Image Calculations with a Numerical Frequency-Modulation Atomic

Article pubs.acs.org/JPCC

Image Calculations with a Numerical Frequency-Modulation Atomic Force Microscope F. Castanié,† L. Nony,‡ S. Gauthier,§ and X. Bouju*,§ Centre d’Élaboration de Matériaux et d’Études Structurales (CEMES), Nanosciences Group, UPR CNRS 8011, Bât F picoLab, 29 rue Jeanne-Marvig, POB 94347, F-31055 Toulouse Cedex 4, France and CEMES, Université de Toulouse, 29 rue Jeanne-Marvig, POB 94347, F-31055 Toulouse Cedex 4, France ‡ Aix-Marseille Université, IM2NP, Centre Scientifique de Saint-Jérôme, Avenue Escadrille Normandie-Niemen, Case 151, F-13397 Marseille, Cedex 20, France and CNRS, IM2NP (UMR 6242), Marseille-Toulon, France § Centre d’Élaboration de Matériaux et d’Études Structurales (CEMES), Nanosciences Group, UPR CNRS 8011, Bât F picoLab, 29 rue Jeanne-Marvig, POB 94347, F-31055 Toulouse Cedex 4, France †

ABSTRACT: We investigated the implementation of a numerical tool able to mimic an experimental noncontact atomic force microscope (nc-AFM). Main parts of an experimental setup are modeled and are implemented inside a computer code. The goal was to build a numerical AFM (nAFM) as versatile, efficient, and powerful as possible. In particular, the n-AFM can be used in the two working regimes, that is, in attractive and repulsive regimes, with settings for a standard AFM cantilever oscillating with a large amplitude (typically, 10 nm) or for a tuning-fork probe with ultrasmall amplitudes (∼0.01 nm). We present various tests to show the reliability of the n-AFM used as a frequency-modulation AFM (FM-AFM). As an example, we calculated FM-AFM images of adsorbed molecular systems, which range from two-dimensional planar molecules to corrugated systems with a three-dimensional molecule. The submolecular resolution of the FM-AFM is confirmed to originate from repulsive Pauli-like interactions between the tip and the sample. The versatility of the n-AFM is finally discussed in the perspective of new functionalities that will be included in the future.

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INTRODUCTION The family of near-field microscopes evolves continuously through technical improvements to achieve higher sensitivity and/or higher speed to follow dynamical phenomena. Some breakthroughs have sped up this evolution. Among them, the dynamical noncontact atomic force microscopy (nc-AFM), also called frequency-modulation atomic force microscopy (FMAFM) is becoming an indispensable tool for surface science, especially at the single molecule level.1−7 Unlike the scanning tunneling microscope (STM), which can be only used on conducting or semiconducting substrates, and on thin insulating films, nc-AFM is able to image a broader variety of surfaces, with an emphasis on insulating substrates without8,9 or with molecular layers on it.10,11 Furthermore, the ability to obtain submolecular features on single objects is remarkably achieved with both techniques.12,13 Nevertheless, an accurate interpretation of the images cannot be reached without the help of calculations. Even though it is more critical for STM due to the ambiguous relation between the electronic structure and the spatial structure of adsorbed molecules,14−17 the case of AFM remains acute, especially when one considers the central role of the tip.18,19 This is one of the reasons why various numerical tools have been developed for the last decades for STM and AFM.20−30 Images can be calculated at different levels of complexity. In the simplest case, one may calculate the tip− © XXXX American Chemical Society

sample force at different locations above the surface in a static situation.31,32 Although this approach is adequate in many cases, it cannot describe situations where the tip−sample interactions are strong enough to induce important structural relaxations.33 One well-understood example is the imaging of the surface of alkali halide or fluoride crystals, where it has been established that the atomic corrugation observed in constant frequency shift images is dominated by the vertical displacements that the surface ions undergo under the influence of the tip.22,34,35 These calculations were performed by constraining the trajectory of the tip to its ideal sinusoidal trajectory. In other words, one considers that the control system of the microscope performs ideally. However, this is not always the case in real life. This approximation becomes limited when the tip−sample interaction becomes so strong that the back-action of the tip−sample force on the cantilever can no longer be ignored. This is the case in particular when abrupt phenomena, such as atom displacements, occur under the tip, leading to dissipation of energy via the adhesion hysteresis mechanism or to vertical and/or lateral manipulations.29,36,37 It is then necessary to include the response of the microscope control Received: January 28, 2013 Revised: April 30, 2013

