Magnetic excitations in pure, lightly doped, and weakly metallic<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant="normal">La</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant="normal">CuO</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math>

We report a comprehensive neutron-scattering study of the evolution of the magnetic excitations in ${\mathrm{La}}_{2\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Sr}}_{\mathit{x}}$${\mathrm{CuO}}_{4}$ for 0\ensuremath{\le}x\ensuremath{\le}0.04. We first present accurate measurements of the magnetic correlation length and the sublattice magnetization of a carrier-free ${\mathrm{La}}_{2}$${\mathrm{CuO}}_{4}$ crystal and analyze these in the context of recent theoretical predictions. We then systematically investigate the influence of different dopants on the magnetism: Our measurements indicate that static vacancies in the ${\mathrm{La}}_{2}$${\mathrm{Cu}}_{1\mathrm{\ensuremath{-}}\mathit{y}}$${\mathrm{Zn}}_{\mathit{y}}$${\mathrm{O}}_{4}$ system affect the magnetic correlations in a similar manner as electrons in ${\mathrm{Pr}}_{2\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Ce}}_{\mathit{x}}$${\mathrm{CuO}}_{4}$. The magnetic correlation length is much more rapidly suppressed as a function of x in ${\mathrm{La}}_{2\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Sr}}_{\mathit{x}}$${\mathrm{CuO}}_{4}$, and for x\ensuremath{\le}0.04 we find that it obeys the empirical relation ${\ensuremath{\xi}}^{\mathrm{\ensuremath{-}}1}$(x,T)=${\ensuremath{\xi}}^{\mathrm{\ensuremath{-}}1}$(x,0)+${\ensuremath{\xi}}^{\mathrm{\ensuremath{-}}1}$(0,T), where \ensuremath{\xi}(0,T) is the measured correlation length of the carrier-free sample. We also report an extensive set of measurements of the dynamical magnetic response function of a crystal of composition ${\mathrm{La}}_{1.96}$${\mathrm{Sr}}_{0.04}$${\mathrm{CuO}}_{4}$ for excitation energies 0.75\ensuremath{\le}\ensuremath{\omega}\ensuremath{\le}45 meV and temperatures 1.5\ensuremath{\le}T\ensuremath{\le}500 K.The dc conductivity of this crystal exhibits three different regimes: metallic for T\ensuremath{\ge}100 K, weakly localized for 100\ensuremath{\ge}T\ensuremath{\ge}10 K, and strongly localized below \ensuremath{\sim}1 K. Our neutron measurements show that the generalized susceptibility of this sample follows a surprisingly simple scaling function in the variable \ensuremath{\omega}/T. This observation allows us to relate our data to a variety of normal-state properties of the layered copper oxides, in particular the dc and ac conductivities. Finally, at temperatures below \ensuremath{\sim}20 K a ``central peak'' with a characteristic energy scale of less than 0.1 meV becomes prominent. Its relation to the localization of the charge carriers at low temperatures remains speculative.

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