CC BY
13Journal of Siberian Federal University. Engineering & Technologies, 2018, 11(6), 645-658
y^K 53.086
Dynamic Modeling of Multimode Resonance Measuring Mode in Atomic-Force Microscopy with Piezoresistive, Self-Actuating Cantilevers
Pavel S. Marinushkin*a, Alexey A. Levitskiya, Tzvetan Ivanovb and Ivo W. Rangelowb
aSiberian Federal University 79 Svobodny, Krasnoyarsk, 660041, Russia hTechnische Universität Ilmena Institut für Mikro- und Nanoelektronik 1 Gustav-Kirchhoff Str., Ilmenau, 98693, Germany
Received 14.06.2018, received in revised form 19.06.2018, accepted 19.07.2018
The development of fast, qualitative and quantitative material characterization methods is one of the most important current issues in the field of nanosystems metrology. On this evidence it seems to he important to conduct a research on the capabilities of multimode resonance imaging mode in atomic-force microscopy (AFM) that allows broadening AFM capabilities in quality of nanonscale structures metrology and nano-object image quantitative analysis. The subject of this paper is modeling ofphysical phenomena that arise during the creation of such systems that describes coherent mechanic and electric phenomena in self-sensing and self-actuating cantilevers operating in multi-frequency resonance mode. The outcome of the research is represented by a virtual dynamic AFM model that allows understanding the signal generation process in AFM control and measuring circuits during sample scanning in multi-frequency mode.
Keywords: Atomic Force Microscope (AFM), Nano- and Microelectromechanical Systems (NEMS/ MEMS), nanometrology, self-actuating and self-sensing cantilever, thermomechanical actuation.
Citation: Marinushkin P.S., Levitskiy A.A., Ivanov T., Rangelow I.W. Dynamic modeling of multimode resonance measuring mode in atomic-force microscopy with piezoresistive, self-actuating cantilevers, J. Sib. Fed. Univ. Eng. technol., 2018, 11(6), 645-658. DOI: 10.17516/1999-494X-0082.
© Siberian Federal University. All rights reserved
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License (CC BY-NC 4.0). Corresponding author E-mail address: [email protected]
Динамическое моделирование многомодового резонансного режима измерений в атомно-силовой микроскопии с пьезорезистивными активными кантилеверами
П.С. Маринушкин", А.А. Левицкий", Т. Иванов6, И.В. Рангелов6
аСибирский федеральный университет Россия, 660041, Красноярск, пр. Свободный, 79 бТехнический университет Ильменау
Институт микро- и наноэлектроники Германия, 98693, Ильменау, Густав Кирхгоф штрассе, 1
Проблема разработки высокоскоростных методов качественной и количественной характеризации материалов является одной из важнейших задач метрологии наносистем. В связи с этим представляет интерес исследование возможностей многомодового резонансного режима атомно-силовой микроскопии (АСМ), позволяющего расширить возможности АСМ в направлении повышения качества и достоверности количественного анализа изображений наноразмерных структур и нанообъектов. Предметом данной работы является рассмотрение вопросов создания динамической модели, описывающей взаимосвязанные механические и электрические процессы в саморегистрирующих активных кантилеверах, функционирующих в многочастотном резонансном режиме. Итогом работы является компьютерная динамическая модель АСМ, позволяющая исследовать процессы формирования сигналов, в управляющем и измерительных контурах АСМ при сканировании образцов в многочастотном режиме.
Ключевые слова: атомно-силовой микроскоп (АСМ), нано- и микроэлектромеханические системы (НЭМС/МЭМС), нанометрология, саморегистрирующий активный кантилевер, термомеханический привод.
Introduction
Since the AFM development thirty years ago [1-3], it became wide known as a high-performance tool for surface topography investigation of wide range of samples. Nevertheless the search for the optimum methods, which would describe samples properties and structure in the most complete way, drives the further development of AFM's. Methods for better surface visualization (sensitivity and resolution), scanning speed, and an ability to provide quantitative analysis of nano-mechanical properties and expanding the AFM applicability fields.
