A notable aspect of high-temperature superconductivity in the copper oxides is the unconventional nature of the underlying paired-electron state. A direct manifestation of the unconventional state is a pairing energy—that is, the energy required to remove one electron from the superconductor—that varies (between zero and a maximum value) as a function of momentum, or wavevector1,2: the pairing energy for conventional superconductors is wavevector-independent3,4. The wavefunction describing the superconducting state will include the pairing not only of charges, but also of the spins of the paired charges. Each pair is usually in the form of a spin singlet5, so there will also be a pairing energy associated with transforming the spin singlet into the higher-energy spin triplet form without necessarily unbinding the charges. Here we use inelastic neutron scattering to determine thewavevector-dependence of spin pairing in La2−xSrxCuO4, the simplest high-temperature superconductor. We find that the spin pairing energy (or ‘spin gap’) is wavevector independent, even though superconductivity significantly alters the wavevector dependence of the spin fluctuations at higher energies.
The experimental technique that we use is inelastic neutron scattering, for which the cross-section is directly proportional to the magnetic excitation spectrum and can be used to probe it as a function of wavevector and energy transfer. In addition, we have selected La2−xSrxCuO4, the simplest of the high-temperature (high-Tc) superconductors. The materials consist of nearly square CuO2 lattices with Cu atoms at the vertices and O atoms on the edges alternating with LaSrO charge reservoir layers. In the absence of Sr doping, the compound is an antiferromagnetic insulator, where the spin on each Cu2+ ion is antiparallel to those on its four nearest neighbours. Because of the unit cell doubling, magnetic Bragg reflections appear at wavevectors such as (1/2, 1/2) (sometimes called (π,π) in the two-dimensional reciprocal space of the CuO2 planes6. Doping yields a superconductor without long-range magnetic order but which has low-energy magnetic excitations peaked at the quartet of wavevectors Qδ = (1/2(1 ± δ),)1/2) and (1/2,1/2(1 ± δ), shown in Fig. 1a. The recentdiscovery of nearly identical fluctuations in the high-Tc YBa2Cu3O7−y bilayer materials7 clearly indicates their relevance to the larger issue of high-Tc superconductivity, and validates the continued study of La2−xSrxCuO4 as the copper oxide with the least structural and electronic complexity.
The samples are single-crystal rods grown in an optical image furnace. The most reliable measure of the quality of bulk superconductors is the specific heat C. For our samples, there is a jump of ΔC/kBTc = 7 mJ mol−1 at Tc = 38.5 K. As T → 0, C = γST where γs is proportional to the electronic density of states at the Fermi level and has the value γS < 0.8 mJ mol−1 K−2. This, together with an estimate of 10 mJ mol−1 K−2 for the corresponding normal state γN, indicates that the bulk superconducting volume fraction 1− γS/γN of our samples is greater than 0.9. This, as well as the high value of Tc and the narrowness of the transition, is evidence for the very high quality of our large, single crystals. The basic experimental configurations are similar to those employed previously8,9. Figure 1a shows the reciprocal space regions probed. A series of scans like those indicated in the figure, performed for a range of energy transfers, were used to build up the Q–E maps in Fig. 1b and c, which show the scattering around the incommensurate peaks in the normal and superconducting states (here E is the energy transfer).
Figure 1b shows that the normal-state excitations at 38.5 K are localized near Qδ but are entirely delocalized in E. In other words, the magnetic fluctuations which are favoured are those with a particular spatial period 1/δ corresponding to Qδ, but no particular temporal period. Cooling below Tc produces a very different imagein Q–E space. In Fig. 1c all low-frequency excitations (E ⩽ 5 meV) seem to be eliminated, and there is an enhancement of the signal above 8 meV at the incommensurate wavevectors. The signal now has obvious peaks at around E = 11 meV and δ= 0.29 ± 0.03 reciprocal lattice units (r.l.u.). We can thus visualize the zero point fluctuations in the superconductor as magnetic density waves undergoing (damped) oscillation with a frequency of 2.75 THz. In the normal paramagnetic state, the motion of the density waves becomes entirely incoherent.
