The calculations presented in this study were performed by using the DFT module of the CP2K code [12
] and employing the PBE exchange-correlation functional [13
]. Gaussian basis sets of DZVP quality were used with semicore GTH pseudopotentials [14
]. The pseudopotentials included 9, 10, 14 and 18 valence electrons for Na, Mg, W and Pt. The auxiliary plane-wave basis, used to calculate the Hartree energy, had an energy cutoff of 4000 eV. To account for the metallic nature of the tip (i.e., a very small band gap) in the simulation, we also employ Fermi–Dirac smearing of the molecular-orbital occupation numbers, with an electronic temperature of 2500 K.
Both the NaCl(001) and MgO(001) surfaces were modeled using a periodic slab, 6 × 6 atoms in area and three atomic layers deep, where the bottom-most layer is frozen in bulk-like positions. For a direct comparison with the results of previous plane-wave calculations employing a periodic-tip model, the NaCl(001) surface was also modeled with a 5 × 5 primitive unit cell surface area, three atomic layers deep, which was chosen to match the x-y periodicity of the periodic tip model. The slabs are periodic in the x-y directions, and there is a vacuum gap of 30 Å in the z-direction. The lattice separation in the NaCl slab is 2.78 Å and in the MgO slab is 2.12 Å. When the geometries of the surface slabs are optimized they exhibit rumpling, with the anions protruding from the surface plane. The corrugation of the NaCl surface is approximately 0.1 Å and 0.04 Å in MgO. The one-electron band gaps for the NaCl surface at 4.9 eV, and for the MgO surface at 3.6 eV, are underestimated, which is typical for PBE calculations.
The tip models are shown in . The cluster Cr and W tips consist of four-layered pyramids, cut from the body-centered-cubic (BCC) structure of the bulk crystals. The top two layers of the 30 atom tips are frozen, and the lower two layers are free to relax. For a direct comparison with the plane-wave calculations presented in [7
], a periodic-tip model consisting of a three-layer BCC slab of Cr with symmetric pyramidal protrusions () was also employed. It is well-known that the structure and morphology of the tip has a significant effect on the tip–surface interaction [17
]; however, this type of pyramidal protrusion was shown to be the best match to the experimental measurements reported for this system [7
]. The work functions for the Cr tips are calculated as being approximately 3.7 eV for both tip models, which is similar to previous calculations for the Cr surface [19
] but slightly less (by 0.2–0.6 eV) than the experimental values [20
]. For all of the tip models the Fermi energy lies well within the band gap of the ionic surface slabs.
(a) Side-on view of the structure of the Cr and W cluster tip models. (b) The structure of the periodic Cr tip model.
To calculate the tip–surface force field, the frozen part of the tip is fixed at a position above the surface, the system relaxed, and the total energy calculated. The tip is then moved a small distance closer to the surface, and this is repeated to map out the energy as a function of the tip position. The gradient of this energy in the z
-direction is then used to determine the tip force. The tip height is defined as the separation that would exist between the front atom of the tip and the surface plane if there were no relaxation in the tip (i.e., with the tip away from the surface). The DFT method is known to underestimate atomistic dispersion forces; however, these are not expected to contribute to the atomic-scale variation of the force on the tip above different atomic sites [3
]. A macroscopic van der Waals attraction is added to the total force on the tip for simulated image calculations, as stated in the Experimental section.
To correct for the basis-set-superposition error (BSSE), which acts to increase the force on the tip originating from the interaction with the surface, due to the overlap of the basis functions of the surface and tip, we employ the counterpoise method to correct the total system energy for different tip positions relative to the surface [22
]. Our calculations demonstrate that the BSSE is similar at a given tip height above both anions and cations (approximately 0.1 eV at 4 Å), and is therefore not likely to contribute to atomic-scale contrast. Furthermore, the BSSE is only present at tip–surface separations below 4.5 Å, as above this height there is no orbital overlap.
