Development of spin-wave-like dispersive excitations below the pseudogap temperature in the high-temperature superconductor ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$

Inelastic neutron scattering measurements of magnetic excitations in hole-doped high-$T_c$ cuprates ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ ($x=0$ and 0.075) have been performed over a wide temperature range of 5–500 K extending beyond the pseudogap temperature of $\sim 360$ K for $x=0.075$. For $x=0$, the two-dimensional spin-wave excitations at low temperatures are thermally robust and the dispersion is slightly softened on heating up to $T=400$ K. For $x=0.075$, the spin-wave-like excitations in the upper part of the hourglass-shaped dispersion remain at $T=300$ K. However, further heating above $T=400$ K induces a broad ridge centered at $(1/2,1/2)$; thereby the hourglass-shaped dispersion becomes blurred above the pseudogap temperature. The localized spin nature of magnetic excitations below $T^{*}$ suggests that the pseudogap is related to the proximity to a Mott insulator rather than being a precursor of Cooper pairs.

Figure 1

(a)–(c) Constant-energy slices of ${\chi }^{\prime \prime }(q,\omega )$ for $x=0$ at $T=5$ K. (d)–(i) Constant-energy cuts of the raw data along the in-plane wave vector $Q=(h,1/2)$ measured at $T=5$ and 400 K. The white dashed line in (a) shows the trajectory of the cuts, whereas the white dots in (a)–(c) are due to absence of data. Neutron scattering data were collected by utilizing multiple incident energies of 14.3, 22.6, 40.7, and 404.0 meV. Two-dimensional data at constant energies were fitted to a ring or a peak around $(1/2,1/2)$ with Gaussian peak width, as shown in the insets in (a)–(c).

Figure 2

(a)–(c) Constant-energy slices of ${\chi }^{\prime \prime }(q,\omega )$ for LSCO ($x=0.075$) at $T=5$ K. (d)–(o) Constant-energy cuts for the raw data along $(h,1/2)$ as in Fig. 1 measured at $T=5$, 300, 400, and 500 K. Neutron scattering data were collected by utilizing multiple incident energies of 17.2, 25.9, 43.4, and 256.8 meV. Two-dimensional data at constant energies were fitted to a ring, a peak, or four incommensurate peaks around $(1/2,1/2)$ with a Gaussian peak width, as shown in the insets in (a)–(c).

Figure 3

Contour maps of ${\chi }^{\prime \prime }(q,\omega )$ for (b), (c) $x=0$ and (d)–(g) $x=0.075$ plotted in the energy momentum space. Measured temperatures and compositions are plotted in (a) the schematic phase diagram of LSCO. The intensities were reproduced from the fitting parameters determined in Figs. 1 and 2. The inset in Fig. 3 shows the enlarged plot around the neck of the hourglass on a double intensity scale. The inset in Fig. 3 shows calculated ${\chi }^{\prime \prime }(q,\omega )$ based on the random-phase approximation (${\chi }_{\mathrm{RPA}}^{\prime \prime }$) for the normal state of underdoped LSCO [20]. The intensities of the ${\chi }_{\mathrm{RPA}}^{\prime \prime }$ are plotted on a logarithmic scale.

Figure 4

The $q$ dependence of the magnon intensity $I\left(q\right)$ for (a) $x=0$ and (b) $x=0.075$. $I\left(q\right)$ was obtained by integrating ${\chi }^{\prime \prime }(q,\omega )$ in the energy range of 40–200 meV. Insets in (a) and (b) show the differences in $I\left(q\right)$ $\left[\mathrm{\Delta }I\right(q\left)\right]$ between 5 K and high temperatures (300, 400, and 500 K). Dashed lines in (a) and (b) and their insets show cutoff $q$ values at which the upward dispersion at 5 K intersects $E=200$ meV. Constant-$Q$ cuts of ${\chi }^{\prime \prime }(Q,\omega )$ of $\mathrm{LSCO}(x=0.075)$ at (c) $(0.58,0.5)$ and (d) $(1/2,1/2)$. The solid lines in (c) are Lorentzian fits for the broad peak at $E=40–60$ meV. The solid lines in (d) are Lorentzian fits for the peak at $E=35–50$ meV in addition to the peak at 20 meV.