Tip-Sample Interaction Force Modeling for AFM Simulation, Control Design, and Material Property Measurement

Transcription

1 11 erian Control Conferene on O'Farrell Street, San Franiso, C, US June 9 - July 1, 11 Tip-Saple Interation Fore Modeling for FM Siulation, Control Design, and Material roperty Measureent Sergey Belikov and Sergei Magonov NT-MDT Developent bstrat Tip-saple interation fore is the key feature easured and anipulated by toi Fore Mirosopy (FM). It is the ain reason why this interation ust be the ajor oponent of quasistati and dynai ontrols used in FM instruentation. Many dynai ontrol odels are available for FM but only few expliitly ontain the tip-saple fores. One of the is based on asyptoti dynais using Krylov-Bogoliubov-Mitropolsky (KBM) averaging. In the latter the tip-saple fores ating on approah and retration are onsidered. Conservative Hertz odel was applied in the first FM siulations with KMB averaging. This siple and useful odel does not over any aspets of FM tip-saple interations (adhesion, energy dissipation, et.) vital for aurate ontrol of the instruent. The purpose of this paper is to provide adequate interation fore odels for FM ontrol syste and illustrate speifi features, suh as juping between aplitude branhes, adhesive avalanhe, et. These features are routinely observed in FM experients but ostly disarded in the ontrol syste. We suggest a hybrid odel for the ontrol syste design to aount for these phenoena. The odel is based on athing the Maugis s JKR-DMT transition that desribes elasto-adhesive interation after geoetrial ontat (penetration) and the Integrated Lennard-Jones odel with adhesive avalanhe that desribes the oleular level interation before the ontat. Mathing paraeters an be alulated based on the assuption that both urves desribe the sae physial interations and ust ath soothly at the oon point of geoetrial ontat. Hybrid odel algoriths are developed that an be used in FM real-tie adaptive ontrol systes with paraeter estiation. I. INTRODUCTION toi fore irosopy (FM) is the haraterization tehnique for high-resolution visualization of topography and loal probing of ehanial and eletrial properties of aterials. The key of FM is tip-saple fore interations, and their theoretial desription is rather opliated. So far there is no rational ath between arosopi indentation and atoi tip-saple fore odels that interplay in the experient due to inute tip-saple separations and ontat diensions. This proble is addressed below, and athing between Maugis JKR-DMT transition and Integrated Lennard-Jones odels with adhesive avalanhe (LJ-) is suggested. We are using Euler-Bernoulli desription of the probe dynais and asyptoti KBM approah to solution of the equation otion of the osillating probe interating with a saple [1]. Most of the known treatents of the FM fore spetrosopy studies are based on oversiplified effetive point-ass odels due to oplexity of the proble [-4]. syptoti FM dynais [1] with the athing interative fore odel desribed below allows eeting higher FM quantitative easureent standards. II. TI-SMLE INTERCTION FORCES IN FM CONTROL ND MESUREMENT ROBLEMS In FM easureents tip-saple interation fores are ontrolled and utilized in every experiental step starting fro the tip engageent to a saple. Control of all FM odes ipliitly or expliitly involves the fores and all quantitative easureents are based on deriving paraeters of fore odels fro experiental data. lthough this topi is extensively overed in literature, only siplified odels were applied. In this setion we illustrate how tip-saple interation is used in the ontrol odels for dynai FM odes (e.g. plitude and Frequeny Modulation). syptoti equations of FM dynai odes near resonane and related analytial lassifiation of the odes were derived and studied in [1]: 1 1 x N 1 1 x N Fa Fr x os y Fa Fr x os y sin ydy os ydy G 1 G 1 G sin os where the following notations (SI units are shown in brakets) are used: x [] and [rad]-aplitude and phase of the vertial defletion signal; [1]-sall paraeter equal to inverse of quality fator Q; ω 1 [1/se]-the 1 st Eigen-frequeny of the antilever; []-the aplitude of free osillation of the antilever; N[N]-noralizing fore defined at [1, Eq.