Role of image forces in non-contact scanning force microscope images of ionic surfaces

Transcription

1 Surface Science 445 (2000) Role of image force in non-contact canning force microcope image of ionic urface L.N. Kantorovich, A.S. Foter *, A.L. Shluger, A.M. Stoneham Department of Phyic and Atronomy, Univerity College London, Gower Street, London, WC1E 6BT, UK Received 20 Augut 1999; accepted for publication 13 October 1999 Abtract We conider the effect of the image interaction on the force acting between tip and urface in non-contact canning force microcope experiment. Thi interaction i relevant when a conducting tip interact with either a polar bulk ample or with a thick film grown on a conducting ubtrate. We compare the atomitic contribution due to the interaction between the microcopic tip apex and the ample with the macrocopic van der Waal and image contribution to the force on the tip for everal repreentative NaCl cluter adorbed on a metal ubtrate. We how that the microcopic force dominate above the plain (001) terrace ite and i olely reponible for image contrat. However, the image force become comparable to the microcopic force above the urface di-vacancy and dominate the interaction above a charged tep Publihed by Elevier Science B.V. All right reerved. Keyword: AFM; Contruction and ue of effective interatomic interaction; Inulating film; Metal inulator interface; NaCl; Surface defect 1. Introduction metal tip are employed. However, recent application of canning tunnelling microcopy (STM ) to Although the importance of image interaction oxide [2] and inulating film grown on metal ha been recognied in metal inulator interface, ubtrate [3], and combined application of STM electrochemitry and other area, their role in and non-contact SFM (NC-SFM ) (e.g. ee Ref. image formation in canning force microcopy [4 6]) bring thee interaction to the forefront. In (SFM) ha not been analyed in detail [1]. The thee application, conducting tip are interacting reaon are mainly related to the mall number of with thin polar film grown on metal ubtrate or experimental ituation where thee force were with conducting oxide. Thi mean that both the evidently important, e.g. where an ionic inulator interaction of the tip with the film and of the film would make a contact with a conducting tip. In with the ubtrate involve the image force and it particular, in mot of the SFM application on hould be taken into account in analyi and inulator, either inulating tip or oxidied Si and interpretation of SFM image. Although the ame i true for contact mode imaging, in thi paper we will focu on a more imple cae of NC-SFM * Correponding author. Fax: +44-(0) in vacuum. addre: adam.foter@ucl.ac.uk (A.S. Foter) In mot NC-SFM etup the cantilever i driven /00/$ - ee front matter 2000 Publihed by Elevier Science B.V. All right reerved. PII: S ( 99 )

2 284 L.N. Kantorovich et al. / Surface Science 445 (2000) with a contant frequency of khz and make ocillation with an amplitude of about Å above the urface [6,7]. When the end of the tip approache the ample, the main frequency of the cantilever ocillation change due to the tip urface interaction. The urface image i obtained a a map of the diplacement of the bae of the cantilever required to maintain a contant frequency change a the tip can the urface. Contrat in an image i produced by making diplacement away from the urface bright, and thoe toward it dark. The mot table imaging ha been obtained in the attractive region of the tip urface interaction near the urface. Experimental [7] and theoretical etimate [8] demontrate that, for table imaging, the ditance between the end of the tip and the urface atom hould be larger than about 4 5 Å. However, thee experimental condition are very difficult to maintain and, in mot cae, the tip crahe into the urface many time during one erie of experi- ment [9], but a change of the tip tructure during one image i a relatively rare event. Therefore, the tructure of the tip apex i dependent not only on vacuum condition and tip preparation but alo on the tability of imaging. Mot of the metallic or doped Si tip ued in SFM experiment are likely to be covered either by iland of a native oxide and/or by the urface material. In thi paper we conider the relative importance of image force in NC-SFM contrat formation uing the example of a finite cluter of NaCl adorbed on a metallic ubtrate. Thi ytem ha been tudied in recent STM [3] and NC-SFM experiment [10,11] and i repreentative of a common cae of an inulating film grown on a metal. Firt, we decribe a numerical method for the effective calculation of the image force between tip and urface and it implementation within an atomitic imulation technique. Then we tudy the relative trength of the image force with repect to the van der Waal and chemical force in different repreentative tip urface ytem. Thee include the urface terrace, neutral and charged tep, and a dipole formed by a vacancy pair at a tep. Finally, we dicu the reult and limitation of our method. 2. Theory 2.1. Theoretical model We conider a conducting pherical tip of radiu R interacting with a conducting emi-infinite ub- trate (ubtrate) with an adorbed finite cluter of ionic material (ample) on it (ee Fig. 1). To model contamination of a conducting tip by an ionic material, we embed a finite cubic MgO cluter oriented by one of it corner down to the NaCl cluter adorbed on the metal ubtrate, a hown in Fig. 1. Thi model i imilar to the one ued in Ref. [8], except that now the tip and ubtrate are conductive. Their conductivity hould be enough to keep their urface at contant potential at each point of low cantilever ocillation. The tip and ubtrate are connected in a joint circuit, a hown in Fig. 1. Bia applied to the tip and ubtrate will pro- duce an external non-uniform electrotatic field and an additional contribution to the ytem energy, which will affect both the geometry of the ample atom and thoe of the tip apex, and the force impoed on the tip. However, thi effect i not very ignificant for typical experimental value of the bia (<1V). Fig. 1. Schematic picture of the microcopic model ued here to imulate the interaction between the tip and the ample. The coordinate axe are aligned with repect to the phere centre (at x=y=0, z=z ) and the metal plane (at z=0) for convenience.

