A tom ic F orce M icroscope:M odeling, Sim ulations,
andE xperim ents
O sam ah M .E lR ifai, and K am alY oucef-Toum i
A bstract| T he quality of atom ic force m icroscope (A F M ) data strongly dependson scan and controllerparam eters. D ata artifacts can result from poordynam ic response ofthe instrum ent. In order to achieve reliable data, dynam ic interactionsbetween A F M com po-nentsneed to be wellunderstood and controlled. In thispaperwe present a sum m ary of ourwork in this direction. It includesm od-elsforthe probe-sam ple interaction, scannerlateraland longitudinal dynam ics, scannercreep, and cantileverdynam ics.T he m odelswere used to study the e®ectofscan param eterson the system dynam ics. Sim ulation resultsforboth frequency response and im aging werepre-sented.E xperim entalresultswere given supporting the sim ulations and dem onstrating the com petence ofthem odels.T heresultswithin will be used to develop algorithm s that allow autom ated choice of key system param eters, guaranteeing reliable and artifact-free data forany given operating condition (sam ple, cantilever, environm ent). C onsequently, expanding theA F M capabilitiesandperm itting itsuse in a widerrange ofapplications.
K eywords| A F M , m odel, experim ents, sim ulation, control
I.Introduction
T he invention ofscanning probe m icroscopy (SPM ), and A F M in particular, hasgreatly contributed to advancing research in nano-science andtechnology to itscurrentstate ofthe art.T he successofA F M asa toolin nano-sciences, liesin itsability to provide controlled nano-levelforce or displacem ent.E xam plesinclude nano-scale studiesofplas-tic deform ation, m icrostructures, and friction [1]. O ther applications require surface topography inform ation of a sam ple, e.g.m icro-fabrication, and nano-defects.
Poor dynam ic interactions between A F M com ponents can resultin corrupting A F M data, [2], andproducing data artifacts.In orderto achieve the bestpossible perform ance one needsto understand these dynam ic interactions. In thisscope, there hasbeen som e work in the literature on developing m odelsthat describe the dynam icsofan A F M cantileverduring interm ittent-m ode operation, [3].T hese m odelsfocuson probe-sam ple interaction at a single lo-cation on the sam ple surface. D ue to the low scanning speedsofthism ode, these m odelsdo not account forthe e®ect ofscanning speed norscannerdynam ics.T herefore, these m odelsfailto capture the overalldynam icsand are not suitable foranalyzing contact-m ode operation, w here scan speed, scannerdynam ics, friction, and controlsystem allneed to be considered.
T hispaperpresentsa sum m ary ofresearch e®orttoward addressing these challenges in A F M technology. In sec-tion II, som e background on A F M isgiven. Section I II, elaborates on the m otivation for this work. M odels for probe-sam ple interactions, scanner lateral and longitudi-naldynam ics, scannercreep, and cantileverdynam icsare presented in section IV .Sourcesofnoise and disturbances are also discussed. B oth experim entaland sim ulation
re-X Y Z Laser Detector py θ Sample Controller Piezo Amplifier Set-point Vz u yPSD + Disturbance Signal
F ig.1. Schem atic diagram ofA F M m ain com ponents. sultsare presented and discussed in section V .Sum m ary and concluding rem arksare given in section V I.
II.A tomic Force M icroscope
A n A F M , F ig.1, hasthree m ain com ponents, nam ely, a scanner, a cantileverw ith a sharp probe , and a cantilever de°ection sensorcom prised ofa lasersource anda position sensitive diode (PSD ).T he scanner, typically a piezoelec-tric tube, providesthree-dim ensionalm otion betw een the probe and a sam ple. Inform ation on sam ple topography orlocalpropertiesisobtained based on probe-sam ple in-teractions. O ne of the m ain operating m odesof A F M is contact m ode. In thism ode, the probe pressesagainst a sam ple, exerting a verticalforce proportional to the can-tileverde°ection. T he probe isthen dragged against the sam ple along each scan line in a rasterfashion.T he slope atthecantileverfree-endism easuredandfedback.D uring scanning, a controlsystem isused to m aintain a constant slope, by adjusting the verticaldisplacem ent of the scan-ner.C hangesin the piezo de°ection are therefore, related to changes in the sam ple topography. T his m ode isthe scope ofthispaper.
