Analytical Study of Noise Spectral Density in Q-controlled Atomic Force Microscopy

Q-controlled atomic force microscopy utilizes an oscillating probe with an adjustable quality factor, which is done by adding a variable damping interaction to the probe. While the Q-control can enhance the probe sensitivity by increasing the Q-factor, the applied interaction increases the noise of the probe motion as well. Thus, the signal-to-noise ratio (SNR) associated with the probe sensitivity varies accordingly. Here, we analytically derive the noise spectral density of the tip motion, and theoretically investigate the SNR in Q-controlled atomic force microscopy. We show that both the overall tip-motion signal and the noise monotonically increase with the effective Q-factor under control, whereas the SNR exhibits a global minimum near the original Q value and asymptotically scales as the square root of the effective Q-factor. Therefore, in general, the Qcontrol technique can be used to alter the quality factor, the magnitudes of signal and noise, and the probe sensitivity in amplitude-modulation atomic force microscopy.