Wave propagation in pyramidal tip-like structures with cubic material properties

The three-dimensional wave propagation in cubic materials like silicon AFM tips is investigated numerically. The used VS-FDTD (Velocity Stress Finite-Difference Time-Domain) code is explained starting with the fundamental equations for the wave propagation in cubic materials. The discretisation, the boundary conditions, and the stability conditions are considered. As a validation of the numerical code, first the simulated wave velocities are compared with analytically calculated results for different polarizations and propagating directions. Second the code is validated by monitoring the internal energy of the simulated structure. Third, the results of the numerical code for long wavelengths agree well with a simplified analytical model for wave propagation in horn geometries. With the presented code pyramidal focusing tips are investigated regarding their focusing and detection properties and regarding the influence of the crystallographic orientation. It turns out that the focusing effect for the primary waves is strong in tips with lateral dimensions at the sharp end which are smaller than the dominant wave length. In contrast, the detection properties of a tip are better if the lateral dimensions at the sharp end of the tip are larger than the dominant wave length.