III.M otivation
A com m ercial A F M was used to scan a Silicon cali-bration steps. T he A F M w as run under PI control. A Silicon N itride cantilever w as used w ith a resonant fre-quency of 13 kH z and sti®ness of 0:2 N =m . Scan re-sultsdem onstrate the high sensitivity of collected im ages to scan and controllerparam eters(Kpand Ki).C om par-ing F ig. 2 (a) (72 ¹m =s;Kp = Ki= 2) to F ig. 2 (b) (96 ¹m =s;Kp= Ki= 20), som e ofthe e®ectsofscanning speedandcontrollergainson theim agecan beseen.H igher gainsresult in oscillationsasthe cantileverfallsalong the right edge ofthe step, w ith peaksprobably indicating m
o-F ig.2. A o-F M im ages:(a)72¹m =s;Kp= Ki= 2, (b)96¹m =s;Kp=
Ki= 20.
F ig.3. A F M im ages:180¹m =s, (a) nom inal contact force, (b)
sm allercontactforce.
m entary lossofcontactbetween the probe and the sam ple. W hile the sharp peak on the left edge ofthe step, F ig.2 (b), can be attributed to a high scan speed com pared to closedloopbandw idth.T he highergainsim prove tracking, asthe sharp left edge of the step isresolved m ore accu-rately. F igures3 (a) and (b) w ere generated w ith a scan speed of 180¹m =s using the sam e controllergains. T he contact force set-pointforF ig.3 (a)isset to the m anufac-turer'srecom m endedvalue, w hile F ig.3(b)a sm allerforce wasused.C hoosing a sm allcontactforce set-pointreduces contact deform ation and friction, however, it reducessta-bility of the contact. A sseen from F ig. 3 (b), the im age generated w ith a sm allcontact force haserroneousheight inform ation, due to lossofcontact between the probe and the sam ple.
Ithasbeen show n thatscan andcontrolparam etersdra-m atically ietersdra-m pact A F M dynam ics. A sa result, im age ar-tifact m ay result because of poordynam ics. In orderto elim inate these typesof artifacts, we need to understand the dynam ic interactionsbetw een di®erent com ponentsof the A F M .T hisisthe objective ofthiswork.
IV .System M odel
W e have reported m odels describing the dynam ics of A F M in [4], [5].Itincludesprobe-sam pleinteraction forces, m odelsforthe piezoelectric scannerlongitudinaland lat-eraldynam icsand coupling betw een them , and cantilever °exuraldynam ics.Sourcesofnoise and disturbancesw ere also discussed. In thissection, the m odelsw illbe brie°y presented and discussed.
A .Probe-sam ple Interaction
A .1 In-contact M odel:V erticalF orces
T hiscontact m echanicsm odel [6], issuitable forA F M operation in airorotherm edia w here adhesion/capillary forcesare dom inant. It describesthe adhesion/capillary contact oftwo elastic spheres. A non-dim
ensionaltransi-2a
2c R
F ig.4. Schem atic ofprobe-sam ple contact.
tion param eter¸, wasde¯ned as, ¸ = ¾o(
9R 2¼wE ¤2)
1
3 (1)
w here, ¾oisthe theoreticalstrength ofthe adhesion junc-tion, R isthe reducedradiusofthe spheres, w isthe D upr¶e w ork ofadhesion, and E¤isthe com bined elastic m odulus ofthe spheres.T histransition param etercan be view ed as the ratio ofelastic deform ation to the e®ective range over w hich surface forcesact.F rom (1), itfollow sthatlargeval-uesfor¸ w ould correspond to com pliant (sm allE ¤), large spheres(R ), and sm alladhesion (w)contacts, w here sm all valuesare forsti® sm allspheresw ith high adhesion.T he m odelcan be used to predict contact force Fcon, contact deform ation ±, and contact radiusa.
T hem odeliscom posedofthreenonlinearalgebraicequa-tion that can be expressed in non-dim ensionalform as,
± = a2¡ 4¸a 3 p m2¡ 1 Fcon = a3¡ ¸a2[ p m2¡ 1+ m2sec¡1(m )] (2) 1 = ¸a 2 2 [(m 2¡ 2)sec¡1(m )+pm2¡ 1]+ 4¸2a 3 [ p m2¡ 1sec¡1(m )¡ m + 1]
w here, m = acasin F ig.4, c isthe radiusoverw hich sur-faceforcearepresent.T heuseofsuch continuum m odelsto describe nano-contactshasbeen supportedby m any exper-im ents.T he levelat w hich continuum m odelsbreak-dow n isnot allclear[7].