3 L.N. Kantorovich et al. / Surface Science 445 (2000) Another contribution, which can be more ig- Finally, the ample atom in region 3 are alo nificant for the procee that take place in an allowed to relax, except in the two layer cloet SFM experiment, i to do with the polariation of to the ubtrate (region 4). The interaction conductive electrode by the charge denity of the between atom in region 1 to 4 are calculated ample. The potential on the tip and ubtrate i uing emi-empirical interatomic potential and maintained by external ource (we hall call them the hell model to treat atomic polariabilitie. a battery). The ample charge denity will change Detail of thee calculation are given in thi potential. To prevent thi from happening, the Section 3.1. battery will move ome charge between the tip and In order to calculate the force impoed on the ubtrate in order to keep the potential contant. tip we ued the following expreion for the total Therefore, there will be ome net ditribution of energy of the ytem: charge denity on their urface that will alo interact with the ample and will affect the force U=1 v +U +U, (1) impoed on the tip. Thi kind of induced charge 2 ij vdw el ij ha been commonly referred to a the image charge and we will call thi interaction the image where v i the interatomic potential between ij interaction. atom i, j in region 1 to 4. Thi interaction energy A preliminary account of thi effect in applicacoordinate r of all atom (and hell) involved. i i a function of both the tip poition z and the tion to SFM ha been given in Ref. [12], where a imple model example ha been conidered. Here The energy U repreent the van der Waal vdw we preent the theoretical microcopic model in interaction between the macrocopic tip and ub- detail and propoe an efficient method that can be trate and depend only on the tip height z with ued to take thi effect into account in a ytematic repect to the ubtrate (ee Fig. 1). Note that in manner. Note that quantum effect in metal polariinteraction doe not affect atomic poition in the preent model the macrocopic van der Waal ation [13 16] are neglected here and our coniderregion 1 to 4. ation i baed entirely on claical electrotatic. According to thi model, the image charge doe The lat term in Eq. (1) repreent the total not penetrate into the bulk of the metal. In reality, electrotatic energy of the whole ytem. We however, there i ome ditribution of the charge aume that the Coulomb interaction between all inide the metal [13 15]. However, thi effect i atom in region 1 4 i included in the firt term not important for procee that occur at ditance in Eq. ( 1) and hence hould be excluded from of the order of everal Ångtröm outide the U. Therefore, the energy U include the inter- el el metal and, therefore, will be neglected in thi tudy. action of the atom in the ytem with the macro- All the atom compriing the microcopic part copic tip and ubtrate. A ha been demontrated of the ytem are plit into three region, a hown in Ref. [12], the correct electrotatic energy hould chematically in Fig. 1. The tip apex i modelled incorporate the work done by the battery to main- by a cluter that i formally divided into two tain the contant potential on the electrode. Thi region. The atom in region 2 are allowed to can be written in the following form [12]: adjut their poition, wherea atom in region 1 are kept fixed in order to keep the hape of the U = 1 Q(0)V+ q w(0)(r )+1 q q w (r, r ). el 2 i i 2 i j ind i j apex. The atom in region 1 move together with 1 i,j the macrocopic part of the tip and, therefore, (2) their poition are determined only by the tip poition along the vertical z-axe, which we will Here V i the potential difference applied to the denote z. Atom in region 2 follow the tip a well, metal electrode. In the etup hown in Fig. 1, ince they are trongly bound to the atom in without lo of generality [12], one can chooe the region 1. However, their poition can change potential w on the metal plane to be zero, o that appreciably due to the tip urface interaction. the potential on the macrocopic part of the tip

4 286 L.N. Kantorovich et al. / Surface Science 445 (2000) will be w=v. We alo note that the ubtrate i conidered in the limit of a phere of a very big radiu R &R, ince the metal electrode formally cannot be infinite [12]. The charge on the tip without charge outide the metal (i.e. when there are only bare electrode and the polarization effect can be neglected) i Q(0) and the electrotatic potential of the bare electrode anywhere outide the metal i w(0)(r). The charge Q(0) and the potential w(0)(r) depend only on the geometry of the capacitor formed by the two electrode and on the bia V. The charge Q(0) can be calculated from the potential w(0)(r) a follow [17]: the bare electrode could be calculated [12] if we knew the exact Green function of the electrotatic problem D r G(r, r )= 4pd(r r ) (4) with the correponding boundary condition [G(r, r )=0 when r or r belong to either the ubtrate or the tip urface] [18]. Therefore, given the applied bia V, the geometric characteritic of the capacitor and the poition {r } of the point j charge {q } between the tip and ample, one can i calculate the electrotatic energy U. The problem el i that the Green function for real tip ample hape and arrangement i difficult to calculate. However, for a number of imple geometrie, exact Q(0)= 1 4p PP w(0) n d, olution of the correponding electrotatic problem exit. The mot common i the planar planar where the integration i performed over the entire geometry [19 22], although ome non-planar urface of the macrocopic part of the tip with the geometrie have alo been conidered, for example integrand being the normal derivative of the poten- a planar hyperboloidal [22,23], a planar pherical tial w(0)(r); the normal n i directed outide the [24] and a planar planar junction with a pherical metal. Summation in the econd term of Eq. (2) bo at one of the electrode [25,26]. i performed over the atom and hell of the In thi tudy we ue the planar pherical geomeample and thoe of the tip apex that are repre- try of the junction, a depicted in Fig. 1. Although ented by point charge q at poition r. Note i i a imilar model ha already been conidered in that only thoe atom and hell of region 2 4 Ref. [24], no detail were given and the author of are conidered explicitly in the energy U and the cited paper claimed that the method of multiple el included in the ummation in Eq. (2) (ee image that they ued did not give a well-converged Section 3.1). expanion. Therefore, in Section 2.2 we give a brief Finally, w (r, r ) in Eq. (2) i the potential at decription of our method, which i alo baed on ind r due to image charge induced on all the metal the method of multiple image, together with ome by a unit point charge at r. Thi function i detail that will be important later to derive the directly related to the Green function G(r, r ) of force acting on the charge and on the tip. Our the Laplace equation, w (r, r )=G(r, r ) method appear to be more efficient than the one ind (1/ r, r ), and i ymmetric [12], i.e. w (r, r )= applied in Ref. [24], a we um the erie of image ind w (r, r), due to the ymmetry of the Green funcfirt term, wherea the ret of the erie i ummed charge explicitly only for a mall number of the ind tion itelf [18]. The total potential at r due to a net charge induced on all conductor preent in analytically up to infinity. Note that a imilar idea the ytem by all the point charge {q }: wa employed in Ref. [20], where the ret of the i erie of image charge for the planar planar juncw (r)= q w (r, r ) (3) ind i ind i tion wa integrated. i i the image potential. Note that the lat double 2.2. Solution of the electrotatic problem of point ummation in Eq. (2) include the i=j term a charge inide the phere plane capacitor well. Thi term correpond to the interaction of the charge q with it own polariation (imilar to i Firt, let u conider the calculation of the the polaronic effect in olid-tate phyic). potential w(0)(r) of the bare electrode, i.e. the The function w (r, r ) and, therefore, the capacitor problem. We note that the potential ind image, w (r), together with the potential w(0)(r) of w(0)(r) atifie the ame boundary condition a ind

