Influence of the tip mass on the tip–sample interactions in TM-AFM

Ultramicroscopy 111 (2011) 1423–1436

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Influence of the tip mass on the tip–sample interactions in TM-AFM Hossein Nejat Pishkenari n, Ali Meghdari Nano-Robotics Laboratory, Center of Excellence in Design, Robotics and Automation, School of Mechanical Engineering, Sharif University of Technology, Tehran, P.O. Box 11365-9465, Iran

a r t i c l e i n f o

abstract

Article history: Received 4 November 2010 Received in revised form 19 May 2011 Accepted 28 May 2011 Available online 6 June 2011

This paper focuses on the influences of the tip mass ratio (the ratio of the tip mass to the cantilever mass), on the excitation of higher oscillation eigenmodes and also on the tip–sample interaction forces in tapping mode atomic force microscopy (TM-AFM). A precise model for the cantilever dynamics capable of accurate simulations is essential for the investigation of the tip mass effects on the interaction forces. In the present work, the finite element method (FEM) is used for modeling the AFM cantilever to consider the oscillations of higher eigenmodes oscillations. In addition, molecular dynamics (MD) is used to calculate precise data for the tip–sample force as a function of tip vertical position with respect to the sample. The results demonstrate that in the presence of nonlinear tip–sample interaction forces, the tip mass ratio plays a significant role in the excitations of higher eigenmodes and also in the normal force applied on the surface. Furthermore, it has been shown that the difference between responses of the FEM and point-mass models in different system operational conditions is highly affected by the tip mass ratio. & 2011 Elsevier B.V. All rights reserved.

Keywords: TM-AFM Finite element method Molecular dynamics simulations Tip mass ratio Higher harmonics

1. Introduction The amplitude-modulation atomic force microscopy (AM-AFM), usually referred to as the tapping mode AFM (TM-AFM), is the most used scanning probe method for the characterization and modification of surfaces at atomic and nanometer scales in air or liquids [1,2]. In this tool, the changes in the AFM cantilever dynamics characteristics caused by the tip–sample interactions are used to detect the surface properties. The main parts of the AFM system are the cantilever and its tip, which is in interaction with the sample surface. Accurate simulation of cantilever dynamics (considering the tip mass) coupled with nonlinear tip–sample interactions necessitates using comprehensive methods capable of precise modeling of these parts for interpretation of the imaging results in TM-AFM. The nonlinear characteristics of the tip–sample interactions in TM-AFM, gives rise to anharmonic oscillations of the cantilever, which was studied earlier [3,4]. Due to influence of cantilever modeling on the image resolution and detection of higher harmonics, an increasing concern is devoted to the distributed characteristic of the structural flexibility in the cantilever. The finite element is a versatile and effective distributed-parameters modeling method, which has been recently used for modeling of AFM cantilever dynamics [5–10]. In the present research, the FEM

n

Corresponding author. Tel.: þ98 21 66165541; fax: þ 98 21 66000021. E-mail address: [email protected] (H.N. Pishkenari).

0304-3991/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ultramic.2011.05.010

method is used to model the cantilever dynamics with the high precision necessary for considering the oscillations of higher eigenmodes of the cantilever. In spite of the significant role of the tip in cantilever dynamic behavior and direct interaction with surface, only a few papers in the literature have been devoted to the study of the effects of mass of the commonly used tips on vibrational motion. Mokhtari-Nezhad et al. [11], have investigated the influences of the tip mass on the sensitivity of all three vibration modes namely lateral excitation, torsional resonant and vertical excitation, of an atomic force microscopy microcantilever. Although their research was not devoted to surface topography measurements, they strongly recommended that the effects of tip mass must be considered in topography measurements. Payam and Fathipour [12], developed a model for the AFM cantilever-tip system based on Euler–Bernoulli beam theory and investigated the variation of the resonant frequencies of the system subject to changes in different parameters such as tip mass and cantilever density. Mahdavi et al. [13], presented a model of the AFM cantilever based on Timoshenko’s beam theory and studied the flexural vibration of the tilted cantilever considering the effects of different parameters such as the tip mass. Kiracofe and Raman [14] demonstrated that tip mass and hydrodynamic loading must be taken into account in optical lever sensitivity calibration and stiffness calibration while considering higher oscillation eigenmodes. Lozano et al. [15] showed that due to the sensitivity of higher oscillation eigenmodes to mass inhomogeneities at the free end, such as tip mass, there are significant limitations to the Sader method.

H.N. Pishkenari, A. Meghdari / Ultramicroscopy 111 (2011) 1423–1436

2. Effect of the tip mass on the cantilever’s resonant frequencies The dispersion relation for the AFM cantilever considering the tip mass is as follows [12]: mtip bn ½sin bn cosh bn cos bn sinh bn rL½1þ cos bn cosh bn  ¼ 0

ð1Þ

where mtip is the tip mass, r and L are the cantilever’s length and the cantilever’s density per length, respectively, and bn is nth solution of the dispersion relation. If the tip mass is neglected, the dispersion relation reduces to 1 þcos bn cosh bn ¼ 0, which is extensively used in the literature [16,29]. Fig. 1(a) shows the ratio bn/bn(0) for n ¼1,2,3,4,5 as a function of the tip mass ratio (mr ¼mtip/mbeam ¼mtip/rL). Here bn(0) denotes the solution of the dispersion relation when the tip mass is neglected. The common tip mass ratio is set in the interval mr r0.7mbeam [11,16,30]. As shown, function bn(mr)/bn(0) is a descending function, and we also have (bi þ 1(mr)/bi þ 1(0)) Z(bi(mr)/bi(0)). The cantilever resonant frequencies can be determined as below:

o2n ðmr Þ ¼

EI

rL4

b4n ðmr Þ

ð2Þ

here on denotes nth resonant frequency of the cantilever, and EI is the cantilever’s rigidity. Here (mr) is used to show the dependency of the resonant frequencies on the mass ratio. Fig. 1(b) shows the ratio on/on(0) for n¼1,2,3,4,5 as a function of the tip mass ratio. In the following calculations, only the first three eigenmodes of the cantilever are taken into consideration, since the frequencies of other eigenmodes are far beyond the bandwidth limit of a commercial AFM. Due to the nonlinear tip–sample interaction the

1 0.99

βi/βi (0)