A .2 In-contact M odel:L ateralF orces
A sthe probe isdraggedagainstthe sam ple w hile in con-tact, a frictionalshearforce develops. A t nano-levelcon-tacts, experim entshave revealedthe dependence offriction force on contact area, [1].F orourpurposes, w e are inter-estedin sim ulating the e®ectofsliding friction force on the cantileverdynam icsduring scanning.A sa ¯rst orderap-proxim ation, wew illassum ethattheinstantaneousfriction force isdirectly proportionalto the instantaneouscontact area (»a2).T hism odeldoesnot considerany explicitde-pendence offriction on scanning speed.A lthough contact m odelsw ere originally developed forstatic loading, it has been show n in [1], that it holdsundersliding conditions w ith not very high sliding speeds. W hen the probe and sam ple are out ofcontact the friction force isset to zero.
Oo X 1 C Oi θ δ θ x δx y y δ V x-Vx+ V y-Vz Ri Ro
F ig.5. C rosssection ofan eccentric piezoelectric tube.
A .3 O ut-of-contact M odel:V erticalF orces
W hen the probe and sam ple are not in contact, van der W aalsforcesbetw een two spheresare assum ed to be the dom inant interaction.
A .4 Point ofC ontact
A t the lim it of lossof contact, the contact radiusa ! 0. Substituting this into eqn. (2), gives the force and separation at that lim it.T hisisused to im pose continuity on the force betw een m odelsofsectionsIV -A .1andIV -A .3 .
B .ScannerL ateralD ynam ics
T here are tw o m ain designsforA F M .In one design, the cantileveris¯xed and a sam ple isplaced on the scanner w hich m ovesitrelative to the cantilever.T hisdesign, gen-erally, lim itsthe m axim um size and weight ofthe sam ple. In the second design, F ig.1, the cantileverisattached to the scannerthat m ovesit relative to a stationary sam ple. In thispaperthe m odelpresented isforthe second, m ore populardesign.
T he scannerisa thin-walled piezoelectric tube that has fourelectrodesofequalsegm entson itsoutersurface, and eithera single orfourelectrodeon itsinnersurface.A pply-ing a voltage Vz, to itsinnerelectrode(s)resultsin exten-sion m otion, (in the Z-axis).M otion in the X orY axisis typically generated by subjecting tw o opposite electrodes to two voltage signals, (Vx+;Vx¡)and (Vy+;Vy¡), w ith the sam e m agnitude but are out ofphase.
In [8], [9], a m odelforan idealuncoupled tube scanner waspresented.D ue to inevitable m achining tolerance, ec-centricity isalwayspresent in the tube.T ypically, a m ax-im um of 50¹m fora 12:7m m diam etertube, [10]. T his seem ingly sm all eccentricity is in fact signi¯cant as the cantilever-de°ection sensorhasA ngstrom rm sresolution. T he m odelisbasedon two eccentric cylinders, F ig.5, w ith eccentricity ±xand±yfrom thegeom etriccenteroftheouter cylinder, O o.T he tube is¯xed at one end and free at the other. A cantileverholderofm assmshisattached to its free end.T he m odelisbased on elem entary bending the-ory forthin-w alled m em bers. T he m ain assum ptionsare sm alldeform ationsand angles, linearelastic m aterial, and
B elow , eqn.(3), givesthe transferfunction betw een the in-putvoltagesandthe lateraldisplacem entuj;(j= 1;2), for i¤num berofm odes.E quations(4), and(5)givetheoutput equationsfordisplacem entsin the X and Y -directions, xp and yp, respectively.T he scanner'sfree end rotations(i.e. slopes)about X and Y -axes, µpxand µpy, respectively, are given in eqn.'s(6)and (7).
