Abstract
The characteristic of an AFM probe scanning the topography on an elastic substrate is investigated. It is different to the conventional rigid substrate. The analytical solution of the novel system is presented. For an elastic substrate the errors of frequency shift determined by using the force gradient methods and the perturbation method are satisfactory only for larger tip-surface distances. The smaller the interacting distance is, the larger the measurement error due to the amplitude of membrane is.
1. Introduction
Advances in electromechanical systems are resulting in new applications ranging from mechanical mass or charge detectors to biological imaging. The conventional atomic force microscopy (AFM) is composed of one elastic probe and the sample on a rigid substrate [1–3]. For the future application, the mechanisms of the creative AFM are studied and used to measure the nanowire or particle coated on an elastic microthin plate (membrane). It is very helpful for developing the new generation of micro/nanoactuator and AFM.
The conventional methods for determining the frequency shift are the force gradient method, perturbation method, and the analytical method. Relevant literatures are reviewed here. Holscher et al. [4] investigated the frequency shift of AFM scanning the sample on a rigid substrate by the force gradient method. For the small amplitude, the force changed linearly with the tip-sample distance D, similar to the case of a spring. Hence the influence of the tip-sample force Fts was described by a spring with a spring constant , which was equal to the negative force gradient, as .
Giessibl [5] investigated the frequency shift in dynamic force microscopy by the perturbation method. A distributed parameter system was simulated by a lumped mass one. When the tip-surface force Fts was described in terms of the tip-surface distance D as follows, , the first frequency shift could be calculated from the integral . Holscher et al. [4] investigated the first frequency shift of noncontact mode by using the lumped mass model and the perturbation method. Lin et al. [6] presented the modified perturbation method for determining the frequency shifts of higher modes.
Lin et al. [7] evaluated the force gradient method, perturbation method, and the proposed analytical method for a rigid substrate. The interpretation of frequency shift determined by the force gradient method was unsatisfactory. For a uniform beam the frequency shift determined by the perturbation method and the analytical method was very consistent. But, for a tapered or V-shaped beam at a small tip-surface distance, the frequency shift determined by the perturbation method was overestimated.
Although the approximated methods are easily used to determine the dynamic behavior of probe to predict the nanotopography of sample, significant error is unavoidable. The analytical method can provide accurate solution without error. Some analytical methods presented by Lin et al. [6–15] were successfully implemented in the study of the dynamic behavior of probe for scanning the sample on the rigid substrate. However, no analytical solution has been given to the response of a nonuniform beam with a tip mass, subjected to a nonlinear interacting force because of its complexity. So far, there is no literature on the dynamic behavior of AFM scanning the sample on elastic substrate.
In this study, the exact frequency shift of an AFM probe scanning a sample on an elastic membrane, subjected to the van der Waal force, is derived. The comparisons of the force gradient method, perturbation method, and the analytical method are presented and assessed. Finally, the effects of several parameters on the sensitivity of measurement are also investigated.
2. Coupled Probe-Membrane Model Subjected to Van Der Waal Force
2.1. Governing Equation and Boundary Conditions of Beam
The governing equation of a AFM cantilever probe, as shown in Figure 1, is [9]The clamped boundary conditions are at s= 0,The free boundary conditions are at s= Lb,The van der Waals force [13] iswhere the tip-surface distanceand AH is the Hamaker constant, D0 is the distance between the tip of the undeformed beam and the surface of sample, and R is the tip radius.

2.2. Governing Equation and Boundary Conditions of Membrane Substrate
2.2.1. Rectangular Membrane
A rectangular membrane as shown in Figure 1 is considered. The governing equation of membrane is [16] where the concentrated interacting force is .
If the membrane is clamped, the boundary condition isIf the membrane is free to deflect transversely at the boundary, the boundary condition iswhere the derivative is taken normal to the boundary.
3. Solution Methods
3.1. Coupled Probe-Spring-Membrane Model
For small amplitude of vibration, the interacting force Fv between the tip and sample around the equilibrium position of the tip can be expressed as a Taylor series. This force is approximated as a linear function of the distance between the tip and sample, which is similar to the spring. The influence of the tip-sample force can be described by a spring with a spring constant [4]. Further, the general system can be simulated in the coupled probe-spring-membrane model as shown in Figure 2.

