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 ponentsneed to be wellunderstood and controlled. In thispaperwe
present a sum m ary of ourwork in this direction. It includesm odelsforthe 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 werepresented.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
X
Detector
Y
Vz
Laser
Piezo Amplifier
θ py
Disturbance
Signal
Set-point
Z
+
u
Sample
Controller
yPSD
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.
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 studiesofplastic deform ation, m icrostructures, and friction [1]. O ther
applicationsrequire 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 location 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 section II, som e background on A F M isgiven. Section III,
elaborateson the m otivation for this work. M odels for
probe-sam ple interactions, scanner lateral and longitudinaldynam ics, scannercreep, and cantileverdynam icsare
presented in section IV .Sourcesofnoise and disturbances
are also discussed. B oth experim entaland sim ulation re-
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 piezoelectric 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 interactions. 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 cantileverde°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 scanner.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 calibration steps. T he A F M w as run under PI control. A
Silicon N itride cantilever w as used w ith a resonant frequency of 13 kH z and sti®ness of 0:
2 N =m . Scan resultsdem onstrate the high sensitivity of collected im ages
to scan and controllerparam eters(K p and K i).C om paring F ig. 2 (a) (72 ¹m =s;K p = K i = 2) to F ig. 2 (b)
(96 ¹m =s;K p = K i = 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-
R
2a
2c
F ig.4. Schem atic ofprobe-sam ple contact.
F ig.2. A F M im ages:(a)72¹m =s;K p = K i = 2, (b)96¹m =s;K p =
K i = 20.
tion param eter¸, wasde¯ned as,
9R
¸ = ¾o(
2¼wE
F ig.3. A F M im ages:180¹m =s, (a) nom inal contact force, (b)
sm allercontactforce.
1
¤2
)3
(1)
w here, ¾o isthe theoreticalstrength ofthe adhesion junction, 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 edas
the ratio ofelastic deform ation to the e®ective range over
w hich surface forcesact.From (1), itfollow sthatlargevaluesfor¸ 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 F con, contact
deform ation ±, and contact radiusa.
T hem odeliscom posedofthreenonlinearalgebraicequation that can be expressed in non-dim ensionalform as,
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
closedloopbandwidth.T he highergainsim prove tracking,
asthe sharp left edge of the step isresolved m ore accurately. F igures3 (a) and (b) w ere generated w ith a scan
speed of 180¹m =s using the sam e controllergains. T he
contactforce set-pointforF ig.3 (a)isset to the m anufacturer'srecom m endedvalue, w hile F ig.3(b)a sm allerforce
wasused.C hoosing a sm allcontactforce set-pointreduces
4¸a p 2
± = a2¡
m ¡1
contact deform ation and friction, however, it reducessta3 p
bility ofthe contact. A sseen from F ig. 3 (b), the im age
F con = a3 ¡ ¸a2[ m 2 ¡ 1+ m 2sec¡1(m )]
(2)
generated w ith a sm allcontact force haserroneousheight
2
p
inform ation, due to lossofcontact between the probe and
¸a
1 =
[(m 2¡ 2)sec¡1(m )+ m 2¡ 1]+
the sam ple.
2
Ithasbeen show n thatscan andcontrolparam etersdra4¸2a p 2
[ m ¡ 1sec¡1(m )¡ m + 1]
m atically im pact A F M dynam ics. A sa result, im age ar3
tifact m ay result because of poordynam ics. In orderto
c
elim inate these typesof artifacts, we need to understand w here, m = a asin F ig.4, cisthe radiusoverw hich surthe dynam ic interactionsbetw een di®erent com ponentsof faceforcearepresent.T heuseofsuch continuum m odelsto
describe nano-contactshasbeen supportedby m any experthe A F M .T hisisthe objective ofthiswork.
im ents.T he levelat w hich continuum m odelsbreak-dow n
isnot
allclear[7].
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 lateraldynam 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.
2 In-contact M odel:L ateralForces
A sthe probe isdraggedagainstthe sam ple w hile in contact, a frictionalshearforce develops. A t nano-levelcontacts, experim entshave revealedthe dependence offriction
force on contact area, [1].Forourpurposes, w e are interestedin sim ulating the e®ectofsliding friction force on the
cantileverdynam icsduring scanning.A sa ¯rst orderapproxim ation, wew illassum ethattheinstantaneousfriction
A .Probe-sam ple Interaction
force isdirectly proportionalto the instantaneouscontact
area (»a2).T hism odeldoesnot considerany explicitdeA .1 In-contact M odel:V erticalF orces
pendence offriction on scanning speed.A lthough contact
T hiscontact m echanicsm odel [6], issuitable forA F M
m odelsw ere originally developed forstatic loading, it has
operation in airorotherm edia w here adhesion/capillary been show n in [1], that it holdsundersliding conditions
forcesare dom inant. It describesthe adhesion/capillary w ith not very high sliding speeds. W hen the probe and
contact oftwo elastic spheres. A non-dim ensionaltransi- sam ple are out ofcontact the friction force isset to zero.
