Determining molecule diffusion coefficients on surfaces from a locally fixed probe: Analysis of signal fluctuations

Methods of determining surface diffusion coefficients of molecules from signal fluctuations of a locally fixed probe are revisited and refined. Particular emphasis is put on the influence of the molecule's extent. In addition to the formerly introduced autocorrelation method and residence time method, we develop a further method based on the distribution of intervals between successive peaks in the signal. The theoretical findings are applied to scanning tunneling microscopy measurements of copper phthalocyanine (CuPc) molecules on the Ag(100) surface. We discuss advantages and disadvantages of each method and suggest a combination to obtain accurate results for diffusion coefficients.

Figure 1

Relaxed structure of the CuPc molecule. Notice that the terminating bonds are connected to hydrogen atoms. The assigned detection area marked in gray is a circle with radius $R=7.6$ Å. It corresponds to the smallest circle that entails all nuclei.

Figure 2

(a) Fluctuating STM tunneling current $I\left(t\right)$ observed during CuPc diffusion on Ag(100) at $T=192$ K ($U_{\mathrm{bias}}=1.7$ V, $I_{\mathrm{setpoint}}=20$ pA). The dashed line marks the threshold value $I_c$, above which the distribution of the current values has no longer a Gaussian shape (see Fig. 3). (b) The rectangular signal $I_{\mathrm{rec}}\left(t\right)=\Theta [I\left(t\right)-I_c]$ associated with $I\left(t\right)$. The widths of the rectangles give the residence times $\tau$ and the distances between the rectangles give the interpeak times ${\tau }^{\prime }$.

Figure 3

Histogram of measured current values for $T=192$ K after shifting the maximum to $I=0$. The solid line marks a Gaussian fit with zero mean and standard deviation determined from the histogram for negative currents. The inset shows an enlargement to demonstrate the determination of the threshold value $I_c$ used for separating the diffusion-induced signal from the noise (see text).

Figure 4

(a) Normalized ACFs $C_{\mathrm{rec}}\left(t\right)/C_{\mathrm{rec}}\left(0\right)$ as a function of time for two temperatures (symbols) and best fits with Eq. (4) for $t<{\tau }_R$ (solid lines). (b) Arrhenius plot of the resulting diffusion coefficients (symbols). A least-square fit (solid lines) according to the Arrhenius law yields $E_a=30$ meV and $D_0={10}^{-9.4}$ cm${}^2$/s.

Figure 5

Illustration of the geometry used for the calculation of (a) the RTD and (b) the ITD. The dashed lines indicate the uniformly distributed initial probabilities on the rings displaced by $\epsilon R$ (a) and $\left({\epsilon }^{\prime }R\right)$ (b) from the absorbing boundary (solid lines). MC stands for the diffusing molecule center.

Figure 6

(a) Residence time distribution for CuPc/Ag(100) at two temperatures (symbols). The solid line marks the result predicted by Eq. (5) after determining the parameters $D$ and $\epsilon$ from the exponential tail region $\tau \gg {\tau }_R$ [cf. Eq. (8) and discussion in text]. (b) Arrhenius plot of the extracted diffusion coefficients. A least-square fit (solid line) with the Arrhenius law yields an activation energy $E_a=30$ meV and a pre-exponential factor $D_0={10}^{-9.4}$ cm${}^2$/s.

Figure 7

(a) ITD for CuPc/Ag(100) for two temperatures. The inset shows the exponential decay at large times, where the straight solid lines (dashed lines in the main log-log plots) are fits with Eq. (13), yielding $D$ (see text). Using these $D$ values, the solid lines in the main plots mark the short-time behavior after fitting with Eq. (11). (b) Arrhenius plot of the extracted diffusion coefficients for all seven investigated temperatures. A least-square fit (solid line) with the Arrhenius law yields an activation energy $E_a=31$ meV and a pre-exponential factor $D_0={10}^{-9.6}$ cm${}^2$/s.

Figure 8

Comparison of the diffusion coefficients obtained from the three methods. Squares, dashed line: ACF method ($E_a=30$ meV, $D_0={10}^{-9.4}$ cm${}^2$/s). Circles, dotted line: RTD method ($E_a=30$ meV, $D_0={10}^{-9.4}$ cm${}^2$/s). Diamonds, dashed-dotted line: ITD method ($E_a=31$ meV, $D_0={10}^{-9.6}$ cm${}^2$/s).