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Evidence for dx2 − y2 symmetry of superconducting order parameter in YBCO from neutron-scattering data
PHYSICA ELSEVIER
Physica C 251 (1995) 348-354
Evidence for dx2_y2 symmetry of superconducting order parameter in YBCO from neutron-scattering data F. Onufrieva Laboratoire Ldon Brillouin, CEA-CNRS CE Saclay, 91191 Gif-sur-Yvette cedex, France Received 15 June 1995
Abstract
Symmetry of the superconducting order parameter in cuprates is analyzed in relation with recent inelastic neutron-scattering data. It is shown that the experimental results for YBa2Cu307_ n rule out the possibility of extended s wave pairing. On the contrary, the neutron scattering data are in a very good agreement with dxz_y2 symmetry of the superconducting order parameter. Since only these two candidates remain after precise recent measurements of Josephson junctions, we conclude that the symmetry of the superconducting order parameter in YBCO is dx2_y2.
Recent discussions of the mechanism of high-T~ superconductivity in cuprates have concentrated on the symmetry of the superconducting order parameter (OP). Many attempts to determine the symmetry based on different kinds of experiments have been done and a sort of consensus is reached: almost all experiments favor a superconductivity of d wave or extended s wave symmetry. It is extremely difficult, however, to distinguish between the latter two: in both cases the electronic spectrum is characterized by nodes in q space and therefore both give powerlaw dependences of the density of states, a power-law low-temperature variation of thermodynamical functions, of NMR relaxation rates, etc. Moreover, in both cases the sign of the order parameter changes on the Fermi surface when the latter has the shape observed experimentally for bilayer cuprates [1,2]. Due to the latter reason very recent Josephson junction experiments [3,4] in which a change of the sign
of the superconducting (SC) order parameter have been discovered also rule out all possibilities except for the two mentioned. In this paper we show that recent inelastic neutron-scattering (INS) measurements for YBa2Cu 3O7_,7 (~/= 0) [5,6] rule out the extended s wave symmetry being, on the contrary, in very good agreement with the dx2_y2 symmetry. These experiments have discovered that the imaginary part of the spin susceptibility, x(k, to), is peaked at QAF and that Im X(QAF, to) as a function of to below Tc has a single-peak structure with the peak around 40 meV and the energy gap around 30 meV. Both the peak and the gap disappear above Tc, Im X(QAF, to) becomes a smooth function of to with an intensity 10 times weaker in comparison with that at low temperature in Ref. [5] (7/= 0.03) and hardly detectable in ReL [6] (~/= 0). The earlier polarized neutron-scattering [7] data show the same
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F. Onufrieva/ Physica C 251 (1995) 348-354 shape of Im X except for the point that instead of the true gap at low energies a magnetic response of very low intensity was observed. The present analysis is based on the theory [8], which like many other theories, is focused on the CuO 2 plane. The theory has been developed within the t - t ' - J model based on the diagrammatic technique (DT) for Hubbard operators that allows one to treat correctly strong electron correlations within the CuO 2 plane. Calculated for metallic hole concentrations the dynamic spin susceptibility is given by the Larkin equation
x ( k , w ) = ,~(k, w ) / ( 1 + J ~ Z ( k , ¢o)),
1 I( k, o g ) = ~ - - ~ q
1 -nq -nk+ q Mkq ~.~q+ ~.~k+q_ ¢o_ iO +
nF q -- n F k+q +Nkqaq--O--k+---q~---~--iO+ I '
(3)
(1)
which formally has an " R P A " structure with irreducible part, ,Y(k, w), instead of zero susceptibility. (In Eq. (1), J k = 2 J ( c o s k x + c o s ky) is the exchange interaction between Cu spins). The irreducible part calculated in the first approximation with respect to 1 / Z (Z is the number of nearest neighbors) includes two different dynamic contributions. The first one, the "localized" contribution (L(k, oJ)), arises from the subsystem of localized spins with quantum short-range antiferromagnetic (AF) correlations. The second one, the "itinerant" contribution (I(k, ~o)), arises from the subsystem of propagating carrier quasiparticles. The first contribution is sensitive to the symmetry of short-range (SR) AF correlations between localized spins, the second is sensitive to the shape of Fermi surface (FS) of itinerant carriers in the normal state and to the symmetry of the SC order parameter. As a result of their competition a rather unusual spin dynamics occurs in the underdoped regime. We have been able to understand [8] on this basis many exotic features observed experimentally by INS in YBaeCu306+ x (YBCO) crystals with oxygen doping from x = 0.52 to x = 0.92. However, an analysis of experimental data is not so transparent for the case of the underdoped regime to give a sure answer about the symmetry of the SC gap. For the overdoped regime, x = 1, the situation is much simpler, since the intensity of the SR AF order parameter, and so of the dynamic "localized" contribution, becomes negligible. Then, the spin susceptibility is described by the Larkin equation (1) with the irreducible part .Y(k, o J ) = L ( ° ) ( ¢ o ) + l ( k , w),
in which the first static and k independent term, L~°~(¢o) = 6(¢o)C(T)/T, is the susceptibility of noninteracting localized spins and is responsible for the high-temperature asymptotics of static susceptibility. The dynamic k dependent part of ,~(k, ¢o) is related only to the "itinerant" contribution which for oJ > 0 is given by
(2)
gkq= 1 -
Nkq = 1 +
ekek+ q + akak+ q O~Ok+q ekek+ q "~-aka~+ q ~k~-~k+q
(3a)