Despite the mentioned progress made in the study of tip mass effects on cantilever stiffness, vibration frequency, and amplitude, there still remain significant matters to be addressed, such as the tip mass effects on the excitation of the higher oscillation eigenmodes, the interaction force, the sample deformation and the measured topography. In addition, as the main challenge of working at the nano-scale, modeling the atomic processes occurring at the tip–sample interface is a fundamental concept to many technological problems. So far, approximate models based on van der Waals and contact forces have played a strong role in modeling of these interfacial interactions [16,17]. However, in most cases, these methods do not consider the individual atomic interactions and hence the realization of imaging with true atomic resolution. One of the simulation tools at the atomic scale is molecular dynamics (MD). Recently, various researches have been devoted to modeling the tip–sample interaction in the imaging process by molecular dynamics [18–27]. For instance, Pishkenari and Meghdari [26,27] investigated the effects of higher order oscillations on the TM-AFM images using MD simulations. Their results demonstrate that in the presence of nonlinear interaction forces, higher eigenmodes of the microcantilever are excited and they play a significant role in the tip and sample elastic deformations. Another challenge in the present model is the combination of MD and FEM in spite of their different time and length scaling. In multi-scale modeling, both methods are run simultaneously and a shaking hand area is utilized to connect them together [28]. As a simpler and more feasible alternative approach for combination, one can consider these two sub-models (MD and FEM) separately. In this hybrid approach, which is used in the present work, force data calculated by MD simulations is used in the FEM simulations. In this way, one can employ the versatile advantages of both modeling schemes. The detail of the proposed technique will be described in the next sections of the paper.

0.98

β1/β1 (0)

0.97

β2/β2 (0)

0.96

β4/β4 (0)

β3/β3 (0) β5/β5 (0)

0.95 0.94 0

0.01

0.02

0.03 0.04 Mass Ratio

0.02

0.03 0.04 Mass Ratio

0.05

0.06

0.07

1 0.98

ωi/ωi (0)

1424

0.96

ω1/ω1 (0)

0.94

ω2/ω2 (0) ω3/ω3 (0)

0.92

ω4/ω4 (0) ω5/ω5 (0)

0.9 0.88

0

0.01

0.05

0.06

0.07

Fig. 1. Effect of the tip mass ratio (mr) on the solutions of the dispersion relation (a) and on the resonant frequencies of the cantilever. Here, only the first five roots of the dispersion relation are considered. bi(0) is ith solution of the dispersion relation when the tip mass is neglected, and bi is ith solution of the dispersion relation when the tip mass is considered. As shown, increasing the tip mass ratio decreases the dispersion relation root values and also the natural frequencies.

higher harmonic components of the cantilever motion near the system eigenmodes are considerably excited, representing the influence of the cantilever intrinsic vibration modes on the harmonics generation. In order to examine the effect of the tip mass ratio on the cantilever vibration, we must calculate the excitation frequency oexc at which the higher harmonic of the excitation force are very close to the second and third resonant frequencies of the cantilever. Because in most common applications of TM-AFM the excitation frequency is very close to the first resonant frequency, it is enough to determine the possible tip mass ratio at which we have o2 ¼n21o1, o3 ¼n31o1 (n21,n31AN), which means the second and third resonant frequencies becomes an integer multiplication of the first natural frequency of the cantilever. It must be noted that the ratios of the second and third eigenmode resonant frequencies to the first resonant frequency of the cantilever (i.e. o2/o1 and o3/o1) are only affected by the tip mass ratio (see Eq. (2)), therefore our study is general and independent from the AFM system specifications. Fig. 2 shows the influence of the tip mass ratio on o2/o1 and o3/o1. According to the plotted figure the following results may be deduced:

 Since o2/o1 and o3/o1 are near integer numbers six and 

eighteen, respectively, we can conclude that 6th and 18th harmonics of the tip motion will be excited significantly. As shown, o2/o1 and o3/o1 are ascending functions of mr and it can be seen that when mr is increased from 0 to 0.07, o2/o1 increases from 6.266 to 6.410, and o3/o1 increases from 17.547 to 18.222.

H.N. Pishkenari, A. Meghdari / Ultramicroscopy 111 (2011) 1423–1436

1425

18 ω2/ω1

Frequency Ratio

16

ω3/ω1

18.2

FR = 18 14

18.1

12

mr = 0.0546

18

10

17.9

8

17.8 0.04

0.045

0.05

0.055

0.06

0.065

0.07

6 0

0.01

0.02

0.03 0.04 Mass Ratio

0.05

0.06

0.07

Fig. 2. Effect of the tip mass ratio on the ratio of the second and third resonant frequencies of the cantilever to its first resonant frequency. As shown in the inset, the value of function o3/o1 becomes 18 at mnr p ¼ 0:0546.

Fig. 3. AFM cantilever model, and a 3D view of the tip–sample configuration used in the MD simulation. A diamond tip made up of 203 carbon atoms is interacting with a five-layer graphite sample made up of 4604 carbon atoms. The sample includes a multiple-atom vacancy at its top most layer. Tip and surface consist of three different kinds of atoms namely frozen atoms, thermostat atoms and Newtonian atoms.

 Using the curve fitting method, the relation between o2/o1 o2 ¼ 6:267 þ40:03m2r 156:5m3r o1

ð3Þ

the tip and sample. The complete description of the FEM and MD method for the modeling of the cantilever dynamics and the tip–sample interactions, respectively, can be found in Ref. [26]. The equations of motions based on the FEM modeling method in the matrix form are as follows [26]:

o3 ¼ 17:55 þ205m2r 978m3r o1

ð4Þ

ðrLM þ mtip NNT Þq€ þ ðBLM þCGÞq_ þ

and mr and also the relation between o3/o1 and mr can be approximated as

EI Kq ¼ ðrLH þmtip NÞ L3

 Aexc o2exc sinðoexc tÞðBLHÞAexc oexc cosðoexc tÞ þNftipsample

 Since increasing mr leads to an increase of value 9o2/o1 69, 

the second eigenmode is more excited at lower values of the tip mass ratio. There exists an intermediate value of mr for which we have o3/o1 ¼18 and this means at this tip mass ratio the excitation of the third eigenmode will become maximum. This tip mass ratio is mrnp ¼ 0:0546 (see inset of Fig. 2).