2
Vy+
Y
Ri
Vx-
δy
y
Oo Oi
C
1
θδ
x δx
θ
X
Ro
Vx+
negligible e®ectsof rotary inertia and sheardeform ation.
B elow , eqn.(3), givesthe transferfunction betw een the inputvoltagesandthe 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, µpx and µpy, respectively, are
given in eqn.
'
s(6)and (7).
Vz
Vy-
F ig.5. C rosssection ofan eccentric piezoelectric tube.
¤
i
X
°ix+ V x+ + °ix¡ V x¡ + °iy+ V y+ + °iy¡ V y¡ + °izV z
uj =
2
s2+ 2³iuj!iujs+ !i
uj
i= 1
(3)
xp =
yp =
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 aalsforcesbetween two spheresare assum ed to be the
dom inant interaction.
µpx =
µpy =
cos(µ±)u1 ¡ sin(µ±)u2
sin(µ±)u1 + cos(µ±)u2
@ u2
@ u1
+ cos(µ±)
sin(µ±)
@z
@z
@ u1
@ u2
cos(µ±)
¡ sin(µ±)
@z
@z
(4)
(5)
(6)
(7)
A .4 PointofC ontact
C .ScannerL ongitudinalD ynam ics
A t the lim it oflossof contact, the contact radiusa !
T he detailsofthe m odelare given in [11].T he m ain as0. Substituting this into eqn. (2), gives the force and
sum ptionsare sim ilarto those ofsection IV -B .T he transseparation atthatlim it.T hisisused to im pose continuity
fer function between input voltages and displacem ent of
on the force between m odelsofsectionsIV -A .
1andIV -A .
3
the tube'
sfree-end zp, forj¤ num berofm odesisgiven as,
.
¤
j
X
°jx+ V x+ + °jx¡ V x¡ + °jy+ V y+ + °jy¡ V y¡ + °jzV z
zp =
2
s2+ 2³jzp !jzp s+ !j
zp
T here are two m ain designsforA F M .In one design, the
j= 1
(8)
cantileveris¯xed and a sam ple isplaced on the scanner
w hich m ovesitrelative to the cantilever.T hisdesign, generally, lim itsthe m axim um size and weight ofthe sam ple. D .ScannerC reep
In the second design, F ig.1, the cantileverisattached to
T heresponseofa piezoelectricactuatorto a rapidchange
the scannerthat m ovesit relative to a stationary sam ple. in input voltage, F ig.6, consistsoftwo m ain parts.T he
In thispaperthe m odelpresented isforthe second, m ore initial part of the response occursovera tim e scale dictated by the m echanical resonance of the actuator. T his
populardesign.
s
T he scannerisa thin-walled piezoelectric tube that has isfollow ed by a slow creeping response occuring over100
fourelectrodesofequalsegm entson itsoutersurface, and ofsecondsand am ounting to asm uch as20% ofthe total
eithera singleorfourelectrodeon itsinnersurface.A pply- response. T he rate and am ount ofcreep strongly depend
ing a voltage V z, to itsinnerelectrode(s)resultsin exten- on the piezoelectric m aterial.E xperim entalfrequency resion m otion, (in the Z-axis).M otion in the X orY axisis sponse ofpiezoelectric actuatorsdisplaysvery little variatypically generated by subjecting tw o opposite electrodes tion in phase at low frequency between input voltage and
to two voltage signals, (V x+ ;V x¡ )and (V y+ ;V y¡ ), w ith the displacem ent.O n the otherhand, a slight decrease in gain
isobservedw ith increasing frequency.Itispossible to sim sam e m agnitude but are out ofphase.
