Direct Relation between the Low-Energy Spin Excitations and Superconductivity of Overdoped High-$T_c$ Superconductors

The dynamic spin susceptibility, ${\chi }^{\prime \prime }(\omega )$, has been measured over the energy range of $2\le \omega \le 10\mathrm{meV}$ for overdoped ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$. Incommensurate (IC) spin excitations are observed at 8 K for all superconducting samples for $0.25\le x\le 0.28$ with ${\chi }^{\prime \prime }$ peaking at $\sim 6\mathrm{meV}$. The IC peaks at 6 meV become smaller in intensity with increasing $x$ and, finally, become unobservable for a sample with $x=0.30$ which has no bulk superconductivity. The maximum ${\chi }^{\prime \prime }$ decreases linearly with $T_c$(onset) in the overdoped region, implying a direct cooperative relation between the spin fluctuations and the superconductivity.

Figure 1

Maximum ${\chi }^{\prime \prime }(\omega )$ at 8 K as a function of $T_c$ for overdoped LSCO. The solid line is the result of a least square fit to a linear function. The inset shows the magnetic shielding measured in 10 Oe after cooling in zero field.

Figure 2

IC peak profiles where the maximum (or near maximum) ${\chi }^{\prime \prime }$ at 8 K has been observed for each sample. ($\omega =6.2\mathrm{meV}$ for $x=0.25$ and 0.28; $\omega =5.2\mathrm{meV}$ for $x=0.27$.)

Figure 3

(a) Energy dependence of ${\chi }^{\prime \prime }(\omega )$ for $x=0.25$ at 8 K (solid circles) and at 30 K (open circles), and for $x=0.27$ at 8 K (squares). The solid and dashed lines are guides to the eye. (b) Temperature dependence of ${\chi }^{\prime \prime }(\omega =6.2\mathrm{meV})$ for $x=0.25$. The solid line is a guide to the eye.

Figure 4

(a) Scan profile for $x=0.25$ along trajectory A shown in the inset. (b) Scan profiles for $x=0.30$ along two trajectories, one across the peak positions (A) and the other through a background position (B). The horizontal axis, $q$, indicates the distance from the $H=K$ line.