surface-tip interactions in noncontact atomic-force ... · surface-tip interactions in noncontact...

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PHYSICAL REVIEW B 15 OCTOBER 1998-IIVOLUME 58, NUMBER 16

Surface-tip interactions in noncontact atomic-force microscopy on reactive surfaces: Si„111…

Ruben PerezDepartamento de Fı´sica Teo´rica de la Materia Condensada, Universidad Auto´noma de Madrid, E-28049, Madrid, Spain

Ivan StichJRCAT, Angstrom Technology Partnership, 1–1–4 Higashi, Tsukuba, Ibaraki 305, Japan

and Slovak Technical University, Department of Physics, Ilkovicˇova 3, SK-812 19 Bratislava, Slovakia

Michael C. PayneTheory of Condensed Matter, Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge CB3 0HE, United K

Kiyoyuki TerakuraJRCAT, National Institute for Advanced Interdisciplinary Research, 1–1–4 Higashi, Tsukuba, Ibaraki 305, Japan

and Institute of Industrial Science, University of Tokyo, Roppongi, Minato–ku, Tokyo 106, Japan~Received 9 March 1998; revised manuscript received 11 May 1998!

Total-energy pseudopotential calculations are used to study the imaging process in noncontact atomic-forcemicroscopy on Si~111! surfaces. At the distance of closest approach between the tip and the surface, there is anonset of covalent chemical bonding between the dangling bonds of the tip and the surface. Displacement curvesand lateral scans on the surface show that this interaction energy and force are comparable to the macroscopicVan der Waals interaction. However, the covalent interaction completely dominates the force gradients probedin the experiments. Hence, this covalent interaction is responsible for the atomic resolution obtained onreactive surfaces and it should play a role in improving the resolution in other systems. Our results provide aclear understanding of a number of issues such as~i! the experimental difficulty in achieving stable operation,~ii ! the quality of the images obtained in different experiments and the role of tip preparations and~iii ! recentlyobserved discontinuities in the force gradient curves.@S0163-1829~98!08336-2#

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I. INTRODUCTION

The atomic force microscope1 ~AFM! has developed astool which allows routine investigation of surface structurby probing the spatial variations of the forces between aand a surface. In its usual mode of operation, theconstantforce mode, the surface structure is determined by measurthe variation of the tip height which yields a constant foras the tip scans across the surface. Unlike the scanningneling microscope~STM!, where the images reflect the eletronic band structure around the Fermi level, the atomicsurface forces probed by AFM are, at least in principle, mdirectly related to the surface atomic structure and heshould also be simpler to interpret. However, experimenproblems with controlling the repulsive forces at the apatom resulting from the long-range attractive forces acton the tip as well as the interpretation of the complex natof the surface-tip forces have made the progress in AFMwell characterized surfaces in ultrahigh vacuum unexpedly slow. In particular, images taken in thecontact moderarely show individual surface defects, which are routinobserved with STM, which raises the question of whetherAFM is really capable of atomic resolution.

Only recently have Giessibl,2 Kitamura and Iwatsuki,3

and Ueyamaet al.4 demonstrated thatnoncontactAFMs op-erating in the attractive regime on reactive surfaces coproduce true atomic resolution. These experiments usealternative ac mode of imaging to reduce the effect oflong-range attractive forces. In this mode the tip movesand out of the interaction region during each oscillati

PRB 580163-1829/98/58~16!/10835~15!/$15.00

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cycle. In its original version, thefrequency shift mode, thefrequency of a tip oscillating at its eigenfrequencyv ismodified by an amountDv, as a result of the change of theffective cantilever spring constantk by Dk due to the tip-sample interaction.Dk is equal to the gradient of the forcbetween the tip and the surface (]Ftip-sample)/]z, and DkandDv are related by

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Hence, in this mode of operation the experiments probeforce gradients and a scan at a constantDv generates a mapof a constant average force gradient. This technique wascessfully used to image a number of reactive semicondusurfaces. Typically, these experiments use stiffer cantilev(;15–50 N/m! than those used in contact mode experime(,1 N/m) with tips of radius,150 Å etched from Si withthe estimated closest approach to the surface of;5 Å andthe attractive force, at that point,;20.15 nN.2 For ex-ample, on Si(111)-737 a clear image of the unit cell withits twelve adatoms has been obtained,2,5 on Si(001)-231 theindividual dimer atoms as well as an SB step and even theA,B,C defects could be resolved,5 as was the atomic structure of the InP(110)-131 surface.6

Force gradient detection has also been combined wstandard STM measurements.7,8 Simultaneous measuremenof the resonance frequency shift of the cantilever andmean tunneling current from the tip showed that stableaging with true atomic resolution is only possible overrestricted range of tip-sample distances. Using the avera

10 835 © 1998 The American Physical Society

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10 836 PRB 58PEREZ, STICH, PAYNE, AND TERAKURA

tunnel current instead of the average force gradient asfeedback parameter, stable operation was achievedatomic images of the Si(111)-737 reconstruction in the vi-cinity of a monoatomic step were obtained.

This technique was later modified by allowing for osclation amplitude dampingwhen the tip comes in close proximity to the surface.9–11 This technique provided very cleaimages of the Si(111)-737 surface, of quality comparablto that of an STM image, showing differences betweenequivalent adatoms. The observed contrast, however, vabetween different experiments: the center adatoms ap0.13 Å higher than the corner adatoms in Ref. 9, whermetallic tip was used, while in Refs. 10 and 11, using atip, the contrast is reversed, with the corner adatoms apping higher than the center adatoms.

It has also been demonstrated that the jump to conmay be avoided by applying a feedback control and thentip-surface forces and energies as a function of tip-surfdistance can be mapped out using a technique that coulcalledAFM spectroscopy.12

All of the experiments outlined above raise a numberquestions, such as the detailed nature of the surface-tip inaction at the distance of closest approach@estimated to be;5 Å, ~Ref. 2!#, which only theory can address. In particlar, very little is known about the mechanism that providatomic resolution in these experiments or about the vations between images taken under different experimeconditions or even between different parts of the sameage. To address some of these questions we have perfoan extensive theoretical study of the tip-surface interactiotaking the Takayanagi-reconstructed Si~111! surface13 as atypical representative of the reactive surfaces probed inexperiments. A theoretical description of the phenomenaare relevant to AFM must properly account for the variatiof the force with varying tip-surface distancez. At largeseparations the tip-surface interaction will be dominatedthe long-range Van der Waals~VdW! dispersive forces whileat small separations the short-range quantum-chemical fohave to be considered. To describe the latter we use defunctional theory~DFT!,14 which has been shown to provida reliable description of the surface-tip interaction at dtances of a few interatomic distances.15,16For the long-rangeforces we adopt a model based on the VdW interactiontween macroscopic bodies.17 Using this description of theinteraction we study in detail both vertical as well as hozontal scans, hereafter referred to as displacement curvelateral scans, respectively. Our work has a rather direct bing on the noncontact frequency shift AFM experiments2–8

and on the AFM spectroscopy experiments.12 Their rel-evance to the amplitude damping experiments9 is less obvi-ous, but differences in the response of the adatoms, sucthose discussed below, are a possible mechanism to exthe observed contrast in this imaging mode. Because ofcomputational cost, accurateab initio studies, similar inspirit to those presented here, exist only for Al surfaces wa high degree of symmetry.15,16

