Revealing the Angular Symmetry of Chemical Bonds by Atomic Force Microscopy

When more than two atoms are involved, the forces that act between atoms generally depend not only on their distance but also on the angles between the atoms. Chemical bonds establish an equilibrium between attractive and repulsive interactions for atoms that is described by the Morse potential (1). When more than two atoms are involved, the bonding energy generally depends on the bonding angle as well as distance, and model potentials with an angular dependence, such as the Stillinger-Weber potential (2), are needed. The directional character of the atomic bonds is reflected in the crystal structure. Most metals condense in a face-centered cubic (fcc), hexagonal close-packed (hcp), or body-centered cubic (bcc) lattice (3). In particular, metals with unfilled p or d shells can show strong directional bonding [chapters 1.4 and 2.5 in (3)]. Although fcc and hcp lattices have the largest possible number of nearest neighbors, the angular dependence of atomic forces leads to slight energetic differences between fcc and hcp lattice structures. Atomic manipulation of cobalt atoms on a copper surface by scanning tunneling microscopy (STM) has shown that a Co adatom on a Cu(111) surface prefers to occupy an fcc site over an hcp site, although the adatom is in both cases sitting directly in a dip between three Cu atoms (4).

Atomic force microscopy (AFM) (5) can measure forces between individual atoms with great precision, and recent progress in three-dimensional force spectroscopy (6–8) helps to investigate the angular dependence of atomic bonding forces. Initial manifestations of an angular dependence of bonding forces led to the observation of subatomic features on the surface of Si (9). The data were explained by the directional dependence of covalent bonds between a Si tip with a Si adatom on the sample [figure 4 in (9)] as predicted by calculating the tip-sample interaction using the Stillinger-Weber potential (2), a classic model potential for diamond structure materials. Density functional theory (DFT) calculations by two groups (10, 11) supported the covalent-bonding theory, but other authors (12) proposed that the data could have been the result of a feedback artifact, and there has been a recent theoretical proposal that the angular dependence could be caused by multiple-atom tips (13). Simultaneous STM and AFM experiments of a W tip imaging a graphite sample showed a single current maximum and multiple force maxima in the image of the W tip atom (14); hence, AFM can address electronic states that are spatially more confined than electronic states that are accessible to STM. In that experiment, the light and small carbon atoms have produced repeated images of the electronic valence states in tungsten atoms [see figures B and C in (15)]. Recent DFT calculations confirmed that the electronic structure of W should show these small features (16). However, the softness and close atomic packing of graphite make it difficult to calculate the experimental observables, and distance-dependent data were not available in the first experiments.

Precise measurements of the interaction of two single and clearly defined atomic bonding partners as a function of distance and angle would allow us to study the evolution of the atomic angular dependencies and provide well-defined experimental data that can serve as a reference for theoretical calculations. Here, we present an experimental study of the interaction of two well-defined bonding partners: a single CO molecule that is bonded to a Cu(111) surface and the metallic front atom of a tip. We analyzed tunneling current and frequency shift in three dimensions and were therefore able to recover the force and potential energy between two atomic bonding partners as a function of angle and distance.

We used STM (17) combined with frequency modulation AFM (18), where a quartz cantilever (qPlus sensor) (19) with a stiffness of k = 1800 N/m, an eigenfrequency of f₀ = 27275.20 Hz, a quality factor of Q = 195870, and a W tip oscillated with a constant amplitude A (here, we use A = 50 pm and A = 100 pm). When the tip of the cantilever was subjected to a tip-sample force F_z with gradient k_z = −dFz/dz, the frequency shifted to f = f₀ + Δf withΔf = f₀ <k_z>/(2k) (1)Our commercial cryogenic scanning probe microscope, operated at 4.3 K (20), measured an average tunneling current <I> in parallel with an average force gradient <k_z> (21). The interaction between tip and molecule was probed by three-dimensional force spectroscopy (6–8), where a set of constant height images spaced by a vertical distance of 10 pm was collected to recover the energy profile. We used the method of Sader and Jarvis (22) to recover forces and energies from frequency shifts (23) (figs. S1 and S2). Because of the small oscillation amplitudes we used, the frequency shift was approximately proportional to the force gradient, so that the atomic short-range contributions (ranging from a few piconewtons up to ≈300 pN) could be measured with a high signal-to-noise ratio on top of the long-range van der Waals forces ranging up to 2 nN (fig. S3).

