###### Abstract  
Electrostatic force microscopy (EFM) is a useful technique when measuring the surface electric potential of a substrate regardless of its topography. Here, we have developed a frequency detection method for alternating current (AC) bias in EFM. Instead of an internal lock-in amplifier (LIA) for EFM that only detects \(\omega_{e}\) and \(2\omega_{e}\), we have used other LIAs that can amplify the amplitude of specific frequency by direct digital synthesizer (DDS), that finds the optimal frequency of surface charge images. In order to confirm the performance of the proposed methods, the electrical properties of lead zirconate titanate (PZT) and triglycine sulfate (TGS) samples were measured. In addition, we compared the performances of the frequency-detection method and the conventional EFM method. Ultimately, enhanced images could be achieved using the frequency-detection method. The optimal modulated frequency-shift for force-gradient measurements was found to be 2 kHz. Additionally, we have shown that it is possible to use a hard cantilever (\(K=42\) N/m, 330 kHz). Therefore, we expect that this technique can be applied to measure the electrical properties of bio-molecular films. electrostatic force microscopy; EFM; direct digital synthesizer; PZT; TGS +
Footnote †: journal: Applied Physics Letters  
_Applied Physics Letters_  
vibrates with a frequency of \(\omega\). Thus, we can assume that the capacitance \(C\), which is dependent on the probe-sample distance \(z\), is between the probe and the sample. In this case, the electrostatic force \(F_{z}^{el}\) can be written as  
\[ \begin{array}{l}F_{z}^{el}=-\frac{1}{2}\frac{\partial C}{\partial z}(V_{DC}-V_ {CPD}+V_{AC}\text{sin}\omega t)^{2}\\ =-\frac{1}{2}\frac{\partial C}{\partial z}\left\{(V_{DC}-V_{CPD})^{2}+2(V_{DC} -V_{CPD})V_{AC}\text{sin}\omega t+V_{AC}^{2}sin^{2}\omega t\right\}\end{array}  \tag{1}\]  
where \(V_{DC}\) is the bias voltage externally applied to the probe, and \(V_{CPD}\) is the contact potential difference (CPD) between the tip and the sample. Given that the electrostatic force is proportional to the square of the voltage, as indicated in the equation above, the vibration contains direct current (DC), \(\omega\), and \(2\omega\) components. First, the DC component is the static attractive force that exists between the electrodes. Second, the \(\omega\) component is a force that is applied to the previously described charges via the AC electric field. Third, the \(2\omega\) component is a force induced onto the capacitors by sole means of the AC voltage. The \(\omega\) component disappears when an appropriate bias voltage is applied to the probe in order to cancel out the CPD; i.e., \(V_{DC}-V_{CPD}=0\). The surface potential, or CPD, can be quantitatively measured using the feedback control of \(V_{DC}\) to ensure that the \(\omega\) component is equivalent to zero [2; 29; 30; 31]. The FM mode refers to the detection method by which the resonance frequency shift of the cantilever is tracked by a self-oscillator or a phase-lock loop (PLL) [2; 5; 29]. Both topographic and electrostatic signals can be detected by the oscillation of a cantilever near the resonance frequency. Although the modulation voltage of the probe is not in resonance frequency, the electrostatic force can be measured and the electrostatic force induces a shift of the natural resonant frequency \(\omega_{o}\). These side-bands are extremely sensitive to the electrostatic forces as a result of the large gradient force from the probe apex.