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dx.doi.org/10.1021/jp400948a | J. Phys. Chem. C XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry C

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where D is the distance between the surface of the substrate and the cantilever when it is at rest.45−48 Most of the time, a phase-locked loop (PLL) is used and has a double purpose. Its first use is to generate an output signal, which is synchronized with the input one, depicting the cantilever motion. Thus, the output signal has the same phase and the same frequency as the input one. However, the input signal is in phase with the cantilever motion. The cantilever motion is −(π/2) phase shifted with the excitation signal when the frequency of this signal equals the resonance frequency of the cantilever. Thus, the output signal has to be shifted by +(π/ 2) to be used as the new normalized excitation signal of the cantilever zts and to keep the cantilever at its resonance frequency. The use of a PLL to generate the excitation signal rather than the use of a self-excited cantilever allows having an excitation signal that is cleaned from the amplitude noise of the cantilever and the deflection measuring setup. Meanwhile, the PLL demodulates the oscillation at f0̃ and calculates the value of Δf. This Δf is the input for the distance controller (DC). Comparing the Δf with a reference frequency shift, the DC allows the regulation of the tip−sample distance. This regulation gives the sample topography. The numerical AFM simulates the behavior of an FM-AFM in ultrahigh vacuum (UHV) and is based on a previous work of Nony et al.28 The code consists of three entangled loops (Figure 1).

system in the calculation to describe the experiment properly. This can be done with different levels of details depending on the analyzed behavior. The work reported here belongs to this last approach. In this paper, we present a new numerical tool able to simulate the behavior of the AFM operated in the frequencymodulation mode. This tool is targeted at the calculation of images and/or spectroscopic-like experiments of molecular adsorbates (including or not charges) on surfaces with the goal to mimic the experimental behavior as closely as possible. The so-called numerical AFM (n-AFM) includes the functional blocks of any standard experimental setup, and the tip−surface interaction is described by molecular-mechanics force fields able to describe the dynamical behavior of the probed system, or alternatively from DFT-derived realistic force fields. To do so, the n-AFM has been made versatile and may be used either in the so-called constant height mode (distance controller disengaged) or in the constant Δf mode (distance controller engaged). However, beyond this rather standard feature, the nAFM has been specifically designed to operate in a very general context. For instance, in the large oscillation amplitude regime (A > 2 nm), the n-AFM virtually describes optical beamdeflection based instruments relying on cantilevers, or it describes tuning-fork-based sensors in the small amplitude regime (A ≃ 10 pm). This former feature, which had not been demonstrated in the reported virtual instruments so far,27−30,38−41 makes the n-AFM a unique tool for investigating submolecular imaging processes, which should bring complementary information to the experiments. Along this line, to demonstrate the performances of the n-AFM, most of the images of single molecules reported here have been obtained in that regime. The paper is organized as follows. In the second section, we briefly described the numerical scheme of the n-AFM and show its dynamical performances in the large- and small-amplitude regime. In the third section, we investigated two examples of individual molecules adsorbed on a graphite and on a silicon carbide surface and discussed the imaging mechanism in relation with the molecular configuration on the surface.

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THEORY AND NUMERICAL METHOD Even though the AFM is widely described,28,42 we recall briefly the main characteristics of how it works. In the FM-AFM mode,43 the cantilever is oscillating and its deflection is driven at its fundamental resonance frequency f0̃ . In this mode, the amplitude of oscillation is kept constant via an amplitude controller (AC), as well as the phase between the drive and the deflection. When the cantilever is far from the sample, it can be considered as a free harmonic oscillator with f0̃ = f 0, with f 0 being the measured frequency. When approaching the sample, an interaction between the tip and the sample appears and disturbs the cantilever motion by introducing a nonlinear dynamics,44 leading to an almost instantaneous frequency shift, Δf = f0̃ − f 0 = −(f 0/(kcA2))⟨Ftsq⟩, where q is the unperturbed motion given by q = A cos(ωt), kc is the stiffness of the cantilever, A is the amplitude of oscillation, and Fts is the force between the tip and the sample. The variation of the frequency shift with respect to the tip−sample distance is a critical feature of the FM-AFM. More explicitly Δf 1 =− 2πkcA f0

Figure 1. Schematic structure of the numerical FM-AFM with the three entangled loops.

