Noise performance of frequency modulation Kelvin force microscopy

Noise performance of a phase-locked loop (PLL) based frequency modulation Kelvin force microscope (FM-KFM) is assessed. Noise propagation is modeled step by step throughout the setup using both exact closed loop noise gains and an approximation known as "noise gain" from operational amplifier (OpAmp) design that offers the advantage of decoupling the noise performance study from considerations of stability and ideal loop response. The bandwidth can be chosen depending on how much noise is acceptable and it is shown that stability is not an issue up to a limit that will be discussed. With thermal and detector noise as the only sources, both approaches yield PLL frequency noise expressions equal to the theoretical value for self-oscillating circuits and in agreement with measurement, demonstrating that the PLL components neither modify nor contribute noise. Kelvin output noise is then investigated by modeling the surrounding bias feedback loop. A design rule is proposed that allows choosing the AC modulation frequency for optimized sharing of the PLL bandwidth between Kelvin and topography loops. A crossover criterion determines as a function of bandwidth, temperature and probe parameters whether thermal or detector noise is the dominating noise source. Probe merit factors for both cases are then established, suggesting how to tackle noise performance by probe design. Typical merit factors of common probe types are compared. This comprehensive study is an encouraging step toward a more integral performance assessment and a remedy against focusing on single aspects and optimizing around randomly chosen key values.