(6)]; []-entral position of the osillating tip (i.e. height); G[1]-relative frequeny shift ultiplied by double quality fator; F a and F r - the approah and retration tip-saple fores, whose odeling is the ain topi of this work. Equation (1) desribes FM dynais for various FM odes lassified in [1]. For exaple, in plitude Modulation (M) ode G=onstant and is typially zero, i.e. antilever is osillating at its 1 st Eigen-frequeny. In M spetrosopy ode the tip s entral position is oving up and down with relatively slow (opared to ω 1 ) speed and two urves are aquired: x vs (aplitude urve) and vs (phase urve). Typial experiental FM aplitude and phase urves are shown in Figure 1. The aplitude and phase plots exhibit respetively a kink and a jup fro negative to positive values as the aplitude drops ~6% fro its initial value. This is a refletion of the bifuration phenoenon desribed below. x x (1) /11/$6. 11 CC 867

2 Figure 1. Typial experiental FM aplitude vs. height (left) and phase vs. height (right) urves. ording to the FM onvention the initial phase is defined as zero. In M iaging ode tip is sanned horizontally (e.g. by X-Y piezo-sanner) and FM ontrol syste traks the aplitude set-point x sp by oving tip s vertial position aording to the ipleented ontrol algorith. quired vs XY and vs XY are displayed as height and phase iages. Height is usually ontrolled by piezo-drive and generi dynais of the drive an be presented in the for H( s, (), u) where uu(x,, ) is the ontrol signal e.g. voltage applied to the drive; s is the differential operator. Linearized odel of the drive is usually utilized where H is equivalent to the transfer funtion. X-Y sanning on the saple is either open-loop or use ontrollers siilar to (). Tip-saple interation fores depend on X-Y position beause of spatial differenes of topography and aterial properties of the saple. In urrent FM ontrollers, usually IDs, aplitude-phase dynais and tip-saple interations are not expliitly utilized. s a result, traking of opliated topography profiles and heterogeneous surfaes ight be hallenging. The resulting FM urves and iages eet only qualitative standards; and there is no odel base for true high-resolution iaging and quantitative easureents of aterial properties of the saple. n adequate FM ontrol syste design requires a nonlinear odel-based approah aounting for tip-saple interation odel with defined and easurable paraeters. These paraeters vary during XY san with the spatial hanges of loal aterial properties. The latter will be refleted in F a and F r behavior, therefore their odeling is ruial for the design. This work provides rigorous derivation of the F a and F r based on solid theory and experiental results of any researhes assebled in Ref. [5], fro LJ- oleular interation (prior to ontat) to elasto-adhesion Maugis s JKR-DMT transition (after penetration). These odels in obination with FM dynais (1), -drive and XY dynais () an be used to design FM ontrol systes that are easier to tune and an eet quantitative easureent standards. artiular ontrol and paraeter estiation designs will be presented in further publiations. III. FM TI-SMLE INTERCTION MODELING. Bakground This setion provides the basi fore interation odels widely used in aterial siene and they are adapted to the purpose of this paper. FM tip-saple interation odeling is based on the synergeti ath of these odels. 1) Integrated Lennard-Jones otential otential per unit area (SI unit [J/ ]) of attration between two half-rystals (planes) is [5, p. 1]: where z is the distane between the plates; [] is the equilibriu distane; and w[j/ ] is the work of adhesion related to Haaker onstant [J] by the forula w 16 ) Derjagin approxiation ording to Derjagin approxiation [5, forula (1.)] the interation fore between a sphere of radius R and a plane is () (4) 1 ) Lennard-Jones Model with dhesive valanhe (LJ-) Maugis [5, pp 9-45] introdued the dissipative odel, alled adhesive avalanhe, of two half-rystals rigidly fixed on one edge and having N atoi planes free to ove. We will reprodue the equations (1.67)-(1.68) of [5] in the for needed for