The most widely used method now is amplitude modulation method where cantilever resonant oscillations are generated at one of its own frequencies (e.g. at the first resonant frequency), while surface visualization is operated at a signal stipulated by oscillations amplitude attenuation as a result of interaction between a sample and a cantilever tip [4]. Nowadays the multimode mode is one of the most prospective ways to broaden the operating modes range, which opens new opportunities for AFM application [5-8].
The wide used method to measure the deflection of the cantilever customs a laser beam reflected from the surface of the cantilever onto a position-photodetector. When the cantilever is bended, the
reflected laser light arise the cantilever at a different angle and provides in a measurable vertical shift of the laser spot on the detector. This method is called as beam detection.
The paper by Viani demonstrated the use of small cantilevers showing a high sensitivity and was used to unfold single molecules using imaging speeds an order of magnitude faster than previously performed with conventional cantilevers [9]. However, the use of small cantilever remains a challenging task using classical optical read-out techniques.
At the same time, during the surface scanning process tip deflection signal is also detected at excitation frequency. The drawback of this approach is the loss of additional useful information about sample that is contained in the deflection signal at frequencies that differ from excitation frequencies. This problem is overcome in the AFM multimode mode that implies cantilever oscillations generation and\or response measurement at one or more frequencies. The received additional information, i.e. amplitude, phase and\or frequency response, may be used for both surface visualization with high resolution [10-12] and simultaneously extraction of additional information about the sample's nano-mechanical properties simultaneously [13, 14].
At the same time, the implementation of AFM multimode methods requires further theoretical discussion, as long as force interaction between the cantilever tip and the sample's surface in multimode mode is complex. As a result, experimental capabilities of AFM multimode mode now requires its theoretical understanding. In particular, the phase images, obtained with this mode, show clearly the detailed structure of the sample, however their physical interpretation is still unclear which obstructs a quantitative analysis of sample properties. Therefore, there is an actual need of research of measured response interpretation features in AFM multimode mode with considering its features and development of model-algorithmic support for near-surface structure properties measurement of materials and thin film in the micro- and nano-range.
1. The features of AFM multimode mode hardware implementation
The main measuring ingredient of atomic-force microscope is cantilever - a tiny force sensor in a form of cantilevered beam, the free end of which has a tip with sharpness at nano-range on it (Fig. 1).
The self-sensing and self-actuating cantilever allows for much easier system integration and significant reduction in its weight. Hence, the microscope provides better controllability for full metrological automation and significant higher scan speeds.
The fabrication and basic characteristics of thermo-mechanically driven cantilevers with integrated resistive readout have been described in detail previously [15-17]. In brief: we use Si cantilevers 300 ^m long, 110 ^m wide, and 3-4 ^m thick. The cantilevers have a piezoresistive Wheatstone-bridge positioned on the base of the cantilever and thermomechanical actuator located at near of the tip.
Probe tip vertical displacement is operated by thermomechanical actuator in form as a resistive heater on cantilever surface. The bridge measuring circuit, located on cantilever surface as well, is used for measuring the displacement.
The capability to record additional responses in amplitude, phase and\or oscillation frequencies is stipulated by complementing a widely used AFM single-frequency circuit by additional generators and synchronous detectors. The described above structural scheme is shown in Fig. 2.
In the presented scheme oscillation generators are used to excite several (in this case, the first three bending) oscillation forms of cantilever by an embedded thermomechanical actuator. A piezo-scanner provides cantilever's displacement relative to the sample along X, Y and Z axis. As the cantilever tip approaches to the sample's surface, the force interaction arises between them, which lead
Fig. 2. The structural scheme ofa multimode AFM. Here/ (7 = 1, 2, 3 )are excitation frequencies;^, c^are cantilever oscillation amplitudes measuredFtexeitatiFnqreqnencisfO
to cantilever amplitude attenuation. When the surface profile of the sample changes, the force acting on the probe become different as well. Thus, there is amplitude modulation of cantilever oscillations by force acting on the probe from the surface. The cantilever mechanical response arising at this point is recorded. After that the acquired signal is amplified and sent to synchronous detectors inputs, reference signals of which are generator output voltages. This is the way to distinguish the envelopes of acquired signals. Then synchronous detectors output voltages are sent to the controller, after that they may be used for feedback while scanning the surface and may be displayed and recorded. Thus, regardless of the main information (topographic) channel cantilever oscillations may be controlled at higher natural frequencies, which allow researching a wider range of tip-sample interactions. Due to this, while scanning there is an opportunity to create a distribution map of other sample surface local properties, apart from topographic ones. The feedback is realized by complementing the scheme with a proportional-integrating link that forms a control signal further amplified and sent to Z-electrode of the scanner. At the output proportional-integrating link the signal is proportional to the sample surface topography height change. Based on this, the image of sample surface characteristics is further formed.