The tips were made from a multicrystalline W wire and cleaned and prepared in situ by electron bombardment and field evaporation at voltages up to 15 kV. Because the tips were prepared and transferred into the low-temperature microscope in an ultrahigh vacuum, we assumed that they expose clean surfaces with a W front atom.

Carbon monoxide is known to chemisorb via its C atom onto a site on top of a Cu atom on the Cu(111) surface (24). It can be manipulated in a controlled fashion on the Cu(111) surface and is observed as a dip in low-bias STM experiments (25). Figure 1 shows the setup of the experiment: A W tip was placed above a CO molecule that is chemisorbed on a Cu(111) surface. Three highly symmetric tip configurations are studied in detail here: Fig. 1A depicts tip 1, a W tip presumably oriented in a [001] direction. This tip was poked into the Cu(111) surface to change its geometry. This tip-shaping procedure used is standard in the low-temperature STM community, and it is commonly assumed that poking the tip in Cu is likely to cover the tip apex with Cu [e.g., (25, 26)]. Further below, we provide evidence that, although tip 1 was poked into the Cu surface, it had a W atom at its front. Figure 1E shows tip 2, a W tip presumably oriented in a [011] direction with a twofold rotational symmetry with respect to the z axis that causes a double dip in the force maps. Tips 2 and 3, however, were treated with great caution after the in situ high-voltage preparation to maintain a safe distance to the sample such that we are sure they expose clean W at the apex. Figure 1I depicts tip 3, a W tip presumably oriented in a [111] direction with a threefold rotational symmetry with respect to the z axis causing a triple force minimum. The transition from tip 2 to tip 3 occurred after a gentle ”poke” (a voltage pulse that caused, among structural changes in the tip apex, single W atoms to drop from the tip onto the Cu surface). As can be seen in fig. S3, B and C, the tip radius was not affected by that transition.

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Fig. 1

Constant height images of tunneling current and deconvoluted force for three different tips over a CO molecule on Cu(111) recorded at three different heights (see axis indicators in top row): (A to D) current and force for a tip that is highly symmetric with respect to rotations around the z axis; (E to H) current and force for a tip with a twofold dip in the force; (I to L) current and force for a tip with a threefold dip in the force; sample bias is +10 mV, and cantilever amplitudes are A = 50 pm (tips 1 and 3) and 100 pm (tip 2). The tunneling current follows an exponential law with I(z) = I_maxexp(−z/λ_I), where λ_I = 55 pm, 56 pm, and 59 pm for tips 1, 2, and 3, respectively. The forces refer to the atomic contributions; long-range interactions are not included here.

A gentle poke event was conducted in the following way: First, the tip was held at a tip height (27) over the Cu surface of about 400 pm at a fixed lateral position. Then, the tip approached the surface by about 700 pm to the surface, so it was driven by about one atomic diameter into the surface. After that, a bias voltage of a few volts was applied, and the tip was slowly retracted within a few seconds. This procedure led to structural changes in the tip apex, and often to dropping one or a few tip atoms onto the Cu surface, whereas the Cu surface remained atomically flat otherwise. This finding is somewhat surprising, because W is much harder than Cu, and one might expect to find depressions in the flat Cu surface after a poke. However, even at the Cu(111) surface layer, a Cu atom still has 9 nearest neighbors (out of 12 in bulk), whereas W atoms at the tip apex have only a fraction of the eight nearest neighbors present in bulk W. (More information about the statistics of poking events is provided in fig. S10.)