uj= i¤ X i= 1 °ix+Vx++ °ix¡Vx¡ + °iy+Vy++ °iy¡Vy¡+ °izVz s2+ 2³ iuj!iujs+ ! 2 iuj (3) xp = cos(µ±)u1¡ sin(µ±)u2 (4) yp = sin(µ±)u1+ cos(µ±)u2 (5) µpx = sin(µ±) @ u1 @ z + cos(µ±) @ u2 @ z (6) µpy = cos(µ±) @ u1 @ z ¡ sin(µ±) @ u2 @ z (7)
C .ScannerL ongitudinalD ynam ics
T he detailsofthe m odelare given in [11].T he m ain as-sum ptionsare sim ilarto those ofsection IV -B .T he trans-fer function between input voltages and displacem ent of the tube'sfree-end zp, forj¤num berofm odesisgiven as, zp= j¤ X j= 1 °jx+Vx++ °jx¡Vx¡+ °jy+Vy++ °jy¡Vy¡ + °jzVz s2+ 2³ jzp!jzps+ ! 2 jzp (8) D .ScannerC reep
T heresponseofa piezoelectricactuatorto a rapidchange in input voltage, F ig. 6, consistsoftwo m ain parts. T he initial part of the response occursovera tim e scale dic-tated by the m echanical resonance of the actuator. T his isfollow ed by a slow creeping response occuring over100s ofsecondsand am ounting to asm uch as20% ofthe total response. T he rate and am ount ofcreep strongly depend on the piezoelectric m aterial.E xperim entalfrequency re-sponse ofpiezoelectric actuatorsdisplaysvery little varia-tion in phase at low frequency between input voltage and displacem ent.O n the otherhand, a slight decrease in gain isobservedw ith increasing frequency.Itispossible to sim -ulate creep behaviorusing a suitable L T I m odel.T he rel-ative degree isnum berofpolesm inusthe num berofzeros of the transfer function. T he transfer function between the input voltage and actuatordisplacem ent should have a relative degree zero at frequenciesm uch low erthan the actuator'sresonance frequency. T hism odelassum esthat the ratio between the am ount ofcreep and the fast scan-nerdisplacem ent isindependent ofinput am plitude.T he assum ption w illbe experim entally tested.
E .C antileverF lexuralD ynam ics
T hisdynam icm odelforthecantileverm otion neglectsef-fectsofsheardeform ation and rotary inertia, and isbased
0 10 20 30 40 50 60 0 50 100 150 200 250 300 350 400 450 Time, [s] Cantilever Displacement, [nm *]
F ig.6. E xperim entalcreep response.
on elem entary bending theory. T he cantileverslope (an-gle), z0
c, ism easured relative to itsbase m otion, zp(t).T he boundary conditionsare taken aszero de°ection and slope relative to the base atthe ¯xed end, and zero m om ent and shearforce at the free-end. F orsim plicity, the cantilever isassum ed to be aligned w ith the X -axis. T he transfer function governing the cantilever'sresponse isgiven by,
zc0(s) = m¤ X m = 1 a2zms2+ a1zms s2+ 2³ cm!cms+ !2cm zp(s) + a2µms 2+ a 1µms s2+ 2³ cm!cms+ !cm2 µpy(s) + afm s2+ 2³ cm!cms+ !cm2 f(zc;zp;µpy;zs) (9) w here, f(zc;zp;µpy;zs)isthe probe-sam ple force, and zs isthe sam ple height.
F .SensorO utput
T he optical-leversensorm easuresthe absolute angle of the cantilever'sfree-end.T herefore, the PSD outputyPS D, isgiven by,
yPS D = µpy¡ zc0 (10)
G .D isturbances
T herm alnoise orB row nian m otion contributesto a fun-dam entalsource ofnoise in A F M .A t therm alequilibrium , the m ean value of the cantileverpotential energy hasto equal12kBT , w here kB= 1:38£ 10¡23J =K isB oltzm ann's constant, and T is the absolute tem perature in K elvin. T he cantilever'sfree-end w illoscillate w ith a R M S value, z0rm s c = 2L3cz rm s c = 2L3c q kBT
kc , w here kcisthe cantilever sti®ness, and Lcisthe cantileverlength. T hisexpression isvalid fora free standing cantilever. If the cantileveris in contact w ith a sam ple, the expression hasto be m odi-¯ed by including the sam ple e®ective sti®nessin kc. A n-othersource ofdisturbance islaserback-action.It isdue to incidence ofphoton °ux from the opticalsensoron the cantilever. B oth therm al and back-action noises w ill be e®ectively m odeled aszero-m ean w hite noise force distur-bancesw ith a com bined constant intensity µ±(t¡ ¿).