In [8], [9], a m odelforan idealuncoupled tube scanner ulate creep behaviorusing a suitable L T I m odel.T he relwaspresented.D ue to inevitable m achining tolerance, ec- ative degree isnum berofpolesm inusthe num berofzeros
centricity isalwayspresent in the tube.T ypically, a m ax- of the transfer function. T he transfer function between
im um of 50¹m fora 12:
7m m diam etertube, [10]. T his the input voltage and actuatordisplacem ent should have
seem ingly sm all eccentricity is in fact signi¯cant as the a relative degree zero at frequenciesm uch low erthan the
sresonance frequency.T hism odelassum esthat
cantilever-de°ection sensorhasA ngstrom rm sresolution. actuator'
T he m odelisbasedon two eccentric cylinders, F ig.5, w ith the ratio between the am ount ofcreep and the fast scaneccentricity ±x and±y from thegeom etriccenteroftheouter nerdisplacem ent isindependent ofinput am plitude.T he
cylinder, O o.T he tube is¯xed at one end and free at the assum ption w illbe experim entally tested.
other. A cantileverholderofm assm sh isattached to its
free end.T he m odelisbased on elem entary bending the- E .C antileverF lexuralD ynam ics
ory forthin-walled m em bers. T he m ain assum ptionsare
T hisdynam icm odelforthecantileverm otion neglectsefsm alldeform ationsand angles, linearelastic m aterial, and fectsofsheardeform ation and rotary inertia, and isbased
B .ScannerL ateralD ynam ics
*
Cantilever Displacement, [nm
]
450
Force
400
Penetration
Region
Out-of-Contact
Region
350
300
250
200
δ
150
Probe-sample
Separation
-kc N/m
100
50
-kc N/m
Pull-off point
Pull-in point
0
0
10
20
30
40
50
60
Time, [s]
F ig.6. E xperim entalcreep response.
F ig.7. Sim ulation:Q uasi-static norm alized force-separation curve.
15
10
*
Force, [nN ]
on elem entary bending theory. T he cantileverslope (angle), zc0
, ism easuredrelative 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
a2zm s2+ a1zm s
zp(s)
2
s + 2³cm !cm s+ !c2m
m=1
a2µm s2 + a1µm s
µpy(s)
2
2
s + 2³cm !cm s+ !c
m
afm
+ 2
f(zc;zp;µpy;zs) (9)
2
s + 2³cm !cm s+ !c
m
5
0
Pull-in Point
+
w here, f(zc;zp;µpy;zs)isthe probe-sam ple force, and zs
isthe sam ple height.
Pull-off Point
-5
0
50
100
150
200
250
300
350
400
450
500
Probe-sample Separation, [nm*]
F ig.8. E xperim entalforce-displacem entcurve.
F .SensorO utput
Feedback m easurem ent noise arising from the optical
T he optical-leversensorm easuresthe absolute angle of sensorcan be due to shot noise, a fundam ental noise for
the cantilever'
sfree-end.T herefore, the PSD outputyPS D , these sensors, in addition to noise from sensorelectronics.
Shot noise can also be m odeled asw hite noise.
isgiven by,
yPS D
=
µpy ¡ zc0
(10)
V .R esults
G .D isturbances
A .Q uasi-static Force-separation C urve
T herm alnoise orB row nian m otion contributesto a fundam entalsource ofnoise in A F M .A t therm alequilibrium ,
the m ean value of the cantileverpotential energy hasto
38£ 10¡23 J =K isB oltzm ann'
s
equal 1
2kBT , w here kB = 1:
constant, and T is the absolute tem perature in K elvin.
T he cantilever'
sfree-end
q w illoscillate w ith a R M S value,
rm s
3 rm s
3
0
BT
zc
= 2L czc = 2L c kk
, w here kc isthe cantilever
c
sti®ness, and L c isthe 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 nothersource 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 disturbanceswith a com bined constant intensity µ±(t¡ ¿).
T he m odelsofsectionsIV -A .
1, IV -A .
3w ere usedto generate the com posite force-separation curve ofF ig.7.Param 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 hisinstability occursw hen an approaching/receding probe jum ps
in/out (pull-in/pull-out 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 calibration 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.