5 L.N. Kantorovich et al. / Surface Science 445 (2000) the original problem, i.e. w(0)=0 and w(0)= V at charge z (ŝ mean reflection with repect to the k the lower and upper electrode repectively. The ubtrate urface z=0). To find the charge Q(0), olution for the plane pherical capacitor i well which i alo needed for the calculation of the known [19] and can be given uing the method of electrotatic energy, Eq. ( 2), one hould calculate image charge (ee alo Ref. [12]). Since we will the normal derivative of the potential w(0)(r) on need thi olution for calculating force later on, the phere and then take the correponding urface we have to give it here in detail. It i convenient integral (ee above). However, it i ueful to recall to chooe the coordinate ytem a hown in Fig. 1. that the total charge induced on the metal phere Then it i eay to check that the following two due to an external charge i equal exactly to the infinite equence of image charge give the poten- image charge inide the phere [17,18]. Therefore, tial at the phere and the metal plane a V and one immediately obtain: zero repectively. The firt equence i given by the image charge z =RV and then z =z /D k 1 k+1 k k Q(0)= 0 z:. (6) for Yk=1, 2,, where the dimenionle k k=1 contant D are defined by the recurrence k Note that the potential w(0)(r) and the charge Q(0) relation D =2l (1/D ) with D =2l and k+1 k 1 depend on the poition z of the phere indirectly l=z /R>1, z being the ditance between the via the charge z and their poition r according phere centre and the plane ( Fig. 1). The point k zk to the recurrent expreion above. Therefore, one charge {z } are all inide the phere along the k ha to be careful when calculating the contribution normal line paing through the phere centre. to the force impoed on the tip due to bia V [i.e. Their z-coordinate are a follow: z =z and 1 when differentiating w(0)(r) and Q(0) in Eq. ( 2)]. z =R[l (1/D )]=R(D +1 l) for Yk= k+1 k k Now we turn to the calculation of the function 1, 2,. The econd equence of charge {z }i k w (r, r ) in Eq. (2). Thi function correpond to formed by the image of the firt equence with ind the image potential at a point r due to a unit repect to the metal plane, i.e. z = z and z = k k k charge at r. Thi potential i to be defined in uch z. An intereting point about the image k a way that, together with the direct potential of charge {z } i that they converge very quickly at k the unit point charge, it hould be zero on both the point z =R l2 1 (i.e. z 2 with k2) 2 k electrode (the boundary condition for the Green and that z <z Yk. Thi i becaue the number k+1 k function). Thu, let u conider a unit charge q= D converge rapidly to the limiting value k 1atr omewhere outide the metal electrode, a D =l+ l2 1, which follow from the original q 2 hown in Fig. 2. We firt create the direct image recurrent relation above, D =2l (1/D ). 2 2 q of thi charge with repect to the plane at the Therefore, while calculating the potential w(0)(r), point r =ŝr to maintain zero potential at the one can conider the charge {z } and {z } explicplane. Then, we create image of the two charge, q q k k itly only up to ome k=k 1 and then um up 0 the ret of the charge to infinity analytically to q=1 and q= 1, with repect to the phere to obtain the effective charge get two image charge j = R/ r R and 1 q f =R/ ŝr R, a hown in Fig. 2, where 1 q 2 2 z R =(0, 0, z ). Thee image charge are both inide z = zk = k0 = z D k k=k 0 n=0 Dn D 1 the phere by contruction and their poition can 2 2 be written down uing a vector function to be placed at z. Thi can be ued intead of the 2 f (r)=r +R2[(r R )/ r R 2] a follow: ret of the erie: r =f (r ) and r =f(ŝr ). Now the potential at j1 q z1 q k the urface will be zero. At the next tep we w(0)(r)= 0 z: k=1 ka 1 r r 1 r ŝr zk zk B, (5) contruct the image j = j and f = f of the charge j and f in the plane, at point ŝr and 1 1 j1 where z: =z for k<k and ŝr repectively, to get the potential at the plane k k 0 z: =z ; then, k0 2 f1 r =(0, 0, z ) i the poition vector of the charge alo zero. Thi proce i continued, and in thi zk k z and r =ŝr =(0, 0, z ) i the poition of the way two infinite equence of image charge are k zk zk k