further developent of this paper. The ain Maugis s idea is to transfor D, rigid interfaial separation, into d, the interfaial separation when the two half-rystals are elastially defored by fores ating between the. D is the input distane of the odel, and d as funtion of D substitutes z in forula () for the potential. Introduing unit-less quantities: D D U z F R d z 6 ; d ; z (5) the equations (1.67)-(1.68) of [5] an be written as D d N d 4d D d Nd 1 4w z z 6 1 1/ 1, if d,if d Figure plots D/ vs. d/ defined by equation (6) for different N. Solution z d D; N of equation (6) ay have several branhes, as illustrated in Figure, that, when substituted to (), adds hysteresis to the potential and related fore odel (4). We will use paraeters N and to ath with the elasto-adhesive odels desribed in the next subsetion. Solution for D d, N N / N 1 (8) 1 Forula in [5] has sign - for the fore direted to the plate (down). In FM onvention, fore is direted up; for interation of two spheres with radii R 1 and R, R=.5R 1R /(R 1+R ). RU 1 4 z z 8 (6) (7) 868

3 will be used below. Figure. lot of D/ vs. d/ defined by equation (6) 4) Elasto-dhesive odels (Maugis s JKR-DMT transition) In this subsetion forulas are suarized for the ase of the fritionless ontat of a sphere of radius R and a half-spae (or two spheres see footnote 1 ). The following notations are used (SI units are shown in brakets): a[]-radius of ontat; K[a]=(4)E *, where E * [a] is redued elasti odulus; [N]-interation fore; h[]-penetration of the sphere into the half-plane. Using unit-less quantities (forulas (4.-4.) in Ref. [5]): a h ; ; 1/ 1/ (9) wr / K wr w R / K Maugis s -paraetri elasto-adhesive odel that onnets JKR and DMT is the following [5, forulas ] 4 1 tan tan (1) (11) (1) Equations (1)-(1), after exluding and, desribe ; between fore and penetration relationship with the Maugis paraeter. This paraeter desribes transition [5, p. 9] between JKR (as ) and DMT (as ) 1 tan 4 (1) (14) This -paraetri odel will be used to ath with LJ- odel in Setion III C. B. Bifurations due to antilever spring onstant in frae of Integrated Lennard-Jones odel In FM as well as other devies easuring defletion of the ; 6 ; antilever, the defletion (z ) due to tip-saple interation is deterined by the fore balane. In the ase of Integrated Lennard-Jones odel () with Derjagin approxiation (4), the fore balane is F z, F z k z (15) R where k [N/] is the antilever spring onstant. Depending on, solutions of equation (15) ay have several branhes, stable and unstable. Condition of stability is Fz, and positions where 9 F 16Rw k (16) z z z are bifuration points. These bifurations hange the fore interations by adding hysteresis as shown in Figure. Figure. Integrated Lennard-Jones tip-saple interation: tip radius R1n; work of adhesion w4j/ ; Integrated Lennard-Jones equilibriu distane n; antilever spring onstant kn/. Values of z (horizontal axis on the plot) are inreasing fro right to left. To ath with elasti solid odels (where penetration is positive) distane fro ontat should be negative by onvention. See forula (17) below. Equation (16) also provides the upper liit on spring onstant k for existene of these bifurations. LHS of (16) is a third order polynoial on y( z). Thus, ondition of existene of bifurations is that the equation k y y 16 Rw has two positive roots whih is the ase if Rw k 9 This an be verified fro the equivalent ondition that loal iniu of the polynoial is negative (fro derivative, loal iniu is at y= -1/ ). For nuerial values of Figure, k is less than [N/]. Illustrated on Integrated Lennard-Jones odel, these bifurations (based on stability hange of fore equilibriu desribed by (15), (16)) are appliable to any fore odel as illustrated below in Figure 5 (green arrows) for LJ- JKR-DMT athing. Figure 4 shows the siulated aplitude urve branhes for the paraeters of Figure. The alulation proedure was 869

