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Free-standing membranes are prepared by the selective water etching of SAO (yellow), followed by the direct/indirect transfer of membranes (orange) onto desired silicon (Si) or silicon nitride (Si 3 N 4 ) supports (blue). Indirect transfer requires an intermediate support such as PMMA (purple) to hold the membranes after water etching. For type-C membranes, a buffer layer made of LAO (green) and ultrathin STO (grey) was also grown ( Methods ). Layer thickness is not to scale.

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