MIM measurements were performed by delivering a small microwave excitation of ~0.1 μW at a fixed frequency in the range 1 to 10 GHz to a chemically etched tungsten tip (25). The reflected signal was analyzed to extract the demodulated output channels, MIM-Im and MIM-Re, which were proportional to the imaginary and real parts of the admittance between the tip and the sample, respectively. To enhance the MIM signal quality, the tip was excited to oscillate at a frequency of ~32 kHz with an amplitude of ~8 nm. The resulting oscillation amplitudes of MIM-Im and MIM-Re were then extracted using a lock-in amplifier to yield d(MIM-Im)/dz and d(MIM-Re)/dz, respectively. The d(MIM)/dz signals are free of fluctuating backgrounds, thus enabling more quantitative analysis, while their behavior is very similar to that of the standard MIM signals (25). Here, we simply refer to d(MIM)/dz as the MIM signal, and the simulations were done accordingly.

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