3. Model of the cantilever dynamics and the interaction force The AFM system configuration is illustrated in Fig. 3. The AFM system includes two main parts: the first part is the cantilever dynamics and the second part is the interaction forces between

ð5Þ where M2n  2n is the global mass matrix, K2n  2n is the global stiffness matrix, G2n  2n is the global structural damping matrix, H2n  1 and N2n  1 are two auxiliary nodal vectors, qT ¼ ½y1 Le y1 y2 Le y2 . . .y2n Le y2n 12n is the nodal displacement vector, n denotes the number of nodal points, B and C are the viscous and structural damping coefficients, respectively, oexc and Aexc are the excitation frequency and amplitude, respectively, mtip is the tip mass, L is the cantilever length, EI is the cantilever rigidity and ftip  sample(t) is the tip–sample interaction force. The precision of the cantilever model can be increased by increasing the number of nodes. The derived relation (5) will be used in the next sections of the paper to simulate the cantilever vibrational motion. The remainder of

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Table 1 System parameters used in simulations. Cantilever rigidity (EI)

1:0125  1011 Nm2 2.097  10  7 kg/m 250 mm mtip r 0.07mbeam ¼3.67  10  12 kg 18  10  4 kg/ms 10  9 kg/s 20 nm 18 nm p ¼0.01, i ¼0.001

Cantilever density per length (q) Cantilever length (L) Tip mass (mtip) Cantilever viscous damping coefficient (B) Cantilever structural damping coefficient (C) Free oscillation amplitude (A0) Amplitude set-point (Aset) The PI controller coefficients

mr = 0.01 mr = 0.03 mr = 0.05 mr = 0.07

0.035 0.03 0.025 0.02 0.015 0.01 0.005 0

0

2

4

6

8

10 12 ω / ω0

14

16

18

0.14

20

0.08 0.06 0.04

9.5

0.06

9

0.04

8.5

0.02 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

8

Mass Ratio

7.5 7 6.5 6 0

0.01

0.02

0.03 0.04 Mass Ratio

0.05

0.06

0.07

0.02

0.03 0.04 Mass Ratio

0.05

0.06

0.07

0.17 0.165 0.16 0.155 0.15

0.02 0

0.08

0.175 Average Force (nN)

An (nm)

0.1

10

5.5

A6 A18

0.12

10.5

dc

0.04

Maximum Repulsive Force (nN)

Amplitude Spectrum (nm)

0.045

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0

Mass Ratio Fig. 4. Effect of the tip mass ratio on the spectrum of the tip motion (a) and on the amplitudes of the 6th and 18th harmonics (b) (when tip taps the sample surface). The FEM model shows some peaks as local maximums at o ¼6 o0 and o ¼18o0. As shown in part (a), the second eigenmode is significantly excited at the lower tip mass ratios (i.e. mr ¼ 0.01), while the third eigenmode is considerably excited when the tip mass ratio is mr ¼0.05. As shown in part (b), the amplitude of the 6th harmonic is a descending function of the tip mass ratio while the amplitude of the 18th harmonic as a function of mr has a sharp peak at mnr ¼ 0:0537. The simulation parameters are set as Aset ¼ 18 nm and oexc ¼ o0.

this section is devoted to a brief description of the force calculation approach. As mentioned earlier, in our simulations, the interactions between the surface and the probe tip are modeled by the MD method. Fig. 3 depicts the configuration of the system including a sample and a tip. In order to compute the tip–sample interface force at a fixed horizontal position (i.e., at a point (x,y)), the tip approaches the sample vertically with a sinusoidal movement of all the tip atoms and in each vertical position the vertical force is determined by the MD simulation code. If this process is done in other horizontal positions, the interaction force can be obtained as a function of x and y, i.e. Ftip  sample(x,y).

0.01

Fig. 5. Effect of the mass ratio on the maximum repulsive force (fmax ) applied to the surface (a), and on the average force (fave) applied to the surface (b). The inset (a) represent that the distance of the closest approach (dc) becomes minimum at mhr ¼ 0:0530 and maximum at mlr ¼ 0:0551, therefore as shown, for function h fmax ðmr Þ the highest value (fmax ¼ 10:3 nN) occurs at mhr ¼ 0:0530 and the l lowest value (fmax ¼ 5:7 nN) occurs at mlr ¼ 0:0551. As shown in part (b), fave(mr) max min has a maximum (fave ¼ 0:176 nN) at mr ¼ 0 and fave(mr) has a minimum (fave ¼ 0:147 nN) at mnr ¼ 0:0537. The simulation parameters are set as Aset ¼ 18 nm and oexc ¼ o0.

4. Results While scanning the surface, the interaction force data are continuously updated to reflect the changes in the relative horizontal position of the tip and the sample. When the tip travels over the vacancy, the sample topography, the maximum repulsive force, and the lowest tip height are recorded. As the cantilever travels over the sample, the cantilever vertical position with respect to the sample surface changes and leads to the change of the oscillation amplitude. In order to maintain the desired oscillation amplitude, the cantilever vertical position in TM-AFM can be controlled through a proportional-integral loop

H.N. Pishkenari, A. Meghdari / Ultramicroscopy 111 (2011) 1423–1436

0

mr = 0

-1 Interaction Force (nN)

1427

mr = 0.0530

-2

mr = 0.0537

-3

mr = 0.0551

0 -2

-4 -5

-4

-6

-6

-7

-8

-8

-10

-9

2.5

2.6

2.7

2.8

2.9

3

-10 -3

-2

-1

0 1 Dimensionless Time

2

3

18 mr = 0

18.1 12

mr = 0.0530 17.9

6 ytip (nm)

Repulsive Region

mr = 0.0551

0

17.7

-6

17.5 2.5

-12 -18

mr = 0.0537

0

2.6

1

2.7

2

2.8

2.9

3

3 4 Dimensionless Time

5

6

Fig. 6. Effect of the mass ratio on the interaction force (a) and the tip motion (b) as a function of time. As shown in the insets, a small change in the tip mass ratio leads to a significant variation of the tip–sample interaction force and also the tip motion during tapping the surface. The simulation parameters are set as Aset ¼18 nm and oexc ¼ o0.