The opportunity of retrieving useful information from the additional data involves the necessity of researching mechanisms of informative signals conditioning during the sample surface scanning as well as it involves the development of reliable and exact amplitude, phase and frequency response interpretations. One of the most efficient solutions for this type of problem is simulation modeling. Unlike other methods, e.g. analytic, simulation modelling is able to describe functioning of the system almost without limitations in terms of detailing. Matlab Simulink has been used for implementation of AFM multi-frequency resonance mode simulation model as it was the most appropriate solution in this ease;.
The further questionsaf creattnghmathematical dascriptian for multi-frequency AFM converters and electronic components in the framework of solving the problem of its simulation model creation will be carried by the example of AFM with active cantilever produced by the Nanoanalytik GmbH Company [°8].
2. Cantilever dynamic model
The sim^eet modf[ deitaMn- caafikvac dinplecemeni (while seonning icone-di mensional model) ccncedaascantieevteasarenonatorwith lumped parameters. In case ot multi-frequency cantilever eucilatian tluimodei ecsaltyinasoste mcd/tOe-ee rantiaieuuaeior^aseollows:
AiO+M)+A z (()=As-t^-9l)+FT.
dt m1 dt m1 m1 m1
d2 z, (() c, dz., (t) k / \ F ■ t \ F.?
+ ~2^ + z2 (t ) = -^sin(co2t-cp2 ) + dt m2 dt m2 m2 m2
d2z (() c dz (() k t\ F . , x F's -sll + --nXl + ^l.z (() = -tt-s-t(cd t-p ) + —
dt mt dt mt mt mt where zt is the vertical cantilever tip displacement at the i-th oscillation mode; mi is the cantilever effective mass at the i-th oscillation mode; ki is the cantilever stiffness at the i-th oscillation mode;
(1)
to,- is the natural frequency at the i-th oscillation mode ( ra , = Jk~Jmi ); F, is the amplitude of the i-th excitatory force; cp, is the initial phase of the i-th oseiïlation; F* tefàe m^b^i^i^c^^^n strength between chntilever tip and the samhle's ^rfaon nt i-Ht oscillationmoУe; c^^et tiiedampinn coefficient of i-th oscillation mode:
ci = 2m =■
e,
where Z is the relative damping coefficient; Qt is the quality coefficient of the i-th oscillation mode.
With the aim to transform model (1) to the form that would be more convenient for the structural modeling, it is considered rational to describe it as a state space. A dynamic object model in the state space is presented as an aggregate of physical variables q^t), ..., qn(t) that determine object's behavior in the further moments of time, on condition that the object's state at the first moment of time and all the applied impacts are known. The connection between input variables u1(t), ..., un(t), output variables Pi(t), ..., pn(t) and state variables q1(t), ..., qn(t) is represented by the first-order differential equations writteninmatrixform.
The followingvariablesareintroducedas thementioned above state parameters:
4](t ) = M ) q(2)(t) =
m, a = dP((t ) = dq(()
dt
dt
qf(t ) = Pn (t )
tn (t) =
dPn (t) s çdt(_ dt dt
(2)
Substituting(2)into(1)results inthefollowing system of equations m state variables:
q = Aq + Bu =
A(
0
h +
B,
0
B n
(3)
where q is the state vector, u is? thu visitor nf nnpul effects, A (Aî>=(î)is )he state matrih,B (w^r)
is fhe sontcol tinput( m^rix.
h]
1i
(2)
t(()
A, =
0
.A
m.
