Abstract
A notable aspect of hightemperature superconductivity in the copper oxides is the unconventional nature of the underlying pairedelectron 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 wavevector^{1},^{2}: the pairing energy for conventional superconductors is wavevectorindependent^{3},^{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 singlet^{5}, so there will also be a pairing energy associated with transforming the spin singlet into the higherenergy spin triplet form without necessarily unbinding the charges. Here we use inelastic neutron scattering to determine thewavevectordependence of spin pairing in La_{2−x}Sr_{x}CuO_{4}, the simplest hightemperature 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.
Main
The experimental technique that we use is inelastic neutron scattering, for which the crosssection 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 La_{2−x}Sr_{x}CuO_{4}, the simplest of the hightemperature (highT_{c}) superconductors. The materials consist of nearly square CuO_{2} 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 Cu^{2+} 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 twodimensional reciprocal space of the CuO_{2} planes^{6}. Doping yields a superconductor without longrange magnetic order but which has lowenergy magnetic excitations peaked at the quartet of wavevectors Q_{δ} = (^{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 highT_{c} YBa_{2}Cu_{3}O_{7−y} bilayer materials^{7} clearly indicates their relevance to the larger issue of highT_{c} superconductivity, and validates the continued study of La_{2−x}Sr_{x}CuO_{4} as the copper oxide with the least structural and electronic complexity.
The samples are singlecrystal 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/k_{B}T_{c} = 7 mJ mol^{−1} at T_{c} = 38.5 K. As T → 0, C = γ_{S}T where γ_{s} is proportional to the electronic density of states at the Fermi level and has the value γ_{S} < 0.8 mJ mol^{−1} K^{−2}. This, together with an estimate of 10 mJ mol^{−1} K^{−2} for the corresponding normal state γ_{N}, indicates that the bulk superconducting volume fraction 1− γ_{S}/γ_{N} of our samples is greater than 0.9. This, as well as the high value of T_{c} 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 previously^{8},^{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 normalstate excitations at 38.5 K are localized near Q_{δ} 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 Q_{δ}, but no particular temporal period. Cooling below T_{c} produces a very different imagein Q–E space. In Fig. 1c all lowfrequency 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 constantE 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 intensitypreserving 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 superconductivityinduced rise in the magnetic coherence lengths (defined as the resolutioncorrected inverse halfwidths at halfmaxima 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 constantQ spectra both away from Q_{δ} (Fig. 3a and b) and at Q_{δ} (Fig. 3c). Superconductivity removes the lowE signal below a threshold energy, while it enhances the higherE signal close to Q_{δ}. The threshold for T < T_{c} 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 (fullwidth at halfmaximum, 2 meV) and
where
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 Γ_{s} 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 Qindependent (Fig. 3d). Thus, the loweramplitude fluctuations with wavevectors different from Q_{δ} 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 La_{2−x}Sr_{x}CuO_{4}.
Figure 4 shows the Q dependence of the spin gap Δ. As anticipated from inspection of the data in Fig. 3, Δ is Qindependent and has the value 6.7 meV. The gap is quite sharp for our sample, with Γ_{s} ⩽ 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(Q_{δ}) is indistinguishable for the present x = 0.163 and the older x = 0.14 samples; the difference in the lowE behaviour is primarily due to the much larger damping (Γ_{x} = 1.2 meV for x = 0.14; ref. 13). In addition, the Q independence of Δ(Q) is consistent with the Qindependent but incomplete suppression of the magnetic fluctuations in the x = 0.14 sample^{14}. 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 Γ_{s} = 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 momentumdependent changes in the magnetic excitations with energies above a momentumindependent spin gap. The data in their entirety do not resemble the predictions^{15},^{16},^{17},^{18},^{19},^{20} for any superconductors, be they swave or dwave. Most notably, all dwave 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, swave 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 weakcoupling dwave approach, which posits nodes and therefore a smaller relative superconductivityinduced 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 nonzero minima. Thus, although our statistics and resolution cannot exclude a small population of spincarrying 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 weakcoupling dwave 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 T_{c} and below 5% of that observed for the incommensurate peaks at 5 K.
Given the overwhelming evidence for dwave superconductivity in the holedoped highT_{c} superconductors^{1},^{2},^{21},^{22}, we see our data not as evidence against dwave 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 swave superconductors. Our measurements, which are sensitive exclusively to the spin sector, taken together with the evidence for dwave superconductivity in the charge sector suggest that the highT_{c} superconductors are in fact Luther–Emery (LE) liquids: that is, they are materials with gapped (triplet) spin excitations and gapless spinzero charge excitations^{23},^{24}. Luther–Emery liquids arise in onedimensional interacting Fermi systems, which formally resemble twodimensional dwave superconductors—the dimensionality (zero) of the nodal points where the gap vanishes in the twodimensional copper oxide is the same as that of the Fermi surface of a onedimensional 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, onedimensional strips of finite widthcut from CuO_{2} planes^{25},^{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 spingap energy. The 6.7meV spin gap whichwemeasure is much closer to the Zeeman energy of theupper critical field measured^{29} for samples similar to ours than toanordinary Bardeen–Cooper–Schrieffer pairing energy ⩾3.5k_{B}T_{c} = 11.6 meV.
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Acknowledgements
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 hightemperature superconductor. Nature 400, 43–46 (1999) doi:10.1038/21840
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