Our previous work, presented in Ref. 18, pointed out tthe covalent chemical interaction between the dangbonds of the adatoms on the surface and of the apex atothe tip do play an important role in the quality of the imagobtained in noncontact AFM. We showed that, even for d

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tances as large as;5 Å, that was the estimated distanceclosest approach in the earlier experiments, this interacgenerates forces that are of comparable magnitude toVdW tip-surface interaction but, more importantly, this cvalent chemical interaction dominates the force gradieprobed in noncontact AFM. Hence, it is this interaction thprovides a mechanism for atomic resolution imaging ofactive surfaces.

In the present work, extensive simulations are usedcompletely characterize that covalent interaction for the cof the Si(111)-737 reconstruction, the system for whicmost of the experiments have been done. These simulatinclude displacement curves over several relevant pointthe unit cell and lateral scans at a constant height of 5 Å overa large area of the surface unit cell. We present a detastudy of the variation of the force and the force gradientsa function of the tip-surface distance on the different atoms and rest atoms. These results confirm that onlyshort-range chemical interaction is able to provide a vation of the force gradient across the surface that is laenough to achieve atomic resolution. They also showimportant role of the relaxation of the atoms in the tip athe surface under the forces present. This atomic relaxatioresponsible for the difference between the tip apex-surfatom distance and the tip displacement, which is the quanaccesible to the experiments. This difference has to be tainto account in the interpretation of AFM scans and aSTM data. Special attention is paid to the possible accuparametrization of our results for the tip-surface interactusing simple model potentials, such as the Morse andRydberg potentials. Such a parametrization would openpossibility of performing realistic simulations of the cantilver dynamics during noncontact AFM operation, still necesary to understand in a quantitative way the relation betwforce gradients and measured frequency shifts. Our latscans provide a direct relation between the main featurethe AFM images and the structural and electronic properof the surface. Particularly relevant are the charge-tranprocesses among the dangling bonds and backbonds indby the tip, which can be related to the changes in the relacontribution of the different dangling bonds to the staclose to the Fermi energy due to the different interactwith the tip.

The rest of the paper is organized as follows. Sectiondescribes the details of the model and computational teniques used, paying special attention to the characterizaof the tip employed in our simulations. The convergencethese calculations with respect to various computationalrameters is discussed in detail in Appendix A. The resultsthe simulations of displacement curves and the discussiothe atomic relaxation effects and the force gradient curand fittings are included in Sec. III A. Lateral scans are psented in III B. Finally, in Sec. IV, our main findings arused to understand a number of experimental issues sucthe difficulty in achieving stable operation, the relation btween the quality of the images and the experimental contions, with special emphasis on the role of the tip prepation, and the origin of the recently observed discontinuitin the force gradient curve.

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PRB 58 10 837SURFACE-TIP INTERACTIONS IN NONCONTACT . . .

II. MODEL AND SIMULATION TECHNIQUES

A. Short-range interactions

As mentioned in the previous section, in this work tshort-range interactions between the AFM tip and the surfare modeled using DFT. Most of the calculations that willpresented were performed on the 535 member of the serie

FIG. 1. Ball-and-stick model of the tips considered.~a! A 4 Siatom tip with dangling bonds of the atoms in the base unsatura~b! the same tip but the dangling bonds in the base are saturatehydrogen atoms;~c! a 10 Si atom tip with the dangling bonds in thbase saturated by hydrogen atoms.

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of Takayanagi reconstructions.13 While computationallymore tractable than the experimentally observed 737 struc-ture, it contains most of the features of the 737 structure, inparticular, the adatoms and rest atoms. For testing purpcalculations have also been performed on the 333 structure.The system that we have considered is a supercell contaia Si~111! slab, two tips, one on each side of the Si slab athe vacuum region, with inversion symmetry imposed onsupercell. The slab consists of eight~111! planes and 200 Satoms for the 535 reconstruction~68 for the 333), with thecentral two layers kept fixed to simulate the bulk crystermination of the surface. The periodic images of tSi~111! slabs are four double layers apart, providingvacuum region of at least 6.86 Å between the tips. Theused in experiments are etched out of single-crystal Si.2,3,5,6

As the natural cleavage planes of Si are the~111! planes, weexpect that the very end of the tip will be bounded by thoplanes. A natural question arises about the finite size oftip to be used in the simulations. Within this model we cosidered three different tips, which are shown in Fig. 1. Treason for considering tips~a! and~b! separately even thougthey both contain four silicon atoms can be seen in Figwhich shows the charge-density differenceDr(r )5r t ip2b(r )2r t ip2a(r ) on a plane through the tips. The satration of the dangling bonds in the base of tip~b! changes thehybridization of the atom in the apex to close to thesp3

hybrid of the bulk, leaving a dangling bond pointing outthe apex atom towards the surface. The third tip~c! with 10Si atoms and the base dangling bonds saturated by hydratoms, has a charge distribution in the apex region similatip ~b!.

In Ref. 18 we showed that although similar in the apgeometry, those differences in the charge density produsignificantly different results for the interaction of the va

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FIG. 2. ~Color! Difference of the electronic charge densities between tip~b! and tip~a! of Fig. 1. Note the presence of asp3-like danglingbond in ~b! protruding out of the apex atom.

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ous tips with the surface when we scan over an adatom infaulted half of the 535 unit cell at a constant height of 5 ÅWe found that the total energy and normal force obtainedthe two tips that have dangling bonds pointing towardssurface@tips ~b! and ~c! in Fig. 1# were very similar, andmuch larger than the values obtained for the unsaturated~a! ~see Fig. 1 in Ref. 18 for details!. These small differencebetween the 4 and 10 Si atom tips@tips ~b! and ~c!, respec-tively# show that the short-range tip-surface interactioncompletely dominated by the interaction of the danglibond of the apex atom with the surface, and thus indicthat using tip ~c! the simulations will be converged witrespect to the size of the model used to represent the Atip. The results presented in the rest of the paper, excepsome of the convergence tests in Appendix A, were obtaiwith the 10-atom tip with saturated dangling bonds in tbase@tip ~c! shown in Fig. 1#.

