AFM dissipation topography of soliton superstructures in adsorbed overlayers

In the atomic force microscope, the nanoscale force topography of even complex surface superstructures is extracted by the changing vibration frequency of a scanning tip. An alternative dissipation topography with similar or even better contrast has been demonstrated recently by mapping the $\left(x,y\right)$-dependent tip damping but the detailed damping mechanism is still unknown. Here we identify two different tip dissipation mechanisms: local mechanical softness and hysteresis. Motivated by recent data, we describe both of them in a one-dimensional model of Moiré superstructures of incommensurate overlayers. Local softness at “soliton” defects yields a dissipation contrast that can be much larger than the corresponding density or corrugation contrast. At realistically low vibration frequencies, however, a much stronger and more effective dissipation is caused by the tip-induced nonlinear jumping of the soliton, naturally developing bistability and hysteresis. Signatures of this mechanism are proposed for experimental identification.

Figure 1

(Color online) AFM Moiré superstructures of incommensurate KBr bilayers islands/NaCl(100), adapted from Ref. 2: (a) topography; (b) dissipation. Note the opposite phase: dissipation is largest at the soliton, where topographic height is minimal (Ref. 8). (c) 1D simulation model with a rigid substrate potential of period $a_{\text{sub}}$, a harmonic chain of rest length $a_0$ (the adsorbate overlayer), and the localized tip potential $u_{\text{T}}$ (here attractive) oscillating horizontally (in experiments the oscillation is usually vertical).

Figure 2

(Color online) The linear-response regime. (a) Overlayer atom density (dashed), rest positions (circles), in the periodic substrate potential (solid). (b) The density of zero-frequency overlayer vibrational modes peaks at the soliton positions. (c) Dissipated power during tip vibration for a tip width ${\sigma }_{\text{T}}={\Delta }_{\text{T}}=a_{\text{sub}}$, weakly attractive potential of amplitude $A_{\text{T}}=-{10}^{-4}$, $\gamma =0.2$ (underdamped regime), $K=5$ (fairly rigid overlayer), and oscillation frequency ${\omega }_{\text{T}}={10}^{-4}\pi$ (roughly 1 GHz). This regime is described well by linear-response theory. Note a huge dissipation peak contrast at the soliton position relative to the terrace between solitons of about ${10}^4$.

Figure 3

(Color online) The strong-interaction hysteretic regime. (a) Mean power dissipated in the steady regime by a strongly interacting tip ($A_{\text{T}}=-0.01$, all other parameters the same as in Fig. 2) scanning the same overlayer $\left(K=5\right)$. (b) Force-displacement response at three typical scanning points marked in the left panel. The strong dissipation at the solitons is now due to hysteretic jumps of the solitons (kinks) across their Peierls-Nabarro barriers. The dissipated energy in a cycle equals the area of the hysteretic loop in the force-displacement plane (shaded). When the potential is strong enough to drag or push a soliton (points A and B), this occurs with hysteresis and the dissipation is large. With the selected oscillation amplitude, the soliton is dragged across two barriers (point A) or a single barrier (point B), depending on the center of tip position ${\overline{x}}_{\text{T}}$. When the tip grabs no soliton (point C) there is no hysteresis and dissipation drops.

Figure 4

(Color online) The dissipated power normalized to the linear-regime scaling, $\overline{P}/{\left|A_{\text{T}}\right|}^2$, versus the strength of the tip-overlayer interaction $\left|A_{\text{T}}\right|$, at point A of Fig. 3 and for the same parameters as in Fig. 2. Notice the sharp onset of the nonlinear regime where the hysteretic mechanisms starts to play a role, leading to a strong enhancement of dissipation.

Figure 5

(Color online) The dissipation reduction due to thermal shrinking of the hysteretic loop, for ${\omega }_{\text{T}}={10}^{-4}\pi$ at the soliton location A of Fig. 3. (a) The frequency dependence of the energy dissipated in one period $\overline{P}2\pi /{\omega }_{\text{T}}$, computed for several temperatures, exhibiting a clear reduction in $\overline{P}2\pi /{\omega }_{\text{T}}$ due to a rise in temperature. (b) Detail of the right-side area of the force-displacement dependency following two typical tip-oscillation periods for $T={10}^{-5}$, solid, and ${10}^{-6}$, dotted, compared to $T=0$, dashed: the hysteretic area is shrinking due to a randomly anticipated thermally activated barrier crossing.