The data sets in rows 2 to 4 in Fig. 1 are tomographic slices of current and force (short-range contribution only) for three different heights, starting with a height (27) of about 340 pm for the top row starting with Fig. 1B, about 220 pm for the center row starting with Fig. 1C, and about 150 pm for the bottom row starting with Fig. 1D (for precise values, see the marks on the z axes in Fig. 1, A, E, and I). At least 40 slices of current and frequency shift, spaced by height decrements of 10 pm, have been probed for each of the tips 1 to 3, and movies S1 to S3 show 40 frames starting at z_max ≈ 550 pm and ending at z_min ≈ 150 pm. Evidently, the force signals show a striking angular dependence that becomes more pronounced with decreasing tip height z.

This pronounced angular dependence of the frequency shift (and force) data, showing dual and triple minima for tips 2 and 3, respectively, is in stark contrast to the z-rotational symmetric force profiles that were found for metallic adatoms and metallic tips (28), presumably Cu-covered Ir tips and CO/Cu(111) (7), and a CO-terminated metal tip interacting with CO/Cu(111) (29). At a first glance, the current images in Fig. 1 and in movies S2 and S3 appear to be symmetric with respect to rotations around the z axis. Binnig suggested that it is possible to search for a potential angular current dependence by subtracting the normalized current data at a large distance from the normalized current data near the surface. The top row of Fig. 2 shows these differential normalized current images for tips 1, 2, and 3. Indeed, the differential images do show an angular dependence that reflects the rotational symmetry of the force images, although the local extrema are closer to the center in the differential current images than in the force images. Thus, some of the angular dependence of the force data also seems to show up directly in the tunneling current.

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Fig. 2

Experimental and simulated differential images of the tunneling current. The tunneling current above a CO molecule at position (x₀, y₀) can be approximated well by a Gaussian dip I(x, y) = I₀(1 − Δexp{−[(x − x₀)² + (y − y₀)²]/ρ²}), where ρ was fitted to the experimental current profiles at large distance, yielding ρ = 387 pm, 401 pm, and 431 pm for tips 1, 2, and 3, respectively, at z ≈ 350 pm. ρ is about 9% smaller for z ≈ 150 pm. The normalized tunneling current is obtained by setting the maximum current (above Cu) to 1 and the minimum (at the center of CO) to 0. The experimental difference between the normalized tunneling current close to the surface (z ≈ 150 pm) and far from the surface (z ≈ 350 pm) is shown in (A), (C), and (E) for tips 1, 2, and 3. The simulated difference between the normalized tunneling current close to the surface (z ≈ 150 pm) and far from the surface (z ≈ 350 pm) shown in (B), (D), and (F) is calculated by assuming a Gaussian dip for the current distribution but allowing a lateral tilt of the CO molecule (x′, y′) = (F_x/k_CO, F_y/k_CO) caused by the experimentally determined lateral forces (figs. S4 and S5).

An influence of atomic forces on current data has been observed with giant corrugations on soft samples (30) and corrugation enhancements on metal surfaces (31). The force field between an STM tip and a molecule can trap hydrogen molecules that modulate the current to improve molecular resolution (32). Although the free CO molecule stands perfectly upright on top of a Cu surface atom, it can be deflected laterally by external forces (29). The softest vibrational mode of CO on Cu(111) is the frustrated translation mode, with an energy of 4 meV as measured by He scattering (33) and by inelastic tunneling spectroscopy (26). The vibrational energy and the molecule’s moment of inertia lead to an estimate of the lateral force constant k_CO (figs. S4 and S5). The lateral forces between tip and CO molecule were obtained from the experimental data by differentiating the energy profile with respect to a lateral direction within the (x, y) plane (fig. S6). For each position (x, y), we calculated the tilt of the CO molecule by dividing the lateral force by k_CO. The tunneling current as a function of (x, y) at constant height could be simulated with a Gaussian dip (see captions of Fig. 2 and fig. S5). Figure 2, B, D, and F, shows the simulated current images with k_CO = 2.1 N/m. The simulated images provide a very good match to the experimental images, demonstrating that the apparent angular dependence in the current images is likely caused by a position-dependent tilt of the CO molecule that modulates the current rather than a direct angular dependence of the tunneling current.