(
,_r=
m„f
b(
0 0 ( (
Forcompfetedescrphioo ef dynamdc model the otate etiuhtkin'p dynamic model has to hccomplrmeitted (3r sei?, itt.ualiIss^s mrhrne. up a sonocrtion botooern the state variablvs
tfU™,
, iK , ttO anP itiPe^ tptpetvariables pb..., pn:
p = Cq + Du,
where p istheoutputvector, C (wxn)istheoutputmatrix, D (mw xr)istheoutputcontrolmatrix.
mi"> i
0
Pi rci 0 " "0 0 0 0 0 0"
p = , C = , C,. =[1 0], D = 0 0 0 0 0 0
Pn 0 C„ 0 0 0 0 0 0
The cantilever frequency characteristics based on the resulting model in accordance with the Matlab Simulinkparameters given inTable1,are presentedin Fig. 3.
Table 1. Cantilever dynamic model parameters
Parameter Value
Effective mass m1, kg 2.81210-10
Effective mass m2, kg 8.072 10-11
Effective mass m3, kg 2.564 10-11
Stiffness ¿1,N/m 53.644
Stiffness ¿2,N/m 652.913
Stiffness ¿3,N/m 1541.808
Parameter Value
Mechanical quality Qx 134
Mechanical quality Q2 253
Mechanical quality Q3 286
Resonance frequency f1,Hz 69509
Resonance frequency f2, Hz 4.527-105
Resonance frequency f3, Hz 1.235 106
1,0E+05 1,0E+06 1,0E+07
Frequency, rad/sec
1,0E+08
-45
-90
-135
-180
l,0E+05
1,0E+06 1,0E+07
Frequency, rad/sec
b
1,0E+08
Fig. 3. Amplitude-frequency)a)and phase -fre qhency(b)cantileverchaeacteristicr
3.Thermomechanical actu atoe motel
Thermomechanical acluator is by deiign o resistive heateron ^lin uarntilevea surface of mass me3ndspeciTcheatu,, Ihelinitiany Tirs reslstencniO. es ambieottemperatuae 70. Wheu thoelaalric cutranU i flows through the conlnctor with rnsistanue OH^ thie power il P. Tint temperaturn of conductor rises by AT. Power that allocates at conductor's resistance, depending on overheating temperatures (relatively to the initial ambient temperaUure), is determined by the following expression:
P(t ) =i i2Rht (l + pA)(=)),
where Ai(() = r(()-ro is the overheat relatively to the initial temperature, p is the temperature coefficient of resistance.
Then the amount of heat accumulated in the conductor is:
- 651 -
&(i)=jAP(/)rf/ .
0
Thedefining e quationfor thermomechanical actuatoristhedifferential equation of heat balance:
c^^QXO-aS (T-Tc), at
cvhere T Js hhe heater temperature, t is the time, a is the reduced heat transfer coefficient, S is the surfacearea ef the heater, Tc iotHaeamlrient tempbrature.
Due Ito ahp defferenee between tle^rmal pxpansionhpeflfpients ofcantilever materials (Si - the baae,A1- metaieizetiant rhe hgat prtducep hy tltt hpater cauptsmechanipat sitresses in cantilever and, as a result, bending. The displacement d of the tip along the axis Z may be calculated by [19]:
2 '
where L i s the centilemer length, pisthecurvature o. tltie cantilever's ^^nt axis [19]:
/2 = - =
1 _ 6bab
Al ESi "A/s^AI (tsi + t^)(aSi
+ (bAlEAltAl ) + 2bsib Al ESi EAltSitAl(2tSi + 3tSi^Al + 2tAl)
where r is the curvature radius, bSi and bAl are the width of silicon and aluminum layers, respectively; aSi h aAl are the coefficients of silicon and aluminum thermal expansion, respectively; ESi h Eai are the ahismnumelasticitymoduhis. Thenequivalent forcedevelope d by the actuator i s:
,, 3EId a »-
where /is ihnmomcnt of section inertia, EI is the equivalent rigidity defined by the following expression [20]:
EI =
ESibSitAl(tSi + ^Al)
4. Model of power interaction between cantilever tip and sample
It is possible to study a sample only due to the variety of powers appearing between the cantilever tip and the sample while surface scanning. Depending on the probe-sample distance different forces may pttran.