0.12

0

0.1 0.08

-1

Δ18 (nm)

Φ18 (rad)

-0.5

0.06 0.04 0.02 0

-1.5

-0.02 -0.04

-2

-0.06

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Mass Ratio

-2.5 -3

0

0.01

0.02

0.03 0.04 Mass Ratio

0.05

0.06

0.07

Fig. 7. Effect of the mass ratio on the phase lag of the 18th harmonic (F18) and on a part of the sample deformation (D18) arisen from excitation of the third eigenmode. As shown, the phase lag is like a step function, which step approximately occurs at mr ¼ 0.0540. As shown in the inset, D18 has a suddenly significant change at mr ¼ 0.0540. The simulation parameters are set as Aset ¼18 nm and oexc ¼ o0.

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based on the following formula [21,26]:

yc ðj þ 1Þ ¼ yc ðjÞpðAðjÞAset Þi

j X

ðAðiÞAset Þ

ð6Þ

Measured Topography (nm)

i ¼ j9

here yc is the tip–sample separation distance and A is the effective tip oscillation amplitude, Aset is the oscillation amplitude setpoint, p and i are the PI controller coefficients. The system parameters used in our simulations are listed in Table 1. The equivalent quality factor and spring coefficient are

0.1 mr = 0.0530

0

mr = 0.0551

-0.1 -0.2 -0.3 -0.4 -0.6

-0.4

-0.2 0 0.2 Horizontal Position (nm)

0.4

0.6

Fig. 8. AM-AFM cross-sectional scan of the vacancy site of the graphite sample obtained by FEM model for two systems with slightly different tip mass ratios mr ¼ 0.0530 , 0.0551. The sample atoms at the scanned cross section before any surface reconstructions due to interaction forces are shown. Due to existence of the observed jump in the phase lag of the third oscillation eigenmode, the measured topography obtained for two systems differs significantly. The simulation parameters are set as Aset ¼ 18 nm and oexc ¼ o0.

Maximum Repulsive Force (nN)

10.5 Lumped Model FEM Model

10 9.5 9 8.5 8 7.5 7 6.5 6 5.5

0

0.01

0.02

0.03 0.04 Mass Ratio

0.02

0.03 0.04 Mass Ratio

0.05

0.06

0.07

0.18

Average Force (nN)

0.175 0.17 0.165 0.16 0.155 Lumped Model FEM Model

0.15 0.145

0

0.01

0.05

0.06

0.07

Fig. 9. Functions fmax ðmr Þ (a) and fave(mr) (b) for FEM and Lumped models. For the lumped model, the dependency of fmax and fave on the mass ratio is negligible, while for the precise FEM model, these forces are highly affected by the tip mass ratio. The difference of fmax obtained by two models becomes maximum at mr ¼ 0.0530 and 0.0551, while the difference of fave obtained by two models is maximum at mr ¼ 0.0537.

H.N. Pishkenari, A. Meghdari / Ultramicroscopy 111 (2011) 1423–1436

(Aset ¼18 nm), can be obtained from the following relations:

approximately Qeq ¼50 and keq ¼ 2 N/m, respectively. In the first simulation, the effect of the tip mass ratio on the cantilever motion in tapping near the surface is investigated. The role of oscillation of the higher cantilever modes is better clarified if the signal is decomposed in its frequency components. Fig. 4(a) shows the effect of the tip mass ratio on the excitation of the higher oscillation eigenmodes. As expected, the FEM model shows some peaks as local maximums at o ¼6 o0 and o ¼18 o0, where o0 ¼ o1 is the first natural frequency of the cantilever as well as the excitation frequency. As expected, the second eigenmode is significantly excited at the lower tip mass ratios (i.e. mr ¼0.01), while the third eigenmode is considerably excited when the tip mass ratio is mr ¼0.0546. In order to study the dependency of the second and third eigenmodes amplitudes on the tip mass ratio more quantitatively, these two functions are plotted in Fig. 4(b). As shown, the amplitude of the 6th eigenmode (A6) is dominant with respect to the amplitude of the 18th harmonic (A18) in the lower tip mass ratios, while A18 is more powerful when the tip mass ratio is in the interval [0.050, 0.059]. A striking feature depicted in Fig. 4(b) is that function A18(mr) has a maximum at mnr ¼ 0:0537, while our prediction was mrnp ¼ 0:0546. The reason for this little difference is that the effective first resonant frequency of the cantilever motion increases slightly when it taps the surface, and therefore as we will see in the next section, excitation with a frequency lower than the resonant frequency of the cantilever leads the maximum of function A18(mr) to shift to the left (i.e. lower values of mnr ¼ 0:0537). The effective resonant frequency of the cantilever motion (o1eff) in this simulation

18oexc ¼ 18o1 ¼ o3eff 9mr ¼ 0:0537

ð7Þ

 o3eff  o 

ð8Þ

) 17:988o1eff ¼ 18o1 ) o1eff ¼ 1:0007o1

It is evident that mr is dependent on many parameters such as the interaction force, the amplitude set-point, the excitation frequency, the equivalent cantilever stiffness and the equivalent quality factor, which will be studied in the next sections of the paper. The maximum tip–sample repulsive force (or sample deformation) is one of the parameters reflecting the influence of mechanical properties on tapping operation as well as the most critical parameter for assessment of sample damage. Due to change of the excitation of the higher harmonics of the cantilever with change of the tip mass ratio, it is expected that the tip mass ratio highly affects the exerted force on the surface. Fig. 5 represents the maximum repulsive force (fmax ) and the average force (fave) applied to the surface as a function of mr. A noticeable behavior of function fmax ðmr Þ is that it produces a quick jump from a maximum to a minimum value with a small change of mr (see Fig. 5a). Since the amplitude of the 18th harmonic affects the tip motion, one may expect that the function fmax ðmr Þ has a behavior similar to the function A18(mr), while as it will be discussed later,

0.1

0.02

0.05 0.052 0.054 ωexc = 0.999ω0 ωexc = ω0 ωexc = 1.001ω0

0.02

Maximum Repulsive Force (nN)

Φ18 (rad)

0.06

0.04

0

ωexc = ω0 ωexc = 1.001ω0

-0.5

0.08 0.06

ωexc = 0.999ω0

0

0

0.01

0.02

0.056

0.058

0.06

-1 -1.5 -2 -2.5

0.03 0.04 Mass Ratio

0.05

0.06

-3

0.07

0

0.01

0.02

0.03 0.04 Mass Ratio

0.05

0.06

0.07

0.25

14 13

ωexc = 0.999ω0 ωexc = ω0 ωexc = 1.001ω0

12 11

Average Force (nN)