In Eqs. (3) and (3a), n~ = nF(12k) is a Fermi function O k is the dispersion law of carriers in the superconducting state, O k = ~ + A~, Ak is the superconducting gap, e k is the dispersion law of itinerant carrier quasiparticles in the normal state. We would like to draw attention to the nature of the itinerant quasiparticles contributing to Eq. (3) in the considered approach. For this we remind that the t-J (or t - t ' - J ) model, being obtained for the case of cuprates from the three-band Hubbard model, treats the CuO 2 plane as a lattice of square plaquettes centered on copper sites with four nearest-neighbor O sites. Three states of the plaquette are taken into consideration in the t-J model: I1) and I - 1 ) , which correspond to the filled p shells on oxygen and a hole on copper with up or down spin, respectively, and 10) which describes the singlet bound state between an oxygen hole and a copper hole (the Zhang-Rice singlet). The hamiltonian of the model is written in terms of so-called X operators (or Hubbard operators (HO's)), X pp'= Ip)(p' I, which describe the transitions between different states of the plaquette. The itinerant carrier quasiparticles are associated with the HO Xi°~ (~r = 1 , - 1), which for the plaquette i creates the singlet bound state between a hole on oxygen and a hole on copper and annihilates simultaneously the spin up or down on Cu. Therefore, the itinerant quasiparticles correspond to propagation of the oxygen hole accompanied by
F. Onufrieva /Physica C 251 (1995) 348-354
350
spin-flip processes of the Cu hole. The dispersion law of these quasiparticles is given by ~q = t(1 + 6)(cos
qx + cos qy)
+ 2t'(1 + 6) cos qx cos
qy --
~,
(4)
where 6 is a concentration of doped holes in the CuO 2 plane. The equation for the chemical potential is such [8] that the FS is large and close to the one observed experimentally. Being the limit of our strong-coupling theory (SCT), the adduced expression for the dynamic spin susceptibility in the overdoped regime on the other hand almost coincides 1 with that of weak-coupling theories (WCT) [9-11]. This coincidence is formal due to the difference in the nature of itinerant quasiparticles. They are true electrons (holes) in the WCT and the specific quasiparticles related to both a hole on Cu and a hole on oxygen in the way discussed above in the considered case. This implies different values of the parameters 2 involved in Eqs. (3,4) in the present approach and in Refs. [9-11] which has important consequences when explicit calculations for the d wave symmetry are performed. However, from the symmetry point of view (important to rule out the extended s wave symmetry as inconsistent with INS data) results of both approaches coincide in the case of the overdoped regime. The same statement is valid with respect to SCT developed on the basis of a SB representation [12,13]. In this case the expression for Xo(k, to) is given by Eqs. (3) and (3a) where some intermediate quasiparticles (spinons) formally play the role of itinerant carriers in our SCT approach and in the WCT and have a similar dispersion law. If in addition the symmetry of the spinonpairing order parameter (OP) is assumed to correspond to that of superconducting pairing OP and the shape of spinon FS to that observed in angle-resolved photoemission spectroscopy (ARPES) (as is done in Refs. [12] and [13]) one arrives at the same (from the symmetry point of view) situation as in the previous two theories. Thus, one can claim that a
sort of consensus exists between different theories about the structure of expression for the spin susceptibility in the overdoped regime. This structure, on the other hand, is supported by different experiments which show that in the limit of the overdoped regime, the behavior of cuprates tends to a normal-metal behavior. Let us analyze now the explicit to and k dependences of the imaginary part of the dynamic spin susceptibility Im
x( k, to) Im Z ( k , to) [1 + J k Re ~ ( k , to)]2 + [Jk Im ~ ( k , to)]2,
(5) with Z(k, to) given by Eqs. (2) and (3) and let us compare them with those measured by INS for YBa2Cu307-,7 (7/= 0). The explicit form of Im x(k, to) depends on the form of Im I(k, to) and Re I(k, to), while both latter functions depend on the shape of Fermi surface of itinerant carriers in the normal state and on the symmetry of the superconducting OP. The former is known from ARPES experiments. For the family of bilayer-cuprates YBCO belongs to, the shape of FS is shown schematically in Fig. 1. We are interested in two symmetries of the superconducting order parameter, d x 2 y 2 symmetry and the extended s wave one, with a superconducting gap given by
qx - c o s q y ) , Aq = A0(cos qx + cos qy). Ad = Ao(C°S
(6)
In both cases nodes exist in q space where Aq = 0 lines intersect the FS, and so Oq = 0. These points are shown in Fig. 1. The most important factor, which determines the behavior of Im g(k, to), is the energy and momentum dependence of Im I(k, to). This function is, roughly speaking, the sum of "two-particle densities of states": Im I = I m I z + I m 12,
1The term S(to) has no influence on the spin dynamics at finite to which we analyze now; however, it is important for the static properties. 2 A s is known, the intensity of the magnetic susceptibility related to itinerant electrons is very week in a normal metal.
F nF+q) Im Ii(k, t o ) = ~ 1 ~,(Mkq(1--nq-q
Xa(aq"F ak+q--to)) ,
(7a)
F. Onufrieva /Physica C 251 (1995) 348-354
qy
351
On the contrary, for the d wave symmetry, Im I(k, to) exhibits a largest gap just for k = QAFIts value is given by A~(QAF) = 2A0~12/z/[(1 + ~)/'] I.
~qx
Fig. 1. Schematic plot of Fermi surface for YBa2Cu306+ x in the metalic state. The points shown as triangles correspond to the nodes (/2 k = 0 ) in the case of dx2_y2 symmetry, as circles to these of the extended s wave one. Nodal wavevectors, qnod,s qnod,d for which Im X is a gapless function are shown as well for both symmetries.
1
Im
12( k, tO) = ~-~ ~ (Ni, q(n ~ - n~+q) q
xs(a,- a,+,- to)}.