4 ipleented using the syste desribed by Equation (1). Two stable branhes are well known to FM pratitioners as lowand high aplitude branhes. The jup fro low to high aplitude branh is visible as the kink in the experiental urve shown in Figure 1. It is iportant to be able to selet the branh that better suits an experient. For exaple, it is lear fro Figure 4 that high-aplitude branh (red) would not be a good hoie beause it is lose to the unstable branh (blak) and is diffiult for traking. In other situations jup (avalanhe) fro low- to high- aplitude branh ay our for uh larger height and only the high-aplitude branh is available for operations. FM ontrol syste with paraeter estiator is able to selet the optial branh for operations. are: and F ;, N, ; N LJ LJ (17) ;, ; (18) Figure 4. Siulated aplitude and phase urves for the tip-saple interation of Figure. Free aplitude free =199 n C. Mathing Maugis JKR-DMT Transition and LJ- Models The athing Maugis JKR-DMT Transition (9)-(1) that desribes elasto-adhesive interation after geoetrial ontat (penetration), and Lennard-Jones odel with adhesive avalanhe (LJ-) that desribes the oleular level interation before the ontat, was suggested in [1, setion IV] and is ipleented below. ll alulations and siulations desribed in [1, 6] were based on siple Hertz odel. The odel athing desribed below for spherial tip and plane saple (or spherial saple see footnote 1 ) overs ajor aspets for this type of interations. It an be siilarly extended to other types, e.g. by using Ref. [7] that desribes odels that extends JKR, DMT and Maugis s transition for arbitrary axi-syetrial tip shape and Ref. [8] that overs analytis of van der Waals fores for ajor pratial situations. Figure 5 shows the priniple of athing. Left part of the Figure shows series of fore urves desribed by equations (4)-(7) with paraeters (N, ) and right part series of urves desribed by equations (9)-(1) with paraeter. ll urves are plotted in unit-less oordinates vs. defined in (5). In Figure 5, is positive for the right part (Elasti solid) and negative for the left part (Lennard-Jones solid). To distinguish the urves, we use notation F for the Lennard-Jones and for Elasti solid. We also use notation LJ for the horizontal oordinate of the Lenard-Jones solid (aording to onvention desribed in Figure ) so that the series of urves Figure 5. Math of Lennard-Jones with adhesive avalanhe (LJ-) and elasti solid odels (JKR-DMT transition) by fitting paraeter N and. Red arrows show jups to ontat (down) during the approah and fro ontat (up) during the retration. Green arrows show bifurations due to soft antilever siilar to those in Fig.. The relationship between D in (5)-(7) and in (17) follows fro their definitions: D LJ w R / K 1/ LJ (19) Eah paraeter in (18) defines the urve and the value ; () Equation () is one-to-one relationship between, 4/,, where the left end defines and DMT urve and the right JKR urve. This allows to selet as the urve paraeter instead of. This paraeter has lear physial eaning the noralized value of the urve at zero-penetration. nother advantage of using is that it an be extended beyond JKR-DMT transition (to the urves that are above the JKR with 4/ in Figure 5). So, instead of (18) we will use the series of urves, ; (1) ; with for DMT, 4/ for JKR and reserve for 4/. uthors are not aware about any theoretial developent of these reserve urves but observed the in FM experients. araeters R (radius of the tip), K (value proportional to the redued elasti odulus), w (work of adhesion), and are geoetrial and aterial properties that an be quantitatively easured and assued to be fixed. Model-based quantitative nanoehanial easureents with FM are desribed in [7, 11]. However, these paraeters ay be unknown and require estiators in adaptive ontrol systes 87