10
-5
Magnitude
k /k = 1
10
10
s
-6
s
10
z´c
k /k = 4
2
Phase, [degree]
200
ks
k /k = 7
c
s
c
3
10
Frequency, [Hz]
100
0
-100
zs
-200
2
10
F ig.9. Schem atic ofprobe-sam ple contact.
c
-8
10
kc
s
c
-7
s
zp
k /k = 2
k /k = 1.5
c
3
10
Frequency, [Hz]
F ig.10. Sim ulation:In-contactfrequency responsefordi®erentratios
ks
.
ofsam ple to cantileversti®ness k
c
4
3
2500
2000
2
1500
1000
1
500
Imaginary Axis
T he probe-sam ple interaction force f(zc;zp;µpy;zs) is
a nonlinearfunction of probe-sam ple separation, and dependson 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,
Imaginary Axis
B .In-contactD ynam ics:Sim ulations
x 10
0
-1
f(zc;zp;µpy;zs) =
gczc + gzzp + gµpyµpy
+kszs + H :
O :
T
0
-500
-1000
-1500
(11)
ks can be considered asa lineare®ective probe-sam ple
contact sti®ness.T he probe-sam ple contact can be represented schem atically asin F ig.9, w here again zc, ism easured 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 equation into eqn. (10) givesthe overallm odel. T he transfer
function ofinterestisbetw een V z andyPS 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 forthe cantilever. T he param eter values
used are given in [11]. T he ratio of sam ple to cantilever
s
sti®nessk
ovedto bean im portantparam eter.C hanges
kc, pr
in thisratio have two m ain e®ectson the m odel, nam ely,
change in thetransferfunction D C gain andchangesin the
frequency ofthe zerosassociatedw ith the scannerbending
m odes, 380H z and2kH z.F igure 10, show sthe sim ulated
frequency response ofthe m odelfordi®erentratiosofsti®s
nesses. F orlarge ratios(e.
g. k
kc = 7), the zeroshave a
higherfrequency than the m ode. F orsm allerratios(e.
g.
s
1· k
·
2
)
,
t
he
f
r
equency
decr
eas
est
o
be
bel
ow
t
hat
of
kc
the m ode.T hischange in pole-zero pattern isreferred to
s
aspole-zero °ipping.M oreover, forsom e value (k
kc ¼ 4),
there ispole-zero cancelation and the bending m odesare
unobservable. F igure 11, presentsa pole-zero m ap ofthe
s
¯rst two m odesfordi®erent valuesof k
ult, as
kc . A sa res
the zerosm ove away from the m ode, the resonance peak
appearsm ore prom inent in the response. F urther, w hen
ks
eachesa
kc iseithertoo large ortoo sm all, the D C gain r
-2
-3
-1400 -1200 -1000 -800 -600 -400 -200
Real Axis
-2000
-2500
0
-200
-195
-190
Real Axis
-185
-180
s
F ig.11. Pole-zero m ap asa function of k
:(left) zoom on 2kH z
kc
m ode, (right)zoom on 380H z m ode.
lim it controlled by kc and ks, respectively.Forinterm ediate values, the D C gain w illdependon both sti®nessesand
changesin ks due to di®erentset-pintsorinputam plitudes
w illchange the D C gain, and zeroslocation.
C .In-contactD ynam ics:E xperim ents
To 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 antileverA hasan estim ated sti®nessof 0:
03N =m and resonance frequency of 13kH z. C antilever B has an estim ated sti®ness of 0:
14N =m and resonance frequency of
14kH z. D i®erent set-points and input am plitudeswere
usedto study the e®ectofthese param eterson the dynam ics.
F igure 12, show s the force-displacem ent curve forthe
G lass sam ple. T he points labeled on the plot are the
force set-points used in the frequency response experim ents.N ote thatthelocalsensitivity aroundtheset-points
(cantileverde°ection perscannerinput voltage)issm aller
forthe largerforce set-point. T hissuggestsreduced contact sti®nesspotentially due to plastic deform ation in the
contact.T he e®ectsofforce set-pointon the dynam icsare
seen in F ig. 13. Forthe largerset-point, 17nN , the D C
20
0
Force Set-points
15
10
Vz = 300 mV
Vz = 500 mV
DC Gain = 0.23
DC Gain = 0.18
-1
10
2
3
10
10
Frequency, [Hz]
5
Phase, [degree]
Force, [nN*]
Magnitude
10
0
-100
0
-200
-5
-0.05
0.25
0.55
0.85
1.15
2
1.45
Scanner Displacement, [µm*]
F ig.12. F orce-displacem ent curve using cantileverA with a G lass
sam ple.
0
200
Magnitude
14 nN
DC Gain = 0.23
180
160
-1
10
17.6 nN
DC Gain = 0.12
140
Force Set-point
120
*
Force, [nN ]
-2
2
10
Phase, [degree]
10
Frequency, [Hz]
F ig.14. In-contact frequency response using cantilever A with a
G lasssam ple:14nN fordi®erentam plitudes.
10
10
3
10
3
10
Frequency, [Hz]
0
100
80
60
40
-100
20
0
-200
2
10
3
10
Frequency, [Hz]
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-pairassociated 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 dissipative propertiesofthe contactw ith changesin the set-point.
