Apparatus and techniques presented combine the features and benefits of amplitude modulated (AM) atomic force microscopy (AFM), sometimes called AC mode AFM, with frequency modulated (FM) AFM. In AM-FM imaging, the topographic feedback from the first resonant drive frequency operates in AM mode while the phase feedback from second resonant drive frequency operates in FM mode. In particular the first or second frequency may be used to measure the loss tangent, a dimensionless parameter which measures the ratio of energy dissipated to energy stored in a cycle of deformation.