Figure 2a–c show a series of constant-E sections through the data in Fig. 1. These graphs show that superconductivity induces a complete loss of signal for E = 2 meV (Fig. 2a), a significant intensity-preserving sharpening of the incommensurate peaks for E = 8 meV (Fig. 2b), and a large enhancement of the peaks for E = 11 meV (Fig. 2c). The peak narrowing in Fig. 2b and c corresponds to a spectacular superconductivity-induced rise in the magnetic coherence lengths (defined as the resolution-corrected inverse half-widths at half-maxima obtained as in ref. 10) from 20.1 ± 0.9 to 33.5 ± 2.0 å, and from 25.5 ± 0.1 to 34.3 ± 0.8 å, respectively.
Figure 3a–c displays constant-Q spectra both away from Qδ (Fig. 3a and b) and at Qδ (Fig. 3c). Superconductivity removes the low-E signal below a threshold energy, while it enhances the higher-E signal close to Qδ. The threshold for T < Tc appears the same for the three wavevectors shown in Fig. 3a–c, with the increase in intensity first visible in all cases at 6 meV. To quantify how superconductivity changes the spectra, we fit the data with the convolution of the instrumental resolution (full-width at half-maximum, 2 meV) and
and A is the amplitude, Δ is the spin gap, Γ is the inverse lifetime of spin fluctuations with E ≫ Δ (if Δ≪ Γ), E′ is an odd function of E which defines the degree to which the spectrum has a gap, and Γs is the inverse lifetime of the fluctuations at the gap edge.
In the normal state, the best fits are obtained for Δ= 0 meV, and the fitted value of Γ is essentially Q-independent (Fig. 3d). Thus, the lower-amplitude fluctuations with wavevectors different from Qδ have lifetimes similar to those at the incommensurate peak positions. The Q dependence of the signal is entirely accounted for by the Q dependence of the real part χ′(Q) of the magnetic susceptibility (Fig. 3e) which, when Δ= 0, is simply the amplitude A. In the superconducting state, Γ (Fig. 3d), which characterizes the shape of the spectrum well above the spin gap, becomes strongly Q dependent. At the same time, χ′(Q) (Fig. 3e), related via a Kramers–Kronig relation to the parameters in equation (1), is suppressed. This explicitly demonstrates that superconductivity reduces the tendency towards static incommensurate magnetic order in La2−xSrxCuO4.
Figure 4 shows the Q dependence of the spin gap Δ. As anticipated from inspection of the data in Fig. 3, Δ is Q-independent and has the value 6.7 meV. The gap is quite sharp for our sample, with Γs ⩽ 0.2 meV for all Q. Also shown in fig. 4 are the results for x = 0.15 (refs 11, 12) and 0.14 (ref. 10). Δ(Qδ) is indistinguishable for the present x = 0.163 and the older x = 0.14 samples; the difference in the low-E behaviour is primarily due to the much larger damping (Γx = 1.2 meV for x = 0.14; ref. 13). In addition, the Q independence of Δ(Q) is consistent with the Q-independent but incomplete suppression of the magnetic fluctuations in the x = 0.14 sample14. In contrast, the results of refs 11 and 12 show a large discrepancy with x = 0.163, where the spin gap quoted in these papers is defined as the threshold for visible scattering. Nevertheless, the results of ref. 11 are consistent with our work if we use thedefinition—advocated here and in ref. 13— of Δ given by equation (2). Fitting the data of ref. 11 to equation (1)with Δ= 6.7 meV yields Γs = 0.5 meV, a value intermediate between our findings of 1.2 and 0.1 meV for x = 0.14 and x = 0.163.
