Muon spin relaxation studies of incommensurate magnetism and superconductivity in stage-4La2CuO4.11andLa1.88Sr0.12CuO4

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We report muon spin relaxation $(\ensuremath{\mu}\mathrm{SR})$ measurements using single crystals of oxygen-intercalated stage-4 ${\mathrm{La}}_{2}{\mathrm{CuO}}_{4.11}$ (LCO:4.11) and ${\mathrm{La}}_{1.88}{\mathrm{Sr}}_{0.12}{\mathrm{CuO}}_{4}$ (LSCO:0.12), in which neutron scattering studies have found incommensurate magnetic Bragg reflections. In both systems, zero-field $\ensuremath{\mu}\mathrm{SR}$ measurements show muon spin precession below the N\'eel temperature ${T}_{N}$ with frequency 3.6 MHz at $\stackrel{\ensuremath{\rightarrow}}{T}0,$ having a Bessel function line shape, characteristic of spin-density-wave systems. The amplitude of the oscillating and relaxing signals of these systems is less than half the value expected for systems with static magnetic order in 100% of the volume. Our results are consistent with a simulation of local fields for a heuristic model with (a) incommensurate spin amplitude modulation with the maximum ordered Cu moment size of $\ensuremath{\sim}0.36{\ensuremath{\mu}}_{B},$ (b) static Cu moments on the ${\mathrm{CuO}}_{2}$ planes forming ``islands'' having typical radius $15--30 \mathrm{\AA{}},$ comparable to the in-plane superconducting coherence length, and (c) the measured volume fraction of magnetic muon sites ${V}_{\ensuremath{\mu}}$ increasing progressively with decreasing temperature below ${T}_{N}$ towards ${V}_{\ensuremath{\mu}}\ensuremath{\sim}40%$ for LCO:4.11 and 18% for LSCO:0.12 at $\stackrel{\ensuremath{\rightarrow}}{T}0.$ These results may be compared with correlation lengths in excess of $600 \mathrm{\AA{}}$ and a long range ordered moment of $0.15\ifmmode\pm\else\textpm\fi{}0.05{\ensuremath{\mu}}_{B}$ measured with neutron scattering techniques. In this paper we discuss a model that reconciles these apparently contradictory results. In transverse magnetic field $\ensuremath{\mu}\mathrm{SR}$ measurements, sensitive to the in-plane magnetic field penetration depth ${\ensuremath{\lambda}}_{\mathrm{ab}},$ the results for LCO:4.11 and LSCO:0.12 follow correlations found for underdoped, overdoped and Zn-doped high-${T}_{c}$ cuprate systems in a plot of ${T}_{c}$ versus the superconducting relaxation rate $\ensuremath{\sigma}(\stackrel{\ensuremath{\rightarrow}}{T}0).$ This indicates that the volume-integrated value of ${n}_{s}{/m}^{*}$ (superconducting carrier density / effective mass) is a determining factor for ${T}_{c},$ not only in high-${T}_{c}$ cuprate systems without static magnetism, but also in the present systems where superconductivity coexists with static spin-density-wave spin order.