(71,)
At low T, the term Im I2(k, to) is negligibly small at finite tO due to the prefactor with Fermi functions, which are equal to zero when /2q contains a gap. The main contribution comes from the term Im Ii(k, to). It is clear from Eqs. (7a) and (7b) that for all wavevectors k, Im It(k , to) exhibits a lowenergy gap, AI(k)= minq(12q + Ok+q), as a result of the existence of a superconducting gap in the one-particle spectrum /2q. The exceptional wavevectors k, for which Im I(k, to) is gapless, are " n o d a l " wavevectors, knod, which join the nodes. For the case of the (1, 1) direction corresponding to the measurements in Refs. [5-7], these wavevectors, d d, for extended s wave and d wave k~od and kno symmetries, respectively, are shown in Fig. 1. The crucial point is that for the extended s wave symmetry and given shape of FS typical for bilayer-cuprates the nodal wavevector coincides with QAF, k~oo = QAF, and therefore Im
(8)
I(QA F, to) is the gapless function.
(9)
As a result, for these two types of symmetries the to dependences of Im I(QAF, to) and Re I(QAF, tO) (and so of Im X(QAF, tO)) are so different that this allows one to distinguish between them rather easily based on experimental data. To demonstrate this we have performed explicit calculations of Im x(k, tO) in the overdoped regime for the considered two symmetries of the SC order parameter. The results are shown in Figs. 2-5. When calculating we have used the same values of the parameters as in the previous calculations for the underdoped regime [8]: t ' = - 0 . 4 5 t and t/J = 1.8. These values on the one hand are close to those predicted theoretically for the t-J model derived for the CuO 2 plane from the realistic three-band Hubbard model (see for example Refs. [14] and [15]) and on the other they have allowed us to fit the experimental INS data for the underdoped YBCO quite well [8]. We have also used the same value of the superconducting gap at T = 0 as in Ref. [8] for = 0.25, namely Ao/J = 0.1, based on the fact of approximative equality of Tc for the overdoped YBa2Cu30 7 and underdoped YBa2Cu306.92 samples (assuming that 8 = 0.25 corresponds to x = 0.92
1.0
T
T
"
r
~
T=O. d - wave
0.8
------
T=O,e~tended S - w ~
.......... T--OA2 (Ao=O) 3. 0 6 IL
< 0
= 0~
E 0.2
0
|
2
3
t.
~/J Fig. 2. Im I(QAF, tO) for d wave and extended s wave symmetries at T = 0 and T above Tc (T = 0.12 J).
352
F. Onufrieva/Physica C251 (1995)348-354
and 6 = 0.35 to x = 1, where ~ is the concentration of doped holes in CuO 2 plane and x is the oxygen doping). The results for Im I(QAF, to) at T = 0 and at T above Tc for both symmetries are shown in Fig. 2. The main features in the superconducting state are the low-energy gap (at to = A~(QAF) ~ 0.38J), the jump of the spectral function at the energy corresponding to the gap and the sharp peak slightly above the gap (at to = 0 . 4 2 J ) for the case of d wave symmetry and the gapless smooth shape for the case of extended s wave one. The second important feature is the strong difference in the shape and intensity below and above Tc in the case of d wave symmetry and the almost unchanged shape in the case of extended s wave symmetry. To understand this we have to keep in mind the tendency of the evolution of the shape of Im I(QAF, to) with decreasing superconducting gap, A0, in the case of the d wave symmetry. The energy positions of the gap, to = zl~, and of the peak, to = Aid + 8vi ~ --/~, move towards low energies as A0 decreases; simultaneously the intensity at the peak decreases proportionally to A0, see Eqs. (9), (7a) and (3a) (~VH--/X is the energy distance between the Van Hove singularity in the one-particle spectral function and the Fermi level). In the limit A 0 ~ 0, the curve for Im I(QA F, to) tends to the line corresponding to the extended s wave symmetry. Thus, the sharp peak and the gap in the case of d wave symmetry are direct consequences of the existence of superconductivity. In the case of extended s wave symmetry, the shape of Im I(QA F, to) at fixed T does not depend on the value of zl0 at all. This difference makes clear the origin of the strong difference in the behavior above and below T~ in the case of d wave symmetry and the absence of a considerable difference above and below Tc in the case of the extended s wave one. In the latter case the difference comes only from the factor of the temperature, whereas for the d wave symmetry, the disappearance of the gap, zl0, above Tc is the factor responsible for the change of shape. The quantity plotted in Fig. 2 is the imaginary part of the " z e r o susceptibility". The behavior of the real part of this function is important as well. On the one hand, Re I(k, to) has an independent influence on Im X (see Eq. (5)) but on the other its
behavior is related to the behavior of Im I(k, to) due to the Kramers-Kronig relations. It is easy to see that the jump in Im I(k, to) at to = A/d in the case of the d wave symmetry is accompanied by the logarithmic singularity in Re I(k, to), whereas in the case of the extended s wave symmetry Re I(QAF, to) is a smooth function of to. Let us note that the mentioned singularity is enough to get a pole-type peak in Im X(QAF, to) at an energy slightly below to = Ad. However, in reality a damping exists even at T = 0, which changes qualitatively the behavior of Re I(QAF, to): the logarithmic singularity transforms to a maximum. As a result the pole type peak in Im X(QAF, to) transforms to a resonance peak at to = A~. These features are demonstrated in Fig. 3 where we plot three functions, Im I(QAF, to), Re I(QAF, to) and Im X(QAF, to) for the two considered symmetries of SC gap. The quasiparticle damping is taken into account in the same way as in Ref. [8] namely by replacing in Eq. (3): to ~ to + i F ( t o ) with F(
to) = [ O~to3, /-',
tO_~<600, to >---toO,
where too is a threshold energy in quasiparticle damping which is proportional to the superconducting gap too ot zl0. As one can see from Fig. 3, a huge difference exists between the to dependences of Im X(QAF, to) in the cases of the d wave and extended s wave symmetries of the SC gap. We would like to emphasize the difference between the natures of the resonance peak discussed above and of the peak in Im X in the theories [11,13]. In the latter theories, the peak in Im X rather corresponds to the peak in Im X0 and is related to a Van Hove singularity. In the case considered, the sharp peak in Im X corresponds to a true resonance which exists as a result of the almost zero denominator of Im X. This nature of the resonance peak in the SC state of d wave symmetry leads to the k shape of Im x(k, to) peaked in QAF for energies close to the energy of the resonance peak (see Fig. 4) as observed experimentally [5,6]. For energies far away, the k shape is slightly different: for low energies it becomes incommensurate, being peaked at k = k,o a.d To summarize the results we show in Fig. 5 the imaginary part of the spin susceptibility for two
F. Onufrieva/ Physica C 251 (1995) 348-354 I
I
0.4
I
~
•. . . . . . . . . . .