Force Probe-sample Separation Penetration Region Out-of-Contact Region -kc N/m -kc N/m Pull-in point Pull-off point δ
F ig.7. Sim ulation:Q uasi-static norm alized force-separation curve.
0 50 100 150 200 250 300 350 400 450 500 -5 0 5 10 15 Probe-sample Separation, [nm*] Force, [ nN ] * Pull-in Point Pull-off Point
F ig.8. E xperim entalforce-displacem entcurve.
F eedback m easurem ent noise arising from the optical sensorcan be due to shot noise, a fundam ental noise for these sensors, in addition to noise from sensorelectronics. Shot noise can also be m odeled asw hite noise.
V .R esults A .Q uasi-static F orce-separation C urve
T he m odelsofsectionsIV -A .1, IV -A .3 w ere usedto gen-erate the com posite force-separation curve ofF ig.7.Pa-ram etersused to generate the curve are given in [4].It is w orth noting that eqn.(2)can predict an instability that hasbeen observed in quasi-static experim ents.T hisinsta-bility occursw hen an approaching/receding probe jum ps in/out (pullin/pullout points), of contact w ith the sam -ple surface corresponding to a sudden jum p in the contact area.T heactualpointofinstability on theforce-separation curve w illdepend on the sti®nessof the cantileverkc, as show n in F ig. 7. A n experim entalforce-separation curve isshow n in F ig.8, w here¤denotesuse ofestim ated cali-bration factors.It show sthe sam e characteristic produced by the m odelin F ig.7, except that hysteresisisobserved in the penetration region asthe approach and retractlines are not the sam e.
zp
z´c
zs kc
ks
F ig.9. Schem atic ofprobe-sam ple contact.
B .In-contactD ynam ics:Sim ulations
T he probe-sam ple interaction force f(zc;zp;µpy;zs) is a nonlinearfunction of probe-sam ple separation, and de-pendson geom etry, environm ent, and probe and sam ple m aterialproperties. T o obtain a linearm odelto be used foranalysis, the force w asexpanded in a T aylorseriesand linearterm swere retained, giving,
f(zc;zp;µpy;zs) = gczc+ gzzp+ gµpyµpy
+kszs+ H :O :T (11)
kscan be considered asa lineare®ective probe-sam ple contact sti®ness.T he probe-sam ple contact can be repre-sented schem atically asin F ig.9, w here again zc, ism ea-sured relative to zp. T he contact and cantileversti®ness kc, are represented astw o springsin series. T he contact sti®nessdoesnot change the orderof the m odel, but has a great im pact on the system 'szeros.Substituting eqn.'s. (3), (7), (8), and(11)into eqn.(9)and the resulting equa-tion into eqn. (10) givesthe overallm odel. T he transfer function ofinterestisbetw een VzandyPS D.T hisdescribes the A F M Z-dynam ics, w hich w e w illfocuson in thiswork. T he m odel used in this study included four bending m odesand two extension m odesforthe scanner, and one bending m ode for the cantilever. T he param eter values used are given in [11]. T he ratio of sam ple to cantilever sti®nessks
kc, provedto bean im portantparam eter.C hanges in thisratio have two m ain e®ectson the m odel, nam ely, change in the transferfunction D C gain andchangesin the frequency ofthe zerosassociatedw ith the scannerbending m odes, 380H z and 2kH z.F igure 10, show sthe sim ulated frequency response ofthe m odelfordi®erentratiosofsti®-nesses. F orlarge ratios(e.g. ks
kc = 7), the zeroshave a higherfrequency than the m ode. F orsm allerratios(e.g. 1· ks
kc · 2), the frequency decreasesto be below that of the m ode.T hischange in pole-zero pattern isreferred to aspole-zero °ipping. M oreover, forsom e value (ks
kc ¼ 4), there ispole-zero cancelation and the bending m odesare unobservable. F igure 11, presentsa pole-zero m ap ofthe ¯rst two m odesfordi®erent valuesofks
kc. A sa result, as the zerosm ove aw ay from the m ode, the resonance peak appearsm ore prom inent in the response. F urther, w hen
ks
kciseithertoo large ortoo sm all, the D C gain reachesa
102 103 10-8 10-7 10-6 Frequency, [Hz] Magnitude 102 103 -200 -100 0 100 200 Frequency, [Hz] Phase, [degree] s c s c s c k s/kc = 4 ks/kc = 7
F ig.10. Sim ulation:In-contactfrequency responsefordi®erentratios ofsam ple to cantileversti®nessks
kc. -1400 -1200 -1000 -800-3 -600 -400 -200 0 -2 -1 0 1 2 3x 10 4 Real Axis Imaginary Axi s -200 -195 -190 -185 -180 -2500 -2000 -1500 -1000 -500 0 500 1000 1500 2000 2500 Real Axis Imaginary Axi s
F ig.11. Pole-zero m ap asa function ofks
kc:(left) zoom on 2kH z m ode, (right)zoom on 380H z m ode.