The total energy as a function of tip height, for the apex of the Cr cluster tip directly above both Cl−
ions in the NaCl surface, and above both O2−
ions in the MgO surface, is shown in . Here the energy change is relative to the energy of the tip and surface when they are completely separated. In each case it is clear that the force is largest directly above anions in the surface, significantly so in the range probed by noncontact imaging, i.e., 3–5 Å. For each tip above an anion in the surface, at close approach (approx. 3–4 Å) the force increases markedly due to a structural change consisting of strong displacement of an anion out of the surface to bond to the tip apex. This jump of a surface ion to the tip apex will result in hysteresis in the tip–surface force field and atomic-scale dissipation being measured by the NC-AFM instrument [23
]. For the Cr tip interacting with the NaCl surface, the total charge on the tip at a separation of 6 Å is less than −0.01 |e
| (from a Mulliken population analysis); however, when the tip comes closer to the surface above a Cl−
ion, there is a small charge transfer to the tip (of −0.03 |e
| at a height of 4 Å and of −0.1 |e
| at 3 Å). For the tip above the MgO surface, a similar transfer occurs, but it is slightly more pronounced (a charge on the tip of −0.16 |e
| at 4 Å above an O2−
ion in the surface).
(a) Energy as a function of cluster Cr tip height above the NaCl(001) surface. (b) Energy as a function of tip height above the MgO(001) surface.
shows the total energy as a function of the tip height for the W tip directly above Cl− and Na+ ions in the NaCl(001) surface. As before, the interaction is strongest above the anion, and increases significantly below 4.5 Å (note this is not due to an instability caused by an atom jump). The charge transfer to the tip at close approach is similar to that in the case of the Cr tip interacting with this surface, which is to be expected due to the similar Fermi energies of the two clusters. In the case of both tips, the origin of the charge transfer at close approach and the increased tip force above the anions is due to the hybridization of the d states in the tip apex atom with the p states in the surface anion.
Energy as a function of tip height for the W tip interacting with the NaCl(001) surface.
In each of the tip–surface combinations, the calculated force fields would result in the anions being imaged as prominent protrusions in a constant-frequency-shift image of the surfaces. To demonstrate this, and show the extent of typical atomic scale corrugation, we simulated the imaging of the NaCl surface with the Cr cluster tip, using typical imaging parameters based on a traditional silicon cantilever (listed in the Experimental section). The force field used for these calculations was calculated on a lateral square grid with a spacing of 1/8 of the lattice constant between points (i.e., four points between adjacent surface ions), and between tip heights of 3 Å and 7 Å. shows a constant-Δf image (Δf = −60 Hz) of the NaCl surface, in which the distance of closest approach is 3.6 Å. The rumpling is approximately 0.6 Å with the protrusions corresponding to Cl− ion lattice positions and depressions to Na+ ion positions.
Constant-frequency-shift image (Δf = −60 Hz) of the NaCl surface imaged with the cluster Cr tip.
To investigate both the contribution of the electronic structure of the tip and the type of simulation method to the interaction between a metallic tip and an ionic surface, we calculated the changes in total energy as a function of tip position for the periodic Cr tip model interacting with the NaCl surface. We used the exact same system configurations as used in previous plane-wave DFT calculations, employing the VASP code [25
] (as described above). The same PBE correlation-exchange functional employed in [7
] was used here. The main difference in the model we apply is in the form of the basis functions, in which the wave function of the system is expanded: Here they are Gaussian and atom-based, as opposed to being plane waves.
shows the total energies (BSSE corrected) as a function of tip position (the exact same positions calculated in [7
]). As in [7
], Morse bond functions were fitted to these energies as a function of tip height for each position, in the noncontact range of 4–7 Å, where no instabilities occur. The derivative of this function gives the force on the tip due to the interaction with the surface, as a function of tip height, which is shown in in the range of 4–6 Å, along with the curves from the plane-wave calculations presented in [7
], and fitted curves for the cluster Cr tip model discussed above. These forces show that the periodic tip model leads to an overall force that is quantitatively smaller than that in the cluster model for a given tip height, by approximately 10% in the 4.5–5.5 Å range.
Total energy changes as a function of tip height for the periodic Cr tip interacting with the NaCl(001) surface, and Morse function curves fitted to the data points.
Figure 6 Tip force as a function of height directly above Cl− (left) and Na+ (right) ions in the NaCl(001) surface, for the cluster tip and periodic tip, and an identical periodic tip but with energies determined from plane-wave (VASP) calculations [7 (more ...)
The absolute forces between the NaCl surface and the periodic tip model above both Cl− and Na+ ions, as calculated in this study, are larger than the forces calculated by using exactly the same setup in the previous plane-wave calculations: The forces are larger by approximately 50–100% in the 4.5–5.5 Å range.