Disentangling superconducting and magnetic orders in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>NaFe</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>x</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>Ni</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:mi>As</mml:mi></mml:mrow></mml:math> using muon spin rotation

Muon spin rotation and relaxation studies have been performed on a ``111'' family of iron-based superconductors, ${\mathrm{NaFe}}_{1\ensuremath{-}x}{\mathrm{Ni}}_{x}\mathrm{As}$, using single crystalline samples with Ni concentrations $x=0$, 0.4, 0.6, 1.0, 1.3, and 1.5%. Static magnetic order was characterized by obtaining the temperature and doping dependences of the local ordered magnetic moment size and the volume fraction of the magnetically ordered regions. For $x=0$ and 0.4%, a transition to a nearly-homogeneous long range magnetically ordered state is observed, while for $x\ensuremath{\gtrsim}0.4%$ magnetic order becomes more disordered and is completely suppressed for $x=1.5%$. The magnetic volume fraction continuously decreases with increasing $x$. Development of superconductivity in the full volume is inferred from Meissner shielding results for $x\ensuremath{\gtrsim}0.4%$. The combination of magnetic and superconducting volumes implies that a spatially-overlapping coexistence of magnetism and superconductivity spans a large region of the $T\ensuremath{-}x$ phase diagram for ${\mathrm{NaFe}}_{1\ensuremath{-}x}{\mathrm{Ni}}_{x}\mathrm{As}$. A strong reduction of both the ordered moment size and the volume fraction is observed below the superconducting ${T}_{\mathrm{C}}$ for $x=0.6$, 1.0, and 1.3%, in contrast to other iron pnictides in which one of these two parameters exhibits a reduction below ${T}_{\mathrm{C}}$, but not both. The suppression of magnetic order is further enhanced with increased Ni doping, leading to a reentrant nonmagnetic state below ${T}_{\mathrm{C}}$ for $x=1.3%$. The reentrant behavior indicates an interplay between antiferromagnetism and superconductivity involving competition for the same electrons. These observations are consistent with the sign-changing ${s}^{\ifmmode\pm\else\textpm\fi{}}$ superconducting state, which is expected to appear on the verge of microscopic coexistence and phase separation with magnetism. We also present a universal linear relationship between the local ordered moment size and the antiferromagnetic ordering temperature ${T}_{\mathrm{N}}$ across a variety of iron-based superconductors. We argue that this linear relationship is consistent with an itinerant-electron approach, in which Fermi surface nesting drives antiferromagnetic ordering. In studies of superconducting properties, we find that the $T=0$ limit of superfluid density follows the linear trend observed in underdoped cuprates when plotted against ${T}_{\mathrm{C}}$. This paper also includes a detailed theoretical prediction of the muon stopping sites and provides comparisons with experimental results.

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