The inner one describes the cantilever motion and its detection. The cantilever motion is described by solving the differential equation of the driven harmonic oscillator z(̈ t ) +

(1)

where ω0, Q, and kc are the angular resonance frequency, the quality factor, and the stiffness of the free cantilever, and z(t), Ξexc(t), and Fts(t) are the instantaneous location of the tip, the excitation signal, and the force of the tip−sample interaction. The differential equation is solved by using a leapfrog algorithm. The time t is incremented at each execution of this loop by Δt, which is related to the sampling frequency fs by Δt = 1/fs.

2π

∫

ω0 ω 2 F (t ) z(̇ t ) + ω02z(t ) = ω02 Ξexc(t ) + 0 ts Q kc

Fts[D + A cos(u)]cos(u) du

0

B



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This technique was already used with force fields generated by density functional theory (DFT) calculations as well as atomistic simulation techniques.26,29,49,50 Nevertheless, the storage of the Fts data table increases with the size of the surface to be imaged, and this method may need a large storage capacity. Finally, this method does not allow the introduction of a molecular dynamics (MD) module in the simulator because of the precomputed force field. • Either the forces are calculated without spatial restriction, without specific predefined points on a grid. The former issue is well adapted for the n-AFM.51 A semiempirical potential is favored to describe the tip−sample interaction because Fts is calculated “at run time”, that is, at each execution of the first loop. For particular molecular systems, one can consider that the imaged system is not perturbed by the tip so that all the coordinates are kept frozen. This is what is done in the following. The forces are then evaluated from molecularmechanics force fields51 or from DFT calculations.52,53 In the present paper, the molecule−surface interactions are calculated according to the second previous item by DFT and dispersion-corrected DFT. Tip−sample forces are evaluated by semiempirical functions because the tip that has been chosen is chemically nonreactive (see description thereafter). This is the reason why one considers that the molecular system is frozen when the tip oscillates above it and that the tip forces do not perturb the molecule conformation. Furthermore, relaxations of the tip and the substrate during the probe scanning are not taken into account in this paper, though they might be included upon needs, because the time needed for a simulation is longer than that for the previous simulation method. Nevertheless, this option shows some advantages. If the sample consists of a molecule−surface system, the various degrees of freedom of the molecule can be activated to reproduce the dynamics of the molecule on the substrate, unlike with the first method, where a new table of force field is needed for each position of the molecule. Thus, the second option allows introducing MD in the simulator. Molecular mechanics (MM) can also be introduced as well. The code of the n-AFM, written in FORTRAN 90, was designed to be as flexible as possible to facilitate the implementation of new options and modules. In the input file, many options are proposed. One of them consists of the choice between the two following modes: constant height mode, that is, recording the Δf variations with the DC disengaged, or constant Δf mode, that is, recording the height variations of the tip with the DC engaged. Second, the nAFM can be used whatever the oscillation amplitude is once appropriate settings for the gains are determined. Here, two distinct regimes are described. The first one is to work with large amplitudes. The n-AFM then virtually describes the cantilevers of an optical beam-deflection based nc-AFM operated in UHV. The following typical settings are used: oscillation amplitude Aset = 5 nm, f 0 = 150 kHz, kc = 30 N/m, Q = 30 000, and fs = 400 MHz. The two gains of the PLL are K0 = 5000 rad V−1 s−1 and KI = 16 s−1, giving a response time of about 0.8 ms. The second operating regime is to work with ultrasmall amplitudes. In this case, the cantilever is virtually replaced by a qPlus tuning-fork sensor.54 Thus, the settings are now: Aset = 0.01 nm, f 0 = 23 165 Hz, kc = 1800 N/m, Q = 50 000, and fs = 60 MHz. The two gains of the PLL are, in this case, K0 = 2000 rad V−1 s−1 and KI = 7 s−1, giving a response