6 288 L.N. Kantorovich et al. / Surface Science 445 (2000) Fig. 2. Contruction of image charge in the phere plane +f: ka 1 capacitor ytem due to one charge q outide the metal. notation, let u aume that the original charge i in the xz-plane. Then it follow from Eq. (7) that x jk+1 <x jk. It i alo een that x jk 0 and z jk z 2 =R l2 1 (ee above) a k2 and the ame for the f-equence. Thi mean that the image charge inide the phere move toward the vertical line paing through the centre of the phere and finally converge at the ame point z 2. Thi i the ame behaviour we oberved for charge in the capacitor problem at the beginning of thi ection (ee Fig. 2). In fact, the calculation clearly how a very fat convergence, o that we can again um up the erie of charge from k=k 0. Thu, the image potential at a point r due to the unit charge at r q i: w (r, r )= 1 k + 0 ind q r ŝr q C j : k=1 ka 1 r r 1 jk r ŝr jk B r r fk 1 r ŝr fk BD, (8) where j: =j for k<k and j: =j = k k 0 k0 2 j (D /D 1), and imilarly for the f-equence. k0 2 2 contructed, which are given by the following Here D i the geometrical characteritic of the 2 recurrence relation: capacitor introduced at the beginning of thi ection. R j =j A ha already been mentioned in Section 2.1, k+1 k ŝr R jk the function w (r, r ) mut be ymmetric with ind q repect to the permutation of it two variable. It r =f(ŝr )=R +R2 ŝr R j i not at all obviou that thi i the cae, ince the fk+1 jk k ŝr R 2, (7) jk meaning of it two argument in Eq. (8) i rather where k=1, 2, and imilarly for the f-equence. different. Neverthele, we how in Appendix A Note, however, that the two equence tart from that the function w (r, r ) i ymmetric. ind q different initial charge. Namely, the j-equence tart from the original charge q and the f-equence from it image in the plane q. The two equence {j } and {f } are to be accompanied by the other k k two equence {j } and {f }, which are the image k k of the former charge with repect to the plane. The four equence of the image charge and the charge q and q provide the correct olution for the problem formulated above ince they produce the potential that i the olution of the correponding Poion equation and, at the ame time, i zero both on the metal phere and the metal plane. It i ueful to tudy the convergence propertie of the equence of image charge. To implify the 2.3. The calculation of the total force acting on the tip In order to calculate the total force acting on the tip, one ha to differentiate the total energy, Eq. (1), with repect to the poition of the phere R. Since we are intereted only in the force acting in the z-direction, it i ufficient to tudy the dependence of the energy on z. There will be three contribution to the force. The force from the electrotatic energy i conidered in ome detail in Appendix B. The econd contribution to the force come from the van der Waal interaction [27].

7 L.N. Kantorovich et al. / Surface Science 445 (2000) Finally, the interatomic interaction in region 1 4 lead to a force that i calculated by differentiating the hell-model energy [the firt term in Eq. (1)]. Therefore: energy U h. Therefore, finally we have: F = [F(h) tip iz ] du vdw x iµ1 0 dz A U el, (12) z B x0 where the firt ummation run only over atom F = du h du vdw du el, (9) tip in region 1. Thu, in order to calculate the force dz dz dz impoed on the tip at a given tip poition z, one ha to relax the poition of atom in region 2 where U =1 S v i the hell model energy. We h 2 ij ij and 3 uing the total energy of the ytem, recall that the ummation here i performed over U +U. Then one calculate the hell-model all atom in region 1 to 4. Only poition of the h el force, F(h), acting on every atom in region 1 in atom in region 1 depend directly on z, ince iz the z-direction a well a the electrotatic contribuatom in region 2 and 3 are allowed to relax. tion to the force given by the lat term in Eq. (12). However, their equilibrium poition, r(0), deteri The van der Waal force between the macrocopic mined by the minimiation of the energy of tip and ample doe not depend on the geometry Eq. (1), will depend indirectly on z at equilibrium, of the atom and can be calculated jut once for r(0) =r(0) (z ). Then, we alo recall that the electroevery given z. i i tatic energy U depend only on poition of el Although the expreion for the force obtained atom in region 2 to 4, a well a on the tip above i exact for the model ued in thi tudy, poition z. uch a calculation i quite demanding ince it Let u denote the poition of atom in region require uing the electrotatic energy U alongide 2 to 4 by a vector x=(r, r, ). The total energy el 1 2 with the hell-model energy U in the optimiation U=U(x, z ), where the direct dependence on z h proce. Mot time i pent in the calculation of come from atom in region 1 of the hell model the force impoed on atom due to the energy energy U and from the electrotatic energy U. h el U. Therefore, in thi work we have adopted the In equilibrium the total energy i a minimum: el following approximate trategy. For every tip poition, all atom in region 2 and 3 were allowed to relax to mechanical equilibrium in accordance with A U =0, (10) x B the hell-model interaction only and we neglected z the effect of the image charge on their geometry. where the derivative are calculated at a given To invetigate the effect of thi, we performed fixed tip poition z. Let x =(r(0) ome fully elf-conitent calculation in the NaCl 0 1, r(0) 2, ) be the olution of Eq. (10). Then, ince x =x (z ), we tep ytem. In thee calculation, ion in region 0 0 have for the force: 2 and 3 are allowed to relax completely with repect to microcopic and image force. We found that the diplacement of ion in the ytem due F = du [x 0 (z ), z ] x = tip dz A U x 0 B 0 z to image force wa le than 0.01 Å and would z not affect our reult. A U z B x0 = A U z B x0, (11) The force acting on the tip wa calculated uing an equation imilar to Eq. (12): where we have ued Eq. (10). Thi reult can be F [F(h) tip iz ] + implified further. Indeed, the partial derivative of x iµ1 0 iµ2 the hell-model, ( U / z ), i equal to the um h (13) of all z-force acting on atom in region 1 due to [F(el) iz ] du vdw x 0 dz A U el, z B x0 all hell-model interaction, ince only thee atom where in the econd ummation we um all z- are reponible for the dependence on z in the force acting on all atom of region 2 due to the