A18 (nm)

0.12

ð9Þ

n

0.5 0.14

0.1

¼ 17:988

1eff mr ¼ 0:0537

0.16 0.14

1429

10 9 8 7 6

ωexc = 0.999ω0 ωexc = ω0 ωexc = 1.001ω0

0.2

0.15

0.1

5 4

0

0.01

0.02

0.03 0.04 Mass Ratio

0.05

0.06

0.07

0.05

0

0.01

0.02

0.03 0.04 Mass Ratio

0.05

0.06

0.07

Fig. 10. Effect of the excitation frequency on the 18th harmonic amplitude (a), the 18th harmonic phase lag (b), the maximum repulsive force and the average interaction force. As shown in part (a), approaching the excitation frequency to the effective first resonant frequency of the cantilever leads to an increase of the 18th harmonic amplitude. As depicted in part (b), a small change of the excitation frequency can significantly affect the phase lag of the third oscillation eigenmode. As shown in parts (c and d), when oexc approaches o1eff, independent of the tip mass ratio value fmax ðmr Þ increases and fave(mr) decreases. The amplitude set-point is Aset ¼18 nm.

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H.N. Pishkenari, A. Meghdari / Ultramicroscopy 111 (2011) 1423–1436

motion to its frequency components we have

in addition to the amplitude of the third eigenmode, the phase lag of the 18th harmonic can significantly influence the maximum force. In contrast with part (a) of Fig. 5 as shown in part (b), the time average of the interaction force as a function of the tip mass ratio, has a minimum value at mnr ¼ 0:0537. This is unpredictable because one may think that function fave(mr) has a behavior similar to function fmax ðmr Þ, while as shown in Fig. 6(a), functions fave and fmax have distinct trends for different tip mass ratios. In other words, at the higher amplitudes of the third eigenmode, the tip motion during tapping the surface in the repulsive region is more complex (see Fig. 6b) and leads to a reduction in the average normal force exerted on the sample (see Fig. 5b). As mentioned earlier, the distance of the closest approach of the tip to the surface and so the maximum interaction force are affected not only by the amplitude of the 18th harmonic but also by the phase lag of the 18th harmonic. The phase lag of the 18th harmonic as a function of the tip mass ratio is depicted in Fig. 7. As shown, function F18(mr) has a quick jump from 2.93 to 0.14 rad, approximately at mr ¼0.0540. In the rest of this subsection we illustrate how the 18th harmonic can influence the tip motion in tapping near the surface. If we decompose the tip

ytip ¼ A0 þ

20 X

Ai sinðio1 t þ Fi Þ ¼ A0 þ

i¼1

20 X

Ai sinðit þ Fi Þ

ð10Þ

i¼1

where Ai and Fi denotes the amplitude of the ith harmonic and t ¼ o t is dimensionless time. Here we have neglected the effects of the harmonics of other eigenmodes because they are beyond the bandwidth limit of an AFM and are negligibly excited. The role of the 18th harmonic on the tip motion can be separated as follows: y18 ðtÞ ¼ A18 sinð18t þ F18 Þ

ð11Þ

This part of motion can shift the maximum value of function ytip(t) (see the inset of Fig. 6(b)). To calculate the contribution of the 18th harmonic on the tip motion when tapping the surface, because of its dominant role on the tip motion, we only consider the effect of this mode on the change of the maximum value of function ytip(t), as following:

tmax ¼ F1 þ

p 2

þ

18A18 cosð18F1 þ 9p þ F18 Þ A1

ð12Þ

4 ωexc = 0.999 ω0

3

ωexc = ω0 ωexc = 1.001 ω0

0

7

6

05 0.

5

05 0.

4

05 0.

05

3

0.02

0.

05

1

0.01

0.

0.

-3

05

-2

05

-1

05

4 3 2 1 0 -1 -2 -3

0.

0

2

1

0.

Δ fmax (nN)

2

0.03 0.04 Mass Ratio

0.05

0.06

0.07

0 ωexc = 0.999 ω0

-0.005

ωexc = ω0 ωexc = 1.001 ω0

Δ fave (nN)

-0.01

0

-0.015 -0.02

-0.01

-0.025

-0.02

-0.03

-0.03

-0.035 0.05

-0.04

0

0.051 0.052 0.053 0.054 0.055 0.056 0.057

0.01

0.02

0.03 0.04 Mass Ratio

0.05

0.06

0.07

Fig. 11. Effect of the excitation frequency on the difference of the maximum interaction force and the average interaction force obtained by the FEM and lumped models FEM Lumped FEM Lumped (Dfmax ¼ fmax  fmax , Dfave ¼ fave  fave ). As shown in the inset, when the tip mass ratio is in the interval 0.05r mr r 0.06, approaching oexc to the effective first resonant frequency, increases the difference of the normal forces obtained by two models.

H.N. Pishkenari, A. Meghdari / Ultramicroscopy 111 (2011) 1423–1436

D18 ¼ A18 sinð18tmax þ F18 Þ

4.2. Influence on the lumped model inaccuracy

ð13Þ

here tmax is an estimation of the time in which the tip is at its minimum distance from the surface (see the inset of Fig. 6(b)), and D18 (not A18) is the contribution of the 18th harmonic on the tip motion at time tmax . In other words, D18 is a part of the sample deformation arisen from excitation of the 18th harmonic. As shown in the inset of Fig. 7, D18(mr) has a quick jump similar to the one obtained in fmax ðmr Þ, which justifies the observed behavior of the maximum interaction force.

