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If magnetic excitations are mediating electron pairing in the high-Tc copper oxides, it is expected that the change in magnetic exchange energy will provide enough energy for superconducting condensation. The condensation energy is known experimentally from specific heat measurements for YBa2Cu3O6.95 (YBCO) to be ∼3 K per formula unit15,16 (f.u.). Within the t–J model, the change in magnetic exchange energy can be calculated from the nearest-neighbour spin correlations3,4,5 between the normal (N) and superconducting (S) states:

where J is the exchange interaction, and Si and Sj are the electron spin operators at nearest-neighbour Cu sites i and j in the CuO2 plane, respectively. Instead of estimating the magnetic resonance’s contribution to the superconducting condensation4,13, here we seek to calculate ΔEex from the entire observable magnetic excitation spectrum. In general, a complete determination of the magnetic excitation spectrum is difficult as spin fluctuations can spread over a large wavevector and energy range. YBCO has two CuO2 planes per unit cell (bilayer) and therefore the magnetic excitations have odd (acoustic) or even (optical) symmetry with respect to the neighbouring planes (Fig. 1). For optimally doped YBCO, the magnetic excitation spectrum is dominated by a resonance mode centred at 41 meV in the acoustic channel6,7, and a mapping of the acoustic and optical magnetic excitations should allow an estimation of ΔEex.

Figure 1: Summary of Q-and ω-dependence of dynamic susceptibility for YBCO.
figure 1

Our experiments were carried out on the ∼117g YBCO single crystal (Tc=92.5 K) used in previous work10. We specify the momentum transfer (qh,qk,ql) (in units of Å−1) as (h,k,l)=(qha/2π,qkb/2π,qlc/2π) in reciprocal lattice units (r.l.u.), where a=3.82,b=3.88 and c=11.68 Å are lattice parameters10. a, Schematic diagram for YBCO. b, The l dependence of acoustic and optical spin fluctuations for Ei=210 meV. c, Dispersion of constant energy peaks in χo′′(Q,ω). The orientation of incommensurate peaks is shown in the inset. d, Dispersion constant energy peaks in χa′′(Q,ω). The solid line shows spin-wave dispersion from undoped YBa2Cu3O6.15 (refs 21, 22). The inset shows the orientation of the incommensurate spin fluctuations below the resonance energy. e,f, Local susceptibility of 100–15 K for optical χo′′(ω) and acoustic χa′′(ω) modes in absolute units. The red circles and black squares are obtained with cuts along the (h,1−h) and (h,h) directions with an integrating width of ±0.15 r.l.u., respectively. The inset in e shows hole-doping dependence of the optical ‘resonance’ with the red circle and the green triangle from refs 24, 25. The vertical error bars in c,d and the horizontal error bars in e,f indicate the energy integration range. The vertical error bars in e and f are statistical uncertainties (1σ).

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Figure 1 summarizes the key conclusions of our work. The optical and acoustic spin fluctuations can be separated by their differences in qz dependence (Fig. 1b). The total magnetic response χ′′(Q,ω) can then be written as χ′′(qx,qy,qz,ω)=χa′′(qx,qy,ω)sin2(qzd/2)+χo′′(qx,qy,ω)cos2(qzd/2), where d=3.342 Å is the spacing between the nearest-neighbour CuO2 planes along the c axis. To probe the entire magnetic spectra in optical and acoustic channels of YBCO, we used the multi-angle position-sensitive spectrometer at the ISIS Facility12,17 and chose incident beam energies of Ei=30,40,62.5,75,90,110,130,138,160,210,280,360 and 450 meV with the incident beam along the c axis. The position-sensitive detectors allow complete determination of the Q-structure of incommensurate spin fluctuations for YBCO in one experimental setting12. This avoids the complication of deconvoluting the instrumental resolution necessary for structure determination of incommensurate peaks using triple-axis spectroscopy18. The temperatures probed were T=15,100 and 290 K. The intensity difference between 15 and 100 K is almost entirely magnetic because of the small value of Q2 probed by the experiment and the small change in the Bose factor for ħω>30 meV (refs 9, 10, 19).

