A perpendicular recording medium having a perpendicular magnetic recording layer and a magnetically soft underlayer structure disposed beneath the recording layer. The soft underlayer structure includes at least first and second soft magnetic layers having different magnetic permeabilities to create a magnetic permeability gradient in the soft underlayer structure. One or more of the soft magnetic layers can be anti-parallel coupled. The soft underlayer structure of the present invention having a magnetic permeability gradient advantageously leads to reduced adjacent track erasure (ATE) while maintaining good overwrite (OW) properties.