As it was demonstrated, the FEM model confirmed that tip motion is not merely a simple single-frequency sinusoidal behavior and tip mass ratio can affect the excitation of the higher cantilever modes. Although the lumped-parameter model is a simple and quick method, it is not able to model the excitation of the higher oscillation eigenmodes. Therefore, it is expected that the inaccuracy of the lumped model increases in the systems with tip mass ratios near the values calculated in the previous sections. Fig. 9 represents the maximum interaction force and the average force for FEM and lumped models. As shown in part (a), the maximum interaction force obtained by the lumped model is

4.1. Influence on the measured topography

Lumped

an approximately constant value fmax ¼ 7:6nN, and its dependency on tip mass ratio is negligible. The greatest difference between the maximum forces obtained by two models occurs at mr ¼ 0.0530 and 0.0551. As shown in part (b), the average force obtained by the lumped model is an approximately constant

Due to the substantial effect of the tip mass ratio on the interaction force, it is expected that the sample deformation and the measured topography of the surface will be dependent on the tip mass ratio. Fig. 8 depicts the cross-sectional scan of the vacancy site of the graphite sample, obtained by FEM models, for two systems with slightly different tip mass ratios mr ¼0.0530 , 0.0551. The image of the system with the tip mass ratio mr ¼0.0530 is below the image of the system with the tip mass ratio mr ¼0.0551, indicating that the tip and sample elastic deformations measured in the system with the tip mass ratio mr ¼0.0530 is larger. The observed behavior arises from the fact that the maximum interaction force measured in the first system is approximately twice the force measured in the second system.

Lumped

value fave ¼ 0:175nN, and the greatest difference between the average forces obtained by two models occurs at mr ¼0.0537. The observed behavior illustrates that when the tip mass ratio is far from the excitation region of the third oscillation eigenmode, the FEM and Lumped model force results approach each other, and when the tip mass ratio is near to mr ¼0.0537 the imprecision of the lumped model is maximized. In the next subsections we will investigate the effects of the system parameters such as the excitation frequency, the amplitude

0.25 Aset = 16 (nm)

14

Aset = 16 (nm)

Aset = 18 (nm)

Aset = 18 (nm)

Aset = 19.7 (nm)

12

Average Force (nN)

Maximum Repulsive Force (nN)

16

10 8 6

0.2

Aset = 19.7 (nm)

0.15

0.1

4 2

0.05 0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0

0.01

0.02

Mass Ratio

0.18

4

0.16 0.14

3

0.14

0.1 0.06 0.08 0.02 0.06 0.052

0.053

0.04

Aset = 16 (nm)

0.02

Aset = 18 (nm)

0

0.054

0.055

0.056

0.01

0.05

0.06

0.07

Aset = 18 (nm) Aset = 19.7 (nm)

1 0 -1

0.057

-2

Aset = 19.7 (nm)

0

0.03 0.04 Mass Ratio

Aset = 16 (nm)

2

0.1

Φ18 (rad)

0.12 A18 (nm)

1431

-3 0.02

0.03 0.04 Mass Ratio

0.05

0.06

0.07

0

0.01

0.02

0.03 0.04 Mass Ratio

0.05

0.06

0.07

Fig. 12. Effect of the amplitude set-point on fmax ðmr Þ, fave(mr), A18(mr) and F18(mr). Reduction of Aset increases of the maximum force, the average force, and the amplitude of 18th harmonic (parts a, b and c), and significantly changes the phase lag function F18(mr)(part d). The excitation frequency is set as oexc ¼ o0.

1432

H.N. Pishkenari, A. Meghdari / Ultramicroscopy 111 (2011) 1423–1436

set-point, the equivalent cantilever stiffness, and the equivalent quality factor on the dependency of cantilever dynamics on tip mass ratio.

ð15Þ

oexc ¼ 1:001o0 ) mnr ¼ 0:0549 and An18 ¼ 0:158

ð16Þ

The above results could be obtained using the relations between the excitation frequency and the critical value of the tip mass ratio. Generally according to relations (4, 15), mnr can be determined as following:

4.3. Dependence on the excitation frequency Since the excitation frequency affect the critical tip mass ratio mnr (the tip mass ratio at which the amplitude of the 18th harmonic becomes maximum), it can change the amplitude and phase lag of the 18th harmonic. On the other hand, approaching the excitation frequency of the effective first resonant frequency of the cantilever leads to an increase in the higher eigenmodes amplitude. As it was calculated, the effective first resonant frequency of the cantilever in our simulation was o1eff ¼1.0007 o1 and therefore if the excitation frequency is set near this effective resonant frequency, firstly mnr approaches its predicted value mnr p ¼ 0:0546, and secondly the maximum value of function A18(mr) increases. Fig. 10(a) depicts the behavior of function A18(mr) for three different excitation frequencies oexc ¼0.999o0, o0 and 1.001o0. According to Fig. 10(a), we have

oexc ¼ 0:999o0 ) mnr ¼ 0:0524 and An18 ¼ 0:125

oexc ¼ o0 ) mnr ¼ 0:0537 and An18 ¼ 0:138

17:55 þ 205mrn2 978mrn3 ¼

18oexc 1:0007o1

ð17Þ

By linearization of relation (23) we have: mr ¼ 1:326668ðoexc =o1 0:959538Þ n

ð18Þ

The above relations is a general characteristic equation which for a given excitation frequency reproduce the tip mass ratio at which the amplitude of the 18th harmonic is maximum. Fig. 10(b) shows the effect of the excitation frequency on the phase lag of the 18th harmonic. A striking feature depicted in this figure is that a slight variation of the excitation frequency leads to a considerable change in function F18(mr). The maximum repulsive force and the average force applied on the surface as a function of the tip mass ratio for three mentioned excitation frequencies are shown in Fig. 10 (c) and (d). At a given

ð14Þ

5 Aset = 16 (nm)

4

Aset = 18 (nm) Aset = 19.7 (nm)

Δ fmax (nN)

3 2 1 0 -1 -2 -3

0

0.01

0.02

0.03 0.04 Mass Ratio

0.05

0.06

0.07

0 Aset = 16 (nm) -0.005

Aset = 18 (nm) Aset = 19.7 (nm)

Δfave (nN)

-0.01

0

-0.015 -0.01

-0.02 -0.02

-0.025 -0.03 -0.03 -0.035

0.05

0

0.052

0.01

0.054

0.02

0.056

0.03

0.058

0.04

0.06

0.05

0.06

0.07

Mass Ratio Fig. 13. Effect of the amplitude set-point on Dfmax and Dfave. As shown in part (a), independent from mr, reduction of Aset leads to an increase of Dfmax value. On the other hand, as shown in the inset of part (b), when Aset varies, the position and value of the minimum point of function Dfave change.