The operation of the AFM in the lateral scanning mowas simulated in a stepwise, quasistatic manner by masmall movements of the tip parallel to the surface at a cstant height above the surface. At each step the atoms inslab and the tip were allowed to relax to their equilibriupositions for that particular tip position. Similarly, for simulating the displacement curves the tip was moved frvacuum towards the surface with the atoms allowed to reafter each tip movement. In each case the Si atoms inbase of the tip and the H atoms attached to them, if thesepresent, are kept fixed in their positions after a tip displament. These simulations are expected to provide an accudescription of both imaging modes since the motion ofAFM tip occurs on a much longer time scale than the atomtimescale. Similarly, finite-temperature effects are notpected to significantly modify the results as typical atomfrequencies~THz! are much larger than the tip oscillatiofrequency~kHz!.

The energies and atomic forces were calculated witDFT ~Ref. 14! in its plane-wave pseudopotentiformulation.19 Optimized nonlocal pseudopotentials20,21wereused for silicon with thep andd components of the potentiamade very similar and optimized to make the pseudopotial rapidly convergent with respect to the cutoff energythe plane-wave expansion. The pseudopotentials wereplied in a separable form, taking thep component as reference and including only thes nonlocal component. The projection of this component was performed in real space.22 TheCoulomb potential was used for the hydrogen atoms. Telectronic orbitals were expanded at theG point of the Bril-louin zone with a cutoff energy of 7 Ry. This set of parameters was found to yield highly accurate results for the toenergies and vibrational frequencies for the 737structure.23,24We used the gradient corrected functional25 forthe exchange-correlation energy in the implementationRef. 26. The density functional was minimized with respto both electronic and ionic degrees of freedom using congate gradient techniques.19 The convergence criteria were sto 531025 eV/atom for the energies and to31022 eV/Å for the forces on the atoms. Since thTakayanagi-reconstructed Si~111! surface is metallic, the occupation of the electronic orbitals around the Fermi level wcarefully checked. The simulations have been done usmassively parallel computation.27 The convergence of the

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B. Long-range interactions

Dispersive VdW interactions are not correctly describby standard DFT. Recently a new long-range nonloexchange-correlation functional has been proposed to exthe aplicability of DFT to the problem of physisorptionsurfaces.28 Although this prescription provides promising results for free-electron-like surfaces, its extension to semicductor surfaces has not been developed yet. We have meled the contribution of the VdW interaction to the totnormal tip-surface force using the Hamaker summatmethod.17 The method is based on summation over the vume of the tip and sample of the asymptotic VdW pair ptentials

WVdW~r !52C

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where the value of C depends on the tip and sample maals. The technique is applicable at distances sufficiently lato make the details of atomic structure unimportant. Forample, for the VdW force acting on an atom over a polarable flat surface of densityr one obtains

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where A is the Hamaker constant, andr the atom-surfacedistance. We used this technique to estimate the norforces and force gradients for the microscopic four Si attip @tip ~a! of Fig. 1# and for a macroscopic spherical tip witthe experimental curvature radius of R540 Å,2 where theinteraction is given by

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with r being the distance from the surface to the center ofsphere. We use the Hamaker constant for the Si-Si intetion through airA51.865310219 J.29 We have consideredthe tip-surface distance in Eqs.~3! and ~4! to be the normaldistance between the tip and the highest adatoms, whheight is taken as a reference for the position of thesurface with densityr„Si(111)… in our calculations. Since theaverage height of the atoms in the surface is lower thanhighest adatoms this model provides an upper bound forinteraction with a 535 reconstructed surface.

It could be argued that by using a sphere we are disgarding the contribution to the VdW interaction of the maroscopic parts of a tip which has a more realistic conishape. A calculation similar to that for the sphere30 showsthat the cone-surface interaction is given by

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with r the distance from the surface to the apex of the coand w the half-angle of aperture of the cone. An estimausing a cone with a half-angle of aperture ofp/4—using thesame Hamaker constant—for the estimated distance of cest approach gives a force of 0.062 nN and a force gradof 0.1 N/m, both of which are much smaller than the valuof 0.50 nN and 2 N/m obtained for the sphere. Ciraciet al.15

found similar conclusions in their study of the VdW interation between metallic tips and surfaces. They considesemi-infinite tips with conical and hemispherical ends ancylindrical shank as a more realistic approximation to theshape. The Lifshitz formula for an atom interacting withflat polarizable surface was integrated over the tip volumeobtain the total VdW interaction. Their results show that themispherical geometry gives contributions to the forceforce gradient an order of magnitude larger than the oobtained with the sharp conical tips~half-angle of apertureequal or less thanp/4), in agreement with our model calculations.

Our calculations, as most of the work in Ref. 17, abased on the additivity of atomic contributions to calculathe total VdW force. As Hartmann30 has shown, a more precise and conceptually satisfactory approach than thaLifshitz31 gives the same results for the regime of small serations, where the nonretarded interaction is the dominterm.

Finally, it should be noticed that the possible error asciated with a continuum description of the surface, insteada detailed microscopic account of the discrete nature oflattice in the areas closer to the tip, should be extremsmall due to the long-range nature of the interaction, a gerous upper bound being given by the order of magnitudethe VdW interaction between the 4 Si atom tip and the sface where the values of the force and the force gradie(20.02 nN and 0.15 N/m, respectively, for a normal dtance of 5 Å! are an order of magnitude smaller than tcorresponding values for the macroscopic tip (20.50 nNand 2 N/m, for the same normal distance!.

III. RESULTS: SIMULATED AFM SCANS

A. Displacement curves: Tip-surface covalent bondand atomic relaxation

The results for the total energies32 and forces due to shortand long-range interactions are summarized in Figs. 3 anrespectively. In Fig. 3 and subsequent figures related toshort-range interaction, the results are plotted as a functiothe ‘‘tip-surface distance,’’ which is defined as the differenin height between the unrelaxed tip apex and the highadatom in the unrelaxed surface. In order to provide a dicomparison, results in Fig. 4 are also shown as a functiothe analogous tip-surface distance; in this case, the differein height between the surface of the sphere and the unrelsubstrate highest adatoms, which defined the position offlat surface in our model for the long-range interactioWith these definitions, variations in the tip-surface distanare then directly related to the relative displacements ofand sample which are measured in the experiments.