The second loop describes the PLL and a phase shifter (PS). Its sampling frequency is smaller than fs. The numerical PLL is composed of a phase detector, a finite impulse response lowpass filter, and a numerically controlled oscillator. An important difference with the previous simulator28 is that the Δf is no longer calculated by a frequency tracker outside the PLL, but inside the PLL via proportional (P) integral (I) gains. In the simulator by Nony et al.,28 the goal of the tracker was to update the center frequency of the PLL at a constant rate (2.5 kHz). In that simulator, the resonance frequencies of the cantilevers were supposed to range only between 100 and 300 kHz. For that range, the update rate of the tracker yielded a stable operation mode. However, this is no longer true if oscillators with significantly different resonance frequencies (like a qPlus tuning-fork sensor, see hereafter) are to be used. Following notations of Nony et al.,28 the time-dependent phase of the numerical controlled oscillator (NCO) is now calculated with ϕNCO(ti) = Σttik[uf(tk)K0 + Σ(KIK0uf(tk)Δts1) + ω0]Δts1, where uf is the control signal, K0 and KI are the P and I gains of the PLL, respectively, and Δts1 is the time step for the NCO. This new implementation avoids instabilities occurring sometimes with the former tracker and allows working whatever the considered frequency range. It is, therefore, more versatile, but requires one to accurately adjust the P and I gains of the PLL upon the oscillator chosen. Finally, the last loop describes the two controllers, AC and DC. It is the slowest one. The AC receives the amplitude of the oscillation from the first loop. With respect to a reference amplitude, and with P and I gains, the AC provides the new amplitude of the excitation, Aexc. Aexc is then multiplied by the normalized excitation signal zts shifted by π/2 to obtain the total excitation signal. The DC receives the Δf from the PLL. As for the AC, it needs three external parameters, which are the reference frequency shift Δfset, the integral, and the proportional gains. The DC provides the distance of the cantilever with which it has to run to keep Δf equal to Δfset. One of the most crucial points is the tip−sample force and the various associated approximations to conduct the calculations as properly as possible. In the case of molecules on a surface, one has to consider first the adsorption. Roughly speaking, this means that the molecule−surface interaction may be calculated by density functional theory (DFT) or by semiempirical functions of molecular mechanics force fields according to the strength of the adsorption, that is, chemisorption or physisorption, or according to the accuracy the physical system requires. Second, a particular attention may be asked for the interaction due to the tip. Here too, if the tip apex is reactive or not, equipped with a CO molecule7 or not, or if the tip is in a strong repulsive regime with the molecule, one may favor the DFT approach rather than force-field interactions. The system to be imaged is relaxed or not, with or without the presence of the tip. Two methods may be distinguished for the use of a numerical AFM. • Either the forces are recorded inside a three-dimensional matrix [according to an (x, y) grid of the scanning process and in the z direction perpendicular to the sample following the number of points requested for the FM-AFM calculation]. At each execution of the first loop, the point corresponding to the position of the tip is selected and the associated force is calculated by interpolation with the precomputed forces of the grid. C



