Moving atoms on surfaces: Impact of external parameters on required lateral force

Combining scanning tunneling microscopy and atomic force spectroscopy the dependence of the lateral force to move single adsorbed atoms on surfaces on the temperature is explored. A decrease of the force with increasing temperature is observed for all atoms and surfaces investigated and may be captured in a simple model. Crossing the critical temperature as well as surmounting the critical magnetic field of a superconductor does not lead to discriminable changes in the lateral force.

Figure 1

(a) Atomically resolved STM image of Pb(111) that was obtained by dragging a single Pb atom with the tip across the surface (bias voltage $V=1\text{mV}$, tunneling current $I=3\text{nA}$). The different contrast is due to on-top ($\circ$), hcp ($◃$), and fcc ($▹$) lattice sites. The $\langle 11\overline{2}\rangle$ direction is indicated. (b) Sketch of first two Pb(111) lattice planes with indication of on-top, hcp, and fcc sites. (c) Illustration of the experimental setup that determines vertical ($F_z$) and lateral ($F_x$) forces. The distance between tip and surface is set by $V$ and $I$. $F_z$ is extracted from changes $\mathrm{\Delta }f$ in the resonance frequency $f_0$ of the oscillating tuning fork. (d) Conductance $G=I/V$ as the tip is scanned across a Pb adatom on Pb(111) at constant height ($100\text{mV},5.8\text{K}$). Prior to scanning the tip height was set at $100\text{mV},56\text{nA}$. Subsequently the tip was approached by $195\text{pm}$. (e) $\mathrm{\Delta }f$, simultaneously recorded with $G$ in (d). The abrupt changes in (d) and (e) at $x\approx 0.11\text{nm}$ indicate the displacement of the adatom. The maxima (minima) of $G$ ($\mathrm{\Delta }f$) are separated by $\approx 0.20\text{nm}$ (horizontal bar), which corresponds to the distance between hcp and fcc sites on Pb(111).

Figure 2

AFM data of a single Pb adatom on Pb(111) as a function of the tip-surface distance ($\mathrm{\Delta }z$) and the lateral position ($x$) acquired at $5.8\text{K}$. Vertical distance $\mathrm{\Delta }z=0\text{pm}$ is defined by feedback loop parameters $100\text{mV}$ and $56\text{pA}$. The tip-surface distance decreases for $-1000\text{pm}\le \mathrm{\Delta }z\le 195\text{pm}$. The adatom is located at $x=0\text{nm}$. (a) Resonance frequency variation $\mathrm{\Delta }f$ and (b) resulting vertical force $F_z$ upon decreasing (from top to bottom) the tip-surface distance. (c) Interaction energy $E=-\int F_z\text{d}z$ and (d) lateral force $F_x=-\partial E/\partial x$. Symbols in (a) depict raw data. For evaluating $F_z,E,F_x$ smoothed $\mathrm{\Delta }f$ data were used. In all panels, black lines are additional data, which are not displayed as symbols for clarity.

Figure 3

Temperature dependence of ${\widehat{F}}_x$ for the indicated adatom-surface combinations. The solid lines represent fits of a simple model to the data (see text). The vertical dashed line indicates the critical temperature of Pb(111), $T_{\text{c}}=7.2\text{K}$. Inset: $\mathit{dI}/\mathit{dV}$ spectra at $T=5.9\text{K}$ (circles) and $T=8.0\text{K}$ (crosses). The feedback loop was disabled at 10 mV and 120 pA (circles), 100 pA (crosses).

Figure 4

(a) Ingredients of the model. The tip moves with velocity $v_x$ across the surface and interacts with the adatom via the energy $E_{\text{t}}$. The adatom at $x=0$ is subject to the superposition of $E_{\text{t}}$ and the periodic lattice potential $E_{\text{s}}$. The adatom hops into the global minimum of $E_{\text{t}}+E_{\text{s}}$ if the barrier $E_{\text{b}}$ is surmounted. The lateral force to move the adatom is obtained from ${\widehat{F}}_x=-\partial E_t(x\approx 0)/\partial x$. (b) Lateral force ${\widehat{F}}_x$ calculated within the model as a function of the lattice constant $a$ and evaluated at the indicated temperatures. For $T=0\text{K}$ the behavior is virtually identical with the variation previously suggested for the maximum lateral force in a sinusoidal lattice potential (top curve) [24].

Figure 5

${\widehat{F}}_x$ as a function of an external magnetic field applied to Pb(111) at $5.8\text{K}$. The vertical dashed line indicates the critical field at $5.8\text{K}$ ($B_{\text{c}}=25\text{mT}$) above which the superconducting state collapses. Inset: $\mathit{dI}/\mathit{dV}$ spectra of the BCS energy gap at external magnetic fields $B=0\text{T}$ (circles) and $B=46\text{mT}>B_{\text{c}}$ (crosses). The feedback loop was disabled at $9.5\text{mV}$ and $100\text{pA}$ (circles), $120\text{pA}$ (crosses).