Probing the geometry of copper and silver adatoms on magnetite: quantitative experiment versus theory† †Electronic supplementary information (ESI) available: Experimental and computational details, as well as further details on the results and analyses. See DOI: 10.1039/c7nr07319d

Benchmarking DFT calculations against precise normal incidence X-ray standing wave measurements.


1
. Ag and Cu were deposited using an Omicron EFM3 evaporator with the sample held at room temperature, and the deposition rate was monitored by a water-cooled quartz crystal microbalance (QCM). One monolayer (ML) is defined as 1 adatom per Fe 3 O 4 (001)-(√2×√2)R45° unit cell, or 1.42x10 14 atoms/cm 2 . These metals (Ag and Cu) were selected for this study due to the high adatom densities that can be achieved with minimal cluster formation 2 , their resistance to adsorption of the residual gas at room temperature, and because they lie at the extremes of the range of geometries based on preliminary DFT+U calculations. The normal incidence X-ray standing wave (NIXSWR) measurements were performed at a normal incidence geometry with respect to the Bragg plane from which the scattering occurred, thus the photons were only incident on the surface normal for the (004) measurements. The coherent positions and fractions over all three reflections are calculated with respect to a Fe 3 O 4 unit cell with the tetrahedral iron atoms at the origin. The apparent height (H ad ) of the adatom, with respect to a bulk-like terminated Fe 3 O 4 (001) surface was calculated from the coherent position of the (004) data, p 004 , by: where d 004 = 2.099 Å is the spacing between (004) scattering planes, and 2 = 0.5 is the coherent position of the Fe oct O 2 plane between the same planes.

Computational details
All the theoretical calculations were performed using the Vienna ab initio Simulation Package (VASP) 3,4 using the Projector Augmented Wave (PAW) approach 5, 6 with a basis set cut-off energy of 550 eV. Calculations were initially performed using the Perdew-Burke-Ernzerhof (PBE) 7 exchange correlation functional with an effective on-site Coulomb repulsion term U eff = U-J = 3.6 eV 8 ; this is the standard procedure for Fe 3 O 4 above the Verwey transition [9][10][11] . We also tested a variant optimized specifically for solids, PBEsol 12 , which accounts for the well-known issue of disfavoured density overlapping present in PBE, using the same U eff . The hybrid functional HSE 13 was investigated with the standard mixing factor 25%, and screening length (0.207 -1 Å -1 ). However, due to the expensive computational cost of such calculations the k-meshes were reduced by a factor 2 along each direction. The kpoints mesh has been optimized such that it delivers total energy with an accuracy of better than 1 meV and reducing the k-mesh by a factor of 2 at PBE level leads to a change in the total energy of only 1 meV. In all cases, structures were relaxed until forces were smaller than 0.02 eV/Å.
The surface calculations utilized an asymmetric surface slab, resulting in a significantly cheaper calculation, with 9 fixed layers and 4 relaxed layers with the subsurface cation vacancy reconstruction 10 . Due to large size of the unit-cell (~100 atoms) the Figure S1: Results of the fitting of the NIXSW data from the a) (113) and b) (044) reflection of Fe3O4. The absorption profiles were obtained from the photoemission yield of Cu 2 3 2 ⁄ and Ag 3 5 2 ⁄ core levels.
Electronic Supplementary Material (ESI) for Nanoscale. This journal is © The Royal Society of Chemistry 2017 adoption of a symmetric setup would be computationally prohibitive for HSE-type calculations. The vacuum "thickness" (separation between adjacent supercells perpendicular to the surface) was set to 14 Å, and the k-mesh 5 × 5 × 1. Initially standard procedure was followed: the surface slabs were based on a theoretical lattice parameter obtained by relaxing the bulk unit cell with the relevant functional (Fd3 ̅ m structure above the Verwey transition with a Γ-centered k-mesh of 5 × 5 × 5). PBE+U and HSE overestimate the lattice by 0.75% and 0.18 %, respectively, whereas PBEsol+U underestimates by 0.61%. Large deviations from the lattice parameter are known to affect calculations of phonon and magnetic properties, but the values obtained here would not be considered problematic, especially for calculations of adsorption geometries and energies. As shown in the main text however, agreement with experiment is only obtained when a lattice parameter within 0.2% of the experimental value (8.396 Å) is used.
Furthermore, under such conditions the PBE+U and HSE functionals produce similar results.
The adsorption energy of the metal adatom on the surface E ad is defined by: i.e. the adsorption energy is the difference between the total energy of the adatom surface slab and the combined total energy of the clean Fe 3 O 4 (001) surface and an isolated metal atom in vacuum.

0.4 ML Cu after annealing
As discussed in the main text, and shown in Table S1, after annealing the Fe 3 O 4 (001)/Cu layer to approximately 550 K a significant increase in the coherent fraction in the Cu 2p XSW was observed. This is attributed to the removal of the metastable Cu adatom site observed in the STM (Cu 1 * in Fig. 1b). No difference is observed in the coherent position, suggesting one of the following three scenarios: (i) the alternative Cu 1 site has a poorly defined adsorption height, centered around that of the regular Cu 1 site; (ii) it has an adsorption height exactly 0.5 layer spacings of the (004) above or below that of the regular Cu 1 site (1.05 Å), which would result in an anti-phase absorption profile to the regular Cu 1 site; or (iii) has a completely random adsorption height over the (004) spacing (2.1 Å).