I
I
JImX(QAF
W)
JRe
I(QAF,
OJ)
Jim
I (QAF,to)
i
0.2
-2001
-3
p
I
,
I
I
'
'
'
F
~11
I
r
' I
.A
loo I- • r. l,.sx • T.~.~
_E "0.2
o0
I
I I~ fi
/ rc=~sK ~f'LI / 0.4
0.6
I
----......... T=O.12(&o=O)
extended S - wave
\~-
n
T = O, d -wave T = O, extended s-wave
0.6
\
353
o '
J' ~! "1 I
,o'/
d- wove
0
0.4
0.6
w/J
'
04
0.2
Fig. 5. Imaginary part of the spin susceptibility for the two considered symmetries of the SC gap for T below and above Tc. In the insert the INS data from Ref. [5] are shown. 0.2
~ ~ ~ / / /
ol 0
0.2
0.4
0.6
C0/J
Fig. 3. Real and imaginary parts of the irreducible part and imaginary part of spin susceptibility at T = 0 for two symmetries of the order parameter (F = 0.03J). considered symmetries of the SC gap for T below and above To. In the inset we show the experimental INS data for YBa2Cu306.97 from Ref. [5]. One can .......... w : 0.16 co : 0.36 co : 0.38 0.6 ......... to = 0.39 A
. . . . to = O.Z,O - - - w : 0.41 ..... ~ = 0.42 /
X
04 x
0.2 0 0
0
.........."..................................
(-g'-7)
i.............................................
(r~,rt) Wavevector
(7 3 )
Fig. 4. Momentum dependence of Im x(k, to) for different energies.
see that the extended s wave symmetry of the SC OP whose main feature (in the case of the dynamic spin susceptibility) is the "normal-state" shape of Im X(QAF, tO) below and above Tc is absolutely inconsistent with the experimental data. On the contrary, in the case of the d wave symmetry the theoretical results are in a very good agreement with neutron-scattering data. This concerns the following points: (1) the existence of the gap and of the resonance peak below Tc, (2) the disappearance of both above To, (3) a negligibly small intensity above Tc, (4) a k dependence of Im X peaked at QAF for co close to the resonance peak. We have to notice that the shape of the k dependence of Im X predicted theoretically for energies far from the resonance peak can hardly be detectable due to low intensity. Summarizing, the d wave symmetry of the superconducting order parameter is strongly supported by INS data for YBCO. What is more remarkable, the extended s wave symmetry can be ruled out, being absolutely inconsistent with the results of neutron scattering. Since only these two candidates remain after precise recent measurements of Josephson junctions [1,2] which have discovered the change of the sign of the SC OP on the FS, one can conclude that the symmetry of the superconducting order parameter in the overdoped YBCO is dx2_y2. One can also extend this statement to the whole doping range
354
F. Onufrieva/ Physica C 251 (1995) 348-354
corresponding to the heavily doped regime in YBCO because of the existence of the low-energy gap and of the high-energy resonance peak in observed by INS for YBCO with x = 0.92 and x = 0.83 [16,17]. The situation for the weakly doped regime is not so clear, for the gap observed experimentally is small and the resonance peak is less pronounced [18,19].
References [1] R. Liu, B.W. Veal, A.P. Paulikas et al., Phys. Rev. B 46 (1992) 11056. [2] A.A. Abrikosov, J.C. Campuzano and K. Gofron, Physica C 214 (1993) 73; D.M. King, Z.-X. Shen, D.S. Dessau et al., Phys. Rev. Lett. 73 (1994) 3298. [3] C.C. Tsuei, J.R. Kirtley, C.C. Chi, L.S. Yu-Jahnes, A. Gupta, T. Shaw, J.Z. Sun and M.B. Ketchen, Phys. Rev. Lett. 73 (1994) 593. [4] D.A. Wollman, D.J. Van Harlingen, J. Giapintzakis and D.M. Ginsberg, Phys. Rev. Lett. 74 (1995) 797. [5] P. Bourges, L.P. Regnault, Y. Sidis, C. Vettier, Phys. Rev. B, to be published. [6] H.F. Fong, B. Keimer, P.W. Anderson et al., Phys. Rev. Lett., 75 (1995) 316.