lim it controlled by kcand ks, respectively.F orinterm edi-ate values, the D C gain w illdependon both sti®nessesand changesin ksdue to di®erentset-pintsorinputam plitudes w illchange the D C gain, and zeroslocation.
C .In-contactD ynam ics:E xperim ents
T o investigate in-contact A F M dynam ics and validate the m odels, tw o sam ples w ere chosen for experim ents, nam ely, a G lassanda Polydim ethylsiloxane (PD M S), sam -ple having Y oung'sm oduli ofelasticity of60and2:5M Pa, respectively. T w o di®erent cantilevers were used. C an-tileverA hasan estim ated sti®nessof 0:03N =m and res-onance frequency of 13kH z. C antilever B has an esti-m ated sti®ness of 0:14N =m and resonance frequency of 14kH z. D i®erent set-points and input am plitudes were usedto study the e®ectofthese param eterson the dynam -ics.
F igure 12, show s the force-displacem ent curve for the G lass sam ple. T he points labeled on the plot are the force set-points used in the frequency response experi-m ents.N ote thatthelocalsensitivity aroundtheset-points (cantileverde°ection perscannerinput voltage)issm aller forthe largerforce set-point. T hissuggestsreduced con-tact sti®nesspotentially due to plastic deform ation in the contact.T he e®ectsofforce set-pointon the dynam icsare seen in F ig. 13. F orthe largerset-point, 17nN , the D C
-0.05-5 0.25 0.55 0.85 1.15 1.45 0 5 10 15 20 Scanner Displacement, [µm*] Force, [nN *] Force Set-points
F ig.12. F orce-displacem ent curve using cantileverA with a G lass sam ple. 102 103 10-2 10-1 100 Frequency, [Hz] Magnitude 102 103 -200 -100 0 Frequency, [Hz] Phase, [degree] 17.6 nN DC Gain = 0.12 14 nN DC Gain = 0.23
F ig.13. In-contact frequency response using cantilever A with a G lasssam ple:sam e am plitude fordi®erentset-points.
gain issm allerandthe 380H z bending m ode hasa sm aller resonance peak.T he decrease in D C gain suggest thatthe e®ective contactsti®nesshasdecreased, hence, the scanner displacem ent (F ig. 9) istransm itted m ore to the sm aller sti®ness;the contact's.T he sm allerresonance peak could bedueto two reasons;thefrequency ofthezero-pairassoci-ated w ith the bending m ode hasslightly decreased forthe largerset-point. H ence, allow ing the contribution of the bending m ode to appearlessprom inent in the response. In addition, it could be a result ofchangesin the dissipa-tive propertiesofthe contactw ith changesin the set-point. Itisim portantto realize thatthe bending m ode resonance frequency did not change.T he contact forcesare ordersof m agnitude sm allerthan the force the scannercan provide, (»100snN vs.»1N ).
