Current-perpendicular-to-plane (CPP) read sensors for magnetic heads having constrained current paths made of lithographically-defined conductive vias, and methods of making the same, are disclosed. In one example, a sensor stack structure which includes an electrically conductive spacer layer is formed over a first shield layer. An insulator layer is deposited over and adjacent the spacer layer, and a resist structure which exposes one or more portions of the insulator layer is formed over the insulator layer. With the resist structure in place, the exposed insulator layer portions are removed by etching to form one or more apertures through the insulator layer down to the spacer layer. Electrically conductive materials are subsequently deposited within the one or more apertures to form one or more lithographically-defined conductive vias of a current-constraining structure.