Next, we ask what we can learn about the physics of the bond from the experimental spatial force dependence. Experimentally, we have access to the net force F in z direction. The semi-empirical Morse potential (1)V_M(d) = E_bond[−2e^{−(d−σ)/λ} + e^{−2(d−σ)/λ}] (2)describes the bonding energy of a diatomic molecule as a function of the atomic distance d. The two terms describe attractive and repulsive interaction components for two atoms with a decay length λ for the attractive and λ/2 for the repulsive branch of the potential, reaching its minimum (bonding energy) at the equilibrium distance σ.

If the force between tip and sample could be fully described by a Morse potential, the force images would show a single attractive minimum with a similar lateral extension as the dip in the current image, and the force images would be symmetric with respect to rotations around the z axis. For tip 1, the force data were approximately symmetric with respect to rotations around the z axis, but the force minimum was laterally markedly more confined (diameter ≈ 330 pm in Fig. 1 D) than the current minimum (diameter ≈ 720 pm). Tips 2 and 3 showed a strong azimuthal dependence in the force image. For tip 2, we found two local minima that became more pronounced with decreasing distance (Fig. 1, F to H). Tip 3 exposed three minima that also became more pronounced with decreasing distance.

Figure 3 shows experimental force versus distance data for tips 1 (panel A), 2 (C), and 3 (E). Tip 1 showed an attractive force that became more attractive with decreasing distance and reached its greatest magnitude at a distance of 145 pm. A closer approach to the surface would have caused a lateral force greater than the moving threshold of 160 pN (7) and would have pulled the CO molecule laterally. In the distance regime from 500 to 200 pm, the force increased exponentially in magnitude with decreasing distance with a decay length of 95 pm. For distances < 200 pm, the attraction increased faster at a decay length of 40 pm.

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Fig. 3

Force versus distance data and fit curves for tip 1 (A), tip 2 (C), and tip 3 (E). For tips 2 and 3, the force curve is fitted at the local maxima (red) and minima (blue); see insets. The fit curves are composed from a Morse potential and an angular dependent attractive exponential term. (G) Representation of the W tip atom as its Wigner-Seitz unit cell and orientation of a CO molecule with azimuthal angle ϕ and inclination angle θ. Here, the [111] orientation is shown with θ = arccos (1/√3) ≈ 54.73° and ϕ = 45°. The [110] orientation corresponds to θ = 45° and ϕ = 0°, and the [100] orientation is obtained for θ = 0° and ϕ = 0°. For clarity, only the front atom of the tip and the CO molecule are shown. (B) Force derived from the model potential at a height of 150 pm and a tip orientation close to [001] with θ = 5° and ϕ = 10°. (D) Force derived from the model potential at a height of 150 pm and a [110] tip orientation with θ = 45° and ϕ = 0°. (F) Force derived from the model potential at a height of 150 pm and a [111] tip orientation with θ ≈ 54.73° and ϕ = 0°. (H) Bonding energy as a function of distance for tips 1 and 3 showing experimental data and the results from our model. Note that the horizontal scales in (A), (C), and (E) and z and z′ in (H) are shifted by 50 pm (fig. S7).

Figure 3C refers to tip 2, where the force curve at the center (red) could be well approximated by a Morse law with E_bond = 5.95 zJ, σ = 217 pm, and λ = 93 pm. At the lateral positions where attraction was strongest (blue graph in Fig. 3C), we did not get close enough to the surface to find an absolute force minimum. The difference between the force curves at the center (red) and the most attractive sites (blue) was well approximated by an exponential function with a decay length of 40 pm.

