NMR studies of high temperature superconductors

1. Phys. Chem. Solids Vol. 53. No. 12. pp. 1651-1656. Printed in Glut Britain.

1992

W22-3697/92 $5.00 + 0.00 0 1992PergamonPfasLtd

NMR STUDIES OF HIGH TEMPERATURE SUPERCONDUCTORS M. TAKIGAWA

IBM T. J. Watson Research Center P.O. Box 21X Yorktown Heights. NY 10598, USA

ARSTRACT: Results of NMR experiments mostly in YBa,CuLLi. are reviewed. The norlnal state data revealed two important aspcc~~ ol’ the magnetic propcrtics. namely. the tcmpcraturc dcpcndcnt antikrmmagnctiu Cu spin correlations and the spin gap behavior. the latter being observed in the reduced oxygen material. These features appear to be the geneml properties ol’ many high T. cupratcs. The Cu nuclear relaxation behavior in the superconducting state show bomc anomalous kaiurc which remains to bc cxplaincd. kqwords:

high Tc superconductors, nuclear magncic rcsonancc, nuclear relaxation. magnetic propcritcs

1INTRODUCTION Nuclear Magnetic Resonance (NMR) experiments have provided valuable microscopic information on the electronic structure and the magnetic properties of high temperature superconductors. Its unique capability to probe the magnetic response at various atomic sites selectively (Cu, 0 and Y) allowed us to examine the results at a very microscopic level. In this article, we briefly summarize the crucial NMR results and try to clarify what has been largely established and what still remains an open question. Generally, a nuclear spin in solid interacts with spin of surrounding electrons (magnetic hyperfine interaction), giving rise to the shift of NMR frequency and nuclear spin-lattice relaxation. In high T, cuprates, particularly in YBazCuJO~-s, detailed shift data at Cu. 0 and Y sites have lead to quite accurate determination of the hypertine interaction, which forms the basis for the analysis of nuclear relaxation rate. In section 2, we describe the hypertine interaction of various atomic sites and the Knight shift results, emphasizing the systematic change of the T-dependence of spin susceptibility in the normal state with the carrier concentration. The nuclear relaxation rate data in the normal state and their implication on the spin correlations are discussed in section 3. It is shown that the peculiar “spin gap” behavior is observed in materials with relatively small carrier concentration. Section 4 is devoted to the results in the superconducting states. 2 HYPERFINE INTERACTION AND KNIGHT SHIFT It is important to have detailed knowledge of the hyperfine interaction for unambiguous interpretation of the NMR results. It is generally believed that Cu-3d, _, and 0-2~~ states are the active orbitals relevant to the low energy properties of the CuOZ planes. However. the Knight shift data at the planar hV~ (Takigawa et al. 1989, Barrett et al.. 1990) and I70 (Takigawa et al. 1989) sites in YBazCuj07 show rather large positive isotropic shift, indicating significant contribution from the Fermi contact hypet-fine interaction with s-electrons (most likely 4s at Cu and 2s at 0). Since the spin density on these s-states results from weak hybridization with the active Cu-3d or 0-2~ states on the neighboring sites, the Knight shift results imply the hyperfine coupling between Cu or 0 nuclei and the electronic spin degrees of freedom on adjuscent sites. This type of hyperfine coupling at the ligand nuclei in magnetic insulators such as transition metal halides has been known for long time and is called the transferred hypertine interaction. Soon after the discovery of the high T, superconductors, it was indicated from high energy spectroscopic experiments that holes doped into the antiferromagnetic insulator go mainly into the 0-2~ states. Then it has been a controversial issue whether the Cu-d holes and doped oxygen holes have dinstinct spin degrees of freedom (two spin tluid) or they are strongly hybridized to form a single spin liquid in the superconducting materials (single spin fluid). All the NMR data obtained so far are consistenl with the single spin tluid model. (Alloul ct

d. 1989. Takignwa

Ct a/. 1990, Mangelschots et d. 1992) In particular. the Knight shift results in the T,=6OK YB;L~CU~O~.~,,material support this picture. Unlike YBa:Cu?O-, (T,=90 K). the magnetic (spin) susceptinility in the 60 K material shows strong temperature dependence in the normal state. Takigawa et al. (1990) have shown that all components of Knight shift at both Cu and 0 sites have the same 1651