T he e®ect ofexcitation am plitude, i.e.sam ple topogra-phy, on the frequency response isshow n in F ig. 14, for set-point of 14nN . It isseen that the largerthe am pli-tude ofexcitation, the sm allerthe D C gain.F orthe larger input am plitude, m ore plastic deform ation m ight occurin the contact due to the largercontact force, w hich in turn reducesthe e®ective contactsti®ness.A sbefore, m ore dis-placem ent istransm itted to the sam ple, hence, reducing the D C gain. T he m ore plastic deform ation there is, the sm allerthe change in contactsti®ness, andhence D C gain. T he resonance peak also changes due to changes in the
102 103 10-1 100 Frequency, [Hz] Magnitude 102 103 -200 -100 0 Frequency, [Hz] Phase, [degree] V z = 300 mV DC Gain = 0.23 V z = 500 mV DC Gain = 0.18
F ig.14. In-contact frequency response using cantilever A with a G lasssam ple:14nN fordi®erentam plitudes.
-1.2 -0.8 -0.4 0 0.4 0.8 1.2 1.6 2 -20 0 20 40 60 80 100 120 140 160 180 200 Displacement, [µm*] Force, [nN *] Force Set-point
F ig.15. F orce-displacem ent curve using cantileverB with a PD M S sam ple.
frequency ofthe bending m ode zero-pair.
F igure 15, show s the force-displacem ent curve for the PD M S sam ple.T he penetration region seem squite nonlin-ear.T hisim pliesthat w hatisbeing m easured by the PSD signal, at least in part, isthe deform ation of the sam ple. T he observationsin F ig.16, are that increasing the input am plitude reducesthe D C gain, the frequency ofbending-m ode zeros, and consequently increasing resonance peak. T hissuggeststhatthecontactsti®nessincreasesw ith larger am plitude, w hich agrees w ith F ig. 15. In addition, in-creasing set-point, F ig.17, resultsin pole-zero °ipping for the ¯rst two bending m odes. T he frequency of the zeros changesfrom being higherto being lowerthan that ofthe associated m ode. T hisisin agreem ent w ith m odelpred-ications in section V -B , (F ig. 10). H ence, contact and cantileversti®nessvaluesare relatively close.
T he m odelcan be furtherim provedon to include nonlin-earitiesin the contact a®ecting the D C gain and dam ping. T hese nonlinearitiesdepend in great part on propertiesof the sam ple. H ence, the form of this dependence is not know n. It ispossible to account forit in the m odel by generalizing the probe-sam ple force to include dissipative term sand retain higherorderterm s.T herefore, eqn.(11), changesto f(zc;_zc;zp;_zp;µpy; _µpy;zs).
102 103 10-2 10-1 Frequency, [Hz] Magnitude 102 103 -200 -100 0 100 200 Frequency, [Hz] Phase, [degree] 4 nm 42 nm 85 nm
F ig.16. In-contact frequency response using cantilever B with a PD M S sam ple:36nN fordi®erentam plitudes.
102 103 10-2 10-1 100 Frequency, [Hz] Magnitude 102 103 -200 -100 0 100 Frequency, [Hz] Phase, [degree] 36 nN 113 nN
F ig.17. In-contact frequency response using cantilever B with a PD M S sam ple:17nm am plitude fordi®erentset-points.
D .Scanning Sim ulation vs.E xperim ents
T he m odelswere used to perform scanning sim ulations. T he sam ple shape used in sim ulation is an experim ental A F M im age ofcalibration steps.F igure 18, show sthe sim -ulated im age vs. the actualsam ple. It can be seen that the sam pled and averaged im age generated from the piezo voltage, Vz, does not correspond well to the actual im -age. T he cantileveroscillationscausesit to loose contact w ith the sam ple and the ham m ering action could in fact be dam aging to the actualsam ple.A F M im age show n in F ig. 19 isof the stepsused in the sim ulation. T he sim -ulations predict the actual response w ell. A lso note the oscillationsobserved in F ig. 19. T hese oscillationsintro-ducesan artifact that could be interpreted incorrectly as surface roughness.
E .Perform ance L im itations D ue to Scanner B ending M ode
T he coupling between scanner bending and extension m odeshasa great im pact on A F M perform ance in several ways.W hen the scanneriscom m anded to m ove up/dow n, there isa slight bending m otion that getsdetected by the PSD .T hescanneristypically calibratedby im aging a stan-dard ofknow n height usually in the 100nm range.D uring im aging, the PSD signal w ill change due to the sam ple topography asw ellasto bending ofthe actuator.Im agin-ing a sam ple of a di®erent height w illresult in a slightly
0 0.5 1 1.5 -10 0 10 20 30 40 50 60 70 Lateral Displacement, [µm] Height, [nm] Averaged Image Sample
F ig.18. Scanning sim ulation resultsofcalibration stepsusing a PI controller. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 -20 0 20 40 60 80 100 120 Position [µm] Height [ nm ]
F ig.19. A F M im age ofcalibration stepsusing a PI controller.
di®erent calibration factor, even ifthe nonlinearity ofthe scannerwasnot a concern.T he change in calibration due to scannerbending istypically lessthan 1% .