Figure 3E shows the force-versus-distance dependence of tip 3, where the force curve at the center (red) was well approximated by a Morse law with E_bond = 6.30 zJ, σ = 222 pm, and λ = 99 pm. The blue curve in Fig. 3E shows the force at the lower one of the three attractive lobes. Again, this curve did not show a minimum within the distance range that was covered, and the difference of the center (red) and attractive lobe curves (blue) could be approximated by an exponential function with a decay length of 40 pm.

The dependence of the force curves as a function of crystal direction and distance allowed us to build a semi-empirical potential V_W-CO/Cu(111), which was composed of a Morse potential and a very short-range, additional attractive component pointing in <100> directions that accounts for the three tip symmetries shown in Fig. 1 as well as the lower-symmetry tips shown in fig. S9. We assumed that the force symmetry was related to the crystal symmetry of the tip material. The crystal structure of W is bcc, and Fig. 3G shows the Wigner-Seitz cell of bcc W. The bcc Wigner-Seitz cell is fourfold symmetric in <001> directions, twofold symmetric in <011> directions, and threefold symmetric in <111> directions. If we assume that the angular dependence is caused by a bonding energy component that is strong along the <001> directions, we can explain the data in Fig. 1 for all three tip orientations. A mathematical function that reflects this angular symmetry is given byα_n(x′/r′,y′/r′,z′/r′) = 1/(1 – 3^1–n)[(x′/r′)²ⁿ + (y′/r′)²ⁿ + (z′/r′)²ⁿ – 3^1–n] (3)with r′ = (x′² + y′² + z′²)^0.5 and an integer n ≥ 2 (fig. S7). This function is constructed to yield α_n = 1 for the <100> directions, α_n = 0 for all <111> directions, and α_n = (1.5ⁿ⁻¹ − 1)/(3ⁿ⁻¹ − 1) for the <110> directions (e.g., α₂ = 1/4, α₃ = 5/64). Because the angular dependent force contribution shows an approximately exponential decay for larger distances (difference of red and blue curves in Fig. 3, C and E), we model the full angular potential byV_W-CO/Cu(111)(x′, y′, z′) = V_M(r′) − α_n(x′/r′,y′/r′,z′/r′) × E_ang exp[σ_ang/λ_ang – (r′⁸ + σ_ang⁸)^1/8/λ_ang] (4)This potential has seven parameters, and a good agreement between the experimental force versus distance data and constant height images is obtained for E_bond = 6.30 zJ, E_ang = 110 zJ, σ = 272 pm, λ = 99 pm, σ_ang = 140 pm, λ_ang = 40 pm, and n = 3. The green lines in Fig. 3, A, C, and E, show the forces as obtained from Eq. 4. Figure 3, B, D, and F, shows the calculated constant height force images obtained by using the model potential. The key features and the signs and magnitudes of the experimental forces are accurately generated by that potential. Even subtle features in V_W-CO/Cu(111), such as the appearance of a small repulsive ring around the attractive center of Fig. 3B, were observed in the experiment. The minima are spaced wider apart in the experimental data than in the calculation according to the model potential V_W-CO/Cu(111), possibly because of the lateral bending of the CO molecule discussed in Fig. 2.

The design of our model potential after Eq. 3 is supported by the physics of chemical bonding. The Morse part of V_W-CO/Cu(111) is treated in most textbooks on quantum mechanics by calculating, for example, the bonding energy of the hydrogen molecular ion by perturbation theory, where the Morse potential is an approximation for the energy as a function of distance for the symmetric pair wave function.

