Calculation and accurate measurement of capacitance of electrically small axi-symmetric microstructures near a probe tip

An efficient calculation method that uses semi closed-form solutions of axisymmetric metallic sub-sections to numerically calculate the capacitance of a metallic probe over stratified dielectric substrates or metallic samples is developed and discussed. The current continuity equation was first used to obtain charge densities at the structure's surface. The capacitance matrix that uses gap and permittivity (CGP) as indexes for specific tip was then numerically estimated using this method. CCP matrix enabled noncontact simultaneous topography and permittivity mapping using a single microwave measurement. This method was subsequently used to calculate the capacitance of spherical and conical tips commonly used in local scanning probe microscopy. To evaluate the accuracy of the this method, the load impedance change of the above structures were measured and compared to their numerical values. The load impedance of the probe tip was experimentally determined from the input impedance of a λ/2 microstrip resonator at 1 GHz The model of the measurement apparatus and a calibration procedure were also developed. The experimental data agreed with the numerical results quite well. Moreover, the experiments also showed that the simple λ/2 microstrip resonator has an excellent capacitance resolution nearing 1.96x10
^{-18 at 1 GHz using a BiCMOS I/Q detector.}