Fields From a Magnetized Conical Shell and Quantitative Magnetic Force Microscopy

A magnetized conical shell has a number of interesting properties. For a very tall uniformly magnetized shell, only the magnetization component directed toward the vertex produced any field. The axial component of the resulting field decays inversely with distance from the vertex. Furthermore, the axial gradient of the axial field has exactly the same form as the field from an isolated charge. This latter property has relevance to magnetic force microscopy (MFM) where sharp pointed tips coated with hard magnetic material can be used to create magnetic images. MFM is often conducted in “dynamic mode” where the vibrating tip senses force gradient. If the tip can be reasonably approximated as a conical shell and is vibrating along the cone-axis perpendicular to the sample, the resulting image will be of the axial (perpendicular) field from the sample sensed at the vertex of the tip. If the surface being imaged is a soft magnetic material, then the attraction of the tip to its own magnetic image must be considered. Conveniently, if the surface contains a discontinuity from soft magnetic material to non-magnetic material, the step change at the boundary provides a useful means of calibrating the properties of the tip. Examples are shown where the soft structures of a hard disk drive write-head are imaged and the actual field strengths quantified.