Planar CuO${}_2$ hole density in high-$T_c$ cuprates determined by NMR Knight shift: ${}^{63}\mathrm{Cu}$ NMR on bilayered Ba${}_2$CaCu${}_2$O${}_4$(F,O)${}_2$ and three-layered Ba${}_2$Ca${}_2$Cu${}_3$O${}_6$(F,O)${}_2$

We report that planar CuO${}_2$ hole densities in high-$T_c$ cuprates are consistently determined by the Cu NMR Knight shift. In single- and bilayered cuprates, it is demonstrated that the spin part of the Knight shift $K_s$(300 K) at room temperature monotonically increases with the hole density $p$ from the underdoped to overdoped regions, suggesting that the relationship of $K_s$(300 K) vs $p$ is a reliable measure to determine $p$. The validity of this $K_s$(300 K)-$p$ relationship is confirmed by the investigation of the $p$ dependencies of hyperfine magnetic fields and of spin susceptibility for single- and bilayered cuprates with tetragonal symmetry. Moreover, the analyses are compared with NMR data on three-layered Ba${}_2$Ca${}_2$Cu${}_3$O${}_6$(F,O)${}_2$ and HgBa${}_2$Ca${}_2$Cu${}_3$O${}_{8+\delta }$ and five-layered HgBa${}_2$Ca${}_4$Cu${}_5$O${}_{12+\delta }$, which suggests the general applicability of the $K_s$(300 K)-$p$ relationship to multilayered compounds with more than three CuO${}_2$ planes. The measurement of $K_s$(300 K) enables us to separately estimate $p$ for each CuO${}_2$ plane in multilayered compounds, where doped hole carriers are inequivalent between outer CuO${}_2$ planes and inner CuO${}_2$ planes.

Figure 1

(a) Crystal structure of bilayered Ba${}_2$CaCu${}_2$O${}_4$(F,O)${}_2$ (0212F). The heterovalent substitution of ${\mathrm{O}}^{2-}$ for ${\mathrm{F}}^{1-}$ increases the hole density. (b) ${}^{63}\mathrm{Cu}$ NMR spectra for 0212F(#1) at $T=60$ K with $H$ perpendicular to the $c$ axis ($H\parallel \mathit{ab}$). The dashed line points to $K=0$. (c) ${}^{63}\mathrm{Cu}$ NMR spectra for 0212F(#1) at $T=60$ K with $H$ parallel to the $c$ axis ($H\parallel c$). The peak marked with an asterisk ($*$) is from unoriented grains with $\theta ~90$${}^{°}$, where $\theta$ is the angle between $H$ and the $c$ axis. If $\theta$ in the unoriented grains were distributed completely randomly, the peak ($*$) could coincide with the spectral peak in (b). (d) Cu NQR spectrum at $T=1.5$ K. The spectrum is composed of ${}^{63}\mathrm{Cu}$ and ${}^{65}\mathrm{Cu}$.

Figure 2

(a) Spin part of the ${}^{63}\mathrm{Cu}$ Knight shift $K_{s,\mathit{ab}}(T)$ with $H\parallel \mathit{ab}$ as a function of temperature. The data are assigned as labeled in the figure. (b) Spin part of the ${}^{63}\mathrm{Cu}$ Knight shift $K_{s,c}(T)$ with $H\parallel c$. $K_{s,c}(T)$ shows a $T$ dependence similar to that of $K_{s,\mathit{ab}}(T)$.

Figure 3

(a) Crystal structure of three-layered Ba${}_2$Ca${}_2$Cu${}_3$O${}_6$(F,O)${}_2$ (0223F). In a unit cell, there are two kinds of CuO${}_2$ planes: outer CuO${}_2$ plane (OP) and inner CuO${}_2$ plane (IP). (b), (c) Cu NMR spectrum of 0223F at $T=200$ K with $H\parallel \mathit{ab}$ and $H\parallel c$. The spectra have two components arising from OP and IP. (d),(e) $T$ dependencies of $K_{s,\mathit{ab}}(T)$ and $K_{s,c}(T)$ for 0223F.

Figure 4

(a) $K_{s,\mathit{ab}}$(300 K) and (b) $B$ plotted as a function of $p$ for 0212F ($\square$), Bi2212 ($+$), Tl2201 ($▵$), and Tl1212 ($◯$). The values of $p$ are estimated from $T_c=T_{c,\text{max}}[1-82.6(p-0.16){}^2]$.[11] The plots denoted by crosses ($\times$) in (b) are for multilayered compounds, which are listed in Table . (c) $p$ dependence of ${\chi }_s$(300 K) evaluated from $K_{s,\mathit{ab}}=(A_{\mathit{ab}}+4B){\chi }_s$ [Eq. (3)]. The solid lines in the figures are to guide the eye.

Figure 5

${}^{63}\mathrm{Cu}$ NQR frequency ${}^{63}{\nu }_Q$ for hole-doped cuprates, 0212F, Bi2212,[36] Tl2201,[49] Tl1212,[21] L214,[18] and Y123.[50, 51] Here, $p$ is evaluated from the Sr-content $x$ for L214 and from $T_{c,\text{max}}$[11] for Y123. The solid lines in the figure are to guide the eye.