For instance, inthe attrcctionmode (Up moving awar fsom the samcPe)She prevcelinptype ef interactirir ls Vaa dea Wnats feece of ^s^ttrn^t^kdhlar intnractirn. In ttra sepulrion mc^ (rip approaching to thn snmnte) elastic and inelsstie intesactions fiitlt rluc sampk psevait. The internetians nre cnlculnted from the Derjagin-Muller-Toropov model [21]:
F (h) =
HR,ir,
--f- in attraction mode (h > a0 )
6h = 0
--=£- + -£* (a°- h)= in repulsion mode,
6a0
where m istheHamaker constant,^ ^tteradmsof cantilevertip ra n1 a^t^i^re^, h = zs + Az is the distance between ths cantiltvar tiptnd thhscm-desurfatc (tm he hliii sentilsver detteettonvhCuc, zs is the distance between the un-benheh cnnheeverandthe eamptef h( is the intermnlecuiam (tnteratomic) distance, E* ea ths elfectivamoduluuof etrrtitityoS the nrobe-samhlesystem:
i _ (i +(i -v,)2
E E, E
where Et and Es are the modulus of tip and sample materials elasticity, respectively, vt and vs are Poisson's ratios of thetipandthesamplematerials,respectively.
H = 0.425e-18; % Hamaker constant R = 20e-9; % Tip radius aO = O.Ss-9; % Inter-atomic distance etip = esi;
vtlp = 0.22; |_
E sample = 130E9; vsample = 0.22;
Pass through input 1 when h?a0 on input 2; otherwise, pass through input 3.
Q>
Ir1
f(u)
Non-Contact mode
f(u)
0ut1
Switch
Contact mode
Fig. 4. Structural model of theblock modelingtip-samdle i^^l^e^^ct^c^n
5. Measuring circuit model
Responses that cume sin» during the scanning of the eampie are recorded by e measuring circuit, ambddrdd enta SUc c;rtipit]ile;s',ee-. Thr cantclenee mreruring deems in ibe Nanoanalytik Gob) aompany's oeom1c-tosfa ii:it(ctoscot)t; it formed fa the srrtem of four piezprerisCsns ige sC3erom ene sideondnu tf frocoi(t]^^ olher)wlththere ^r^^^^c^^i^a SgP8 ohms =nch ouo. AH of the piezoresistors are located so that cantilever deformation causes resistance changes, equal in absahca vcdua ach npposiie in rign, cn the acytccrut stoukteer or thr brkCe.Tte ical reference eeldanc ic et0 = no ]et
At Hi£) (t^^eitrC o° tlocsi maacuring ciiechtsit tie measured voltadsrg° ig nto^ojjnettonr^t Ir Phr d^erence of ifo reintideteslstences:
V = V
e out * 0
RV R3
V R1 + R2 R3 + R4 J
wvliiîi-s; V0 is ttie refvrencp ~e*olrii!£he apphed te the measering tiremt.
While ssanniing, the eantikner peoceives she external forcs action i7 fromihe tuefoce^aermgiïs dsflecteon lees'.
n = kAz,
3EI
wears k = —r- is tge sSiifmess.
je3
It is obvious that the maximum cantilever deflection is observed at the loose end (Y = L coordinate):
v ' 3EI
The maximum bending moment for the cantilever under research loaded at the end by the conc entraterforce F appearb at the attacfmentpoi nt (J= 0) and is expressed by:
M = F{L-Y) ^MmrX= FL .
Mechanical stresses fromthegiven load reach the greatest value on its surface Z = ±tJ2 in the section where Mmax acts, i.e. in the place where the cantilever is clamped [22]:
1 I nsmaxz FLb 3Eb
<3 l-max— l-l-7
I max | j 2/ 2L •
At last, mechanical stresses' impact on each of the; piezoresistors included in the measuring circuit with the resistance R causes the resistanae incra oaent AR,-:
¿h^^neA;
where n is the piezoresistive coefficient, the valup and the sign of which depend on the resistor's locationat the cantilever(thelongitudinal piezoresistive coefficient is ji;= 70eall Pa^tle transrerse piezeeerlstivecoafficirnt iso,= aot).
