Low-energy antiferromagnetic spin fluctuations limit the coherent superconducting gap in cuprates

Motivated by recent attention to a potential antiferromagnetic quantum critical point at $x_c\sim 0.19$, we have used inelastic neutron scattering to investigate the low-energy spin excitations in crystals of ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ bracketing $x_c$. We observe a peak in the normal-state spin-fluctuation weight at $\sim 20\mathrm{meV}$ for both $x=0.21$ and 0.17, inconsistent with quantum critical behavior. The presence of the peak raises the question of whether low-energy spin fluctuations limit the onset of superconducting order. Empirically evaluating the spin gap ${\mathrm{\Delta }}_{\mathrm{spin}}$ in the superconducting state, we find that ${\mathrm{\Delta }}_{\mathrm{spin}}$ is equal to the coherent superconducting gap ${\mathrm{\Delta }}_c$ determined by electronic spectroscopies. To test whether this is a general result for other cuprate families, we have checked through the literature and find that ${\mathrm{\Delta }}_c\le {\mathrm{\Delta }}_{\mathrm{spin}}$ for cuprates with uniform $d$-wave superconductivity. We discuss the implications of this result.

Figure 1

(a) Schematic of electronic gaps around a quadrant of reciprocal space for a square ${\mathrm{CuO}}_2$ plane with lattice constant $a=1$. Green line: below $T_c$, a $d$-wave gap is found near the node, extrapolating to ${\mathrm{\Delta }}_0$, but switches to the pseudogap in the antinodal region, rising to ${\mathrm{\Delta }}_{\mathrm{pg}}$. Blue line: above $T_c$, the coherent gap (with maximum energy of ${\mathrm{\Delta }}_c$) closes to form a Fermi arc. (b) Schematic spin excitation spectrum as a function momentum transfer Q, defining the spin gap ${\mathrm{\Delta }}_{\mathrm{spin}}$ and $E_{\mathrm{cross}}$.

Figure 2

Volume susceptibility measured for samples of LSCO with $x=0.17$ and 0.21. For $x=0.17$, bottom (top) was 4.5 cm (15 cm) from the start of growth; for $x=0.21$, bottom (top) was 6 cm (14 cm) from the start of growth.

Figure 3

Scattered intensity as function of energy and $\mathbf{Q}=(H,0,0)$ for LSCO $x=0.17$ at 36 K. The intensity has been integrated over $-0.2\le K\le 0.2$ and over all measured $L$ within the range $-2\le L\le 5$. Antiferromagnetic scattering is allowed around $H=1,-1,-3$. The range of sample orientations measured was selected so that, for $H=-1$, the $L$ range sampled was within $-2\le L\le 2$; the $L$ ranges at other $H$ values vary considerably. The difference in phonon intensity at $H=-1$ and 1 is due to different sampled $L$ ranges.

Figure 4

Constant-energy slices of magnetic scattering intensity (after subtraction of the phonon contribution) as a function of the in-plane wave vector for the $x=0.17$ (left) and 0.21 (right) samples at base temperature. Excitation energies are (a) and (d) 8, (b) and (e) 18, (c) and (f) 30 meV; energy width is 1 meV and incident energy is 60 meV. White areas correspond to detector gaps.

Figure 5

Q-integrated ${\chi }^{″}\left(\omega \right)$ for (a) $x=0.17$ and (b) $x=0.21$ at base temperature and $T=36$ K; the energy bin width is twice as large for the $x=0.21$ results to compensate for the different sample orientation with respect to the spectrometer. Difference in ${\chi }^{″}\left(\omega \right)$ between base temperature and 36 K for (c) $x=0.17$ and (d) $x=0.21$. (The differences were taken using data without phonon correction, as the phonon contribution cancels almost completely.) In the latter, the experimental ${\mathrm{\Delta }}_{\mathrm{spin}}$, discussed in the text, is indicated in red.

Figure 6

Analysis of $\mathrm{\Delta }{\chi }^{″}={\chi }^{″}\left(5\text{K}\right)-{\chi }^{″}\left(36\text{K}\right)$ for LSCO $x=0.17$. Constant-energy slices of $\mathrm{\Delta }{\chi }^{″}$ for (a) $\hslash \omega =6$ meV and (b) 15 meV. (c) and (d) The corresponding cuts obtained after integrating over $K$ within the range denoted by the dashed white lines in (a) and (b); a constant background difference has been subtracted. The dashed lines in (c) and (d) are fitted Gaussian peaks used to evaluate the magnetic contribution to $\mathrm{\Delta }{\chi }^{″}\left(\omega \right)$.

Figure 7

Plot of ${\mathrm{\Delta }}_c$ and $E_{\mathrm{cross}}$ vs ${\mathrm{\Delta }}_{\mathrm{spin}}$, corresponding to the data and references in Table 1.

Figure 8

Comparison of the antinodal quasiparticle energy at $T<T_c$ in ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ [93] (solid circles), ${\mathrm{\Delta }}_c$ estimated from $T_c$ (open squares), and the reported values of ${\mathrm{\Delta }}_{\mathrm{spin}}$ from Table 1 (open diamonds). Note that we have used the values of $p$ reported in [93], which were determined from the size of the Fermi surface, resulting in an increase in $p$ by 0.02 from typical estimates based on $T_c\left(p\right)$ [96]. To put the spin gap results on the same scale, we have increased the corresponding values of $p$ listed in Table 1 by 0.02. The vertical dashed line indicates the adjusted $p$ value at which an STS study [20] found a crossover in the Bogoliubov quasiparticle distribution from a finite arc to the full Fermi surface.