Figure 2a–c summarizes the temperature dependence of the 34≤ħω≤44 meV scattering at the position of optical spin fluctuations (Ei=90 meV). A clear incommensurate scattering appears at 15 K (Fig. 2a) and is replaced by a broad response at 100 K (Fig. 2b). Figure 2d–f shows the temperature dependence of the scattering around the 41 meV acoustic resonance obtained by using Ei=210 meV. At T=15 K, the scattering shows a sharp resonance centred at (1/2,1/2) (ref. 10). On warming to 100 K, the resonance disappears (Fig. 2e). Further warming to 290 K does not change the scattering significantly (Fig. 2f).

Figure 2: Summary of the temperature dependence of the optical and acoustic scattering at the resonant energy for YBCO.
figure 2

a–f, The temperature dependence of the magnetic scattering around ħω≈40 meV at optical (a)–(c), Ei=90 meV, and acoustic (d)–(f), Ei=210 meV channels. A clear commensurate magnetic scattering is seen at ∼40 meV at 15 K in d, whereas a shows incommensurate scattering. The intensity of phonons increases with increasing temperature and wavevector (b,c).

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Figure 3 summarizes the optical spin fluctuations for 31≤ħω≤75 meV. For 31≤ħω≤34 meV, the scattering shows no difference between normal and superconducting states (Fig. 3a,e). As there is little normal-state magnetic scattering, there must be an optical spin gap around 34 meV. On increasing the energy transfer to ħω=39±5 meV, where the acoustic channel has a commensurate resonance, spin fluctuations in the optical channel form a broad incommensurate structure away from (1/2,1/2) (Fig. 3b). Figure 4f confirms the incommensurate nature of the scattering and shows that the (h,h) and (h,0.5) cut directions are non-equivalent. For ħω=52±5 meV, again we find incommensurate peaks, but this time the scattering is more box-like with enhanced corners (Fig. 3c,g). The orientation of the scattering is rotated 45∘ from that in Fig. 4b, similar to acoustic high-energy spin excitations in YBa2Cu3O6.6 (ref. 12). Finally, on moving the energy to ħω=66±9 meV, the temperature difference spectrum is featureless. Inspection of the unsubtracted data at 15 K reveals no observable magnetic scattering (Fig. 3h).

Figure 3: Temperature difference (15–100 K) at various energies for optical mode defined as scattering with cos2(qzd/2)>0.8.
figure 3

a–d, Images obtained with Ei=75,90,130 and 210 meV, respectively. The image at ħω=45.5±1.5 meV does not have enough statistics to determine the Q-structure. e–h, Cuts with ħω=32.5±1.5,39±5,52±5 and 66±9 meV, respectively. The upper panel of h shows a cut through the unsubtracted data of h at 15 K. The vertical error bars in e–h are statistical uncertainties (1σ).

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Figure 4: Temperature difference spectra (15–100 K) at various energies for acoustic mode defined as any scattering with sin2(qzd/2)>0.8 (see Fig. 1b).
figure 4

a–e, Images obtained with Ei=90, 210, 210, 90 and 110 meV, respectively. f–j, Cuts obtained with ħω=41±5,31±4,40±5,55.5±3.5 and 64.5±4.5 meV, respectively. The upper panel of j shows a cut through the unsubtracted data of e at 15 K. The instrumental Q-resolutions are marked by white circles in a–e. The vertical error bars in f–j are statistical uncertainties (1σ).

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Figure 4a–e shows the intensity difference spectra between 15 K and 100 K at various energies in the acoustic channel. For 20≤ħω≤27 meV, the temperature difference has negative intensity, possibly due to a reduction in the magnetic response on entering the superconducting state19 or large phonon population at 100 K (Fig. 4a). A cut through the image confirms no intensity gain at low temperatures (Fig. 4f). When increasing the energy to 27≤ħω≤35 meV, the difference image shows a quartet of incommensurate peaks at (1/2±δ,1/2) and (1/2,1/2±δ) with δ=0.11±0.02 reciprocal lattice units (r.l.u.). To demonstrate that the incommensurate scattering is not circular around (1/2,1/2), we made two cuts through the image. The cut along the (h,1/2) direction clearly shows two incommensurate peaks around (1/2,1/2). In contrast, a cut along the (h,1−h) direction has no incommensurate peaks. Note that a circular symmetry at ħω=35 meV has been suggested for YBCO from previous triple-axis experiments18.