[7] H.A. Mook M. Yethiraj, G. Aeppli, T.E. Mason and T. Armstrong, Phys. Rev. Lett. 70 (1993) 3490. [8[ F. Onufrieva and J. Rossat-Mignod, Phys. Rev. B 52 (1995), 7572; F. Onufdeva, J. Rossat-Mignod and V.P. Kuslmir, Physica B 206 & 207 (1995) 667. [9] N. Bulut and D.J. Scalapino, Phys. Rev. B 45 (1992) 2371; ibid., 47 (1993) 3419. [10] Q. Si, Y. Zha, K. Levin and J.P. Lu, Rev. B 47 (1993) 9055; Y. Zha, K. Levin and Q. Si, Phys. Rev. B 47 (1993) 9124. [11] K. Maki and H. Won, Phys. Rev. Lett. 72 (1994) 1758. [12] T, Tanamoto, H. Konho and H. Fukuyama, J. Phys. Soc. Jpn. 62 (1993) 717; ibid., 62 (1993) 1455. [13] G. Stemman, C. Pepin and M. Lavagna, Phys. Rev. B 50 (1994) 4075. [14] M. Hybertsen, E.B. Stechel, M. Schluter and D.R. Jennison, Phys. Rev. B 41 (1990) 11068. [15] V.I. Belinicher and A.L. Chernyshev, Phys. Rev. B 47 (1993) 390; ibid., 49 (1994) 9746. [16] J. Rossat-Mignod, L.P. Regnault, C. Vettier, P. Bourges, P. Burlet, J. Bossy, J.Y. Henry and G. Lapertot, Physica C 185-189 (1991) 86. [17] J. Rossat-Mignod, P. Bourges, F. Onufrieva, L.P. Regnanlt, J.Y. Henry, P. Burlet and C. Vettier, Physica B 199 & 200 (1994) 281. [18] J. Rossat-Mignod, L.P. Regnault, P. Bourges, C. Vettier, P. Burlet and J.Y. Henry, Phys. Scripta T 45 (1992) 74. [19] J.M. Tranquada, P.M. Gehring, G. Shirane, S. Shamoto and M. Sato, Phys. Rev. B 46 (1992) 5561.
Physica C 251 (1995) 348-354
Evidence for dx2_y2 symmetry of superconducting order parameter in YBCO from neutron-scattering data F. Onufrieva Laboratoire Ldon Brillouin, CEA-CNRS CE Saclay, 91191 Gif-sur-Yvette cedex, France Received 15 June 1995
Abstract
Symmetry of the superconducting order parameter in cuprates is analyzed in relation with recent inelastic neutron-scattering data. It is shown that the experimental results for YBa2Cu307_ n rule out the possibility of extended s wave pairing. On the contrary, the neutron scattering data are in a very good agreement with dxz_y2 symmetry of the superconducting order parameter. Since only these two candidates remain after precise recent measurements of Josephson junctions, we conclude that the symmetry of the superconducting order parameter in YBCO is dx2_y2.
Recent discussions of the mechanism of high-T~ superconductivity in cuprates have concentrated on the symmetry of the superconducting order parameter (OP). Many attempts to determine the symmetry based on different kinds of experiments have been done and a sort of consensus is reached: almost all experiments favor a superconductivity of d wave or extended s wave symmetry. It is extremely difficult, however, to distinguish between the latter two: in both cases the electronic spectrum is characterized by nodes in q space and therefore both give powerlaw dependences of the density of states, a power-law low-temperature variation of thermodynamical functions, of NMR relaxation rates, etc. Moreover, in both cases the sign of the order parameter changes on the Fermi surface when the latter has the shape observed experimentally for bilayer cuprates [1,2]. Due to the latter reason very recent Josephson junction experiments [3,4] in which a change of the sign
of the superconducting (SC) order parameter have been discovered also rule out all possibilities except for the two mentioned. In this paper we show that recent inelastic neutron-scattering (INS) measurements for YBa2Cu 3O7_,7 (~/= 0) [5,6] rule out the extended s wave symmetry being, on the contrary, in very good agreement with the dx2_y2 symmetry. These experiments have discovered that the imaginary part of the spin susceptibility, x(k, to), is peaked at QAF and that Im X(QAF, to) as a function of to below Tc has a single-peak structure with the peak around 40 meV and the energy gap around 30 meV. Both the peak and the gap disappear above Tc, Im X(QAF, to) becomes a smooth function of to with an intensity 10 times weaker in comparison with that at low temperature in Ref. [5] (7/= 0.03) and hardly detectable in ReL [6] (~/= 0). The earlier polarized neutron-scattering [7] data show the same
0921-4534/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved
SSDI 0 9 2 1 - 4 5 3 4 ( 9 5 ) 0 0 4 1 5 - 7
349
F. Onufrieva/ Physica C 251 (1995) 348-354 shape of Im X except for the point that instead of the true gap at low energies a magnetic response of very low intensity was observed. The present analysis is based on the theory [8], which like many other theories, is focused on the CuO 2 plane. The theory has been developed within the t - t ' - J model based on the diagrammatic technique (DT) for Hubbard operators that allows one to treat correctly strong electron correlations within the CuO 2 plane. Calculated for metallic hole concentrations the dynamic spin susceptibility is given by the Larkin equation
x ( k , w ) = ,~(k, w ) / ( 1 + J ~ Z ( k , ¢o)),
1 I( k, o g ) = ~ - - ~ q
1 -nq -nk+ q Mkq ~.~q+ ~.~k+q_ ¢o_ iO +
nF q -- n F k+q +Nkqaq--O--k+---q~---~--iO+ I '
(3)
(1)
which formally has an " R P A " structure with irreducible part, ,Y(k, w), instead of zero susceptibility. (In Eq. (1), J k = 2 J ( c o s k x + c o s ky) is the exchange interaction between Cu spins). The irreducible part calculated in the first approximation with respect to 1 / Z (Z is the number of nearest neighbors) includes two different dynamic contributions. The first one, the "localized" contribution (L(k, oJ)), arises from the subsystem of localized spins with quantum short-range antiferromagnetic (AF) correlations. The second one, the "itinerant" contribution (I(k, ~o)), arises from the subsystem of propagating carrier quasiparticles. The first contribution is sensitive to the symmetry of short-range (SR) AF correlations between localized spins, the second is sensitive to the shape of Fermi surface (FS) of itinerant carriers in the normal state and to the symmetry of the SC order parameter. As a result of their competition a rather unusual spin dynamics occurs in the underdoped regime. We have been able to understand [8] on this basis many exotic features observed experimentally by INS in YBaeCu306+ x (YBCO) crystals with oxygen doping from x = 0.52 to x = 0.92. However, an analysis of experimental data is not so transparent for the case of the underdoped regime to give a sure answer about the symmetry of the SC gap. For the overdoped regime, x = 1, the situation is much simpler, since the intensity of the SR AF order parameter, and so of the dynamic "localized" contribution, becomes negligible. Then, the spin susceptibility is described by the Larkin equation (1) with the irreducible part .Y(k, o J ) = L ( ° ) ( ¢ o ) + l ( k , w),