The conclusion that emerges from the data in Figs. 34 is that only the short-range interaction at ‘‘near contacdistances (;3 –5 Å! is able to provide a variation in th

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FIG. 3. Short-range interaction energy~middle panel! and nor-mal force~bottom panel!, as a function of the tip-surface distancfor tip ~c! of Fig. 1 over corner hole atom~squares!, over a restatom~triangles! and the adatom on the long diagonal~black circles!in the faulted half of the unit cell, and over the adatom on the lodiagonal in the unfaulted half~white circles!. For comparison, theinteraction between two tips@tip ~b! of Fig. 1 as shown in the insetwhite diamonds# is also shown. The top panel shows a ball-anstick model of the 535 reconstruction, including a top view of thunit cell and a lateral view of the atoms close to the lattice plaalong the long diagonal. The atoms with dangling bondsmarked: corner hole~Ch!, faulted ~F R! and unfaulted~U R! restatoms, faulted diagonal~F Ad! and off-diagonal~NF Ad! adatoms,and unfaulted diagonal~U Ad! and off-diagonal~NU Ad! adatoms.

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bond and tip apex dangling bond. This can be seen by cparing the curves for the scans over an adatom, a rest aand corner hole atom. This conclusion is further corroboraby results for a model system consisting of two interactingtips also shown in Fig. 3. These results, for both the toenergy and the normal force, are very similar to those fortip interacting with a surface adatom, apart from the strikdifferences in the position of the minima. The key to undstanding the difference in the position of the minima is thwhen the atoms in the tip and the surface are allowed to runder the forces present, the relative movemement of theapex and the surface atoms is no longer the same as thmore distant parts of the tip and the sample. Our calculatishow important relaxations even for a tip-surface distanc5 Å, where the actual apex-adatom distance~apex-apex dis-tance for the two interacting tips! after atomic relaxation reduces to 4.72 Å for the 535 reconstruction, 4.77 Å for the333 reconstruction~see Appendix A for details!, and 4.92 Åfor the two Si tips. The effect of atomic relaxation becomincreasingly important as the tip and surface get closer. Tpoint is illustrated in Fig. 5, where the distance betweenapex tip atom and the surface atom just below the tip orelaxation is taken into account is shown as a function oftip displacement~measured in terms of the tip-surface dtance defined above!. Considering the case of the tip dis

FIG. 4. Long-range van der Waals interaction energy~top panel!and force~bottom panel! as a function of the tip-surface distance fa macroscopic spherical tip of radius of 40 Å.

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placement over one of the adatoms as an example and tathe relaxed configuration at 5 Å as thestarting point, if nofurther relaxation were present the apex-adatom distawould follow the dashed line shown in Fig. 5. It can bclearly seen that the real apex-adatom distance rapidly dates from that linear behavior due to the relaxation of bthe tip and surface atoms. Differences between the cuthus reflect the different atomic relaxations due to the diffent bonding of the atoms to the surface on the various stems considered. In all cases the minimum in the total ene~and zero in the normal force! corresponds to the tip positiowhere the apex-surface atom distance~apex-apex distance! isroughly equal to 2.35 Å, which is marked by the horizondotted line in Fig. 5, the Si-Si nearest-neighbor distancebulk Si. This difference between the apex-surface atomtance and the tip displacement, which is the quantity accsible to the experiments, should be taken into accountonly in the AFM scans but also in the interpretation of STdata. Similar effects occur in the STM and have beenvoked to explain the falloff in corrugation on Cu surfaces33

and the constant value of the apparent barrier height at stunnel resistances for several metallic surfaces.34

The vertical scan over the rest atom shows a similarhavior to the scan over the adatom but the minima in bthe total energy and normal force are displaced by aro1.25 Å, which is roughly the difference in height~1.12 Å!between the adatom and rest atom in the 535 reconstruc-tion. The minimum in the total energy also corresponds tdistance between the tip apex atom and the rest atom equ2.35 Å. As the tip approaches the surface the rest amoves up towards the tip apex until the tip-surface distais roughly equal to 2.5 Å, close to the point where the mamum of the normal force is reached and the curvature ofapex-adatom distance versus tip-surface distance chasign ~see Fig. 5!. The same behavior was found in the ad

FIG. 5. Apex-adatom~apex-rest atom! distance as a function otip displacement for tip~c! of Fig. 1 over a diagonal adatom~blackcircles! and a rest atom~white circles! in the faulted half of 535reconstruction. The apex-apex distance for the interaction betwtwo tips @tip ~b! of Fig. 1 is also shown~squares!#. The dashed lineshows the apex-adatom distance for the tip~c! over the adatom fora rigid displacement of the tip if no relaxation were allowed startfrom the tip configuration at 5 Å. The horizontal dotted line corrspond to the nearest-neighbor distance in bulk Si.

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tom scan. In fact, if the rest atom curve in Fig. 5 is displacin the horizontal direction by 1.25 Å, it overlaps almost pefectly with the one for the adatom.

The displacement curves discussed above were madeatoms on the faulted half of the unit cell. Results for tunfaulted half are very similar for both the total energy athe normal force, as the comparison between the scansthe faulted and the unfaulted adatom shows~see Fig. 3!, andfor this reason they will not be discussed further.

It was already pointed out in Ref. 18 that the covalecharacter of the short-range dangling bond interaction psisted at least up to a surface-tip distance of;5 Å. Thecharge-density difference between the surface with the tia distance of 5 Å over one of its adatoms and a superpositiof the charge densities of an isolated surface and tip~see Fig.2 in Ref. 18! showed a pronounced charge transfer toadatom dangling bond under the tip from the dangling boon the rest atom and from the backbond. It should be notithat a similar reverse charge transfer has been experimenobserved after chemisorption of molecules onSi(111)-737 surface.35 This accumulation of charge in thadatom dangling bond~notice that the dangling bond in thtip has already one electron! is precisely what we expect athe onset of the formation of a covalent bond between theand the surface. The work presented here provides moredence of the covalent character of this interaction. In partilar, the fact that this is not just a polarization effect inducby the tip on the surface states but the onset of real covabonding is further confirmed by the spatial distribution of tcharge density of the highest occupied electronic stawhich clearly show that they are a bonding combinationthe original dangling bond states in both the apex atomthe adatom~see the discussion about the character ofstates around the Fermi energy at the end of Sec. III BFig. 13!. Further support comes from the study of the formtion of this bond as the tip approaches the adatom.charge density difference shown in Fig. 6 for a tip-surfadistance of 3.5 Å very much resembles the one in Fig. 1 i nRef. 18, except for the increase in the charge transferrethe adatom. The concept of chemical reactivity betweentip and the surface18 was recently invoked to explain thdifference in the contrast between inequivalent adatomsthe Si(111)-737 surface in the amplitude dampinexperiments.9

In order to perform simulations of the cantilever dynamunder the conditions used in noncontact AFM it woulddesirable to have a simple but accurate description oftip-surface interaction. Morse potentials are known to pvide a reasonable description of covalent bonding in aatomic molecule through a simple analytical function:

V~r !5V0H F12expS 22br 2Rc

RcD G2

21J , ~6!

whereV(r ) represents the total bonding energy as a functof the interatomic distancer , andV0 , b, andRc are param-eters that define the strength and range of the bonding inaction. Due to their simplicity, Morse potentials have beused in different contexts, including AFM simulations,36 tosimulate bonding interactions. We have also consideother functional dependencies used to fit total-energy cur

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like the Rydberg function,37 V(r )52V0@11b(r2Rc)#exp@2b(r2Rc)#. The results obtained for the fits witthis potential are very similar to the ones presented belowthe Morse potential, and for this reason they will notdiscussed further.