Figure 4c shows the data at the resonance energy and its wavevector dependence has a gaussian lineshape centred around (1/2,1/2) (Fig. 4h). For energies above the resonance (52≤ħω≤59 meV), the scattering is incommensurate although the low counting rate does not enable an unambiguous identification of its Q-structure (Fig. 4d,i). For ħω>60 meV, the temperature difference spectra as well as unsubtracted data show no evidence of magnetic scattering around (1/2,1/2) (Fig. 4e,j).

Figure 1e and f summarize the superconductivity-induced (15–100 K) change in the local dynamic susceptibility χ′′(ω) in absolute units for optical and acoustic channels of YBCO, respectively. In the optical channel, χo′′(ω) has a peak at ħω=46 meV, similar to theoretical expectations20. The magnetic spectral weight gradually increases from above the spin-gap value of ∼30 meV (ref. 10), peaks at 46 meV, and finally diminishes for energies above ∼70 meV. The acoustic channel behaves similarly although it peaks at the expected resonance position of 40 meV (ref. 10). The magnitude of the total spectral weight in the optical channel 〈m2〉op=0.078±0.02 μB2 and in the acoustic channel is 〈m2〉ac=0.102±0.02 μB2 f.u.−1 respectively. This value is similar to that for YBa2Cu3O6.6 around the acoustic resonance energy (〈m2〉ac=0.12±0.02μB2 f.u.−1 for 24<ħω<44 meV)12. As the high-energy response in underdoped YBa2Cu3O6+x (x=0.5,0.6) takes up much more spectral weight than the resonance11,12, it is surprising that there is essentially no observed magnetic response for energies above 60 meV in YBCO (Fig. 1f). Compared with undoped YBa2Cu3O6.15 (refs 21, 22), which has a total integrated moment of ∼0.4μB2 f.u.−1 when integrated up to 120 meV, the total integrated moment in optimally doped YBCO has only about 26% of the spectral weight in the same energy range only in the acoustic channel.

Using the spin excitation spectra in Figs 3 and 4, we have calculated the changes in the magnetic excitations from the normal to the superconducting state and estimated δ〈Si·Sj〉=−0.020±0.008 f.u.−1, where 〈Si·Sj〉 is the spin–spin correlation function for nearest-neighbour copper atoms (see Supplementary Information). This estimate neglects contributions from energies below 24 meV and above 70 meV, where magnetic scattering is difficult to resolve. Also, in equation (1) the difference between the normal and superconducting states is meant to be determined at the same temperature, whereas here we had to take normal-state data at 100 K and superconducting-state data at 15 K, neglecting a possible temperature dependence of the normal-state magnetic excitations. To assess the error introduced by these neglections, we have fitted an RPA–BCS (random phase approximation–Bardeen–Cooper–Schrieffer) model calculation of the spin excitation spectrum23 to our data and calculated the missing contributions within this model. This calculation indicates that our value for ΔEex could be of the order 30% too large due to these neglections (see Supplementary Information).

Assuming an exchange coupling of J=100 meV, the change in exchange energy would be ΔEex=2J δ〈Si·Sj〉=−4.1 meV f.u.−1=−24 K per planar Cu. This value is a factor of 1.3 times larger than the 18 K per Cu estimated from the acoustic resonance alone in previous work4. Even if we consider that our estimation may be too large by 30%, the change in the exchange energy is still much larger than the condensation energy15,16.

Our results reveal two important conclusions for spin excitations of optimally doped YBCO. First, the optical resonance reported earlier24,25 is actually incommensurate and this naturally explains the large Q widths previously reported. Second, our determination of the dynamical susceptibility in absolute units enables an estimation of the change of the magnetic excitation energy available to the superconducting condensation energy 4,14,26. We find that the magnetic exchange energy is about 15 times larger than that of the superconducting condensation energy, thus indicating that magnetism can be the driving force for electron pairing and superconductivity.