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We investigate Cu-doped Ba(Fe$_{1-x}$Cu$_x$)$_2$As$_2$ with transport,magnetic susceptibility, and elastic neutron scattering measurements. In theheavily Cu-doped regime where long-range stripe-type antiferromagnetic order inBaFe$_2$As$_2$ is suppressed, Ba(Fe$_{1-x}$Cu$_x$)$_2$As$_2$ (0.145 $\leq x\leq$ 0.553) samples exhibit spin-glass-like behavior in magneticsusceptibility and insulating-like temperature dependence in electricaltransport. Using elastic neutron scattering, we find stripe-type short-rangemagnetic order in the spin-glass region identified by susceptibilitymeasurements. The persistence of short-range magnetic order over a large dopingrange in Ba(Fe$_{1-x}$Cu$_x$)$_2$As$_2$ likely arises from local arrangementsof Fe and Cu that favor magnetic order, with Cu acting as vacancies relievingmagnetic frustration and degeneracy. These results indicate locally brokenfour-fold rotational symmetry, suggesting that stripe-type magnetism isubiquitous in iron pnictides.

Local breaking of four-fold rotational symmetry by short-range magnetic order in heavily overdoped Ba(Fe$_{1-x}$Cu$_{x}$)$_{2}$As$_{2}$