Methods and media structures are provided for increasing writability and reducing unintentional erasure of perpendicular magnetic recording media. Variable permeability is controlled within a thin soft underlayer (SUL) structure, independent of bulk SUL material properties such as magnetic moment (Bs) and magnetic anisotropy (Hk). Media with an improved combination of easier writability on the recorded track and difficult erasure off-track (between tracks and on neighboring tracks) is achieved, in part, by an unbalanced antiferromagnetically coupled (AFC) SUL structure. A permeability gradient is established within the soft underlayer with layers having different values of permeability and magnetic thickness (Bs*t). In an aspect, a first SUL layer includes a high permeability region and an overlying low permeability region. A second layer antiferromagnetically couples the first layer to a low permeability third SUL layer. The present invention may be utilized with high density perpendicular recording media requiring carefully balanced magnetic properties.