Correlation length in cuprate superconductors deduced from impurity-induced magnetization

We report a new multinuclei based nuclear magnetic resonance method which allows us to image the staggered polarization induced by nonmagnetic $\mathrm{Li}$ impurities in underdoped $\left({\mathrm{O}}_{6.6}\right)$ and slightly overdoped $\left({\mathrm{O}}_7\right)$ ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+y}$ above $T_C$. The spatial extension of the polarization ${\xi }_{\mathrm{imp}}$ approximately follows a Curie law, increasing up to six lattice constants at $80\mathrm{K}$ at ${\mathrm{O}}_{6.6}$ in the pseudogap regime. Near optimal doping, the staggered magnetization has the same shape, with ${\xi }_{\mathrm{imp}}$ reduced by a factor 2. ${\xi }_{\mathrm{imp}}$ is argued to reveal the intrinsic magnetic correlation length of the pure system. It is found to display a smooth evolution through the pseudogap regime.

Figure 1

Magnitude of the local shifts $\mid \mathrm{K}(m,n)\mid$ at the ${}^{17}\mathrm{O}$ (b), ${}^{89}\mathrm{Y}$ (c), and ${}^{63}\mathrm{Cu}$ (d) nuclei for $H∕∕c$ from an impurity-induced polarization $S_n^m$ as represented in absolute values in (a) as a function of the distance to the impurity along the (1,0) direction. The checkerboards (insets) show the same quantities on the bidimensional ${\mathrm{CuO}}_2$ layer. Black color is for the maximum intensity and each square corresponds to one nucleus. In (b), the ${}^{17}\mathrm{O}$ neighbor sites to the impurity (hatched area)—not observed up to now—are not considered here as they only sense one $\mathrm{Cu}$ site, and should undergo a large shift.

Figure 2

Data for the shift with respect to the main ${}^{89}\mathrm{Y}$ line of the two satellites resonances detected at ${\mathrm{O}}_{6.6}$. A reference spectrum taken at $100\mathrm{K}$ is displayed. The computed satellite shifts with the parameters values given in Fig. 6 are shown as gray lines. The inset displays the equivalent figure for the ${\mathrm{O}}_7$ case, and shows that the satellites cannot be resolved.

Figure 3

Broadening of the NMR spectrum of the planar oxygens induced by $\mathrm{Li}$ impurities with $x_N=2\%$ for ${\mathrm{O}}_{6.6}$ (triangles) and ${\mathrm{O}}_7$ (circles). The broadening of the line in pure compound has been substracted. In the inset is displayed a typical Lorentzian spectrum for ${\mathrm{O}}_{6.6}$ with $x_N=2\%$ $\mathrm{Li}$ at $80\mathrm{K}$. The small “bump” in the low frequency tail corresponds to the resonance line of the apical site, almost completely attenuated by dynamical contrast (fast repetition time).

Figure 4

Comparison of the experimental ${}^{89}\mathrm{Y}$ spectrum at $80\mathrm{K}$ in ${\mathrm{O}}_{6.6}$ to simulated ones and their corresponding polarization (inset). Dotted line indicates the experimental second satellite shift. Polarizations $A$, $B$, and $C$ have the same amplitude at $r=1$ but different curvatures. They lead to the computed spectra $A$, $B$, and $C$. For this given amplitude, only the $B$ exponential decay account for the experimental features. For a different $r=1$ amplitude, the polarization $D$ leads to simulated spectrum $D$ which can also account for the experimental shape. On these simulated spectra, the first satellite position does not coincide with the experimental, as we use the value ${}^{89}\mathrm{A}_{\mathrm{hf}}$of the pure compound for the hyperfine coupling between ${}^{89}\mathrm{Y}$ and ${}^{63}\mathrm{Cu}$ nn of the $\mathrm{Li}$.

Figure 5

Comparison of the experimental ${}^{17}\mathrm{O}$ spectrum (thick gray line) at $80\mathrm{K}$ in ${\mathrm{O}}_{6.6}$ with $x_N=2\%$ of $\mathrm{Li}$ to the simulated ones with the corresponding polarizations shown in the inset, with the same symbols. The short range polarization is fixed to the $B$ case of Fig. 4. The shape of the polarization for $r>\sqrt{5}$ is varied to fit the experimental spectrum. A convolution of the computed spectra by a Lorentzian function corresponding to the broadening of the pure sample has been performed.

Figure 6

Comparison of the experimental ${}^{17}\mathrm{O}$ spectrum at $80\mathrm{K}$ in ${\mathrm{O}}_{6.6}$ with $x_N=2\%$ of $\mathrm{Li}$ to simulated ones (b) with the corresponding polarizations in (a), with the same symbols. For these different amplitudes $D$, $B$, and $E$, the best fit of the experimental spectrum is obtained with an exponential curvature. Among these three cases, only $B$ can account both for the shape and the broadening of the spectrum.

Figure 7

Polarization extension ${\xi }_{\mathrm{imp}}$ (as defined in the text) as function of temperature for underdoped ${\mathrm{O}}_{6.6}$ (closed circles) and slightly overdoped ${\mathrm{O}}_7$ (open circles). The inset shows the polarization envelope at $80\mathrm{K}$ for ${\mathrm{O}}_{6.6}$ and ${\mathrm{O}}_7$.

Figure 8

Comparison of our result for the polarization extension ${\xi }_{\mathrm{imp}}$ to the correlation length $\xi$ in pure compounds extracted from INS experiments (Ref. 2) and the MMP analysis (Ref. 3). We plot also at ${\mathrm{O}}_7$ the characteristic length ${\xi }_n$ of the additional AF fluctuations observed in the presence of $\mathrm{Zn}$ on the INS spectra (Ref. 15).