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where a, b, A, and B are constants depending on the type of atoms and are chosen in the data file of the MM4 force field.58,59 The variable rij is the distance between an atom of the substrate and one of the tip apex. Numerous tests were performed to validate the numerical code. The first one consisted of checking whether the n-AFM describes properly the behavior of a driven harmonic oscillator. In this case, the free oscillator is driven in(out of) phase with respect to the excitation at low(high) frequency, and the oscillation amplitude was recorded and successfully compared with the theory. When applying excitation pulses at the resonant frequency, it was made sure that both excitation and oscillations were in quadrature and that the oscillation amplitude was maximum. In a second series of tests, we checked whether Δf ∝ A−(3/2) in the large amplitudes regime A ≫ d, where d is the minimum tip−sample distance during oscillation, as established by Giessibl.45 The sample used in this test is a graphene surface, and the tip cluster is composed of carbon atoms as described above. The tip is positioned exactly above a top site (above an atom of the graphene surface). The distance d is kept constant during the simulation at d = 0.4 nm, and the amplitude is varied from 14 nm down to 0.01 nm. Figure 3a shows that the curve Δf(A−(3/2)) is linear down to A = 2.2 nm, which fulfills the condition A ≫d if A > 5d.45 Notice that, following ref 38 and with eq 2 and H = 1 eV, one calculates a slope equal to −2.4 × 10−11 Hz·m3/2 that corresponds well with the slope in Figure 3a. The setting corresponds to a cantilever-like probe, and it does not allow experimentally decreasing the amplitude below 1 nm. To go below with smaller amplitudes, one needs to use parameters for a tuning fork, as shown in Figure 3b. Another finding of Giessibl45 is that Δf no longer varies as A−(3/2) when A ≪ d. The n-AFM is able to confirm this finding. One can see in Figure 3b that Δf becomes constant with respect to A−(3/2) when A < 0.02 nm. Thus, the condition A ≪ d is satisfied when A < d/20. The numerical controllers AC and DC have also to be tested. For AC, the test consists of changing the reference amplitude at time t while measuring the required time for the amplitude to reach this new reference. This test allows the user to set the integral and proportional gains of AC. One can see in Figure 4 an inappropriate setting for the gains, which can lead to a large overshoot (red curve). On the other hand, the green curve in Figure 4 is the response with a suitable setting for the P and I −4 gains (following notations in Nony et al.,28 Kac and p = 15 × 10 −1 Kac = 0.05 s ). Even though a small overshoot (about 1% of i

time of less than 2.0 ms. Options regarding the way the scan is performed are available as well. One can choose a twodimensional scan [Δf(x) and d(x), where d is the tip−sample distance], or a three-dimensional image can be chosen [Δf(x, y) and d(x, y)]. The last option is a false four-dimensional image [Δf(x, y, z) and d(x, y, z)].55,56 Finally, one has to mention that the power of the n-AFM lies in its capability to mimic an experimental setup as closely as possible. This means that a careful adjustment has to be done on the values of gains (PLL, AC, and DC) when one changes the working settings, in particular, the oscillation amplitude. These features make the nAFM a unique tool to simulate the instrumentation in the qPlus tuning-fork mode with realistic force fields. To test and validate the n-AFM, we consider a physical system consisting of a carbon tip facing a carbonated sample. The tip is composed of two parts (Figure 2): a nanometric

Figure 2. (a) The probe consists of an atomic cluster with 29 atoms supported by a sphere with a radius of 4 nm. (b, c) Views of the tip apex structure.

sphere and a cluster of atoms for the tip apex. The sphere has a radius R of 4 nm. The force of interaction with a surface Fsphere−surface is well described by the standard form vdW HR 1 sphere − surface FvdW (r ) = − 6 (r − R )2 (2) where H is the Hamaker constant and r the sphere−surface distance. The cluster shows a pyramidal diamond-like structure. It is composed of 29 atoms.57 The force of interaction between the cluster and the sample is described by the derivative of the following Buckingham pair potential cluster − surface EvdW (r ) =

⎡

∑ ⎢⎢A exp(−rija) − B i,j

⎣

⎤ b6 ⎥ rij6 ⎥⎦

(3)

Figure 3. Variation of Δf with respect to the oscillation amplitude to the power −3/2. The tip−surface distance d is kept constant at d = 0.4 nm. (a) Typical parameters of a cantilever are used, and the amplitude varies from 14 to 2.2 nm. (b) Typical parameters corresponding to a tuning fork where the amplitude varies between 0.4 and 0.01 nm. For red, blue, and green curves, the tip−surface distance d is equal to 0.50, 0.45, and 0.40 nm, respectively. D



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0.07 Å for coronene and 0.22 Å for HBC, due to the environment underneath (Figure 5). We have calculated spectroscopic Δf(d) curves above particular carbon atoms on each molecule (Figure 6) with settings corresponding to a

Figure 4. Responses of the AC to a pulse signal with inappropriate (red curve) and correct (green curve) proportional and integral gains.