8 290 L.N. Kantorovich et al. / Surface Science 445 (2000) electrotatic energy U. The calculation of the el electrotatic contribution to the force acting on atom i conidered in detail in Appendix C. 3. Reult In order to demontrate the relative ignificance of the image force with repect to the other force, we applied the interaction model decribed above to calculate the tip urface force of everal characteritic ytem. Thee ytem were choen a they repreent a urface ubtrate cla that ha been tudied extenively in STM [3] and NC-SFM experiment [10,11]. Specifically, each of the ytem repreent a feature that i likely to be found in experiment, and in which image force may play a ignificant role in the interaction Detail of the calculation The ytem ued to calculate the force are all et up a hown in Fig. 1, with only the exact tructure of the NaCl cluter changing between ytem. For thee calculation the tip conit of a phere of radiu 100 Å with a 64-atom MgO cube embedded at the apex. The cube i orientated o that it i ymmetric about the z-axi with a ingle oxygen ion at the lowet point of the tip. The top three layer of the cube fall within the phere radiu and contitute region 1, a hown in Fig. 1. The exact number of ion in region 1 i et o a to keep the nano-tip attached to the phere neutral. The remaining ion of the cube contitute region 2. The cluter ued conit of four layer of NaCl, with the top two layer deignated region 3 and the bottom two layer region 4, a in Fig. 1. The metal plate i 2 Å below the bottom of the cluter and the bia i held at 1.0 V in all calculation. Thi etup i conitent through all ytem calculated and, where appropriate, for all interaction calculated. Thi give u confidence that we can compare the relative value of force in the ame ytem and between different ytem. The interaction between ion in region 1 4 wa calculated uing a tatic atomitic imulation technique a implemented in the MARVIN computer code [28,29]. We will refer to thi interaction a microcopic in further dicuion. The nano-tip and the NaCl cluter are each divided into two region, I and II. In term of Fig. 1, region I conit of region 2 and the top two layer of the cluter (region 3), and region II conit of region 1 and the remaining bottom two layer of the cluter (region 4). The region I ion are relaxed explicitly, whilt the region II ion are kept fixed to reproduce the potential of the bulk lattice and the remaining tip ion in the relaxed ion. The calculation i periodic, o that the infinite urface i repreented; however, thi mean large urface unit cell mut be ued to avoid interaction between tip image in different cell. We made ure that the hell-model contribution to the force i completely converged with repect to the ize of the periodically tranlated imulation cell. Electronic polariation of the ion i implemented via the Dick Overhauer hell model [30]. Buckingham two-body potential were ued to repreent the non-coulombic interaction between the ion. The parameter for thee interaction are well teted and are fully decribed in Ref. [31]. To calculate the hell-model contribution to the microcopic force between tip and urface, S [F(m)] [ee Eq. (13)], we firt calculate the iµ1 iz x0 total hell-model energy of the ytem at a range of tip urface eparation, and then differentiate it numerically to find the force a a function of eparation. The image force i calculated by taking the relaxed geometry from the hell-model calculation at each tip urface eparation. Any ion within the phere (i.e. region 1) are not conidered in the image force calculation, a it i impoible to have ion within the conducting phere. Alo, any ion cloer than 2.5 Å to the phere did not produce any image within the phere, a thee would produce an unrealitically large interaction. The force on each atom in the ytem due to the image interaction i then calculated, and the force on the tip atom i ummed to find the contribution of the image force to the tip urface force, ee Eq. (13). Thi calculation i not periodic; the NaCl cluter i now a finite body. However, ince our NaCl periodic cell i large enough that we can neglect the interaction between the tip image, we

9 L.N. Kantorovich et al. / Surface Science 445 (2000) can effectively conider the NaCl ample to be a block of atom. We found that thi interaction finite cluter at all tage of the calculation. The converge to a contant value when the radiu of only difference i that the periodic boundary condi- the phere exceed about 30 Å. The force exerted tion in atomitic imulation do not allow the on one atom due to thi interaction at characteri- atom at the cluter border to relax a in a free tic tip ample ditance i everal order of magnitude cluter. However, thi effect i mall and doe not maller than the force between macrocopic affect our concluion. The approach decribed tip and ubtrate. Alo, the force decay with allow u to enure the conitency of our model ditance a r 4. Thi effectively mean that only hown in Fig. 1 throughout the whole modelling the top layer of the ample contribute to the proce. van der Waal force, and there are not enough The final contribution to the force i the van der ion in that layer for it to be ignificant. Therefore, Waal interaction. It include the following contribution: we neglected thi interaction in further (i) between the macrocopic Si tip of calculation. conical hape with the phere of radiu R at the end [27] and emi-infinite ubtrate; (ii) the diper NaCl tep ion force between the atom in the ample treated atomitically; and (iii) the interaction between the The firt ytem tudied wa a tepped NaCl macrocopic part of the tip and the ample atom. cluter produced by placing a (in term The firt contribution i calculated analytically of an eight-atom cubic unit cell ) block on top of [27]. In fact, the macrocopic contribution to the a5 5 2 block o that two corner are aligned. van der Waal force i the ame in each of the A chematic for the calculation cell of thi ytem three ytem decribed below, a it depend only i hown in Fig. 3. Thi ytem give u a good on the tip urface eparation, macrocopic phere opportunity to tudy the interaction over ite of radiu, cone-angle and Hamaker contant of the different coordination. The upper terrace of the ytem [ 27]. All thee quantitie are identical in tep i a good repreentation of the ideal (001) each ytem we look at, o that the van der Waal urface of NaCl; a long a we remain at leat force act a a background attractive force inde- three row from the edge the force are converged pendent of the microcopic propertie of the ytem with repect to row choice. However, the ion of [8]. The Hamaker contant needed for the calcula- the tep-edge have a coordination of four, compared tion of the macrocopic van der Waal force i with a coordination of five for the terrace etimated to be 0.5 ev [32]. ion. To etimate the importance of the third contribution, To tudy the difference between thee type of we have calculated explicitly the diperion ite we calculated the interaction over an Na ion interaction between one atom and a pherical in the terrace and a Cl ion at the tep edge, a Fig. 3. Sytem chematic for NaCl tep.