Direct imaging of adatom adsorption site
The coherent fraction, f H , and coherent position, P H , introduced in the main text with a colloquial definition, can be more specifically interpreted as the amplitude and phase (respectively) of the H th -order Fourier component, ℱ , of the element specific geometrical structure factor, ℱ.
As mentioned in the text, the NIXSW experiments presented here exploited the (004), (113) and (044) Table S1) for Fe tet in bulk sites. Panel (d) has a ball and stick schematic, with the same colour scheme as Fig. 2, of the first Fe oct O and Fe tet layers below the Fe tet layer that this two dimensional slice is transecting. In all cases the slice is taken at a position in the [001] direction where the maximum atomic density is found and is specifically, as a fraction of the (001)   layers in the bulk unit cell. For simplicity, only the results for adsorption of the adatoms above one of these layers are presented as 2D maps. The slice corresponds to a plane that is parallel to the Fe 3 O 4 (001) surface, at the determined height of the respective adatom, obtained directly from the (004) reflection. The atomic density maps, display two global maxima (at the centre and the corners) as well as multiple local maxima. These local maxima are assigned to artifacts originating from utilising a finite set of crystallographic planes resulting in a truncation error in the Fourier expansion, and can be discounted as the global maxima are 40% more intense than these local ones, and, as discussed below, a theoretical reconstruction of the Fe tet exhibits identical artifacts. Thus the Cu and Ag adatoms are assumed to occupy only the sites corresponding to the global maxima, which are all located in sites bridging two oxygen atoms. In the bulkterminated Fe 3 O 4 (001) surface there are two such sites, specifically one site with a Fe tet atom directly below and one site without; the global maxima, and thus the adatoms, lie in the latter. However, due to the (√2×√2)R45° reconstruction of the surface half of these sites contain an interstitial Fe tet directly below it 10 .
As the NIXSW determines the adsorption site with respect to the bulk lattice, it is insensitive to the surface reconstruction and cannot discriminate between these two different surface sites. It is likely, though, that the adatom sits in the site without any Fe atom directly below it, as is favoured by our DFT-based calculations.
As mentioned above, the Fe 3 O 4 (001) surface has four possible terminations, each a symmetrically identical Fe oct O layer. These four layers are related by a 90° rotation and a translation of magnitude √2 × 0.25 along the [11 ̅ 0] direction, where a is the bulk lattice parameter. All four terminations are equivalent, therefore all will be present on the sample surface and the adatoms will occupy symmetrically identical sites at all four terminations. In Fig. 1 the atomic density map, obtained from the expansion of eqn. (3), is shown for a single one of these terminations. Figure S2 shows a three dimensional isosurface map, of the atomic density, over the whole unit cell for Ag and Cu. For comparison an atomic density map was calculated from the theoretical coherent fractions and positions for a tetrahedral Fe atom (shown in Table S1) and included in Figure S2. The two dimensional slice of the atomic density map that is comparable to Fig. 1f for the ideal tetrahedral Fe atoms and the Ag adatom, is shown in Fig. S3b-d. Figure S3a shows Fig. 1f without the overlayed atomic positions for clarity.

Cu 2p 3/2 and Ag 3d 5/2 X-ray photoelectron spectroscopy
Representative Cu 2p 3/2 and Ag 3d 5/2 X-ray photoelectron spectra are shown in Figure S4. The Cu 2p 3/2 spectra were measured at a photon energy of 1100 eV, and the Ag 3d 5/2 spectra at 600 eV. In both cases the absolute binding energy was calibrated against the Fermi energy acquired at the same photon energy. The Cu 2 3 2 ⁄ and Ag 3 5 2 ⁄ XPS measurements exhibit a binding energy of 932.8 eV and 368.8 eV, respectively, consistent with a 1+ oxidation state 19,20 (Note that the absolute binding energies of Ag oxidation states are controversial in the literature 20 , however 368.8 eV is a somewhat higher binding energy than the welldefined Ag 3 5 2 ⁄ binding energy of bulk metallic silver at 368.3 eV [21][22][23] ).

PBE+U calculations: U (eff)
The PBE+U calculations presented in the main body of the text utilize a U (eff) of 3.61 eV, which was chosen because it accurately models the electronic structure of the bulk magnetite 24,25 . Specifically a U (eff) of 3.61 eV reproduces the magnetic moments well, predicts a small gap in the majority spin contribution resulting in a bulk semi-metal and reproduces the charge disproportion when performing a Bader charge analysis. However, as discussed in the main text, this U (eff) fails to model the adsorption structure of the metal adatoms accurately. This raises the question of whether there is a U (eff) that would model the structure properly, at the expense of modelling the electronic properties improperly. To this end we ran a series of calculations over a range of U (eff) values between 2 and 5 eV. These calculations were performed with a basis set cut-off energy of 350 eV, with a k-mesh of 2x2x1. From these calculations the resulting H Ag , H Cu (as described in eqn. (1) in the main text) and lattice parameter (a) were compared against the experimental values, shown in Fig. S5. When the U (eff) values are set significantly lower than 3.61 eV then the lattice parameter tends towards the experimental value; above 3.61 eV the lattice parameter plateaus. A U (eff) value of 3.61 eV corresponds to the minimum difference, between experiment and theory, in the adsorption height of the copper adatom, but varying U (eff) had little to no effect on the adsorption height of the Ag adatom. Note that in all cases the adsorption height of both adatoms is underestimated and the lattice parameter is overestimated. Thus is not possible to "fudge" the correct result by variation of this parameter.