Technical Comments

Subatomic Features in Atomic Force Microscopy Images

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Science  30 Mar 2001:
Vol. 291, Issue 5513, pp. 2509
DOI: 10.1126/science.291.5513.2509a

Recently, Giessibl et al. (1) reported a distinct substructure in atomic force microscopy (AFM) images of individual adatoms on the Si(111)-(7×7) surface. A cross section along the fast scanning direction showed two peaks per adatom, which were interpreted as images of two unsaturated dangling bonds at the apex of the silicon tip interacting with the dangling bond of the adatom.

To support this interpretation, Giessibl et al. presented calculated frequency shift (Δf) images at a constant tip-to-sample distance, z < 300 pm. Their simulations were based on the Stillinger-Weber (SW) potential and assumed a tip apex atom that had two dangling bonds directed toward the sample. Our calculations using the same simple model and experimental parameters have shown similar substructures, but atz = 240 pm instead of z = 285 pm.

Unfortunately, neither of these results can be directly compared with the experiment, which was performed at a constant Δfrather than a constant z. To simulate both constant-Δf and constant-z scanlines, we have calculated Δf contours in our simulations (Fig. 1). In these plots, the variation of Δf as a function of x at constant z can be extracted using the Δf color bar next to the figure; the variation of z at constant Δf is shown by the solid contour lines. In the case that considers short-range forces only (Fig. 1A), at z = 240 pm, Δf(x) does indeed exhibit two distinct peaks (cross section in Fig 1B). This distance, however, is close to the silicon bond length of 235 pm and to the computed maximum in the binding energy between the tip and sample—that is, it is close to the point at which covalent attraction and Pauli repulsion balance each other. Around x = 0, this value of z lies in an area in which Δf becomes less negative with decreasing distance, a situation incompatible with stable feedback operation, which requires an opposite slope for Δf. Thus, if only short-range forces are considered, the calculated substructure could in principle be observed in a constant-z scan but not in a constant-Δf scan.

Figure 1

Simulated Δf(x,z) scans. (A) Simulation considering only short-range forces. (B) Constant-height scanline from Fig. 1A, atz = 240 pm. (C) Simulation considering both short-range chemical forces and long-range van der Waals forces.

Without long-range forces, we calculate the maximum Δf below 285 pm to be −5 Hz, which corresponds well to the −8 fNm calculated by Giessibl et al. (1). Astonishingly, however, their experiment was performed at a Δf between −140 Hz and −160 Hz, which suggests that more than 95% of Δf is due to long-range van der Waals and electrostatic forces, at variance with the claim in the abstract that the experimental technique has an enhanced sensitivity to short-range forces. The experimental Δf can be reproduced if a van der Waals interaction of a spherical tip with a radius of 162 nm with a flat surface is added to the short-range force (Fig. 1C). In this simulation, a substructure was indeed found in a constant-Δf scan at −160 Hz. The z-variation was extremely small, however, at around 5 pm, much less than the 180 pm found in the experiment by Giessibl et al. (1). More important, the substructure is not due to two bonds at the tip apex but rather to short-range repulsion above the adatom. To keep Δf constant, z must be reduced to increase the overall attraction in order to compensate for the localized repulsion. A simulation using a tip with only one dangling bond and a short-range interaction fitted to ab initio computations predicts a ringlike substructure under the same conditions. The observation of a double-peak structure in a constant-Δf scan thus does not prove the existence of a tip with two dangling bonds. Furthermore, molecular-dynamics simulations (2) assuming either type of interaction predict that nondestructive imaging is not possible at distances comparable to the Si bond length.

A much simpler interpretation of the observed substructure arises if one recalls that the maximum Δf due to short-range forces, −5 Hz, is much smaller than the experimental setpoint value of −160 Hz and is also significantly smaller than the ±18 Hz Δffeedback error signal reported by Giessibl et al. in supplemental figure 4 (3). Based on the magnitude of the error signal compared with the contribution to Δfresponsible for atomic-scale contrast, we conclude that the crescent-like structures observed by Giessibl et al. are due to overshoot and ringing of the feedback. This is supported by the fact that the crescents are perpendicular to the fast scan direction and that their asymmetry reverses with the scan direction.

Although we find the idea of detecting more than one bond per atom appealing, it is unlikely that the SW potential, which is fitted to the properties of silicon in the sp3-bonded diamond structure, would properly describe the rehybridization expected at the tip apex either far from or close to the sample adatoms. More refined calculations are required on the modeling side, whereas on the experimental side, such effects would be best observed in a constant-height mode and should be confirmed by scans in different directions.


Response: Hug et al. question neither the theoretical findings nor the extremely low noise of the data reported by us (1). Instead, they offer an alternative explanation of our measured images: They propose that the images are caused by a feedback artifact. In the following discussion we show that this is not the case, and explain why the arguments put forward by Huget al. do not hold.

Figure 1

Error analysis of topographic data from (1). All three images represent unfiltered data and are displayed with the same z scale. (A) Raw topographic data at a nominally constant frequency shift of –160 Hz. (B) Topography error signal corresponding to Fig. 1A. Topography error is derived from the frequency error signal (the deviation between the actual frequency shift and the setpoint of –160 Hz), which, as we have noted (9), has a magnitude of ±18 Hz (peak) with an rms value of 9 Hz. The frequency error signal translates into a topography error signal through the variation of the frequency shift with distance ∂Δf/∂z ≈ 1 Hz/pm (10), which corresponds to an rms topography of ≈ 9 pm. (C) Corrected topographic data, given by sum of the topographic data and the topography error signal. The difference between the uncorrected and corrected data is very minor.