M. TAKIGAWA

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T-dependence, indicating that the spins on the Cu-d states and the O-p states have the same TSimilar results were obtained in YBazCu40x by Mangelschots e’t ul. dependence of susceptibility. (1992). A simple ionic model of the hyperfine interaction in the 0.10~ planes was proposed by Mila and Rice (1989). based on the single spin fluid picture and successfully explained the anisotropy of the Knights shift and nuclear relaxation rate at Cu sites in YBa$Ju~O~. This model includes the coupling of (‘%Zu nuclei to the electronic spins both on the same sites and on the nearest neighbor Cu sites. It is straight foward to extend the model to include the “0 and x”Y sites.

where A and B are the on-site and the transferred hyperiine coupling at Cu sites and C (D) is the tmsferred hyperfine coupling at 0 (Y) sites. One can see that the spin part of the Knight shift at all sites is simply proportional to the spin susceptibility x
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3NUCLEARSPINRELAXATIONINTHENORMALSTATE While the Knight shift measurements provide microscopic information about the static spin susceptibility, the nuclear spin-lattice relaxation rate l/T, is related to the dynamical spin fluctuations. The Mila-Rice hyperfine Hamiltonian eq. (1) leads to the following expression

where o, is the NMR frequency and x”(q, o) is the imaginary part of the dynamic susceptibility which we assume to be isotropic. A,is the on-site (ri = 0) or transferred (r, f 0) hyperfine coupling tensor for the nuclei under consideration and a is the direction perpendicular to the external magnetic field. The hypertine form factor for nuclear relaxation Flj(q) = c 1AyUaI2 at Cu and 0 sites are shown in a4, ’ as a function of q. Fig. ,_ One of the most significant NMR results in YBazCu~07 is that W at the planar Cu and 0 sites (hereafter denoted as h3W and “W, respectively) exhibit very different T-dependence and magnitude in the normal state, while they show a similar T-dependence below about I IOK (Hammel et al. 1989). While I%’ is T-inde~~ent and enhanced only by factor 2 over the Korringa value, a value calculated and si~niticantly for non-interacting electrons using the measued Knight shift, “‘W is Tde~ndent (about I6 times) enhanced over the Korrinpa value at 100 K. It has been argued by many authors (Shastry 1989. Bulut er rrl. 1990, Millis et al. 1990) and is now fairly well established that such difference is due to the antiferromagnetic short range correlations among Cu spins which grows with decreasing temperature. Since this has been discussed in another article (Takigawa 1992). we do not repeat it here. But the essential point is that within the single spin tluid picture. the different behavior at various sites is solely due to the difference in the hypertine form factor 1A? 1 and the form factor has a maximum at the antiferromagnetic wave vector Q=(tr, A) for Cu sites but goes to zero at Q for 0 sites. The relaxation behavior is quite different in the reduced oxygen material YBa,CutOhnj. The T-dependence of 17W in these two materials are shown in Fig. 3. In YB~zCU,~O~,.~,~, “W continues to decrease from room temperture and this T-dependece is nearly the same as that of xsvi,,. The loss of low frequency amplitude of x”(q, o) as indicated by the decrease of “W should be transferred to higher energies as discussed by Millis and Monien 11992) in order to satisfy the total moment sum rule. This then implies that a (pseudo) gap develops in the spin excitation spectrum at typical q at low temperatures. However, the ratio ‘“‘WP7W shown in Fig. 4, which gives a measure of the enhancement of x”(q, co) at q - Q relative to q - 0, continues to increase down to near T,.. This suggests that the antiferromagnetic correlations continues to grow in spite of the spin gap formation. 0.4 Fig.

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M. TAKIGAWA

1654

While the nuclear spin-lattice relaxation is caused by low frequency spin excitutions (x”), the static susceptibility (x’(q, co = 0)) gives rise to the indirect nuclear spin-spin coupling which causes the transverse nuclear spin relaxation (Pennington er ul. 1989. 1991). Such indirect (RKKY) coupling is written as ha(r;j)I!‘Ip, where

cj is the vector connecting the two nuclear spins Ii and ii and xq is the static q-dependent Such coupling causes the transverse nuclear spin relaxation which can be approximated exp 1 - (REt)Z/2], where

