The Electrostatic Field of CO Functionalized Metal Tips

Abstract

This chapter conclusively shows that the electric field created by CO functionalized metal tips cannot be described by a single dipole. It is necessary to take into account both the positive dipole that describes the electric field created by the metal tip and the negative charge cloud strongly localized in front of the oxygen atom. We have incorporated this insight into a theoretical model that allows the efficient simulation of AFM measurements retaining a first-principles accuracy. Using this model, we have identified the contrast formation mechanisms for localized ionic defects (Cl vacancies on a metal–supported NaCl bilayer). The opposite sign and different spatial extension of the associated electric fields explain the rich contrast observed. While both terms compete to determine the contrast of uncompensated, extended defects like the Cl vacancy, atomic–scale resolution of the ionic lattice arises mainly from the CO electric field as the more extended field created by the metal apex averages out the contribution coming from those periodic and rapidly varying charge distributions. The insight gained from our analysis is used to address the apparent contradiction in the interpretation of previous experiments involving CO molecules either as a tip on a metallic apex probing ionic surfaces or as an adsorbate probed with a pure metallic tip.

Appendices

A DFT Calculation Details

All density functional theory (DFT) calculations were carried out using the projector augmented wave (PAW) method as implemented in VASP [35]. We have used the PBE XC functional supplemented by semi-empirical DFT-D3 van der Waals (vdW) interaction [31], a plane wave cutoff of 400 eV, and fine electronic convergence ($E_{\text {SCF}} = 10^{-4}$ eV) on all calculations. Furthermore,

• All volumetric data was calculated on an uniform mesh with 0.075 Å grid spacing with the dipole correction applied to the z-direction. For the electrostatic potentials, a uniform 1D filter in the z-direction was applied to the volumetric data in order to eliminate high frequency noise ($\lambda = 2$ grid points). The z-component of the E-fields shown in Sects. 15.2, 15.3, 15.4 and 15.7 was calculated from the gradient of the electrostatic potential.

• Calculations used to fit the model (Sect. 15.5) used a $3\times 3\times 1$ grid for sampling the Brillouin zone.

• Sect. 15.5.2 calculations used the $\varGamma$ point for the sampling of the Brillouin zone and ionic relaxations were considered converged when forces were less than $10^{-2}$ eV/Å.

• Sect. 15.5.3 calculations used a $7\times 7\times 1$ grid for the sampling of the Brillouin zone and ionic relaxations were considered converged when forces were less than $10^{-2}$ eV/Å.

B Parametrization of the Short Range Interactions

In order to parametrize the Morse potentials of the SR interaction, we perform static DFT force calculations on a clean NaCl bilayer on a 4-layer Cu(100) slab probed by a CO molecule on a $2\sqrt{2} \times 2\sqrt{2}$ unit cell with a large vacuum (total cell size 15.9 Å  $\times$ 15.9 Å $\times$ 42 Å) (see Fig. 15.19a, b). Calculations were carried out in VASP [35], using the PBE XC functional supplemented by semi-empirical DFT-D3 van der Waals (vdW) interaction [31], a plane wave cutoff of 400 eV, a fine electronic convergence ($E_{\text {SCF}} = 10^{-4}$ eV), and a $7 \times 7 \times 1$ grid for the sampling of the Brillouin zone. Force curves were calculated with a 25 pm interval on 3 different sites: Cl, Na, and bridge (defined as the midpoint between a Na and Cl site). Figure 15.19c, d show the total and vdW forces obtained for those three sites (red, blue, and yellow markers correspond to the Cl, Na, and bridge sites in Fig. 15.19c, d. The electrostatic interaction is calculated, as in the model, from,

$$V_{\text {ES}}=\int {\rho _{\text {CO}}\varPhi _{\text {sample}} d\mathbf {r}^3}$$

(see Fig. 15.19e). Finally, the short range (SR) contribution (Fig. 15.19f) is obtained from

$$V^\text {DFT}_\text {SR} = V^\text {DFT}_\text {total} + V^\text {DFT}_\text {vdW} + V_\text {ES}$$

and fitted, through a least-squares method, to a sum of Morse potentials,

$$V_{\text {SR}} = \sum _{i=\text {Na,Cl,ions}}{D^i_e \left( (1-\exp {[-a^i(|\mathbf {x}-\mathbf {x}^i|-r^i_e)]})^2-1\right) },$$

where $| \mathbf {x} - \mathbf {x}^i |$ is the distance between the O atom in the CO probe and the corresponding ion, $D_e^i$ (well depth), $a^i$ (that controls the inverse of the width of the potential), and $r_e^i$ (equilibrium bond distance) are the species dependent parameters determined by the fitting, and the sum extends to all the atoms of the ionic surface.

Fig. 15.19

a Front view of the relaxed clean NaCl/Cu(100) surface used in the DFT spectroscopy calculations. b Lateral view of the surface along with the CO probe. c Total, d vdW, e electrostatics (ES), and f short range (SR) forces for Cl (blue), Na (red), bridge (yellow) and hollow (gray) sites. Markers correspond to DFT data while the lines to calculations with the model. The Cl, Na, and bridge sites were used in the parametrization of the SR interaction, while the hollow site is calculated to show the ability of the model to reproduce the DFT results on any point of the surface

Table 15.1 Morse potential parameters fitted from DFT calculations. These parameters provide an excellent fit to the DFT force curves

Results for the total, vdW, electrostatic, and short range forces on the 3 sites are plotted on Fig. 15.19c–f. Bullets correspond to the values obtained from DFT calculations and lines represent the results from the model. Note that the DFT-D3 theory is used to estimate the vdW interaction both in the DFT calculations and in the model; hence, markers and lines of Fig. 15.19d are identical. For the three sites, forces calculated with DFT and the model are in excellent agreement. Table 15.1 shows the fitted parameters.

In order to assess the transferability of our model to sites different from the ones included in the SR fitting, we have tested the predictions of the model for a new site: a hollow position (defined as the midpoint between two Cl atoms). Figure 15.19 shows the excellent agreement between the DFT calculations (grey markers) and the model (grey lines) on this site.