Itisim portantto realize thatthe bending m ode resonance
frequency did notchange.T he contact forcesare ordersof
m agnitude sm allerthan the force the scannercan provide,
(»100
snN vs.»1N ).
T he e®ect ofexcitation am plitude, i.
e.sam ple topography, on the frequency response isshow n in F ig. 14, for
set-point of 14nN . It isseen that the largerthe am plitude 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 displacem 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 changein contactsti®ness, andhence D C gain.
T he resonance peak also changes due to changes in the
-20
-1.2
-0.8
-0.4
0
0.4
0.8
*
Displacement, [µm ]
1.2
1.6
2
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 forthe
PD M S sam ple.T he penetration region seem squite nonlinear.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 ofbendingm ode zeros, and consequently increasing resonance peak.
T hissuggeststhatthecontactsti®nessincreasesw ith larger
am plitude, w hich agrees w ith F ig. 15. In addition, increasing 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 odelpredications in section V -B , (F ig. 10). H ence, contact and
cantileversti®nessvaluesare relatively close.
T he m odelcan be furtherim provedon to include nonlinearitiesin 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;µp_
y;zs).
90
0
Magnitude
10
4 nm
17 nm
Actual cantilever trace
80
70
-1
10
60
42 nm
10
2
10
3
10
Frequency, [Hz]
200
Height, [nm]
85 nm
-2
50
Averaged Image
40
30
20
0
10
-100
0
Phase, [degree]
100
-200 2
10
-10
3
0
10
Frequency, [Hz]
F ig.16. In-contact frequency response using cantilever B with a
PD M S sam ple:36nN fordi®erentam plitudes.
Sample
0.5
1
1.5
Lateral Displacement, [µm]
F ig.18. Scanning sim ulation resultsofcalibration stepsusing a PI
controller.
120
0
10
Magnitude
36 nN
100
-1
10
80
10
2
10
3
10
Frequency, [Hz]
Phase, [degree]
100
Height [nm]
113 nN
-2
60
40
0
20
-100
0
-200 2
10
3
10
Frequency, [Hz]
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, V z, 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 ulationspredict the actual response w ell. A lso note the
oscillationsobserved in F ig. 19. T hese oscillationsintroducesan artifact that could be interpreted incorrectly as
surface roughness.
E .Perform ance L im itations D ue to Scanner B ending
M ode
-20
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Position [µm]
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 resonance at 380H z wouldbe wellw ithin the feedback bandw idth.Tuning the controllergainsto achieve a high bandw idth 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 odelessobservable 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.Further, 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.
T he coupling between scanner bending and extension
m odeshasa greatim 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 standardofknow n height usually in the 100nm range.D uring
F .C reep C om pensation
im aging, the PSD signal w ill change due to the sam ple
A 3rdorderL T I ¯lterwasused to com pensate forcreep
topography aswellasto bending ofthe actuator.Im agining a sam ple ofa di®erent height w illresult in a slightly in the A F M piezoelectric scanner, [12]. T he perform ance
1200
Displacement, [µm]
0.56
1.68
1.12
2.24
the m odels. T he results w ithin w ill be used to develop
algorithm sthat allow autom ated choice ofkey system param eters, guaranteeing reliable and artifact-free data for
any given operating condition (sam ple, cantilever, environm ent).C onsequently, expanding the A F M capabilitiesand
perm itting itsuse in a w iderrange ofapplications.
Height, [nm]
1000
380 Hz Scanner
Bending Mode
800
600
600
400
550
200
500
R eferences
0.34
0
0
0.2
0.4
0.36
0.6
0.38
0.8
1
Time, [s]
F ig.20. O scillationsdue to scannerbending m ode in experim ental
A F M im age ofa 1046 nm Silicon step.
Displacement, [µm]
700
1.4
Height, [nm]
600
4.2
2.8
5.6
Compensated
Uncompensated
500
400
300
200
100
0
0
0.5
1
1.5
2
Time, [s]
F ig.21. A F M im ageof530nm Silicon steps, with and withoutcreep
com pensation, 2:
8¹m =s.
ofthe ¯lterwastested, 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
sensorsata 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 ponentsneed to be wellunderstood and controlled. In this
paperwe presented a sum m ary ofourwork in thisdirection.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 resultsforboth frequency response and im aging
were presented.E xperim entalresultswere given supporting the sim ulationsand dem onstrating the com petence of
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