We have tried to fit our total-energy curves for the shorange tip-surface interaction and the interaction betweentwo tips shown in Fig. 3 with a Morse potential, takingr asthe tip-surface distance andV0 , b, andRc as fitting param-eters. However, it has not been possible to find a good fitboth the attractive and the repulsive parts of the interactand the fits obtained taking just the attractive part and fpoints in the repulsive region close to the minimum provia very poor fit to the calculated force derivatives. Considing the differences between the tip-surface distance andapex-surface atom distance when relaxation is taken intocount as discussed above, and the fact that the Morse potial should describe the interaction of two atoms as a fution of their real interatomic distance, we have also tried ataking r as the apex-surface atom distance. Although wetained a little improvement, the quality of the fit was still fafrom satisfactory.

A direct fit to the calculated forces using the derivativethe Morse potential, takingr as the real apex-surface atodistance proved to be much more successful. The fits fortip-adatom, tip-rest atom, and tip-tip force are shown in F7. It should be noticed that although the fit is done with treal interatomic distance, the plot is presented with respecthe corresponding tip-surface distance defined above. Th

FIG. 6. Charge-density difference between the self-consiselectronic density of the interacting system and the superpositiothe densities of the isolated tip and surface. The plot is taken oplane perpendicular to the surface along the long diagonal infaulted half of the unit cell. The tip is on top of the F Ad. Thtip-surface distance is 3.5 Å. The squares indicate the positiothe Si atoms in that plane both in the tip and in the Si surface.solid ~dotted! lines indicate an increase~decrease! of electroniccharge; the contours correspond to60.4, 61, 62 ~as in Fig. 2in Ref. 18!, 64, 66, 68, 610, 612, 114 and 11631022 electrons/Å3. Notice the transfer of charge to the adatodangling bond from the backbond and the dangling bond in the F

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fits reproduce the behavior of the force close to its minimuand the integration of those curves provide a reasonablethough not very accurate, description of the total energythe different systems. The quality of the fits deteriorateswe try to include more fitting points in the repulsive partthe force, showing that, as one can expect, the tip-surinteraction deviates from the pure ‘‘diatomic’’ case as theapproaches the surface and the response of the adatomrest atom becomes more influenced by the atoms in theers below. It must be noted that good fits are only obtainwhen we use the real apex-surface atom distance, furconfirming the importance of the relaxation effects describabove. Typical values for the fitting parameters are showTable I. The quality of the fits for the force and the totenergy seems to be better in the case of the interaction otwo tips than for the tip-surface interaction. In particular, tfits for the adatom and rest atom on the 535 reconstructiondo not reproduce the calculated values for the largesttances~around 5 Å!, which are well described in the casethe tip-tip interaction. The convergence tests discusseAppendix A seem to rule out large inaccuracies in our cculated forces and one can speculate as to the origin of tdifferences. One possible explanation is simply that a Mortype function is too simple to describe the details of tinteraction, and a fit that tends to better reproduce the awith larger values around the minimum will fail to describthe behavior accurately for large distances. In this resp

TABLE I. Parameters obtained in the fit of the forces with tderivative of the Morse potential@Eq. ~6!# for tip ~c! interactingwith an adatom and a rest atom on a Si(111)-535 surface, and forthe interaction between two tips@tip ~b! in Fig. 1#.

V0 (eV) b Rc(Å)

Adatom 2.273 1.497 2.357Rest atom 2.636 1.526 2.357Two tips 3.094 1.328 2.292

FIG. 7. Morse fits to the normal force for the tip-diagonal adtom ~black circles! and the tip-rest atom~white circles! interaction.The fit for the tip-tip interaction~squares! is shown for comparisonIn each case the symbols correspond to the calculated values oforce while the lines represent the fit.

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the tip-tip interaction should be easier to describe ascharge in the dangling bonds of the two tips is the same othe whole distance range. There are no large changes ineigenstate distribution around the Fermi level since onlystates associated with the two unsaturated dangling bondpresent. The situation is much more complicated for thesurface interaction where several states associated withdifferent dangling bonds on the surface are close toFermi level. In this case, charge piles up in the danglbond interacting with the tip and is depleted from othereas, the amount of charge depending on the strength ointeraction between the adatom- and tip-dangling bonThis is described in detail at the end of next section.summary, the fitting process provides further support tocentral role of covalent bonding in the tip-surface interactiobut, at the same time, it reveals some of the subtle difences from a simple model of diatomic bonding.

As pointed out in the Introduction, force gradients playcrucial role in noncontact AFM. The force gradients are na direct output of our total-energy calculations, but we ccalculate them from the force versus distance curvescussed above. The most accurate way to do this would btake a numerical derivative of the calculated force curve othe whole distance range. However, that would requirefiner grid than the one we can afford due to the computional cost of the calculations. One possible solution istake a derivative of the force fits obtained above. Figurecompares the force gradients for the tip-tip, tip-adatom, atip-rest atom interaction, obtained as a derivative of the cresponding force fits, with the force gradients associated wthe long-range VdW interaction for the macroscopic tip, otained as well as a derivative of Eq.~4!. This figure clearlyshows the similarities between all the covalent bondingteractions as opposed to the VdW interaction. Obviously,values for the force gradients are not very accurate for lardistances due to the limitations of the fits. Our force fits teto underestimate the force gradients for large distances w

-

the

FIG. 8. Force gradients for the the tip-diagonal adatom~blackcircles!, the tip-rest atom~white circles!, and the tip-tip~squares!interaction~obtained as a derivative of the corresponding force fi!.Force gradients associated with the long-range VdW interactionthe macroscopic spherical tip of radius 40 Å are indicated bydashed line.