5 Fro the other hand, paraeters N and are not diretly easurable. These paraeters an be alulated based on the assuption that left (Lennard-Jones) and right (elasti) urves desribe the sae physial syste and as a onsequene ust ath soothly at the oon point LJ. This leads to the following equations for the paraeter N ; F ; N () 4 and, after finding N, the equation for paraeter, or where ; F ;, N F D LJ LJ () D; N w R / K D LJ w R K 1/ / 1/ 1 (4) F D; N D D (5) ; and N is the solution of equation (). We now disuss oputational proedures of finding N and fro equations () and (4), (5). ording to (8), for D, z d N / N 1 (6) Substituting this to (4), using (5) and (9), obtain F ; N FR z wr 4 z z, 1 4 N 1 x 8x where x N Then the solution of equation () is x 1 1 (7) N (8) where x is a root of the polynoial equation 4 x 8x, x 1 (9) Table 1 desribes the regions of paraeter and orresponding ranges of N (solution of equation ()) and. The Table and Figure 5 deonstrate that Lennard-Jones solid with adhesive avalanhe (LJ-) overs the whole ranges of and N, while there is a gap for 4 /, N 5. 97, ( with 4 / aking JKR a liiting ase) where no theoretial odel of interation after penetration is available for 4 /. This ust be filled beause the urves fro the gap do our in FM experients. 4 t zero, F ;, N of (17) does not depend on N Coents - DMT after penetration long-range onstant attrative fore before the ontat (-, -4/) (, 5.97) (,) Maugis s DMT-JKR transition LJ- -4/ 5.97 JKR LJ- with N=5.97 (-4/,) (5.97, ) na No odel available after penetration LJ- before the ontat na Classial Lennard-Jones fore (no adhesive avalanhe) goes to infinity approahing the saple penetration is not possible Table 1. Ranges of paraeter, orresponding and N solution of (); () an be found by algorith (8)-(9). fter athing N is found, an be alulated by forula (4) to ake the transition sooth. depends on geoetry (R), aterial properties (w, K) and, aording to forula (5), on derivatives of well defined funtions at zero with a given paraeter. Calulations of the derivatives ay be tie onsuing for real-tie operations. In this ase the unit-less an be tabulated, e.g. for the range of funtion, 4/ DMT-JKR transition. to be used for the urves desribed by IV. EXERIMENTL CSES Figure 6-C shows aplitude vs. distane and fore vs. distane urves, whih were reorded on polystyrene (S) with three Si probes having different stiffness 4N/, 4N/ and.4n/. Initially, the probes were osillating with the aplitude of ~ n that dereases to zero and the tip and saple oe into interittent ontat. On further saple exursion the daped probes defleted due to strong repulsive interations. The fore urves reorded with the stiffer probe does a oderate attrative interation and a non-linear repulsive part that indiates on elasti deforation of the saple. Suh urves an be used for alulation of elasti odulus of polyer aterials as shown in Ref. [7,11]. The adhesion effet is best deteted in the fore urves reorded with softest probe, Figure 6C. The linear repulsive fore response indiates that this probe is not stiff enough for aking indentation. On approah, the soft probe snaps into the surfae when the attrative fore gradient beoes larger than probe stiffness. The pull-out fore is stronger on the probe retration fro the surfae. The aplitude urve reorded with the probe of interediate stiffness exhibits a kink oon to the bifuration effet. t the sae tie the probe exerises the attrative interation that followed by sall snap into ontat. The retration trae is only slight different fro the approah one. This ase study reveals that a ost sensitive detetion of the tip-indued indentation and adhesive effets ight require the use of a partiular probe. This does not ean that adhesion an be disregarded in the analysis of the indentation results. 871

6 Figure 6. plitude and fore urves obtained on polystyrene fils using Si probes with stiffness 4 N/ (), 4 N/ (B) and.4 N/ (C). The blue and red traes orrespond to the probe approah and retration. The aplitude and defletion axis is in arbitrary units. It is worth noting that these fore urves were obtained at relatively low frequenies, e.g. one approah-retration yle per seond. In osillatory odes, where either an aplitude daping or a frequeny hange are eployed for onitoring the tip-saple fore interations, the approah-retrat yle is in the 5-4 khz range. The reording of the fore urve during the fast yle is hallenging but already feasible with the use of speially-shaped probes [9-1]. The analysis of these urves, (urrently liited to repulsive part of the approah urve using Hertz, JKR or DMT odels [7,11]) will eet better quantitative standards by using JKR-DMT - LJ- athing odel applied to repulsive and attrative regions of both approah and retration experiental