The angular dependent part is more complicated. In his paper (34), Feynman demonstrated that forces in molecules (and thus in tip-sample interactions) can be calculated from electrostatics once the wave functions have been found by solving Schrödinger’s equation. Because the valence electrons of CO form lone pairs, we do not expect that CO develops a strong electronic overlap with the W tip atom. In bulk W, the atoms form bonds to their next neighbors across the hexagons in Fig. 3G that point toward the corners of the cubic unit cells in the eight <111> directions. Even for W atoms at the surface, calculations show that the valence charge density has maxima in the <111> directions (16, 35, 36), so each W atom can be viewed as an electric multipole with a positively charged center and eight negative charges pointing in the <111> directions (fig. S8). Because the lone pairs in CO cannot form molecular orbitals with the W electrons, we modeled the interaction by an electrostatic dipole of the CO molecule interacting with the multipoles of the W atoms. If the CO dipole has a negative charge on the oxygen atom, attraction occurs along the <100> directions (fig. S8) and repulsion along the <111> directions. The exponential decay of this strongly directional force is also motivated by electrostatics. The electrostatic potential at the surface of a periodic array of charges with lattice constant a decays proportional to exp(−2πz/a) (37). The lattice constant of W is 273 pm, and the corresponding decay length is, at 273 pm/2π = 43 pm, very close to our fit parameter, λ_ang = 40 pm.

The precise measurement of forces and currents between a metallic tip and CO/Cu(111) allows us to identify the tip. Initially, we assumed that tip 1 in Fig. 1A was covered by Cu because it has been prepared by poking into the Cu surface. To check this assumption, we prepared a Cu surface with a Cu adatom and picked up a CO molecule with the tip; if the short-range force depends only on the bonding partners, the force spectrum should be the same for a CO-terminated tip probing a Cu adatom as for a Cu-covered tip probing a CO molecule adsorbed on Cu. Experimentally, the attractive force between a CO-terminated tip and a Cu adatom was below 30 pN, only one-tenth of the attraction of tip 1 with CO. Our model potential V_W-CO/Cu(111) explains the magnitudes of forces and the geometric details of the force with the same parameters as for tips 2 and 3 very well for a [001] orientation. Thus, we are confident that tips 1, 2, and 3 are all W front atom tips oriented in [001], [011], and [111] directions, respectively. Our model potential from Eq. 4 even provides a very good agreement with experimental images recorded with tips that are oriented at odd angles—that is, tips that are not aligned with high-symmetry crystal directions (as shown in fig. S9).

Tip 1 exerted an attractive force of 300 pN at a distance of 150 pm, and this tip worked reasonably well for atomic manipulation. Lateral forces are known to be relevant in atomic manipulation (7), and the lateral force field that was generated from tip 1 (fig. S4A) allowed reliable lateral dragging of adsorbed CO molecules (or other adatoms) across the surface. Tips 2 and 3 did not provide enough attractive force for reliable manipulation of CO and tended to push adsorbed CO away from the tip in a less-controlled fashion (fig. S4, A and B). Probing the tips with a CO molecule allowed us to assess the effectiveness of a given tip for atomic manipulation and provided immediate feedback about the success of a tip-poking event to form a suitable tip (see fig. S10 for a histogram of obtaining tips of type 1 to 3 by poking). We note that figures 1D and 3A in (7) showed a force profile with a slightly repulsive crescent around the wide attractive valley, similar to our tip 1.

Lastly, we discuss the origin of the difference of appearance in force data versus tunneling current data. Stoll (38) has discussed how the magnitude of the decay length of the tunneling current connects to spatial resolution and found that the smaller the spatial decay length, the greater the spatial resolution can become (fig. S11). The angular part of the interaction potential has a decay length of only about 40 pm, whereas the decay length of the tunneling current ranges from 53 to 60 pm. Probing the short-range interactions at very small distance also increased spatial resolution from molecular resolution in the attractive regime to atomic resolution in the repulsive regime when imaging pentacene by Gross et al. (39). A density-functional study (40) has found that the origin of the submolecular contrast is Pauli repulsion with a very small decay length.

• Received for publication 31 January 2012.
• Accepted for publication 20 March 2012.