susceptibility. by a Gaussian

and a is the direction of the magnstic tield. One can see that R, is determined by xl, near y=Q, The measured reluxution rate contains the ~dditionul contribution from the direct dipolar coupling. We have measured the fiX’u trunsverse relaxation rate in YBazCu,&~,ht with the field of 35 kOe along the c-axis as shown in Fig. 5. Since the broad resonance line from chain Cu sites overlapped with the sharp planat Cu resonance, we had to subtract the long decay component coming from the chain Cu sites from the measured spin echo decay in order to fit the data to a single Gaussian. The transveres reluxution rate R, increase smoothly with decreasing tem~ruture down to about IO0 K, implying the monotonic increase of xQ in contrast to the shurp decrease of hjW, i.e. the imaginary part x”. This result reminds us of the situation in superconductors where there is a real singlet spin gap 2A for all q. Since the superconducting gap is usually much smaller than the spin tluctuation energy vq at typical q, the static susceptibility x,, is not affected by the onset of superconducting state except for small q < A/y: - l/g, as one can easily understand from the Kmmers-Kronig relation,

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The result of R, indicates that the spin gap is smaller than the energy of the antiferromagnctic fluctuations, the latter being estimated to be smaller than 10 meV by Monien, Pines and Tnkigawa (1991) from the analysis of $>W and I70 based on the model of Millis. Monien and Pines ( 1990). On the other hand X
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NMR studies of high temperature superconductors

1655

It should be mentioned that a similar monotonic increase of R, has been observed in YBa2Cu,0K by ltoh ef ul (1992), where the nuclear spin-lattice relaxation in this material clearly shows the spin gap behavior similar to 60 K YBCO material (Zimmermann er al 1989, Machi et al 1990). The spin gap has been observed directly in recent neutron scattering experiments. Nuetron scattering probes the spin fluctuations near Q. Therefore its low energy data is closely related to h3W Detailed experiments on YBa2CujOh.h (T,=S3 K) were reported by Tranquada cr al (preprint) and Sterklieb er al (preprint). At co=5 and 8 meV, x”(Q o) measured by neutrons shows a broad peak near 60 - 70 K. Although this T-dependence is similar to h3W the peak temperature is much lower than that of 63W (150 K). At 10 K. x”(q, o) as a function of w shows a gap feature aroud 9 meV, which is much smaller than the typical frequency of antiferromagnetic spin fluctuations at high temperatures (- 20 meV). As mentioned before, the T-dependence of q=O spin susceptibility does not seem to be compatible with such a small gap. Thus although both NMR and neutron scattering data revealed the spin gap phenomena. they do not seem to be quantitatively consistent. 4 NUCLEAR RELAXATION IN THE SUPERCONDUCTING STATE NMR measurements in the superconducting state provide usuful information about the nature of pairing state. Extensive measurements have been done at the Cu sites in YB~?CUJO, and revealed the following anomalous features. I). Absence of the Hebel-Slichter peak just below T,, which is common for almost all high T, materials, and non-exponential T-dependence of l/r,. 2). Unusual T-dependence of the anisotropy of h3W. Since the first point was extensively discussed in another paper (Takigawa. 1992) here we discuss the second result (Takigawa, Smith and Hults 1991, Martindale et al. 1992). Fig. 6 shows the T-dependence of the ratio of Cu relaxation rates with the magnetic field paralell th3W,) and perpendicular (e.3W.lb)to the c-axis (Takigawa, Smith and Hults 199 1). h3W.lhwas measured at a field of 0.44 T. h3WEwas measured at both zero field (NQR) and 0.44 T. In order to eliminate the effect of fast relaxation from vortex core which is anisotropic, it is necessary to perform measurements at low magnetic field. The anisotropy is nearly T-independent in the normal state but decreases rapidly below T<. Then it shows a minimum at 70 - 80 K and increases again at lower temperature. Since one can see in Fig. 2 that the form factor for 63W.,bhas more weight near Q than that for h3Wc, this T-dependence can be explained if x”(q - Q, w,,)/x”(q - 0. w,,) has the similar T-dependence. Bulut and Scalapino (1992) have shown that an anisotropic d-wave gap A,, = A(T)( cos q, - cos qy) reproduces such T-dependence by using the random-phase-approximation to calculate x(q, co). Their model has the following two aspects. First is the q-dependence of the coherence factor I + (Ek+qEk+ Ak+yAk)/Ek+qEk which appears in the expression for x”(q. 63) (Ek @k) is the quasiparticle (band) energy and Ek +q - Ek = 0). This factor is close to 2 for q - 0 but nearly vanishes at q=Q for small 0 because Ak+V = - Ak. This q-dependence of the coherence factor makes the anisotropy decrease below T,. The second feature is that x”(q - 0, o,,) decreases much more rapidly at low temperatures than x”(q - Q. o,,) because the wave vectors which connect nodes of the gap on the Fermi surface a~ located near Q. This fact is responsible for the upturn of the anisotropy at low temperatures. A similar conclusion has been reached by Lu (preprint).