urves. Visoelasti nature of polyers results in frequeny dependene of their ehanial properties that is not onsidered by the above odels. Further theoretial and experiental work is needed in this respet. V. CONTROL, SIMULTION ND MESUREMENT STRTEGY FM asyptoti dynais (1) with states (x, ) = (aplitude, phase) along with the JKR-DMT LJ- athing odel for F a and F r an be used for advaned ontrol of the FM experient, siulation of the iages and fore urves, and loal easureents of aterials properties. ratial syste designs for these appliations will be desribed in forthoing publiations while soe short notes are presented here. Control. Both experiental (Fig. 1) and siulated (Fig. ) aplitude/phase urves indiate that the ontrol syste (1) is a hybrid syste with three disrete states: Low plitude; High plitude; and Bifuration. Forally there is also the fourth state orresponding to the unstable branh in Figure 4. FM ontrol syste traking the set-point aplitude should identify and selet a disrete state and avoid jup to another state (through bifuration). This an be ahieved by real-tie paraeter estiation and adaptive traking of the state based on the estiated paraeters. When the disrete state is seleted, any of available adaptive ontrol tehniques with paraeter estiation are appliable. Stable traking of the set-point aplitude is the key feature of the preise profiling of the oplex surfae strutures with inial tip wear. Knowledge about tip-saple fores is also invaluable for hoosing sanning onditions with a inial ontat area (that an be alulated fro the paraetri fore odel) leading to high-resolution iaging. Using the odel and the disrete state, aplitude set-point an be paired with the phase set-point (as in Figure 4). Then syste (1) an be linearized near this set-point pair and linear ontrol design with paraeter adaptation an be utilized. Siulation. Referene [6] desribes an FM siulator based on Hertz odel. Ipleenting the JKR-DMT LJ- athing desribed in this paper allows siulation of uh ore opliated and realisti FM experients. This is also true for siulation of frequeny odulation ode experients that are perfored in different environents. Measureent. Outputs of paraeter estiator used in adaptive ontrol an be used for real-tie aterial properties apping. nother and ore aurate interative analysis an be based on off-line fitting of experiental fore urves. Suessful quantitative FM easureents based on JKR odel were reported in Refs. [7,11] and will benefit fro using athing odel desribed in this paper to easure ore opliated non-jkr aterials. The odel athing tehnique an be extended to other fores sensed prior the tip-saple ontat, e.g. eletrostati. t this tie the desribed athing odels are under experiental verifiation. Of partiular interest are the ases where eletrostati tip-saple interations are enhaned by using the onduting probes and easureents are perfored in different huidity. In the latter pratially-iportant ase a enisus fore beoes the additional onstituent of the overall tip-saple fore interations. Results of verifiation and use of the odel will be topis for forthoing publiations. VI. SUMMRY The hybrid odel of tip-saple fore interations that inludes arosopi deforation and atoi fore potential with adhesive avalanhe is a substantial step in extending the ontrol of FM easureents and siulation of FM experients for ore realisti non-onservative ases. rational theoretial desription of tip-saple fores needs to address the loal easureents of eletrostati fores that represent a fast inreasing part of FM appliations. REFERENCES [1] S. Belikov, S. Magonov, roeed.. Control Conf. 9, 979. [] B. Capella and G. Dietler, Surf. Si. Rep. 1999, 4, 1. [] R. Garia, and R. erez, Surf. Si. Rep., 47, 197. [4] X. Xu, J. Meher, and. Raan, hys. Rev. B 1, 81, 547. [5] D. Maugis, Contat, dhesion and Rapture of Elasti Solids, Springer,. [6] S. Belikov, S. Magonov, roeed.. Control Conf. 1, [7] S. Belikov et al, JVST B 9, 7(), 984. [8]V.. arsegian, Van der Waals Fores, Cabridge Univ. ress, 6. [9] O. Sahin, et. al. Nature Nanotehnol. 7,, 57. [1]. Sarioglu, and O. Solgard, ppl. hys. Lett. 8,9, 114. [11] H. Lakrout et. al., roeed. dhesion So. Meeting, 8. 87