Aset) is observed, the set point with ±2% is reached after 9 ms where the response is stable. Notice that the same test has been done with DC. In this case, it is the reference Δf, which is changed instead of the reference amplitude. As already mentioned, finding good parameters for the gains is of fundamental importance and needs special care in order to avoid artifacts. Moreover, contrary to some other simulators,27,60 the various sources of noise61 are not considered here, even though the n-AFM is able to deal with such effects. The n-AFM has already shown its ability to image bare surfaces,51 and we now present calculated images of molecular systems adsorbed on a surface.

Figure 6. (a, b) Δf(d) curves when the tip is above three carbon atoms of the coronene and the HBC molecule, respectively. The green, blue, and red curves correspond to positions above the carbon atoms circled in green, blue, and red in Figure 5.

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RESULTS The main purpose, here, is to demonstrate the reliability of the n-AFM to calculate FM-AFM images of molecular systems in various working regimes. The first system consists of carbonated molecules deposited on a graphite surface. The molecules are coronene C24H12 and hexabenzocoronene (HBC) C42H18 (Figure 5). The adsorption state has been

tuning-fork sensor simulation with a cantilever of high stiffness and with an oscillation amplitude of 10 pm.1 It appears clearly that there is an edge effect. The tip−molecule interaction is more attractive above the center of the molecule than at the border of the molecule. Another point is that the correspondence between the z position of the carbon atom and the Δf(d) curve is not straightforward. For instance, the atom circled in red in Figure 5a is higher than the one circled in green, whereas the red and green curves in Figure 6a at d = 6.065 Å coincide. This is due to the effect of the close environment beneath and near the considered atom. Figure 7a,b shows calculated images in the constant height mode with a tip−graphite surface distance of 6.065 Å. Here, in Figure 7a,b, the minimal distance between the last atom of the tip and the molecule is 2.91 and 3.00 Å for coronene and HBC, which, however, corresponds to the repulsive regime. This is confirmed by the positive Δf measurement. Notice that the graphite surface is not resolved due to the low attractive interaction between the tip and the substrate at this distance (blue background with negative Δf in Figure 7). One recognizes very well the honeycomb structure of the molecule where one can distinguish the C−C bonds. This observation of corrugation at the atomic scale in the regime of repulsive forces is similar to previous experimental results where the skeletons of organic molecules are clearly identified with an impressive resolution.1−5,7,52 In these results, the atomic resolution on molecules is obtained by using a CO-modified tip. This modification of the tip termination generates an acute apex and leads to a chemical inertness of the probe with respect to the sample. Similar to the tip used here, a nonreactive tip allows entering the regime of repulsive forces with stable imaging conditions. Additionally, Gross et al. performed DFT studies to

Figure 5. Models of a single (a) coronene and (b) hexabenzocoronene molecule adsorbed on a graphite surface. For clarity, only a single surface layer is represented. The color scale on the molecules represents the z position of the atoms above the surface at z = 0 Å. From red to blue, z ranges between 3.10 and 3.19 Å in (a), and between 2.9 and 3.23 Å in (b).

calculated using a local orbital occupancy approach and secondorder perturbation theory [LCAO-S2 + van der Waals (vdW) formalism].62−65 To show the sensitivity of the n-AFM, we have not considered the molecule in its most favorable adsorption state, which intuitively follows the ABAB graphite stacking with the molecule lying flat on the surface at 3.11 Å for coronene and at 3.04 Å for HBC. We have chosen a metastable position (Y. Dappe, personal communication) where some atoms are slightly higher than the mean plane of the molecule, with a corrugation for the z position of the carbon atoms of E



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Figure 8. (a) Calculated FM-AFM images in the constant frequency shift mode (Δf = −6 Hz) of a single adsorbed coronene molecule. The size of the image is 2.6 × 2.6 nm2. (b) Calculated FM-AFM images in the constant frequency shift mode (Δf = −8 Hz) of a single adsorbed hexabenzocoronene molecule. The size of the image is 1.74 × 2.28 nm2.

