Rapid reconstruction of a strong nonlinear property by a multiple lock-in technique

We propose and discuss a rapid reconstruction procedure for strongly nonlinear signals and validate it for dynamic force microscopy. Harmonics of the cantilever resonance frequency shift, generated by a low-frequency modulation of the tip-sample distance, are detected by a phase-locked loop followed by 12 lock-in amplifiers. The distance dependence of the frequency shift can be reconstructed by summing up the sampled Fourier components with judiciously assigned phase shifts. Following a successful test with a model potential, we report a measurement of the frequency shifts induced by the force field above a KBr(001) surface at room temperature in ultrahigh vacuum. Experimental spectra justify the neglect of harmonics beyond tenth order in the range where clear atomic-scale contrast appears in images of the lower harmonic intensities. A high-resolution three-dimensional frequency shift dataset was measured in $400\mathrm{s}$. The method can in general be applied to any single-valued physical quantity with a smooth nonlinear dependence on a control variable.

Figure 1

(a) Modeled distance dependence of the frequency shift $\Delta f$ and real-time $Z$ scanner modulation. (b) $\Delta f$ signal in the time domain and (c) its fast Fourier transform amplitudes in the frequency domain. (d) Calculated distance dependence of the time-averaged $\Delta f$. (e) Reconstructed real-time $\Delta f$ signals and (f) their distance dependencies, including up to 6, 12, and 18 harmonics. Calculation parameters: resonance frequency, $f=1\mathrm{M}\mathrm{Hz}$; cantilever oscillation amplitude, $A=100\mathrm{p}\mathrm{m}$; effective mode stiffness, $k=1400\mathrm{N}/\mathrm{m}$; $Z$ scanner modulation frequency, $f_Z=30\mathrm{Hz}$; and amplitude, $A_Z=500\mathrm{p}\mathrm{m}$.

Figure 2

Schematic diagram of the measurement setup based on the multiple lock-in technique.

Figure 3

(a) Examples of power spectral densities of the modulated resonance frequency $\Delta f\left(t\right)$ at three different atomic sites on KBr(001). (b) Topographic image obtained at a constant time-averaged $\overline{\Delta f}=-120\mathrm{Hz}$. (c) Series of 12 harmonic intensity maps simultaneously recorded while scanning the surface. Imaging parameters: $A_{\text{2nd}}=100\mathrm{p}\mathrm{m}$; $f_{\text{2nd}}=1011520\mathrm{Hz}$; $Q_{\text{2nd}}=17920$; $V_{\text{bias}}=0\mathrm{V}$; $f_Z=46\mathrm{Hz}$; and $A_Z=500\mathrm{p}\mathrm{m}$.

Figure 4

(a) Reconstructed 1D-$\Delta f$ vs. distance curve at a particular site. (b) Constant height images of $\Delta f$ extracted from the reconstructed 3D-$\Delta f$ dataset at different $Z_c$. The location of (a) is indicated by circles in (b). The cross markers indicate reference points.