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We present a combined neutron diffraction and high field muon spin rotation($\mu$SR) study of the magnetically ordered and superconducting phases of thehigh-temperature superconductor La$_{1.94}$Sr$_{0.06}$CuO$_{4+y}$ ($T_{\rm c} =37.5(2)$~K) in a magnetic field applied perpendicular to the CuO$_2$ planes. Weobserve a linear field-dependence of the intensity of the neutron diffractionpeak that reflects the modulated antiferromagnetic stripe order. The magneticvolume fraction extracted from $\mu$SR data likewise increases linearly withapplied magnetic field. The combination of these two observations allows us tounambiguously conclude that stripe-ordered regions grow in an applied field,whereas the stripe-ordered magnetic moment itself is field-independent. Thiscontrasts with earlier suggestions that the field-induced neutron diffractionintensity in La-based cuprates is due to an increase in the ordered moment. Wediscuss a microscopic picture that is capable of reconciling these conflictingviewpoints.

Field-induced electronic phase separation in the high-temperature superconductor La$_{1.94}$Sr$_{0.06}$CuO$_{4+y}$