TVe caniiiever mecsurieo circuit modelcreated in conformity with the ratio described above is presented at iig. 5.
Parameters
RO
VO
PiLong PiTrans
H?>-
PiLongitudinal
PiLong ¡ludinall
R4
R2
1—1
R2 - X -
Xl
PfTransversdl
Fig. 5. The cantilever measuringcircuitmodel
f{U)
6. Simiim^^ion model of atomic-force micro scop e
The Seveloped ma(hemaCica(modelsof AFMconvertersandelectronicacsemblies, ims(emented as algoeitTmiebkccks, allowewitchingto a muiti-frequeneyAdMsimuloüon modsl. TTemicrctcope
- 654 —
Fig. 6. The simulation model of the multi-frequency atomic-force microscope
model described by a differential equations system has been implemented by means of Matlab Simulink package as a structural scheme (Fig. 6).
As the basis for the model the parameters matching the Nanoanalytik Gmbh company's atomic-force microscope cantilever were taken: cantilever length L = 350 ^m, cantilever width w = 140 ^m, aluminum layer thickness tAl = 0.7 |im, silicon oxide layer thickness tSiO2 = 0.5 |im, silicon layer thickness tSi = 2 ^m, silicon elasticity modulus ESi = 170 GPa, aluminum elasticity modulus EAl = 70 GPa, coefficient of aluminum linear expansion aAl = 23.3T0"6 °C-1, coefficient of silicon linear expansion aSi = 2.61610-6 °C-1, temperature coefficient of aluminum resistance pAl = 4.310-3 °C-1, heat capacity cAl = 753 J/(kgK), heat capacity cSi = 713 J/(kgK), radius of cantilever tip Rtip = 20 nm, Poisson's ratio of the tip material vtip = 0.22, thermomechanical actuator resistance Rh0 = 24 Ohm.
Examples of exciting impacts and measuring responses obtained by simulation modeling in Simulink are shown in Fig. 7.
Examples of measuring responses to the surface topography in the form of a periodic structure with a trapezoidal profle50 nm high,received after srmulatton mnthe Simutlnk enviroement, are presented in Fig. C.Theenvelope;li of tie emplltudemodurafedrignareireefiryt resomanf rcequency ct uted for feedbackducing the ocapmlngoCtherueface. Second and the third modes of the cantilever provide two more channels of extra information A2, and A3, Values plotted on the graphs along the vertical axes can give an estimate of the magnitude of informative signals. Noticeable in Fig. 8, c artifacts expressed in signals jumps can be attributed to insufficiently accurate setting of the feedback loop parameters. Overall, the results indicate that the model correctly reflects the nature of the relationships between the input impacts and the response signals and corresponds to theoretical concepts.
Conclusion
In this paper we have presented a model describing the operation of self-actuating and self- sensing cantilevers and their mechanical and electrical characteristics. This paper presents also particularly a
Time (seconds) » jo"'
Fig. 7. Examplesofmodeling: a-cantilever oscillation electricexcitingsignal (sate ofsigealseatth frequencies fi, f JO, S - time dependence oeshe forcitde-eioped lee/ the iheemomechanical aeteistor; e - vertical disvlacement of canti lrvectip
Fig. 8. Examples of simulation response results: a, c, e- amplitude responses of A1; A2 and A3, respectively; b, d, f - phaseresponses 4>1; 4>2 and^3, respectively.The sizeoffieldunderresearch is15 x 15 ^m
dynamical model of the multimode AFM that can be applied for modeling measuring responses of the multi-frequency atomic-force microscopy. The model considers the input parameters in the form of surface topography of an arbitrary profile, including the materials with heterogeneity of mechanical properties.
The developed model allows researching and interpreting the measured responses in the multimode mode AFM taking into account its natural features. Also it is of interest as a basis for development of model-algorithmic support for measurements of the surface properties of materials and thin film structures in micro- and nanoscales using self-sensing and self-actuating cantilevers.
Acknowledgments
This work was supported by the German Academic Exchange Service (DAAD) and the Ministry of Education and Science of the Russian Federation under funding programme "Mikhail-Lomonosov-Programme-Linie A, 2016".
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