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Using muon spin rotation and infrared spectroscopy we study the relationbetween magnetism and superconductivity in Ba$ _{1-x} $K$ _{x} $Fe$ _{2} $As$_{2} $ single crystals from the underdoped to the slightly overdoped regime. Wefind that the Fe magnetic moment is only moderately suppressed in most of theunderdoped region where it decreases more slowly than the N\'{e}el-temperature,$ T^{\mathrm{N}} $. This applies for both the total Fe moment obtained frommuon spin rotation and for the itinerant component that is deduced from thespectral weight of the spin-density-wave pair breaking peak in the infraredresponse. In the moderately underdoped region, superconducting and staticmagnetic orders co-exist on the nano-scale and compete for the same electronicstates. The static magnetic moment disappears rather sharply near optimaldoping, however, in the slightly overdoped region there is still an enhancementor slowing down of spin fluctuations in the superconducting state. Similar tothe gap magnitude reported from specific heat measurements, the superconductingcondensate density is nearly constant in the optimally- and slightly overdopedregion, but exhibits a rather pronounced decrease on the underdoped side.Several of these observations are similar to the phenomenology in the electrondoped counterpart Ba(Fe$ _{1-y} $Co$ _{y} $)$ _{2} $As$ _{2} $.

Muon spin rotation and infrared spectroscopy study of magnetism and superconductivity in Ba$ _{1-x} $K$ _{x} $Fe$ _{2} $As$ _{2} $