in which the first static and k independent term, L~°~(¢o) = 6(¢o)C(T)/T, is the susceptibility of noninteracting localized spins and is responsible for the high-temperature asymptotics of static susceptibility. The dynamic k dependent part of ,~(k, ¢o) is related only to the "itinerant" contribution which for oJ > 0 is given by
(2)
gkq= 1 -
Nkq = 1 +
ekek+ q + akak+ q O~Ok+q ekek+ q "~-aka~+ q ~k~-~k+q
(3a)
In Eqs. (3) and (3a), n~ = nF(12k) is a Fermi function O k is the dispersion law of carriers in the superconducting state, O k = ~ + A~, Ak is the superconducting gap, e k is the dispersion law of itinerant carrier quasiparticles in the normal state. We would like to draw attention to the nature of the itinerant quasiparticles contributing to Eq. (3) in the considered approach. For this we remind that the t-J (or t - t ' - J ) model, being obtained for the case of cuprates from the three-band Hubbard model, treats the CuO 2 plane as a lattice of square plaquettes centered on copper sites with four nearest-neighbor O sites. Three states of the plaquette are taken into consideration in the t-J model: I1) and I - 1 ) , which correspond to the filled p shells on oxygen and a hole on copper with up or down spin, respectively, and 10) which describes the singlet bound state between an oxygen hole and a copper hole (the Zhang-Rice singlet). The hamiltonian of the model is written in terms of so-called X operators (or Hubbard operators (HO's)), X pp'= Ip)(p' I, which describe the transitions between different states of the plaquette. The itinerant carrier quasiparticles are associated with the HO Xi°~ (~r = 1 , - 1), which for the plaquette i creates the singlet bound state between a hole on oxygen and a hole on copper and annihilates simultaneously the spin up or down on Cu. Therefore, the itinerant quasiparticles correspond to propagation of the oxygen hole accompanied by
F. Onufrieva /Physica C 251 (1995) 348-354
350
spin-flip processes of the Cu hole. The dispersion law of these quasiparticles is given by ~q = t(1 + 6)(cos
qx + cos qy)
+ 2t'(1 + 6) cos qx cos
qy --
~,
(4)
where 6 is a concentration of doped holes in the CuO 2 plane. The equation for the chemical potential is such [8] that the FS is large and close to the one observed experimentally. Being the limit of our strong-coupling theory (SCT), the adduced expression for the dynamic spin susceptibility in the overdoped regime on the other hand almost coincides 1 with that of weak-coupling theories (WCT) [9-11]. This coincidence is formal due to the difference in the nature of itinerant quasiparticles. They are true electrons (holes) in the WCT and the specific quasiparticles related to both a hole on Cu and a hole on oxygen in the way discussed above in the considered case. This implies different values of the parameters 2 involved in Eqs. (3,4) in the present approach and in Refs. [9-11] which has important consequences when explicit calculations for the d wave symmetry are performed. However, from the symmetry point of view (important to rule out the extended s wave symmetry as inconsistent with INS data) results of both approaches coincide in the case of the overdoped regime. The same statement is valid with respect to SCT developed on the basis of a SB representation [12,13]. In this case the expression for Xo(k, to) is given by Eqs. (3) and (3a) where some intermediate quasiparticles (spinons) formally play the role of itinerant carriers in our SCT approach and in the WCT and have a similar dispersion law. If in addition the symmetry of the spinonpairing order parameter (OP) is assumed to correspond to that of superconducting pairing OP and the shape of spinon FS to that observed in angle-resolved photoemission spectroscopy (ARPES) (as is done in Refs. [12] and [13]) one arrives at the same (from the symmetry point of view) situation as in the previous two theories. Thus, one can claim that a
sort of consensus exists between different theories about the structure of expression for the spin susceptibility in the overdoped regime. This structure, on the other hand, is supported by different experiments which show that in the limit of the overdoped regime, the behavior of cuprates tends to a normal-metal behavior. Let us analyze now the explicit to and k dependences of the imaginary part of the dynamic spin susceptibility Im
x( k, to) Im Z ( k , to) [1 + J k Re ~ ( k , to)]2 + [Jk Im ~ ( k , to)]2,
(5) with Z(k, to) given by Eqs. (2) and (3) and let us compare them with those measured by INS for YBa2Cu307-,7 (7/= 0). The explicit form of Im x(k, to) depends on the form of Im I(k, to) and Re I(k, to), while both latter functions depend on the shape of Fermi surface of itinerant carriers in the normal state and on the symmetry of the superconducting OP. The former is known from ARPES experiments. For the family of bilayer-cuprates YBCO belongs to, the shape of FS is shown schematically in Fig. 1. We are interested in two symmetries of the superconducting order parameter, d x 2 y 2 symmetry and the extended s wave one, with a superconducting gap given by
qx - c o s q y ) , Aq = A0(cos qx + cos qy). Ad = Ao(C°S
(6)
In both cases nodes exist in q space where Aq = 0 lines intersect the FS, and so Oq = 0. These points are shown in Fig. 1. The most important factor, which determines the behavior of Im g(k, to), is the energy and momentum dependence of Im I(k, to). This function is, roughly speaking, the sum of "two-particle densities of states": Im I = I m I z + I m 12,
1The term S(to) has no influence on the spin dynamics at finite to which we analyze now; however, it is important for the static properties. 2 A s is known, the intensity of the magnetic susceptibility related to itinerant electrons is very week in a normal metal.