10 292 L.N. Kantorovich et al. / Surface Science 445 (2000) Thi mean that for thi ytem the image force act in a imilar way to the macrocopic van der Waal force, i.e. a an attractive background force that i blind to atomic identity and, therefore, doe not contribute to the SFM image contrat Pair vacancy Fig. 4. Force over NaCl tep. The econd ytem tudied i formed by taking the NaCl tep dicued in Section 3.2 and removing an Na Cl pair of nearet-neighbour ion from the upper terrace, a hown in Fig. 5. Thi effectively create a dipole on the upper terrace of the tep, but doe not greatly affect the original geometry of the tep. In thi ytem we calculated the total force contribution over the ion at the edge of the pair vacancy: the Na ion labelled 1 in Fig. 5 and Cl ion labelled 2. The force contribution a a function of tip urface eparation are hown in Fig. 6. The macrocopic van der Waal force i obviouly identical to the previou example, and i here only for comparion. The microcopic force i alo very imilar to the previou example. Thi i expected, a we are till looking at the interaction over the ame ion and the double vacancy of oppoitely charged ion ha little ignificant effect on thi force. The removal of the vacancy ion doe change the local coordination of the ion at the edge of the vacancy, but thi i compenated by relaxation of thee ion away from the vacancy. Thi compenation mean that the microcopic van der Waal and electrotatic force directly over the edge ion are imilar to the force over the ion in the defect- hown in Fig. 3. Thi allow u to compare the effect of coordination and chemical identity with the force between tip and urface. The firt point to note i that the macrocopic van der Waal force i the ame for both anion and cation; thi i an obviou effect from the way in which we calculate thi interaction. It i alo the leat ignificant force at tip urface eparation le than 6 Å, a can be een in Fig. 4. The behaviour of the microcopic force i a would be expected for the interaction of an oxygen ion (which imulate the end of the tip apex, ee above) with the ion in the urface. The force i attractive over the poitive Na in the terrace and repulive over the negative Cl at the tep edge. The microcopic force become the dominant interaction in the ytem at around 5.5 Å. A the tip get very cloe to the Na ion in the terrace the ion begin to diplace toward the tip oxygen, greatly increaing the attractive force. When the eparation i cloer than around 4.7 Å the diplacement of the Na ion toward the tip exceed 1 Å. Thi ion intability can be een clearly both in the microcopic and image force curve for Na in Fig. 4 and ha already been decribed in the literature in the context of AFM [33]. The image force itelf i the dominant interaction at longer range, but it i fairly conitent over cation and anion in the NaCl terrace. The difference in image force over the Cl and Na ion i le than 0.01 ev/å until the intability at 4.7 Å. Fig. 5. Sytem chematic for pair vacancy on NaCl terrace.

11 L.N. Kantorovich et al. / Surface Science 445 (2000) attractive and repulive over the Na ion. The difference in magnitude of the image force over the two ion i due to the aymmetry of the nanotip ion at the end of the conducting tip. Although thi reult implie that the image force i omewhat enitive to the geometry of the interacting feature, it doe demontrate that the image force would feel a defect in the urface Charged tep In the final ytem tudied, we looked at a charged tep where imilar ion run along each edge of the tep. Fig. 7 how a chematic of the Fig. 6. Force over pair vacancy in NaCl tep. calculation cell ued. The ytem i created by taking the neutral tep etup from Section 3.2 and free terrace. Thi i een clearly by the onet of jut removing ion from four row of the upper Na ion intability at the ame tip urface epara- terrace. Thi charged row of ion i imilar to the tion of about 4.7 Å. bridging oxygen row een in the TiO (110) 2 The image force over the edge ion demontrate urface, a urface that ha recently been tudied very different behaviour to the plain tep ytem experimentally by NC-SFM [34]. tudied in Section 3.2. The only imilarity i that We calculated the contribution to the total the microcopic force become the dominant interaction force over a Cl ion at one edge of the tep, labelled at the ame ditance of 5.5 Å. The image 1 in Fig. 7, and over an Na ion at the other edge, force over the Cl vacancy edge ion i attractive labelled 2. The force contribution are hown and almot twice a large ( 0.05 ev/å) a the in Fig. 8. force over the Cl at the plain tep edge The macrocopic van der Waal force for the ( 0.03 ev/å). The image force clearly feel the charged tep i again identical to previou example defect in the terrace and the increae in force and plotted only for comparion. The microcopic reflect the change in the local charge environment van der Waal and electrotatic force over both of the Cl ion. Thi i even more clearly hown by the ion are increaed compared with the previou the image force over the Na ion at the vacancy two ytem. Over the Na ion at the tep edge the edge. The force i much maller than the microcopic force i 0.1 ev/å at 5.5 Å compared with force at all eparation and actually become 0.05 ev/å at 5.5 Å for the Na ion in the terrace repulive at around 6.2 Å. Thi mean that the of the plain tep. Thi doubling of the force i alo induced potential in the conducting tip reflect the change in local charge environment produced by the vacancy. The net interaction over the Cl ion i een over the Cl ion, where the force at 5.5 Å i ev/å over the charged tep and ev/å over the plain tep. Thi i a conequence of the Fig. 7. Sytem chematic for charged NaCl tep.

12 294 L.N. Kantorovich et al. / Surface Science 445 (2000) i very enitive to the charge of the ytem between the electrode. 4. Dicuion In thi paper we conidered the contribution to the force acting on a tip in a typical etup of an NC-SFM, including: (i) a macrocopic van der Waal interaction between the macrocopic tip and ubtrate that i alway attractive and give no image contrat; (ii) a microcopic force between the tip apex and the ample that we decribed uing the hell model; for the firt time (iii) the Fig. 8. Force over charged NaCl tep. image interaction due to macrocopic polariation of the metal electrode (both tip and ubtrate). The latter two contribution may be either attractive change of geometry of the charged tep ytem: or repulive depending on the ditance to the both ion have lower coordination than the ion urface and the nature of the urface ite that the in the plain tep ytem. Lowering the coordination tip i above. They both play a role in image increae the gradient of the electrotatic potential contrat in the NC-SFM experiment. We clearly around thee ion and, therefore, increae the demontrated in thi paper the ignificance of the microcopic force between tip and urface. image interaction, which in ome cae i the The image force dominate the interaction over dominant force. It i epecially intereting to note the charged tep at nearly all tip urface epara- the ability of the image force to reolve the charged tion; only after the onet of Na ion intability at di-vacancy and charged tep. Thi how that the 4.7 Å doe the microcopic attraction between the image force could be the dominant ource of tip and the tep Na overcome the image force. contrat in NC-AFM image of charged feature. Over the Cl ion the image force i completely In thi tudy we have adopted everal approximation dominant, and i approximately an order of magnitude that we are now going to dicu. Firt, larger than the microcopic force. At the edge note that we have aumed a particular etup for of the charged tep, the row of imilar ion produce our model, in term of the way the bia i applied a row of imilar image charge of oppoite to the ytem, a can be een in Fig. 1. Thi type ign in the conducting tip. In the other ytem of etup i commonly ued [35], but other poi- thi effect i effectively compenated by the bilitie may exit. For example, a etup where the alternating ion pecie. A in the previou ytem, tip and ubtrate are decoupled and their potential the image force doe feel the difference between are changed independently i equivalent, ince only one ide of the tep and the other; thi i manifeted the abolute magnitude of the potential difference in the oppoition of the interaction at either ide between the tip and the metal ubtrate matter. of the tep. Over the Cl ion the image force i Another limitation of the preent model i that, attractive and over the Na ion the force i repul- for the calculation of the image interaction, we ive. Thi i a reflection of the image charge tudied finite cluter of a particular hape. ditribution produced in the tip by the charged However, it i important to undertand how the tep ytem, an exaggerated verion of the effect reult of the calculation would depend on the ize een over the vacancy. The ignificance of the of the cluter if we wih to make any general image force above the ion of the charged tep i in agreement with our preliminary reult of Ref. [12], where it wa found that the image force concluion about image force in thee ytem. A ha been mentioned in previou ection, the image force over a particular ion i omewhat