Our experiments show that superconductivity produces strongly momentum-dependent changes in the magnetic excitations with energies above a momentum-independent spin gap. The data in their entirety do not resemble the predictions15,16,17,18,19,20 for any superconductors, be they s-wave or d-wave. Most notably, all d-wave theories anticipate dispersion in the spin gap which would have been observed over the wavevector range and for the energy resolution of the present experiment. At the same time, s-wave theory cannot account for the value of the spin gap. We are unaware of calculations which yield the dramatic incommensurate peak sharpening and enhancements that we see above the spin gap while at the same time showing a large reduction in the real part of the magnetic susceptibility.
There are other difficulties with the conventional weak-coupling d-wave approach, which posits nodes and therefore a smaller relative superconductivity-induced reduction in scattering between, rather than at, the incommensurate peaks. Figure 2b shows the opposite—just above the gap energy, the incommensurate peak intensities are preserved while the scattering between the peaks is suppressed. Furthermore, the peak sharpening in momentum space for ℏω > Δ finds a precedence only in quantum systems, such as S = 1 antiferromagnetic (Haldane) spin chains and rotons in superfluid helium, which have well defined gaps with non-zero minima. Thus, although our statistics and resolution cannot exclude a small population of spin-carrying subgap quasiparticles, the systematics of the signal found near the gap energy make such quasiparticles improbable.
As for any other spectroscopic experiment, we can only place an upper bound on the signal below the dispersionless gap. Inspection of Fig. 3 shows that in between the incommensurate peaks at Q = (1/2(1 + δ/2),1/2(1 − δ/2), where ordinary weak-coupling d-wave theories generally anticipate nodes in the spin gap, the intensity for 2 meV at 5 K is less than 14% of what was seen at Tc and below 5% of that observed for the incommensurate peaks at 5 K.
Given the overwhelming evidence for d-wave superconductivity in the hole-doped high-Tc superconductors1,2,21,22, we see our data not as evidence against d-wave superconductivity but as proof that the spin excitations in the superconducting state do not parallel the charge excitations in the fashion assumed for ordinary d- and s-wave superconductors. Our measurements, which are sensitive exclusively to the spin sector, taken together with the evidence for d-wave superconductivity in the charge sector suggest that the high-Tc superconductors are in fact Luther–Emery (LE) liquids: that is, they are materials with gapped (triplet) spin excitations and gapless spin-zero charge excitations23,24. Luther–Emery liquids arise in one-dimensional interacting Fermi systems, which formally resemble two-dimensional d-wave superconductors—the dimensionality (zero) of the nodal points where the gap vanishes in the two-dimensional copper oxide is the same as that of the Fermi surface of a one-dimensional metal. There are other arguments forthe applicability of the concept of Luther–Emery liquids. The first is that theory indicates that such liquids are the ground states of ladder compounds, one-dimensional strips of finite widthcut from CuO2 planes25,26,27,28. The second involves the breakdown of spin–charge separation when the spin gap collapses to zero, which can be brought about by a magnetic field whose Zeeman energy matches the spin-gap energy. The 6.7-meV spin gap whichwemeasure is much closer to the Zeeman energy of theupper critical field measured29 for samples similar to ours than toanordinary Bardeen–Cooper–Schrieffer pairing energy ⩾3.5kBTc = 11.6 meV.
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We thank K. N. Clausen for help and support during the experiments, and B.Batlogg, G. Boebinger, V. Emery, S. Kivelson, H. Mook, D. Morr, D. Pines, Z.-X. Shen, C.-C. Tsuei and J. Zaanen for discussions. Work done at the University of Toronto was sponsored by the Natural Sciences and Engineering Research Council and the Canadian Institute for Advanced Research, while work done at Oak Ridge was supported by the US DOE. T.E.M. was supported by the Alfred P. Sloan Foundation, and A.S. was supported by the TMR program.
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Lake, B., Aeppli, G., Mason, T. et al. Spin gap and magnetic coherence in a clean high-temperature superconductor. Nature 400, 43–46 (1999) doi:10.1038/21840
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