F nF+q) Im Ii(k, t o ) = ~ 1 ~,(Mkq(1--nq-q
Xa(aq"F ak+q--to)) ,
(7a)
F. Onufrieva /Physica C 251 (1995) 348-354
qy
351
On the contrary, for the d wave symmetry, Im I(k, to) exhibits a largest gap just for k = QAFIts value is given by A~(QAF) = 2A0~12/z/[(1 + ~)/'] I.
~qx
Fig. 1. Schematic plot of Fermi surface for YBa2Cu306+ x in the metalic state. The points shown as triangles correspond to the nodes (/2 k = 0 ) in the case of dx2_y2 symmetry, as circles to these of the extended s wave one. Nodal wavevectors, qnod,s qnod,d for which Im X is a gapless function are shown as well for both symmetries.
1
Im
12( k, tO) = ~-~ ~ (Ni, q(n ~ - n~+q) q
xs(a,- a,+,- to)}.
(71,)
At low T, the term Im I2(k, to) is negligibly small at finite tO due to the prefactor with Fermi functions, which are equal to zero when /2q contains a gap. The main contribution comes from the term Im Ii(k, to). It is clear from Eqs. (7a) and (7b) that for all wavevectors k, Im It(k , to) exhibits a lowenergy gap, AI(k)= minq(12q + Ok+q), as a result of the existence of a superconducting gap in the one-particle spectrum /2q. The exceptional wavevectors k, for which Im I(k, to) is gapless, are " n o d a l " wavevectors, knod, which join the nodes. For the case of the (1, 1) direction corresponding to the measurements in Refs. [5-7], these wavevectors, d d, for extended s wave and d wave k~od and kno symmetries, respectively, are shown in Fig. 1. The crucial point is that for the extended s wave symmetry and given shape of FS typical for bilayer-cuprates the nodal wavevector coincides with QAF, k~oo = QAF, and therefore Im
(8)
I(QA F, to) is the gapless function.
(9)
As a result, for these two types of symmetries the to dependences of Im I(QAF, to) and Re I(QAF, tO) (and so of Im X(QAF, tO)) are so different that this allows one to distinguish between them rather easily based on experimental data. To demonstrate this we have performed explicit calculations of Im x(k, tO) in the overdoped regime for the considered two symmetries of the SC order parameter. The results are shown in Figs. 2-5. When calculating we have used the same values of the parameters as in the previous calculations for the underdoped regime [8]: t ' = - 0 . 4 5 t and t/J = 1.8. These values on the one hand are close to those predicted theoretically for the t-J model derived for the CuO 2 plane from the realistic three-band Hubbard model (see for example Refs. [14] and [15]) and on the other they have allowed us to fit the experimental INS data for the underdoped YBCO quite well [8]. We have also used the same value of the superconducting gap at T = 0 as in Ref. [8] for = 0.25, namely Ao/J = 0.1, based on the fact of approximative equality of Tc for the overdoped YBa2Cu30 7 and underdoped YBa2Cu306.92 samples (assuming that 8 = 0.25 corresponds to x = 0.92
1.0
T
T
"
r
~
T=O. d - wave
0.8
------
T=O,e~tended S - w ~
.......... T--OA2 (Ao=O) 3. 0 6 IL
< 0
= 0~
E 0.2
0
|
2
3
t.
~/J Fig. 2. Im I(QAF, tO) for d wave and extended s wave symmetries at T = 0 and T above Tc (T = 0.12 J).
352
F. Onufrieva/Physica C251 (1995)348-354
and 6 = 0.35 to x = 1, where ~ is the concentration of doped holes in CuO 2 plane and x is the oxygen doping). The results for Im I(QAF, to) at T = 0 and at T above Tc for both symmetries are shown in Fig. 2. The main features in the superconducting state are the low-energy gap (at to = A~(QAF) ~ 0.38J), the jump of the spectral function at the energy corresponding to the gap and the sharp peak slightly above the gap (at to = 0 . 4 2 J ) for the case of d wave symmetry and the gapless smooth shape for the case of extended s wave one. The second important feature is the strong difference in the shape and intensity below and above Tc in the case of d wave symmetry and the almost unchanged shape in the case of extended s wave symmetry. To understand this we have to keep in mind the tendency of the evolution of the shape of Im I(QAF, to) with decreasing superconducting gap, A0, in the case of the d wave symmetry. The energy positions of the gap, to = zl~, and of the peak, to = Aid + 8vi ~ --/~, move towards low energies as A0 decreases; simultaneously the intensity at the peak decreases proportionally to A0, see Eqs. (9), (7a) and (3a) (~VH--/X is the energy distance between the Van Hove singularity in the one-particle spectral function and the Fermi level). In the limit A 0 ~ 0, the curve for Im I(QA F, to) tends to the line corresponding to the extended s wave symmetry. Thus, the sharp peak and the gap in the case of d wave symmetry are direct consequences of the existence of superconductivity. In the case of extended s wave symmetry, the shape of Im I(QA F, to) at fixed T does not depend on the value of zl0 at all. This difference makes clear the origin of the strong difference in the behavior above and below T~ in the case of d wave symmetry and the absence of a considerable difference above and below Tc in the case of the extended s wave one. In the latter case the difference comes only from the factor of the temperature, whereas for the d wave symmetry, the disappearance of the gap, zl0, above Tc is the factor responsible for the change of shape. The quantity plotted in Fig. 2 is the imaginary part of the " z e r o susceptibility". The behavior of the real part of this function is important as well. On the one hand, Re I(k, to) has an independent influence on Im X (see Eq. (5)) but on the other its