Scannerextension m ode istypically around 4to 8kH z. T he required feedback bandw idth during scanning can be estim ated based on scan rate and im age resolution (num -ber of data points per scan line). F or a scan rate of 2H z and a resolution of 512, the bandw idth would be 2£ 2£ 512=1000= 2:048kH z. H ence, the bending res-onance at 380H z wouldbe wellw ithin the feedback w idth.T uning the controllergainsto achieve a high band-w idth band-wouldresultin oscillationsin the im age.T hiscan be seen in F ig.20, w here oscillationsdue to the 380H z m ode is observed in an experim ental A F M im age. Increasing forceset-pointtendsto m akethebending m odelessobserv-able by m oving the zeroscloserto thatpoles.H ow ever, the high contactforce m ay cause dam age to the sam ple and/or reduce im age resolution.A lternatively, the bandw idth can be reduced along w ith scan speed.T hisresultsin a longer scan tim e m aking scannercreepm ore observable in the im -age.F urther, sincethe bending m odeistheclosestm odeto the j! axis, F ig.11, ithasa greate®ecton the robustness ofthe feedback system .Pole-zero °ipping can cause closed looppolesto crossthe j! axiscausing feedback instability. F .C reep C om pensation
A 3rdorderL T I ¯lterwasused to com pensate forcreep in the A F M piezoelectric scanner, [12]. T he perform ance
0 0.2 0.4 0.6 0.8 1 0 200 400 600 800 1000 1200 Time, [s] Height, [nm] 0.34 0.36 0.38 500 550 600 Displacement, [µm] 0.56 1.12 1.68 2.24 380 Hz Scanner Bending Mode
F ig.20. O scillationsdue to scannerbending m ode in experim ental A F M im age ofa 1046 nm Silicon step.
0 0.5 1 1.5 2 0 100 200 300 400 500 600 700 Time, [s] Height, [nm] 1.4 2.8 4.2 5.6 Uncompensated Compensated Displacement, [µm]
F ig.21. A F M im ageof530nm Silicon steps, with and withoutcreep com pensation, 2:8¹m =s.
ofthe ¯lterw astested, e.g.F ig.21, by im aging 530and 1590nm steps.W ith com pensation, thecreepresponsewas reduced to 2:6 % w ith com pensation forim agestaken over 6:67 m inutes. C om pensation did not degrade for larger steps, suggesting that the a linearm odelforcreep m ay be justi¯ed. N onlinearitiesin the scannerfast displacem ent can be dealt w ith separately. C losed loop operation can o®erbettercreep com pensation but is a m ore expensive option. In addition, it reducesim age resolution forsm all scans/sam ple featuresdue to lim ited dynam ic range ofthe sensorsat a high bandw idth.
V I.Summary and C onclusion
T he quality ofA F M data strongly dependson scan and controllerparam eters.D ata artifactscan result from poor dynam ic response of the instrum ent. In orderto achieve reliable data, dynam ic interactionsbetw een A F M com po-nentsneed to be wellunderstood and controlled. In this paperwe presented a sum m ary ofourwork in thisdirec-tion.It included m odelsforthe probe-sam ple interaction, scannerlateraland longitudinaldynam ics, scannercreep, and cantileverdynam ics. T he m odelsw ere used to study thee®ectofscan param eterson thesystem dynam ics.Sim -ulation results for both frequency response and im aging were presented.E xperim entalresultswere given support-ing the sim ulationsand dem onstratsupport-ing the com petence of
the m odels. T he results w ithin w ill be used to develop algorithm sthat allow autom ated choice ofkey system pa-ram eters, guaranteeing reliable and artifact-free data for any given operating condition (sam ple, cantilever, environ-m ent).C onsequently, expanding the A F M capabilitiesand perm itting itsuse in a w iderrange ofapplications.
R eferences
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