13 L.N. Kantorovich et al. / Surface Science 445 (2000) dependent on the geometry of the ytem being the image force could exaggerate the effect of ion tudied. Note that we cannot increae the cluter diplacement on imaging of conducting material ize indefinitely, ince it i limited to roughly twice or thin film with contact AFM. the radiu of the phere ued to model the tip. The image force alo depend on the charge Beyond thi limit the top of the phere affect the ditribution and ionicity of the nano-tip ued, a interaction with the urface; thi i an unphyical well a the charge itelf. A ha been dicued in reult a, in reality, the tip ha eentially infinite our recent paper [37], a more realitic model of height at thi cale. In order to invetigate the an NC-SFM tip would probably conit of an effect of the ize of the cluter, we calculated the ionic oxide layer covering a emiconducting ilicon image force over the ame terrace Na ion a in the tip. The implet model of thi tip i an oxygen firt ytem tudied, but increaed the ize of atom or a hydroxyl group adorbed on a ilicon the cluter by everal hundred atom ( being till cluter [37]. Thi model repreent a nano-tip with within the limit of the maximum cluter ize). Thi a decaying ionicity a you move away from the effectively mean that the local geometry and apex. The reduction of the magnitude of the charge environment of the ion under the tip apex charge in the nano-tip would reduce the magnitude remain the ame, but the total number of charge of the image charge in the conducting tip in the ytem change ignificantly. The calcula- and, therefore, the image force. tion how that the image force over the cluter doe increae a the number of atom i increaed, but then converge. The increae in image force i Acknowledgement due to the interaction of the extra charge in the ytem, but the difference i an order of magnitude LNK and ASF are upported by EPSRC. We maller than the image force itelf and o would are grateful to A. Baratoff, R. Bennewitz, E. Meyer not affect our reult ignificantly. and A.I. Livhit for ueful dicuion and to For conitency, throughout thi tudy we have A.L. Rohl for help in MARVIN calculation. ued a neutral nano-tip. It i known, however, that the tip ued in real experiment can be highly contaminated by external material, which may lead Appendix A to tip charging. Therefore, it i relevant to tudy the effect of a charged tip (i.e. a nano-tip with a In thi appendix we how explicitly that the different number of anion and cation) on the function w (r, r ) of Eq. (8) i ymmetric with ind A B image force. To imulate thi, we added four repect to it variable. Firt of all, one ha to be uncompenated oxygen ion to the original nanotip clear about the notation. While conidering the o that it net charge became 8e. We found function w (r, r ), we imply that the polariing ind A B that the image force more than double over the unit charge i located at r and, therefore, all B charged tep. Thi ytem i an extreme example image charge and their poition entering Eq. (8) of thi effect, but an increae in image force can will be deignated by the letter B, e.g. f (B), k be een when uing a charged tip in all the ytem j (B), r (B), etc. Conequently, while conidering k jk that have been tudied here. The effect of tip the function w (r, r ) we imply that the ource ind B A contamination could be even more ignificant in unit charge i at r and the image charge and A the cae of contact SFM. In previou tudie of their poition will be marked by the letter A, e.g. contact SFM imaging [36], the importance of ion j (A), r (A). k jk diplacement and ion exchange between tip and Let u now write down eparately Eq. (8) for urface ha been demontrated. Our reult here w (r, r ) and w (r, r ) and compare them term ind A B ind B A how that diplacement of ion caue a large by term. For implicity we aume that the expan- increae in the image force between the tip and ion are infinite. The firt term in both expreion urface. Ion exchange may alo charge the tip, again changing the image force. Thi mean that are obviouly identical ince r ŝr = A B r ŝr. Let u now prove that the firt term in B A

