica C 172:423–426.
41. Timofeev YA, Struzhkin VV, Hemley RJ, Mao HK, Gregoryanz EA (2002) Rev
Sci Instr 73:371–377.
Chen et al.
Conclusions
We have obtained a clear picture of the oxygen isotope effects in
high-temperature superconductors based on a phenomenological
phonon-mediated BCS-like model. We demonstrate that the variation of ␣ among various optimally doped cuprates is controlled by
the effective next-nearest-neighbor hopping. The obtained oxygen
isotope effect resembles the pressure effect both at ambient conditions and under high pressure. The good agreement between
theory and experiment strongly suggests that the role of phonons
should be taken into account for explaining the high-temperature
superconducting properties in cuprates.
PNAS 兩 March 6, 2007 兩 vol. 104 兩 no. 10 兩 3735
PHYSICS
Materials and Methods
Samples of single crystal YBa2Cu3O7⫺␦ were grown by a self-flux
technique. We performed the measurements of Tc under high
pressures by using a highly sensitive magnetic susceptibility technique with diamond anvil cells (41). The technique is based on the
quenching of the superconductivity and suppression of the Meissner effect in the superconducting sample by an external magnetic
field. The magnetic susceptibility of the metallic parts of the
high-pressure cells is essentially independent of the external field.
Therefore, the magnetic field applied to the sample inserted in the
diamond cell mainly affects the change of the signal coming from
the sample. Our high pressure Raman system and experiments are
detailed in ref. 38. We have used synthetic ultrapure anvils to reduce
diamond fluorescence. Samples were loaded into diamond anvil
cells with helium as a pressure medium, and pressure was determined by using the ruby fluorescence technique.