to play a crucial role in future electronic devices used in highpower, high-frequency, and high-temperature conditions.66,67 This material is also interesting because the electronic properties of an adsorbed molecule may be decoupled from those of the substrate due to its wide gap. Among the rich variety of reconstructions of the SiC,68,69 the particular surface chosen here is the 6H-SiC(0001)-3 × 3 exhibiting silicon tetramers regularly disposed with a hexagonal geometry (lattice parameter is 9.24 Å) on a silicon adlayer.70−72 In the prospect of working out the behavior of functionalized molecules adsorbed on a semiconductor surface, such as the silicon carbide surface, FM-AFM is well-adapted for characterization purposes. Some studies of organic molecules on this surface were already reported73−77 that take advantage of the dangling bond mesh provided by the silicon pyramids (Figure 9). Here,

Figure 7. (a, b) Calculated FM-AFM images in the constant height mode (d = 6.065 Å corresponding to a repulsive regime) of a single adsorbed coronene and hexabenzocoronene molecule, respectively. The size of the image is 2.59 × 2.59 nm2. (c, d) Calculated FM-AFM images in the constant height mode at d = 7 Å of a single adsorbed coronene and hexabenzocoronene molecule, respectively. The size of the image is 1.74 × 2.28 nm2. All the images are calculated with a tuning-fork sensor.

explain that the atomic contrast originates from the Pauli repulsion force.1,2,52 Notice that, following what these authors did, we have considered that the atoms of the molecule are frozen in their position during the tip scanning. Indeed, it is clear from Figure 7b that the sensitivity is important in this regime that appears in very slight displacement in the z direction of some atoms on one edge of HBC. Similar results were proposed recently by the same group for a HBC adsorbed on Cu(111).7 On the contrary, long-range vdW forces do not permit getting a clear atomic contrast, as observed in calculated images at d = 7.0 Å in Figure 7c,d. In this case, there is a contrast reversal giving rise to small depressions at the center of the carbon rings. These features appear especially with HBC, where seven small holes are situated on the central part of the rings inner to the molecule. As the coronene is less extended than HBC, only a featureless depression around −14 Hz appears. This effect of geometrical density of the carbon atoms on the periphery of the molecule is seen on the edges of the HBC image in yellow where the tip probes less attractive areas (see Figure 3A in ref 7). To go through the attractive regime, we have calculated images (Figure 8) where the coronene and the HBC molecules appear as a large circular and hexagonal bump without corrugation inside. The minimal tip−molecule distance at the center is about 4.5 Å. Moreover, the Δz value between the signals above the graphite surface and above the HBC molecule is 2.99 Å, which corresponds closely to the adsorption height of the molecule. For flat hydrocarbonated adsorbates as coronene and HBC, the atomic contrast becomes more and more apparent when one goes from the attractive to the repulsive regime, which is also confirmed experimentally.1,7 We address now the case of a more corrugated sample consisting of a C60 molecule adsorbed on a silicon carbide (SiC) surface. Silicon carbide is a wide-band-gap semiconductor (from 2.4 to 3.3 eV according to the polytype) and is expected

Figure 9. Relaxed atomic structure of a C60 molecule adsorbed on a silicon pyramid of the 6H-SiC(0001)-3 × 3 surface.

the case of C60/6H-SiC(0001)-3 × 3 is tackled.78,79 The relaxed structure in Figure 9 was obtained by DFT calculations (F. Spillebout and P. Sonnet, personal communication) in the framework of the Vienna Ab initio Simulation Package.77 It was already reported experimentally and theoretically that a fullerene molecule may adopt orientations exhibiting a pentagon, a hexagon, or a carbon−carbon bond on various surfaces.80−97 The rotation barrier of a C60 molecule on a single adsorption site may be considered as rather small, and there is a variety of situations for rotation with rolling between surface sites.98−100 Here, we do not discuss the energy of adsorption as well as the rotation barrier because it is out of the scope of the paper and we just consider that the C60 molecule stays fixed in its relaxed orientation on the surface. In the attractive regime at constant Δf, the calculated image of the C60 molecule is shown in Figure 10a. The image exhibits a triangular shape with an F