In some respects, the comment by Hug et al. actually confirms our work. Hug's calculations, like ours, use the SW potential (2); indeed, they use routines developed by our group. Their work confirms that in imaging silicon adatoms with a (001)-oriented Si tip, a transition from a single peak in the adatom image to a double peak occurs at a distance close to the next-neighbor distance in silicon. Their calculations suggest that the transition should take place at a distance ≈ 260 pm (their figure 1C); when the distance is ≈ 243 pm, a 5-pm-deep dip in the adatom image occurs. Several details of their presentation need to be corrected, however: for example, the short-range force is attractive for z > 238 pm, not repulsive as they state, and the sign in the frequency shift data in their figure 1B should be negative. Moreover, the experimental depth that we found for the dip in the adatom image was ≈ 20 pm [figure 1D of (1)], not 180 pm as stated in the comment; thus, the qualitative calculation differs from experiment by only a factor of 4, not 36. The remaining difference presumably traces to inaccuracies of the calculation, which takes neither laterally dependent electrostatic forces nor elastic deformations of tip and sample into account (3).

Although Hug et al. find that the two tip orbitals should be visible at a tip-sample distance of 240 pm, they speculate that the tip might become unstable at such a small distance, because molecular-dynamics calculations (4) predict that nondestructive imaging is not possible at distances comparable to the Si bond length. However, the same molecular-dynamics calculations [page 357 of (4)] also predict a very high surface diffusion rate already for the bare Si surface, in contrast to experimental evidence, so their prediction of a tip instability is not a strong argument against the experimental findings.

Correctly, Hug et al. point out that imaging the two tip orbitals in the topographic mode is only possible if other attractive forces such as electrostatic- or van der Waals forces are present. These background forces are necessary to ensure that ∂γ/∂z > 0, even for z close to the equilibrium next-neighbor distance. In our experiment, this background force is mainly supplied by electrostatic interaction, caused by a voltage bias of 1.6 V between tip and sample, as stated in (1). Hug et al. also question the enhanced sensitivity achieved to short-range forces in our experiments. This enhancement, however, follows straight from classical mechanics; in the limit of small amplitudes, the frequency shift is proportional to the force gradient, whereas, for amplitudes large compared with the range of the tip sample force F ts, the frequency shift is much more sensitive to long-range forces (5).

Hug et al. explicitly speculate that our data resulted from overshooting or ringing of the feedback loop. Both theoretical and experimental considerations argue against this, however. Feedback oscillations at the lowest resonance of the feedback system occur if the gain of the instrument's feedback loop is set too high, because the additional phase shifts above resonance would turn the intended negative feedback into positive feedback. Actual feedback oscillations have three necessary attributes: (i) an oscillation at or above the lowest resonance frequency of the system, (ii) a slowly decaying or an exponentially increasing oscillation, and (iii) a phase between the error signal and the feedback output of roughly 180° (6). None of these characteristics are present in our data. In our AFM, the lowest eigenfrequency in the zdirection is ≈ 6 kHz. To follow the subatomic peaks (distance ≈ 0.25 nm) at a scanning speed of 4 lines per second and an image size of 10 nm, the feedback tracks the surface at a speed of ≈ 320 Hz, 20 times slower than the resonance frequency of our microscope. Feedback oscillations at this frequency are impossible; they would produce “artifacts” with an apparent lateral size of 10 pm—25 times smaller than an atom. The topography and the error signal likewise show clearly that feedback tracking errors are negligible (Figs. 1 and 2).

Figure 2

Line profiles corresponding to the highlighted scan lines in Fig. 1, A, B, and C: raw topographyz(x), black; experimental error signal, red; error-corrected topography, blue; calculated error signal, gray (11). The error-corrected profiles shown in blue are almost identical to the raw topography data; a vertical offset was applied to the error-corrected topography data to distinguish it from the raw topography data. The measured error and the calculated error are virtually identical (rms deviation 1.4 pm), which shows that feedback oscillations are not present.

We agree that modeling the tip sample interaction using the SW potential can only serve as an approximation. We have chosen the SW potential because it yields a much better agreement with the elastic properties of bulk silicon [table 1 in (7)] than competing model potentials with a similar complexity. Using the SW potential also results in excellent agreement between experiment and theory in the analysis of recently published data on frequency shift versus distance (8). The SW potential thus appears to offer a realistic model for the interactions between Si atoms not only for the bulk state, but also for conditions far from equilibrium, such as the interaction of a Si tip with a Si surface.

Finally, we emphasize that in refined calculations of the adatom images (Fig. 3), the experimental and the calculated image show a striking similarity. These data provide compelling evidence that our interpretation is correct, and we remain convinced that our findings will also be reproduced in other laboratories.

Figure 3

(A) Experimental image of a single adatom [magnification from a corner adatom in the faulted half of figure 1C of (1)]. The shape of the adatom images and the depth of the depression between the two crescents varies slightly between the four types of adatoms. (B) Calculation of the normalized frequency shift γ (contribution of SW potential only) for z = 225 pm and the tip geometry defined in figure 4A of (1). Unlike in figure 4B of (1), the plots in this figure were generated using the more accurate equation 4 of (6) instead of equation 12 of (6) for γ, and nearest- and next-nearest-neighbor interactions are taken into account. We expect that strong electrostatic long- and short-range forces act in addition to the SW forces.



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