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compared to the values obtained from a numerical derivaon a finer grid. The force gradient on the faulted adatomdistance of 5 Å obtained from the calculation of the force oa fine grid centered at that distance and the use of a numcal derivative is almost a factor of 2 larger than the oobtained from the force fit~see Table I in Ref. 18!. Theresults in Fig. 8 clearly show that the covalent bondingteraction dominates the force gradients that cause thequency shifts used to create the experimental imagWe note that similar results have also been obtained for333 surface.38

B. Lateral scans

The conclusion from the simulations of the vertical scais that the tip-surface force and force gradients for distanlarger than 4 Å are only significant close to the adatomAlthough a direct calculation of a force gradient map of tsurface is not possible due to the computational cost, Figand 8 show the direct proportionality between forces agradients for large distances. It should be noticed thatfunctional dependence such as the Morse potential issumed, the behavior at large distances would be dominby an exponential term, and the force gradient wouldessentially the force multiplied by a constant factor. In tcase of a power-law dependence~such as the one obtainedfor example, for the VdW interaction!, the gradient would beessentially the force divided by a factor proportional to tdistance, which in a lateral scan at a constant height woagain be a constant. Therefore using the lateral force swe can map out the AFM ‘‘image’’ of the adatoms.

Lateral force and total-energy scans at a tip-surfacetance of 5 Å are presented in Figs. 9 and 10. In Fig. 9 wshow the total energy32 and the normal force for a tip scanning along the long diagonal and for a scan parallel tolong diagonal over the off-diagonal adatoms. Figureshows the values of the forces calculated for a set of scparallel to the long diagonal, which probe intermediate potions between the off-diagonal adatoms and the long dianal. Each intermediate scan is 0.5 Å closer to the long donal, starting from the off-diagonal adatom scan, which3.84 Å off the long diagonal.

The differences between the scans over the diagonaloff-diagonal adatoms are related to a symmetry breakingtortion; there is a Jahn-Teller distortion that breaks theC3vsymmetry of the surface and leaves the off-diagonal adatat lower heights than the diagonal ones. A similar symmebreaking was found in Ref. 39 for this reconstruction. In trespect, the behavior of the 535 reconstruction seems to bdifferent from the one calculated for the 737 reconstructionusing the same theoretical approach,40,41 where no distortionwas found. This results in a different response of the adatin the two scans. The adatom response for the scan alonlong diagonal can be seen in Fig. 11, which shows that astip approaches the diagonal adatom in one half of thecell this adatom moves upwards and electronic chargtransferred into the dangling bond of that adatom, whilethe same time the adatom in the other half of the unit cmoves downwards and its dangling bond is depletedcharge. As the tip reaches the point midway betweenadatoms, both of them have come back to their original

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sitions, and the process reverses with the faulted adamoving upwards and taking charge, as the tip approacthis adatom. The normal displacements of the adatomsvery different for the diagonal and off-diagonal scans athis is reflected in differences in the strengths of the ccomitant tip-surface covalent bonds that are formed. We nthat adjustment of the adatom-apex distance to the svalue yields energies and forces that are identical for thedifferent scans. The interaction between dangling bondsthe adatoms and the dangling bond on the apex atom gdeep minima in the total-energy curve over the adatomsa barely detectable signal at the positions of the rest ato

FIG. 9. Short-range interaction energy~eV, medium panel! andforce ~nN, bottom panel! for tip ~c! of Fig. 1 scanning over the longdiagonal~black circles! and for a scan parallel to the long diagonover the off-diagonal adatoms~white circles! at a tip-surface dis-tance of 5 Å. The labels indicate the position of the atoms wdangling bonds on the surface~F Ad, U Ad, R, Ch, NF Ad, and NUAd. The top panel shows the two scans on a top view of the unitof the 535 reconstruction.

-

nr


FIG. 10. ~Color! Normal force~nN! for tip ~c! scanning over anarea of the unit cell~shown in thetop panel! at a tip-surface heightof 5 Å. The scan direction is parallel to the long diagonal. Theminima correspond to the positioof the adatoms, with the deepeone over the F Ad.

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Similar results are obtained also for the normal force, albwith an enhancement of the contrast so that the rest atare better resolved with the normal force. The differenbetween the faulted and unfaulted halves of the unit cellin line with the different amount of charge in the danglinbonds in the two halves of the unit cell.42 The lowest bindingenergy and normal forces are obtained around the cohole. From the half-width of the peaks in the force scansestimate the lateral resolution of the order of;3 Å. Thecalculations performed on the 333 structure38 yield broadlysimilar results.

These results suggest that the tip may induce imporlong-range charge-transfer processes among the danbonds and backbonds that are directly correlated withnormal displacements of the surface atoms. This in turn

its

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be traced back to the way the tip interaction changesrelative contribution of the different dangling bonds on tsurface to the electronic states close to the Fermi eneWhen the tip is placed in a position midway between the tdiagonal adatoms, our calculation shows four electrostates, associated with the different adatoms and the cohole, within a region of;0.1 eV around the Fermi energyBelow them, in a region of 0.2 eV, we find the states asciated with the rest atoms. Two of the states associated toadatoms~located at20.018 and20.089 eV with respect tothe last occupied eigenstate! have a large weight on thefaulted adatom. The charge densities associated with thstates are shown in Fig. 12. These states show the fauadatom dangling bond mixed with the dangling bonds frothe unfaulted diagonal adatom, the corner hole and the n

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diagonal adatoms~not shown on the figure!. A comparisonwith the local density of states obtained for the case whthe tip is on top of the faulted adatom shows that theinteraction singles out the faulted adatom from the rest ofadatoms and lowers these two states by 0.12 eV. The inaction ~see Fig. 13! has removed the contributions from thcorner hole and the other adatoms to these states and osmall contribution from the unfaulted diagonal adatommains. The faulted adatom dangling bond now appestrongly mixed with the tip dangling bond and nonnegligibcontributions from the rest atoms are observed due toproximity in energy of those states.

IV. DISCUSSION AND CONCLUSIONS

We have presented above a detailed study of thesurface interactions involved in the noncontact AFM imaing of reactive surfaces. Our displacement curves showfor ‘‘near contact’’ distances~below 5 Å!, the short-rangecovalent chemical bonding between the dangling bondsthe tip and the surface provides interaction energiesforces comparable to the macroscopic tip-surface VdW inaction, and completely dominates the force gradients proby the experiments. More importantly, both the displacemcurves and the lateral scans show that the covalent intetion has a strong dependence on the tip-surface distanceprovides a clear contrast among different sites in the u

FIG. 11. Lower panel: normal displacements of the adatomstip ~c! of Fig. 1 scanning over the long diagonal. The two uppcurves correspond to the diagonal adatoms, the two intermediathe off-diagonal faulted adatoms and the two lower to the odiagonal unfaulted adatoms. Notice that the symmetry betweentwo halves of the unit cell is preserved during the scan. Uppanel: charge-transfer process accompanying the normal addisplacements, charge-density difference between theconsistent electronic density for the tip on the faulted and thefaulted diagonal adatom~details as in Fig. 6, labels as in Fig. 9!.

