Imaging Acoustic Waves by Microwave Microscopy: Microwave Impedance Microscopy for Visualizing Gigahertz Acoustic Waves

Atoms in all materials are constantly shaking, which is largely responsible for the transfer of heat and sound. In a uniform crystalline solid, the motion of the lattice can be decomposed into just a handful of normal modes of vibration or, in the language of quantum mechanics, a set of quantized eigenmodes of the elastic structure known as phonons [1]. In the frequency (f) regime of interest for microwave engineers, there exist three (one longitudinal, two shear) branches of vibrational modes with long wavelengths-"long" when compared with the atomic spacing. Their frequencies, which represent the energy of each quantum, are linearly proportional to the inverse wavelengths, which represent the momentum of each quantum. The ratio between the two is typically several kilometers per second. In good crystals with few imperfections, these vibrations travel a very long distance-"long" when compared with the wavelength-with little decay of the amplitude under the ambient temperature and pressure. At audio frequencies, these sound waves can propagate in solids. They are commonly known as acoustic waves (see "Nomenclature Used Throughout").