Localization of the phantom force induced by the tunneling current

The phantom force is an apparently repulsive force, which can dominate the atomic contrast of an AFM image when a tunneling current is present. We described this effect with a simple resistive model, in which the tunneling current causes a voltage drop at the sample area underneath the probe tip. Because tunneling is a highly local process, the areal current density is quite high, which leads to an appreciable local voltage drop that in turn changes the electrostatic attraction between tip and sample. However, Si(111)-7$\times$7 has a metallic surface state and it might be proposed that electrons should instead propagate along the surface state, as through a thin metal film on a semiconducting surface, before propagating into the bulk. In this paper, we first measure the phantom force on a sample that displays a metallic surface state [here, Si(111)-7$\times$7] using tips with various radii. If the metallic surface state would lead to a constant electrostatic potential on the surface, we would expect a direct dependence of the phantom force with tip radius. In a second set of experiments, we study H/Si(100), a surface that does not have a metallic surface state. We conclude that a metallic surface state does not suppress the phantom force, but that the local resistance $R_s$ has a strong effect on the magnitude of the phantom force.

Figure 1

Finite-element analysis of a voltage drop in a plain bulk material (a) and a bulk material covered with a thin metal sheet on top (b)–(d) are shown. The bulk material has a constant conductivity of ${\sigma }_b=10\frac{\mathrm{S}}{\mathrm{m}}$ [equal to the Si(111) sample used in the experiment]. In (b)–(d) the metallic surface state of Si(111)-7$\times$7 is modelled by a metal sheet with a thickness of 100 pm. The conductivities of the metal sheet increase with ${\sigma }_s={10}^2\frac{\mathrm{S}}{\mathrm{m}}$, (b), ${\sigma }_s={10}^4\frac{\mathrm{S}}{\mathrm{m}}$, (c), to ${\sigma }_s={10}^6\frac{\mathrm{S}}{\mathrm{m}}$ in (d). Without the conductive surface layer, (a), the voltage drop amounts to 230 mV and is highly localized, while the conductive surface layer leads to a reduction and a lateral spreading of the voltage drop.

Figure 2

Qualitative distance dependence of the force $F_{ts}$ according to a Morse potential, the corresponding force gradient $k_{ts}$, and the tunneling current $I$. For small cantilever amplitudes, the frequency shift $\Delta f$ is proportional to $k_{ts}$. In region I, $\Delta f$ decreases, whereas $I$ increases. In region II, $\Delta f$ starts to increase.

Figure 3

(a) Schematic of the experimental setup. With an implemented switch, the preamp can be used for the $I$ signal (switch in position A-B) or can be shorted to ground (switch in position A-C). In (b), simultaneous acquired $I$ and $\Delta f$ data in constant height mode are shown. The images were scanned from top to bottom. In the middle section the $I$ signal was shorted to ground to check if the phantom force effect is real (i.e., caused by a voltage drop at the sample surface) or caused by a preamp artifact.

Figure 4

(a) Schematic of a tip with a large macroscopic radius in front of a surface with a highly localized change of the surface potential induced by the tunneling current. (b) The same situation, but with a smaller tip.

Figure 5

(a) Tunneling current $I$ and (b) $\Delta f$ data taken during constant height scanning. A Si$($111$)$-7$\times$7 unit cell is outlined by a white diamond in (a). In (b) the adatoms appear more bright, which is due to less attraction induced by the tunneling current. (c) Phantom force contribution as a linear dependence between the current and $\Delta f$ data. The phantom force slope $\Xi$ of the line is $0.73\frac{\mathrm{Hz}}{\mathrm{nA}}$. Images are 10 nm$\times$10 nm, $A\approx 400$ pm, $k=1800\frac{\mathrm{N}}{\mathrm{m}}$, $f_0=19130$ Hz and $V_{\mathrm{bias}}=-1.5$ V.

Figure 6

(a) Spectrum of $\Delta f\left(V_{\mathrm{bias}}\right)$ for determining the $V_{\mathrm{CPD}}$ between tip and sample to 0.4 V. (b) Spectrum of $\Delta f\left(z\right)$ taken at $V_{\mathrm{CPD}}$. The macroscopic tip radius $R$ was extracted by fitting the curve to a long-range van der Waals force contribution, for (b) the extracted $R=600$ nm. The inset shows the tip radius for a parabolic tip shape.

Figure 7

The phantom force slope $\Xi$ in $\frac{\mathrm{Hz}}{\mathrm{nA}}$ characterizing the phantom force is plotted versus the macroscopic tip radii $R$. Two different sensors were used: sensor 1 (red triangles) and sensor 2 (blue dots). The phantom force slopes $\Xi$ show no dependence on the macroscopic tip radii $R$.

Figure 8

Phantom force slopes $\Xi$ plotted versus the defect density on the Si(111)-7$\times$7 surface. The phantom force slopes $\Xi$ seem small for less defects on the Si(111)-7$\times$7 and more defects point to increased $\Xi$.

Figure 9

(a) At tip-sample distances characteristic for STM and with a tunneling current, the phantom force effect can be observed. Simultaneous acquired $I$ and $\Delta f$ data with atomic contrast are shown in (b) and (c) for $V_{\mathrm{bias}}=1.5$ V. In (d), the relation between $\Delta f$ and $I$ data is plotted. Data points with increased $I$ data, as the defect outlined in red, show a stronger decrease (less attraction) of the $\Delta f$ values. However, at close tip-sample distances, (e), and low bias, in this case 200 mV, the defect appears darker (more attractive), (f). Image (g) shows (f) with low-pass filtering and plane substraction applied for clarity. Images are 2 nm $\times$ 2 nm, $A=100$ pm, $k=1800\frac{\mathrm{N}}{\mathrm{m}}$, $f_0=19131$ Hz.