14 296 L.N. Kantorovich et al. / Surface Science 445 (2000) the firt um in both expanion are alo identical Then, we notice that the denominator in the lefthand ide can be tranformed a follow: term by term, i.e. that for any k r(k) A f (r ) = f (r(k 1)) ŝf(r ) = f [r(k 1)] r(1) j (A) B A B A B k+1 and imilarly for the denominator in the right- r r (A) = j k+1 (B) r r (B). (14) hand ide. In addition, the product x(r )x(r )i B jk+1 A jk+1 A B cancelled out on both ide, o that we are left Firt of all, a direct calculation how that they with: are identical for k=0, which can be written down in a ymbolic form a x[r(1) A ],x[r(k 2) ] G x[r(k 1) ] A A r(1) f [r(k 1)] H x(x) B A y f(x) = x(y) x f(y), (15) =x[r(1) B ],x[r(k 2) ] G x[r(k 1) ] B B r(1) f [r(k 1)] H. (18) where x(r)= R/ r R and the function f(r) ha A B been introduced in Eq. (7). It i implied in the Notice that the expreion obtained i imilar to above identity that x=r Eq. (16) with r r(1) B B and r A r(1) and the order k A and y=r ; however, it B A obviouly hold for any choice of vector x and y. reduced by one. Repeating the procedure, we Note that thi reult imply correpond to the reduce the order again and get Eq. (18) with cae of a ingle phere for which the potential r(1) B r(2) and r(1) B A r(2). If k=2p i even, thi proce i repeated until A w (r, r ) i known to be ymmetric [18]. we get exactly Eq. (15) ind A B Conider now the cae of any k. Let u denote with x=r(p) and y=r(p), which i true. If k=2p+1 A B r(n) i odd, however, then we will get the ame expre- A =ŝr (A) and r(n) j n B =ŝr (B) for Yn=1, 2,. Then it follow from j n Eq. (7) that ion r(p+1) =f [r(p)] 1 on both ide. Thu, Eq. (14) i B proven Yk A r (A)=f [r(n)] and r(n+1) =ŝf [r(n+1)] for any o that the firt term in the firt jn+1 A A A n 1 [while r (A)=f (r ) and r(0) um in Eq. (8) are identical in the two expanion. j1 A A =r for n=0]; A imilar formulae can be written for the charge Note again that Eq. (14) i valid for any two generated by the ource unit charge at r. Uing vector r and r. B A B thi notation, we have j (A)=( 1)k Compare now the econd term in the firt um k x(r )x[r(1) in the expanion of w (r, r ) with the firt term A A ] x[r(k) A ] and the ame for j k (B), o that ind A B Eq. (14) can be rewritten a: in the econd um in the expanion of w (r, r ), ind B A ee Eq. (8). They are identical a well for any k, term by term: x(r )x[r(1) A A ],x[r(k 1) A =x(r )x[r(1) B B ],x[r(k 1) B ] G x[r(k) A ] r B f [r(k) A ] H The expreion in the quare bracket in either ide of Eq. (16) can be implified by mean of the identity in Eq. (15). Namely, we ue x=r(k) and A y=r to implify the expreion in the quare B bracket in the left hand ide of Eq. (16) and x=r(k) and y=r in the right-hand ide, giving B A x(r )x[r(1) A A ],x[r(k 1) A =x(r )x[r(1) B B ],x[r(k 1) B j k+1 (B) r A ŝr jk+1 (B) = f k+1 (A) r B r fk+1 (A). (19) ] G x[r(k) B ] r A f [r(k) B ] H. (16) Indeed, by contruction, the image charge j k (A) are built due to the poitive ource unit charge at r, wherea the image charge f (A) are built due A k to the ource negative unit charge at r =ŝr,o A9 A that f (A) j (A9 ) and r (A) r (A9 ). Therefore, k k fk jk Eq. (19) appear to be exactly the ame a Eq. (14) correponding to one of the ource charge at r rather than at r. Thi alo mean that the A9 A econd term in the firt um in the expanion of w (r, r ) i, term by term, identical to the firt ind B A term in the econd um in the expanion of ] C x(r A ) w (r, r ). Similarly, one can prove that the ind A B r(k) D B f (r A ). (17) econd term in the econd um of the two expan- ] C x(r B ) r(k) A f(r B ) D

15 L.N. Kantorovich et al. / Surface Science 445 (2000) ion alo coincide term by term. The proof i The calculation of the derivative of the econd complete. part of the energy that i to do with charge {q } i proceed in a imilar manner, although the calculation i more cumberome. We need the derivative with repect to z of the induced potential, w (r, r ). It follow from Eq. (8) that for every Appendix B ind j i charge q we hould, therefore, conider the derivative of the charge j, j a well a of their i In thi appendix we explain how the contribupoition r and ŝr ; after that, the ame calcula- k k tion ( U / z ) to the total force acting on the jk jk el tion hould be repeated for the f-equence. Note tip i calculated. The electrotatic energy U i el that the actual dependence on z come from given by Eq. (2), (6) and (8). A ha been already r =(0, 0, z ) in Eq. (7). Let u fix ome charge q. mentioned in Section 2, the calculation of the i We firt define the derivative derivative i not imple ince the dependence of U on z i not only explicit but alo contain el C(j k ) = r a ome implicit dependencie. It i the purpoe of j a k, (20) z thi appendix to conider how thi derivative i calculated in ome detail. where Greek indice a, b will be ued to deignate The energy U conit of two part: that due el the Carteian component of vector. Uing Eq. to bare electrode and that due to charge. We (7) one can get the following recurrent relation firt of all conider the firt part, i.e. the derivative for the quantitie C(j k ): with repect to z of the charge Q(0) and the a potential w(0)(r). Uing definition of charge z k C(j k+1 ) =d + R2 given in Section 2.2, one ha the following et of a az R2 k G [h a C(j a k ) d ] az recurrent equation for the derivative of the charge with repect to z : z k+1 z = 1 D k A z k z z k D k D k z B, 2 R2 k b [h C(j b k ) d ]R R b bz ka kbh, (21) where the vector R =ŝr R, d i Kronecker k jk ab delta ymbol and we alo defined h a h =1 for a a a=x, y and h = 1. Note that thi relation can z be ued tarting for k=0 if we et R =r R and 0 i C(j 0 ) =0. a Finally, the derivative of the charge j can be k calculated uing their definition in Eq. (7) and the quantitie in Eq. (20) a follow: where / z =0 and the derivative D / z are z1 k in turn obtained from D / z =(2/R)+ k+1 [D 2( D / z )], which tart from D / z =2/R. k k 1 The poition of image charge z alo depend on k z and the correponding derivative are eaily expreed via the derivative D / z a follow: k z / z =R( D / z ) 1. The calculation of the k k j derivative of z and z with repect to z i k+1 = R 2 2 z R k G j k j k [h C(j z R2 b k ) dbz]r b k b kbh, calculated eaily owing to their explicit dependence on it (ee Section 2.2). Thu, the calculation pro- (22) ceed a follow. Firt of all, the derivative D / z for all needed value of k k are calcu- where again thi expreion can be ued tarting k 0 lated uing the recurrent relation above, then the from k=0 if we et: j / z = 1. Firt, one calcu- 0 derivative of the charge and their poition are late the derivative C(j k ) uing the recurrent equation in Eq. ( 21); then the derivative of the a alo calculated. Thi make it poible to calculate the derivative of Q(0) and w(0)(r) with repect to charge j are eaily calculated from Eq. (22). k z in a imple fahion in accordance with Eq. (6) Then thi procedure i repeated for the f-equence. and ( 5). The formulae obtained above allow one to