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cell. Thus, this interaction satisfies all the conditions for pviding atomic resolution on reactive surfaces and for enhaing the resolution in other systems.

These conclusions have received a large amount ofperimental support. The large differences in normal forcand force gradients between different atomic sites in thecell obtained in the simulations, with clear maxima over tadatoms, correlates with the contrast observed in all theperimental images taken in the constant frequency smode, where maxima in the tip height occur at the positioof the adatoms. Our finding of atomic resolution inducedtip-surface reactivity18 has recently been invoked to explathe observed differences in imaging inequivalent adatomsthe Si(111)-737 surface.9 Our present work lends furthesupport to this interpretation. The discontinuity in the forgradient curve found on Si(111)-737 and its interpretationin terms of a crossover between van der Waals and chembonding6,11 is consistent with our results. The role of thdangling bond interactions is also supported by the expmental observation that the onset of tunneling current andrapid variation of the frequency shift, recorded simultneously in Ref. 7, take place at very similar tip-surface dtances. Our displacement curves~see Sec. III A and Figs. 3and 8! explain the difficulty in achieving stable operationthe frequency shift mode reported in early experiments. Tis due to the change in slope of the normal force, andassociated change of sign for the force gradients, which taplace over a small range of distance.

Our results can also be used to discuss the qualitynoncontact AFM images of Si(111)-737. The early image

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FIG. 12. Charge density associated with the two eigenstcloser to the Fermi energy, which have a larger weight onfaulted adatom. The tip is placed in a position midway betweentwo diagonal adatoms. The contours correspond to 0.2, 0.4, 0.6,1.0, 1.2, 1.4, 1.6, 1.8, 2.0~continuous lines! and 3.0, 4.0, 5.0~dashed lines! 31022 electrons/Å3.

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of Giessibl2 shows a low-quality image except for the widof one unit cell where the characteristic protrusions of theadatoms suddenly appear and then disappear again. Thfect could be induced either by pickup of a surface atomthe tip or by losing a contaminant previously attached toapex of the tip apex. In either case, if a dangling bond poing towards the surface is formed we would expect a dmatic improvement in resolution. A similar scenario is posible in the case of the metallic tip used in the dampexperiments of Ref. 9, which provided images of extremgood quality. In the experimental setup of Kitamura aIwatsuki,3 using a stiffer cantilever than Giessibl~37 N/mcompared to 17 N/m! these processes are much less likelyhappen and the original images were more fuzzy. Howevesimilarly dramatic improvement in resolution was achievafter tip depassivation.5 The tip depassivation is likely toexpose the tip dangling bonds and so this change in expmental resolution could be due to the effects discussed inpaper.

A pronounced enhancement of the image contrast w

FIG. 13. As Fig. 12 when the tip is on top of the faulted diaonal adatom. The bottom panel shows the distribution of occupeigenstates close to the Fermi level for the two positions of thewith the arrows marking the evolution of the two eigenstates csidered in Fig. 12 and this figure.

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scanning with a Si tip on a Si(111)-737 surface has beenrecently reported.11 The enhancement correlates with the oservation of a new type of force gradient curves which cotain a discontinuity. Although, for the sake of brevity, wshall refer to these observed rapid changes in the forcedient as a discontinuity, it is necessary to point out that thare no real, physical force discontinuities. They are simchanges that occur over such a small distance as not tresolvable in the experiment. In contrast to other expements, the sample was prepared in this case in a diffechamber to avoid the deposition of Si atoms on the tip apAt the beginning of the AFM measurements only force gdient curves that did not have a discontinuity were observHowever, after accidental contact between the tip andsample during the measurements force gradient curves wdiscontinuity began to be observed. The discontinuity incates a rapid increase of the attractive force acting on thewhich, as pointed out by the authors, can be naturallyplained in terms of the onset of a chemical bonding intertion between the dangling bond out of the Si atoms pickedon the tip apex and the adatom dangling bond. In additionthe previous evidence that tip happens to pick up Si atofrom the surface,43 atomic-scale removal of Si adatomsactually observed in this experiment. It should be noticthat the contrast enhancement in the scans is mainly duthe discontinuity in the interaction, with the dark parts of timage corresponding to the smaller~lower! value of the forcegradient below the discontinuity and the bright ones tolarger ~upper! value. This is consistent with the stronger iteraction due to the onset of covalent bonding occurring othe adatoms and the much weaker interaction occurring othe rest atoms and the corner hole as we obtain in our silations.

The model we considered here is in many respectsoversimplification. In particular, we consider only tips withsingle unsaturated dangling bond, scanning a perfect surparallel to the long diagonal at a constant height. Despitethese simplifications our model yields energies, forcesforce gradients that are qualitatively comparable to the emated experimental values. For example, the short-ranormal forces shown in Fig. 9 are comparable to estimaexperimental force of20.14 nN~Ref. 2! and the calculatedforce gradient of;10 N/m at over the adatoms is closethe force constant of the piezolever~17 N/m! in that experi-ment making the scenario of an accidental tip-surface conand the concomitant apex atom pickup quite likely.

A direct quantitative comparison of calculated force gdients and measured frequency shifts is still precluded bycomplicated dynamics of the cantilever: noncontact measments often use a large amplitude of oscillation of the tipthe force due to interaction with the surface acts on thefor only a small part of the oscillation cycle and the chanin frequency is relatively small. It is worth noting, howevethat the ratio between the force gradients calculated fortips with and without dangling bonds is roughly the samethe ratio between the frequency shifts measured duringeral scans for the tips with and without a discontinuitytheir force gradient curves.11 Recently Giessibl has used canonical perturbation theory to derive an analytical expressfor the frequency change due to tip-sample forces whichbe described by an inverse power law.44 This opens the pos

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sibility to determine the frequency change due to the vdinteraction—which is naturally described by these invepower laws—and to estimate the contribution to the fquency change coming from the chemical bonding forusing a simple Lennard-Jones pair potential to describebonding interaction. After setting the parameters forLennard-Jones potential using the bulk silicon propertieshas fitted the theoretical frequency change to differentperimental results in the literature, assuming a pyramidaand using as a fitting parameter tan2(a/2)AH , with a the tipangle andAH the Hamaker constant. The fitted theoreticcurves provide a reasonable description of the experimedata over a range of distances. In particular, they confirmpronounced effect of the chemical forces on the frequeshift at tip-sample distances around 5 Å in the experimentswith Si tips and Si~111! surfaces. These chemical bondinforces are responsible for the important departure frombehavior associated with a pure VdW interaction observethe experiments and qualitatively described by the simplifimodel for covalent bonding included in the theoretical cur

On the other hand, our estimates for the forces and fogradients differ by about an order of magnitude from thoobtained in AFM spectroscopy experiments12 even at theheights where the short-range chemical interactions shdominate. One possible explanation for these discrepanmight be the difficulty in relating the measured frequenshifts with the force gradients pointed out above. Somethese effects might be addressed experimentally and/orefinement of our present model. Application of the theoryother systems studied experimentally, such as the InP~110!surface are under way.