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1656

M. TAKEAWA

The isotropic s-wave model also gives rise to decrease of the anisotropy below T, due to the loss of Stoner enhancement near Q rather than the coherence effect. However, it does not explain the upturn at lower temperatures. Although the anomalous T-dependence of the anisotopy is explained by a dwave model, other results such as the T-dependence of the penetration depth seem to support the swave state. The symmetry of the gap thus remains to be an open question. 5 CONCLUDING REMARKS NMR and neutron scattering experiments have established the antifeerromagnetic spin correlations in high T, cuprates. Both the Kights shift and the nuclear relaxation dnta in many of the high T, cuprates indicate the formation of pseudo spin gap at temperatures much higher than T,. However, the quantitative character of the spin gap as well as its relevance to superconductivity remain to be clarified. Some of the NMR data in the su~rconductin~ state of YBqCut07 seem to favor the d-wave pairing state. which is the natural consequence of the pairing interaction mediated by antife~oma~netic spin tluctuations. The symmetry of the gap seems to be a still open question. REFERENCES: M. Takigawtl, PC Harnmcl. R.H. Hcll’ncr xnd 2. Fisk, Phys. Rev. B39t 198(3)7371. S.E. Barrel1 catrrl.. Phys. Rev. I%411IYYO)62X3. M. Takigawa PI trl.. Phys. Rev. Lett. fdl989)lXbS. H. Alloul. T. Ohno and P. Mcndcls. Phys. Rev. Lctt. ~19~Y)l7(Ml. M. Takigawa rr 4.. Phys. Rev. R& 1991 ,247. I. Magclschms et d., Phybica Cl94 lYY2)277. F. Mila and T.M. Rice. Physica Cl57( 19891561. R.E. Walx~cd~.R.F. Bell and D.B. Mitzi. Phys. Rev. B44t l9Y 117760. K. Ishida. Y. Kitaoka G. Zheng and K. Asayama. J. Phys. Sot. Japan 6& 1991 )X16. P.C. Hammcl ef d.., preprint. P.C. Hamt~i or rd.. Phys. Rev. Lctt. 6~I~~~Y)I~2~ B.S. Shasuy, Phys. Rev. Lett. 63(19X9)1288. N. B&II. D. Hone, D.J. Scalapino and NE Bickers. Phys. Rev. Lea. 64 iYYtX723. A.J. Millis. H. Monicn and D. Pines. Phys. Rev. B42( 19YO)l67. M. Takigawa. Lkewrric~ Proyc~rrics mt/ Meclrtr~~i.snrs irtHi&Tc Sy~f,r~~~~f~l~(.f~~r.~ (4s. K. Kadowaki. T. Oguchi antI T. .%;aki. North-Holland. Anuiterdam. IYY2). A.J. Millis and H. Monicn. Phys. Rev. B&(lYY2)305Y. C.H. Pennin~ton cl d.. Phys. Rev. R3Yt I989t274. C.H. Pcru~iIl~tonand C.P. Slichtcr, Phys. Rev. Lctt. 66t lY91)3XIZ. H. htonicu. D. Pints and M. Takicigdwa.Phys. Rev. B43t I991 OSX. Y. itoh et (I/., J. Phys. Str. Japan 61f IY92)1287. H. Zimmcrmann L’Ird.. Physica C159t 19x9 )6X1. T. Machi UI rd.. Physica C173( I9YO132. J.M. Tranquada. P.M. Gchrinp. C. Shirans. S. Shamao and M. Sate. preprint. B.J. SternI&. M. Sate. S. Sh&oto. G. Shirune and J.M. Tr~quatla. pre&ni. M. Takieawa. J.L. Smith and W.L. Hulth. Phvs. Rcs. U&it iY9117764. J.A. M~~i,,~~~ rr d., Phys. Rev. Lett. tilk 1492j702. N. Buiut and D.J. SealqGno. Phys. Rev. LCII. fi&lk1YY2)706. J.P. Lu. preprint.