H.N. Pishkenari, A. Meghdari / Ultramicroscopy 111 (2011) 1423–1436

tip mass ratio, the maximum force applied to the surface increases when oexc approaches o1eff, while the average force decreases when oexc approaches o1eff. Fig. 11 depicts the difference between the normal forces obtained by the FEM and lumped models. As shown, the inaccuracy of the lumped model in determination of the normal forces increases when the excitation frequency approaches the effective first resonant frequency particularly at tip mass ratios near mnr . 4.4. Dependence on the amplitude set-point

4.5. Dependence on the quality factor During TM-AFM operation, the response of the cantilever probe to the nonlinear interaction forces is significantly influenced by its quality factor (Qeq). The cantilever quality factor is a measure of the dissipation in the system and it affects the scanning conditions. A probe with a high Qeq factor represents a lower rate of energy loss with respect to the stored energy of the cantilever, and so it requires a smaller value of the excitation signal to achieve the desired amplitude set-point. On the other, at the same amplitude set-point, a probe with a low Qeq factor exerts greater forces to the sample surface (see Fig. 14(a) and (b)) and therefore leads to larger values of the sample deformation. Hence an enhancement of Qeq factor leads to a decrease of the amplitude of the 18th harmonic. See Fig. 14(c) and (d) depict the effect of the Qeq factor on the dependency of the 18th harmonic amplitude and its phase lag on the tip mass ratio. As shown, the quality factor does not affect mnr or the effective first resonant frequency. Since the major part of the discrepancy between the lumped and FEM models arises from the excitation of the third oscillation eigenmode, a reduction of the Qeq factor leads to an increase of the difference between the force obtained by these two models (see Fig. 15).

0.18

11 9

Average Force (nN)

Qeq = 50 Qeq = 75 Qeq = 100

10 8 7 6 5 4 2

0

0.01

0.02

0.03 0.04 Mass Ratio

0.05

0.14 0.12 0.1

0.06

0.07

0.1

0.06 0.04

0.06

0.02 0.052

0.04

0.053

0.054

0.055

0.056

0.02

0.03 0.04 Mass Ratio

0.05

-0.5

-1

-1

-1.5

-3 0

0.01

0.02

0.03 0.04 Mass Ratio

0.05

0.06

0.07

0.07

-2

-1.5

-2.5 0.052

0.053

0.054

0.055

0.056

0.057

-2.5

0.02

0.06

-0.5

-2

0.057

0.01

0

0

Qeq = 100

0.08

0.08

0

0.5

Qeq = 75

0.12

0.1

0.06

Qeq = 50

0.14

0.12

0

Qeq = 50 Qeq = 75 Qeq = 100

0.16

0.08

3

0.14

A18 (nm)

lumped models increases when the amplitude set-point decreases (see Fig. 13(a) and (b)).

Φ18 (rad)

Maximum Repulsive Force (nN)

At a constant value of the free oscillation amplitude, decreasing the amplitude set-point leads to an increase of interaction force between tip and sample (see Fig. 12(a) and (b)). Due to the larger nonlinear forces in the smaller amplitude set-points, the excitation of the higher oscillation eigenmodes is intensified. Furthermore, when the amplitude set-point decreases, due to increase of the normal force, the effective first resonant frequency of the cantilever increases and therefore when the excitation frequency is set as oexc ¼ o0 ¼ o1, mnr decreases slightly. Fig. 12(c) shows the influence of the amplitude set-point on the amplitude of the third oscillation eigenmode of the cantilever. As the amplitude set-point decreases, function A18(mr) is scaled up and shifted slightly to the left (see inset of Fig. 12(c)) and in addition the phase lag of the 18th harmonic as a function of the tip mass ratio, shifts in the vertical direction (see Fig. 12(d)). Due to intensification of the 18th harmonic amplitude, the difference of the interaction forces obtained by the FEM and

1433

Qeq = 50 Qeq = 75 Qeq = 100

0

0.01

0.02

0.03 0.04 Mass Ratio

0.05

0.06

0.07

Fig. 14. Effect of the cantilever quality factor on fmax ðmr Þ, fave(mr), A18(mr) and F18(mr). As shown, enhancement of Qeq decreases the maximum repulsive force, the average force and the amplitude of 18th harmonic (parts a, b and c). On the other hand, as shown in part (d), enhancement of Qeq does not affect function F18(mr) considerably. The simulation parameters are set as Aset ¼ 18 nm and oexc ¼ o0.

1434

H.N. Pishkenari, A. Meghdari / Ultramicroscopy 111 (2011) 1423–1436

3 Qeq = 50 Qeq = 75 Qeq = 100

2.5 2

Δfmax (nN)

1.5 1 0.5 0 -0.5 -1 -1.5 -2

0

0.01

0.02

0.03 0.04 Mass Ratio

0.05

0.06

0.07

0 Qeq = 50

0

-0.005

Qeq = 75

-0.005

Δfave (nN)

-0.01 -0.015

Qeq = 100

-0.01 -0.015 -0.02

-0.02

-0.025

0.05

0.052

0.054

0.056

-0.025 -0.03

0

0.01

0.02

0.03 0.04 Mass Ratio

0.05

0.06

0.07

Fig. 15. Effect of the quality factor on Dfmax ðmr Þ and Dfave(mr). As shown, independent from mr, enhancement of Qeq leads to reductions of Dfmax and Dfave(mr) values.

4.6. Dependence on the cantilever stiffness The equivalent cantilever stiffness (keq) affects the relationship between the normal force exerted on the sample surface and the deflection of the cantilever in the normal direction. A cantilever with a large stiffness represents more rigidity in the tip motion, and applies greater forces to the sample surface to reach the desired amplitude set-point. Consequently it is expected in systems with a large value of cantilever stiffness, during scanning in tapping mode, the normal force applied to the surface and also the sample deformation will become greater (see Fig. 16(a) and (b)). Due to increase of the interaction force in systems with larger values of the cantilever stiffness, it may be deduced that the enhancement of keq leads to an increase of the excitations of higher oscillation modes. However, a more precise investigation of keq effect on the excitation of the higher oscillation eigenmodes demonstrates that the main factor in the excitation of the higher oscillation eigenmodes is not the interaction force but the forcestiffness ratio. In fact, the parameter directly affecting the excitation of higher oscillation eigenmodes is the force-stiffness ratio and enhancement of this parameter leads to an intensification of the amplitudes of the higher oscillation eigenmodes. Although enhancement of the cantilever stiffness increases the normal force applied to the surface, it also reduces the force-stiffness ratio (see the inset of Fig. 16(a)). Hence an enchantment of the

cantilever spring constant reduces the amplitude of the higher oscillation modes. Fig. 16(c) and (d) depict the effect of the cantilever stiffness on functions A18(mr) and F18(mr). Due to the amplitude reduction at the larger cantilever stiffness values, one may deduce the difference between forces calculated by the FEM and Lumped models is also reduced. However as shown in Fig. 17(a) and (b), the difference between forces obtained by the two models is greater at the larger values of the cantilever stiffness. This happens because of the parameter affecting the inaccuracy of the lumped model is as following:

Dfmax  keq D18 ¼ keq A18 sinð18tmax þ F18 Þ

ð19Þ

indicating in addition to the amplitude the force difference is also dependent on the cantilever stiffness.