ACKNOWLEDGMENTS

The calculations were performed on the JRCAT supcomputer system. This work was partly supported byNew Energy and Industrial Technology Development Ornization~NEDO!. R.P. acknowledges financial support frothe HCM Program~European Union! under Contract NosERBCHBICT30779 and ERBCHRXCT930369, and frothe CICYT ~Spain! under Project No. PB92-0168.

APPENDIX A: CONVERGENCE TESTS

We have performed several tests for both the 333 andthe 535 reconstructions in order to check the accuracyour calculations with respect to the thek-point sampling andthe energy cutoff used. These tests support the main consion of our paper: the strong contrast found in both the to

TABLE II. k-point sampling convergence of the total enerand normal force for the interaction of a 4 Siatom tip@tip ~b! in Fig.1# on two positions on the Si(111)-333 surface.

Adatom Mid position

Energy (G) 20.088 eV 20.020 eV’’ ~3k points! 20.142 eV 20.041 eV

Force (G) 20.194 nN 20.013 nN’’ ~3k points! 20.322 nN 20.017 nN

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energy and the normal force between the adatoms and opositions in the unit cell is due to the onset of a covalechemical interaction between the tip- and adatom-dangbonds. In summary, test calculations at 11 Ry for the 535reconstruction with the 10 Si atom tip@tip ~c! in Fig. 1# showa very small reduction in the corrugation both in the ene~by 0.001 eV! and the normal force~by 0.012 nN! with re-spect to the 7-Ry cutoff used in the calculations presenabove, and further tests at 20 Ry on the 333 reconstructionsconfirm these findings with reductions of 0.016 eV and 0.0nN. Tests on the 333 reconstruction show that the apparecorrugation increases with the accuracy of thek-point sam-pling and, with a slightly higher sampling density than usin the 535 calculations, gives results that are very similarthose presented above. What follows is a more detailedcount of these tests.

1. k-point sampling tests

The interaction of the Si(111)-333 surface with a 4 Siatom tip with a dangling bond pointing out of the apex ato@tip ~b! in Fig. 1# has been calculated using the same 7-cutoff and two different sets of points: one set is just tG point, while the other contains threek points: ~0, 1/3!,

FIG. 14. Short-range interaction energy~top panel! and force~bottom panel! for tip ~c! of Fig. 1 over an adatom on the 535reconstruction~black circles! and tip~b! on the 333 reconstruction~white circles!.

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~1/3, 0!, and~21/3, 1/3!. We have fully relaxed the structurfor two different positions of the tip: on top of one of thadatoms~the one in the unfaulted half of the unit cell inposition midway between the adatoms on the long diagoThe convergence to the equilibrium configuration was pformed with the same criteria as in the rest of our calcutions, in particular, so that the forces on individual atowere all less than 0.01 eV/A. We have also calculatedsurface and tip total energies with the samek-point samplingto get the proper reference for the total energy of the inacting system.

Our results are summarized in Table II. Both the toenergy and the normal force show that the corrugationcreases significantly when we use the denser sampling.results for this improved sampling are very close to the oobtained for the 535 reconstruction using theG point onlyinteracting with the same tip (20.23 eV and20.34 nN forthe diagonal adatom on the unfaulted half of the unit ce!.Adatom displacements also compared very well with the vues obtained in the 535 reconstruction. Displacemencurves~see Fig. 14!, with the denser sampling for the 333andG only for the 535, provide very similar results, furtheconfirming the accuracy of theG sampling for the 535.

2. Energy cutoff tests

We have performed a full relaxation of the ionic coordnates for two different positions of the tip over thsurface—on top of an adatom, and in a position midwbetween the adatoms on the main diagonal—for both th

TABLE III. Cutoff convergence of the total energy and normforce for the interaction of a 10 Si atom tip@tip ~c! in Fig. 1# on twopositions on the Si(111)-535 surface.

Adatom ~faulted! Mid position

Energy~7 Ry! 20.129 eV 0.000 eV~11 Ry! 20.128 eV 0.000 eV

Force~7 Ry! 20.395 nN 20.052 nN~11 Ry! 20.402 nN 20.071 nN

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33 and the 535 reconstructions. In the case of the 535reconstruction we compare our results for the 7-Ry cutwith the ones obtained using 11 Ry. For the 333 reconstruc-tion we have also considered a 20-Ry cutoff.G sampling isused in all these calculations. In this case we have notculated the energies of both the surface and the tip indepdently for each cutoff, so only differences in total enerbetween the two positions, taking the mid position as a rerence, have been considered.

Our results are summarized in Tables III and IV In tcase of the 535 reconstruction there is a very small redution in the corrugation both in the total energy and normforce. The change in the normal force is of the order ofconvergence criteria for the individual forces on the equilrium configuration. The 333 reconstruction shows a largereduction in the corrugation when moving to the 20-Ry coff, but still exhibits a very clear contrast between the twpositions of the tip. As both the 333 and 535 calculationsshow a similar trend when moving to larger cutoffs, we epect a similar behavior for the 535 system with a 20-Rycutoff.

Finally, we have to mention that the structure and engetics of other silicon surfaces are well reproduced and ptically converged with our optimized pseudopotential and7-Ry cutoff used in this work. As an example, our pseudpotential shows the correct buckling on the dimers ofSi(111)-231 surface, a quantity that is not recovered bystandard~not optimized! pseudopotential until a cutoff largethan 12 Ry is used.

TABLE IV. Cutoff convergence of the total energy and normforce for the interaction of a 4 Siatom tip@tip ~b! in Fig. 1# on twopositions on the Si(111)-333 surface.

Adatom ~unfaulted! Mid position

Energy~7 Ry! 20.070 eV 0.000 eV~11 Ry! 20.052 eV 0.000 eV~20 Ry! 20.054 eV 0.000 eV

Force~7 Ry! 20.194 nN 20.013 nN~11 Ry! 20.194 nN 20.015 nN~20 Ry! 20.178 nN 20.023 nN

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