A current-perpendicular-to-the-plane magnetoresistive sensor has magnetic damping material located adjacent either or both of the sensor side edges and back edge to reduce the effect of spin transfer torque. The damping material may be Pt, Pd, Os, or a rare earth metal from the 15 lanthanoid elements. The damping material may be an ultrathin layer in contact with the sensor edges. An insulating layer is deposited on the damping layer and isolates the sensor's ferromagnetic biasing layer from the damping layer. Instead of being a separate layer, the damping material may be formed adjacent the sensor edges by being incorporated into the material of the insulating layer.