Screening of Point Charge Impurities in Highly Anisotropic Metals: Application to ${\mu }^{+}$-Spin Relaxation in Underdoped Cuprate Superconductors

We calculate the screening charge density distribution due to a point charge, such as that of a positive muon (${\mu }^{+}$), placed between the planes of a highly anisotropic layered metal. In underdoped hole cuprates the screening charge converts the charge density in the metallic-plane unit cells in the vicinity of the ${\mu }^{+}$ to nearly its value in the insulating state. The current-loop-ordered state observed by polarized neutron diffraction then vanishes in such cells, and also in nearby cells over a distance of order the intrinsic correlation length of the loop-ordered state. This strongly suppresses the magnetic field at the ${\mu }^{+}$ site. We estimate this suppressed field in underdoped ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+x}$ and ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$, and find consistency with the observed $\sim 0.2\mathrm{G}$ field in the former case and the observed upper bound of $\sim 0.2\mathrm{G}$ in the latter case. This resolves the controversy between the neutron diffraction and mu-spin relaxation experiments.

Figure 1

The orbital current pattern in the ${\mathrm{CuO}}_2$ plane.

Figure 2

Schematic of ${\mathrm{CuO}}_2$ planes and the ${\mu }^{+}$ site.

Figure 3

The screening charge density in the planes in the Thomas-Fermi approximation for a muon centered between planes ($d=c/2$). (a) Nearest-neighbor plane. (b) Next-nearest-neighbor plane.

Figure 4

The screening charge density in nearby planes in the Thomas-Fermi approximation, with a large asymmetry in the placement of the muon ($d=0.3c$). (a) Nearest-neighbor plane. (b) Next-nearest-neighbor plane.