NMR studies of pseudogap and electronic inhomogeneity in Bi${}_2$Sr${}_2$CaCu${}_2$O${}_{8+\delta }$

We report ${}^{17}$O NMR measurements in single crystals of overdoped Bi${}_2$Sr${}_2$CaCu${}_2$O${}_{8+\delta }$ with $T_c=82$ K. We measure the full anisotropy of the planar oxygen Knight shift, electric field gradient, and spin-lattice-relaxation rate tensors, and show that the entire temperature dependence is determined by the suppression of the density of states in the pseudogap below $T^{*}\sim 94$ K. The linewidth can be explained by a combination of magnetic and quadrupolar broadening as a result of an inhomogeneous distribution of local hole concentrations that is consistent with scanning tunneling microscopy measurements. This distribution is temperature independent, does not break $C_4$ symmetry, and exhibits no change below $T^{*}$ or $T_c$.

Figure 1

Single-crystal ${}^{17}$O NMR spectra of Bi${}_2$Sr${}_2$CaCu${}_2$O${}_{8+\delta }$ in 9 T for fields ${\mathbf{H}}_0$ applied along the $c$ axis and in the $ab$ plane at 110 K. Solid lines are fits as described in the text.

Figure 2

The magnetic shift $K_{\alpha }$ (upper panel) and the electric field gradient ${\nu }_{\alpha }$ (lower panel) as a function of temperature for $\alpha =a,b,c$. The colors correspond to the same field orientations as in Fig. 1. The dotted line in the upper panel indicated $T_c$.

Figure 3

$K_{\alpha }$ versus ${\chi }_{\alpha }$ for $\alpha =a,b$. The colors match those in Fig. 1. As described in the text, the best fits to the data give the hyperfine coupling constants $2C_{aa}=102\left(40\right)$ kOe/${\mu }_B$ and $2C_{bb}=160\left(40\right)$ kOe/${\mu }_B$, and $K_{0a}=-0.5\left(4\right)\%$ and $K_{0b}=-0.8\left(4\right)\%$.

Figure 4

(a) The average of the rms second moments of the first satellite transitions ($\frac{3}{2}\leftrightarrow \frac{1}{2}$, $-\frac{3}{2}\leftrightarrow -\frac{1}{2}$) for the field parallel and perpendicular to the Cu–O bond axis. (b) The rms second moment $\delta n_p$ of the local hole concentration distribution of the two oxygen sites (see text for details), and (c) the difference between $\delta n_{p,a}$ and $\delta n_{p,b}$ versus temperature. The dashed line indicates $T^{*}=94$ K and the dotted line indicates $T_c=82$ K.

Figure 5

(a) The spin-lattice-relaxation rate $T_1^{-1}$ for the field along each direction $a$ ($•$, red), $b$ ($\blacksquare$, blue), and $c$ ($▴$, green). (b) The relaxation rates $R_{\alpha }$ determined from the $T_{1\alpha }$ data as described in the text. (c) $R_{\alpha }/T$ and ${\left(K^s\right)}^2$ (solid lines) versus $T$. Here, $K^s=K-K_0$, where $K_0$ is determined in Fig. 7.

Figure 6

The spin component of the Knight shift $K_{\alpha }^s$ versus $T/T^m$, where $T^m=125$ K. The solid line is the Johnston-Nakano scaling form (Ref.39).

Figure 7

$\sqrt{R_{\alpha }/T}$ versus $K_{\alpha }$ for $\alpha =a,b,c$. The colors match those in Fig. 1. The solid line is the best fit to the data as described in the text with slope 103(11) (s K)${}^{1/2}$. The dotted line has a slope given by the Korringa value (see text).