behavior is related to the behavior of Im I(k, to) due to the Kramers-Kronig relations. It is easy to see that the jump in Im I(k, to) at to = A/d in the case of the d wave symmetry is accompanied by the logarithmic singularity in Re I(k, to), whereas in the case of the extended s wave symmetry Re I(QAF, to) is a smooth function of to. Let us note that the mentioned singularity is enough to get a pole-type peak in Im X(QAF, to) at an energy slightly below to = Ad. However, in reality a damping exists even at T = 0, which changes qualitatively the behavior of Re I(QAF, to): the logarithmic singularity transforms to a maximum. As a result the pole type peak in Im X(QAF, to) transforms to a resonance peak at to = A~. These features are demonstrated in Fig. 3 where we plot three functions, Im I(QAF, to), Re I(QAF, to) and Im X(QAF, to) for the two considered symmetries of SC gap. The quasiparticle damping is taken into account in the same way as in Ref. [8] namely by replacing in Eq. (3): to ~ to + i F ( t o ) with F(
to) = [ O~to3, /-',
tO_~<600, to >---toO,
where too is a threshold energy in quasiparticle damping which is proportional to the superconducting gap too ot zl0. As one can see from Fig. 3, a huge difference exists between the to dependences of Im X(QAF, to) in the cases of the d wave and extended s wave symmetries of the SC gap. We would like to emphasize the difference between the natures of the resonance peak discussed above and of the peak in Im X in the theories [11,13]. In the latter theories, the peak in Im X rather corresponds to the peak in Im X0 and is related to a Van Hove singularity. In the case considered, the sharp peak in Im X corresponds to a true resonance which exists as a result of the almost zero denominator of Im X. This nature of the resonance peak in the SC state of d wave symmetry leads to the k shape of Im x(k, to) peaked in QAF for energies close to the energy of the resonance peak (see Fig. 4) as observed experimentally [5,6]. For energies far away, the k shape is slightly different: for low energies it becomes incommensurate, being peaked at k = k,o a.d To summarize the results we show in Fig. 5 the imaginary part of the spin susceptibility for two
F. Onufrieva/ Physica C 251 (1995) 348-354 I
I
0.4
I
~
•. . . . . . . . . . .
I
I
JImX(QAF
W)
JRe
I(QAF,
OJ)
Jim
I (QAF,to)
i
0.2
-2001
-3
p
I
,
I
I
'
'
'
F
~11
I
r
' I
.A
loo I- • r. l,.sx • T.~.~
_E "0.2
o0
I
I I~ fi
/ rc=~sK ~f'LI / 0.4
0.6
I
----......... T=O.12(&o=O)
extended S - wave
\~-
n
T = O, d -wave T = O, extended s-wave
0.6
\
353
o '
J' ~! "1 I
,o'/
d- wove
0
0.4
0.6
w/J
'
04
0.2
Fig. 5. Imaginary part of the spin susceptibility for the two considered symmetries of the SC gap for T below and above Tc. In the insert the INS data from Ref. [5] are shown. 0.2
~ ~ ~ / / /
ol 0
0.2
0.4
0.6
C0/J
Fig. 3. Real and imaginary parts of the irreducible part and imaginary part of spin susceptibility at T = 0 for two symmetries of the order parameter (F = 0.03J). considered symmetries of the SC gap for T below and above To. In the inset we show the experimental INS data for YBa2Cu306.97 from Ref. [5]. One can .......... w : 0.16 co : 0.36 co : 0.38 0.6 ......... to = 0.39 A
. . . . to = O.Z,O - - - w : 0.41 ..... ~ = 0.42 /
X
04 x
0.2 0 0
0
.........."..................................
(-g'-7)
i.............................................
(r~,rt) Wavevector
(7 3 )
Fig. 4. Momentum dependence of Im x(k, to) for different energies.
see that the extended s wave symmetry of the SC OP whose main feature (in the case of the dynamic spin susceptibility) is the "normal-state" shape of Im X(QAF, tO) below and above Tc is absolutely inconsistent with the experimental data. On the contrary, in the case of the d wave symmetry the theoretical results are in a very good agreement with neutron-scattering data. This concerns the following points: (1) the existence of the gap and of the resonance peak below Tc, (2) the disappearance of both above To, (3) a negligibly small intensity above Tc, (4) a k dependence of Im X peaked at QAF for co close to the resonance peak. We have to notice that the shape of the k dependence of Im X predicted theoretically for energies far from the resonance peak can hardly be detectable due to low intensity. Summarizing, the d wave symmetry of the superconducting order parameter is strongly supported by INS data for YBCO. What is more remarkable, the extended s wave symmetry can be ruled out, being absolutely inconsistent with the results of neutron scattering. Since only these two candidates remain after precise recent measurements of Josephson junctions [1,2] which have discovered the change of the sign of the SC OP on the FS, one can conclude that the symmetry of the superconducting order parameter in the overdoped YBCO is dx2_y2. One can also extend this statement to the whole doping range
354
F. Onufrieva/ Physica C 251 (1995) 348-354
corresponding to the heavily doped regime in YBCO because of the existence of the low-energy gap and of the high-energy resonance peak in observed by INS for YBCO with x = 0.92 and x = 0.83 [16,17]. The situation for the weakly doped regime is not so clear, for the gap observed experimentally is small and the resonance peak is less pronounced [18,19].
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