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Figure 10. (a) Calculated image of a C60 molecule on the SiC 3 × 3 surface in the Δf constant mode (Δf = −18 Hz) with an amplitude of 5 nm. The image size is 3 × 3 nm2. (b) Two scan lines above the molecule (from left to right) when the orientation of the tip is either a facet or an edge (green and red curves, respectively). The scan lines have been acquired at a middle position of the image.

that the molecule may diffuse during the tip approach and that it is deformed due to the tip interaction. This means that similar results could have been obtained using one of the simpler approaches described in the Introduction. Nevertheless, we think that this intermediate validation step of the n-AFM was necessary before proceeding further. Now that the efficiency of the n-AFM is demonstrated, the next step will be to implement a molecular dynamics approach for the tip + molecule + surface system, based on a previous work.51 As mentioned in the Introduction, this aspect will be important if one wants to address the molecular manipulation process, as demonstrated recently,101 which needs to take into account the time varying force on the tip in MD calculations.102 Additionally, the n-aFM can be used in amplitude modulation mode (AM-AFM) without important modification; that is to say, this option is naturally implemented in the present version. This allows calculations in various environments, including the ultrahigh vacuum, as it is shown here, or at the solid−liquid interface, as it will be described in a forthcoming paper.

almost 15 Å width. We have adopted a large amplitude of 5 nm with the corresponding settings to get a performing use of the n-AFM. In this regime, the image process is sensitive to the structure of the tip apex. As it was already mentioned, the sharp tip apex follows a [111]-oriented diamond cluster where the atoms are piled up with six layers to generate a C3v symmetry. The interaction between the triangular second layer of the apex and the spherical molecule contributes to the triangular aspect of the image. In consequence, according to the incoming orientation of the tip, that is to say, either with an edge (red curve in Figure 10b) or with a facet (green curve in Figure 10b), the long-range interactions do not favor an isotropic response. We have checked this tip effect by considering the case where the tip apex consists of a C60 molecule oriented with a pentagon or a hexagon parallel to the surface. It is found that the image exhibits a cylindrical symmetry showing the wellknown convolution between the tip apex structure and the imaged system.

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CONCLUSION

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In this paper, we have demonstrated that the n-AFM is reliable to provide accurate calculated images of AFM in the frequencymodulation mode. Both attractive and repulsive regimes were addressed on two types of systems. The first one considers the case of flat hydrocarbon molecules, and the second one deals with a three-dimensional system consisting of a fullerene molecule deposited on a corrugated silicon surface. The oscillation amplitudes range from large (tens of nanometers) to ultrasmall (typically 0.01 nm) in order to simulate a probe like a cantilever or a tuning fork. Even if small oscillation amplitudes could overcome the description of the overall electronics, having in mind that all information is contained in the force fields, it remains dynamic processes and selfconsistent local relaxations between the tip and the surface that the n-AFM can describe, and that static force fields will not be able to do. This aspect will be particularly important if one tackles problems with charged molecules. Moreover, our results show that an acute tip apex, similar somehow to a tip with a CO molecule in the apical position, is responsible for the impressive submolecular resolution obtained in the repulsive regime. Indeed, whereas long-range attractive interactions bring a background force, Pauli repulsion has a more localized interaction extension, revealing the atomic structure of the sample. Prior to the image calculations, both molecular systems on the surface and the tip apex were relaxed. During the imaging process, the tip−sample interaction is calculated between parts in their frozen structure and only the dynamical process of the FM-AFM machine is running. Thus, one neglects

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel: +33 562 25 78 12. Fax: +33 562 25 79 00. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support from the Agence nationale de la recherche (ANR) under the MolSiC project (ANR-08-P058-36) within the PNano program is gratefully acknowledged. The authors warmly thank Y. Dappe, F. Spillebout, and P. Sonnet for the DFT calculated structures of the adsorbed molecules. Part of this work was performed using High Performance Computing resources from the CALcul en MIdi-Pyrénées (CALMIP) facilities (Grant No. 2011-[P0832]).

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Image Calculations with a Numerical Frequency-Modulation Atomic

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