5. Conclusions The analysis of cantilever dynamics with a concentrated tip mass in TM-AFM has been carried out. Some of the main conclusions demonstrating the influences of the tip mass ratio on TM-AFM measurements are as follows:

 A substantial correlation between the tip mass ratio, the 18th harmonic amplitude and the normal force applied to the

12 10 8

6

keq = 1 (N/m) keq = 3 (N/m)

4 3 0

Mass Ratio

6

keq = 1 (N/m) keq = 2 (N/m)

0.22

keq = 3 (N/m)

0.18 0.14 0.1

4 0.06 0

0.14 0.12 0.1

0.01

0.02

0.03 0.04 Mass Ratio

0.05

0.06

0.07

keq = 2 (N/m)

0.14

0.08 0.08 0.06 0.06 0.052

0.053

0.054

0.02

0.03 0.04 Mass Ratio

0.05

-1.5

-2

-2

0.055

0.053

-2.5

keq = 1 (N/m)

0.054

keq = 2 (N/m) keq = 3 (N/m)

-3 0

0.01

0.02

0.03 0.04 Mass Ratio

0.05

0.06

0.07

-1 -1.5

0.02 0

0.06

-1

-0.5

0.1

0.01

0

keq = 3 (N/m)

0.12

0.04

0

0.5

keq = 1 (N/m)

0.16

Φ18 (rad)

0.16

A18 (nm)

0.01 0.02 0.03 0.04 0.05 0.06 0.07

1435

0.26

keq = 2 (N/m)

5

Average Force (nN)

14

Force-Stiffness Ratio (nm)

Maximum Repulsive Force (nN)

H.N. Pishkenari, A. Meghdari / Ultramicroscopy 111 (2011) 1423–1436

0

0.07

0.01

0.02

0.03 0.04 Mass Ratio

0.05

0.06

0.07

Fig. 16. Effect of the cantilever stiffness on fmax ðmr Þ, fave(mr), A18(mr) and F18(mr). As shown, enhancement of keq decreases the maximum repulsive force, and the average force (parts an and b). As shown in the inset of part (a), the force-stiffness ratio decreases with increasing the cantilever stiffness. On the other hand, enhancement of keq leads to a small decrease of the 18th harmonic amplitude (part c), and does not change the phase lag of the 18th harmonic significantly (part d). The simulation parameters are set as Aset ¼ 18 nm and oexc ¼ o0.









surface implies that the tip mass ratio is useful to optimize the imaging conditions. For any given cantilever and operation condition, there is a critical tip mass ratio (near mnr ¼ 0:0537) at which a small change in the probe mass can lead to a significant variation of the normal force, the sample deformation and the measured topography and hence can damage the sample. In other words, near this tip mass ratio the system sensitivity to tip mass variation due to probe contamination during scanning or tip-particle attachment during manipulation is considerable. For the tip mass ratio where the frequency of the third cantilever eigenmode and the 18th oscillation harmonic coincide with each other, in the presence of the nonlinear interaction forces, the 18th harmonic is substantially excited, and therefore at this tip mass ratio the cantilever vibration, the exerted normal force and the sample deformation are highly influenced by the oscillation of the 18th harmonic. The amplitude and phase lag of the 18th harmonic can be recorded to map the surface properties. Inaccuracy of the lumped model depends significantly on the tip mass ratio. In the other words, the difference between the results obtained by the FEM and lumped models increases when the tip mass ratio approaches to its critical value. If the excitation frequency is set near the effective first resonant frequency of the system, the critical tip mass ratio







approaches its predicted value mrnp ¼ 0:0546, and the maximum amplitude of the 18th harmonic increases. Hence when oexc-o1eff, the maximum normal force and the difference between forces obtained by the FEM and lumped models increases. The amplitude set-point reduction (at constant free oscillation amplitude) increases the normal force exerted on the surface and consequently magnifies the amplitude of the higher harmonics. A striking result observed in the conducted simulations is that at the lower amplitude set-points the critical value of the tip mass ratio decreases slightly. In addition, the reduction of the amplitude set-point increases the inaccuracy of the lumped model particularly when the tip mass ratio is near to mnr . When the system environment has a low quality factor, the interaction force, the sample deformation, the excitation of the higher oscillation modes and the discrepancy between the lumped and FEM models become greater particularly when the tip mass ratio is near to mnr . For system with large cantilever stiffness, the force-stiffness ratio becomes small leading to a reduction in the excitation of the higher oscillation modes. However with an enhancement of the cantilever stiffness the difference between forces obtained by the FEM and lumped models increases particularly when the tip mass ratio is near to mnr .

1436

H.N. Pishkenari, A. Meghdari / Ultramicroscopy 111 (2011) 1423–1436

4 keq = 1 (N/m)

3

keq = 2 (N/m) keq = 3 (N/m)

Δ fmax (nN)

2 1 0 -1 -2 -3

0

0.01

0.02

0.03 0.04 Mass Ratio

0.05

0.06

0.07

0 keq = 1 (N/m) -0.005

keq = 2 (N/m) keq = 3 (N/m)

Δfave (nN)

-0.01

0

-0.015 -0.01 -0.02 -0.025

-0.02

-0.03 -0.03 -0.035

0.05 0

0.052 0.01

0.054 0.02

0.056

0.058

0.03 0.04 Mass Ratio

0.06 0.05

0.06

0.07

Fig. 17. Effect of the cantilever stiffness on Dfmax and Dfave. As shown, for mr 40.03, enhancement of keq leads to a considerable increase of Dfmax value, and for mr o 0.03, enhancement of keq leads to a slight decrease of Dfmax value. On the other hand, independent from mr, enhancement of keq leads to an increase of Dfave value.

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