3.1.1. Governing Equation and Boundary Conditions of Probe
The governing equation of a cantilever beam as shown in Figure 1 is [9]The clamped boundary conditions are at s= 0,The free boundary conditions are at s= Lb,where the effective spring constant and d0 is the initial interacting distance between the probe tip and the membrane.
3.1.2. Governing Equation and Boundary Conditions of Rectangular Membrane
A rectangular membrane as shown in Figure 1 is considered. The governing equation of membrane is [16]
3.1.3. Displacement of Beam and Membrane
Considering the harmonic motion of the system, the displacement of beam and membrane areSubstituting (15) into (9)–(13), one obtainswhere . The clamped boundary conditions are at s= 0,The free boundary conditions are at s= Lb,The general solution of (16) can be written aswhere the normalized fundamental solution iswhich satisfies the following normalization condition:Substituting (21) into (17)–(20), one obtainsThe amplitude of vibration of the clamped membrane can be expressed as [10]where , . Substituting (15) into (14), the coefficients are derived asSubstituting (25) into (24), the frequency equation is obtained:where (/((/σh)Ω2−)), .
The natural frequency of the system can be determined via (27).
3.2. Force Gradient Method
Holscher et al. [4] investigated the frequency shift of AFM scanning the sample on a rigid substrate by the force gradient method. The effective spring constant iswhere the effective spring constant of the cantilever uniform beam . The natural frequency under the Van der Waal force is or , where f0 is the natural frequency of the cantilever. Because , the frequency shift becomesWhen the membrane substrate is elastic, the stiffness of the membrane must be considered, but the effect of the inertia of membrane is neglected. The corresponding effective spring constant is , where the composite spring constant , as shown in Figure 3, in which the effective spring constant of the membrane is By neglecting the damping effect, the natural frequency under the van der Waals force is obtained: Further, considering the damping effect between the probe and sample in the viscous model, the damped natural frequency can be approximated by [8]where Q is the quality factor of the system and γ is the damping ratio. Obviously, the less the quality factor is, the larger the frequency shift is.

3.3. Generalized Perturbation Method
Giessibl [3] investigated the frequency shift in dynamic force microscopy by the perturbation method. The distributed parameter system is simulated by a concentrated mass one. The frequency shift without the damping effect can be calculated from the integralwhere . The displacement of membrane is derived as follows:The displacement of membrane can be determined via (34a). If , the displacement becomesIt is found from (34b) that if , .
Based on (33), the frequency of the system without the damping effect is obtained:Further, substituting (35) into (32), the damped natural frequency is obtained.
3.4. Analytical Method
3.4.1. Equation and Conditions of Probe
The dimensionless governing equation and boundary conditions of beam areAt ,At , where
3.4.2. Dimensionless Equation and Conditions of Rectangular Membrane
The dimensionless governing equation and boundary conditions of membrane areThe clamped boundary condition is
3.4.3. Solution Method
In general, it is difficult to derive the analytical solution of the coupled system with a nonlinear and time-dependent boundary condition. If the amplitude is small, the motion of the system is assumed to be harmonic [2, 12].The dynamic solutions of beam and plate can be expressed asSubstituting (17) into (6b), the corresponding dynamic interacting distance becomeswhere the dimensionless interacting amplitude isThe minimum interacting distance .
(a) Displacement Amplitude of Probe. Substituting (17) into the governing equation (9) and the boundary conditions (10)–(13), one obtainsas ,as ,Because condition (49) is time-dependent, the balanced method is considered here [10]. Multiplying (49) by and integrating it from 0 to the period T, , (49) becomesThe general solution of (45) can be written aswhere are the normalized fundamental solution which satisfies the following normalization condition:Four normalized fundamental solutions of (51) arewhere . Substituting (51) into (46)–(48), (50a), and (50b), the beam displacement iswhereIt should be noted that “” is the frequency equation of the independent beam.
(b) Displacement Amplitude of Membrane. Substituting (43) into (41) and multiplying it by and finally integrating from 0 to the period T, , (41) becomesSubstituting (43) into the clamped boundary condition (42), the associated boundary condition isThe amplitude of vibration can be derived [16]:It is found from (57) that the dimensionless natural frequency of an independent clamped membrane is
(c) Frequency Equation. Substituting the probe-membrane amplitudes (54a), (54b), and (57) into the interacting amplitude (44b), the frequency equation of the coupled subsystem is obtained: It is observed from (59) that the frequencies of oscillation are dependent on the amplitude of oscillation at the tip. Given the interacting distance , the exact corresponding frequencies can be easily determined via (59) by using the numerical method proposed by Lin [10].
4. Dynamic Results and Discussion
Considering a rigid substrate, comparison of the frequencies determined by different methods is made and listed in Table 1. It is found that the results determined by the analytical method for are the same as the exact solutions determined by Lin [9]. The results determined by the perturbation method for are slightly different from the exact solutions. When the tip-surface distance is small, the results determined by the force gradient method and the beam-spring-membrane model are greatly different from the exact solutions. The errors of frequency shift determined by using the force gradient methods are satisfactory only for larger tip-surface distances. The errors of frequency shift determined by using the perturbation method are smaller than those by the force gradient method for small tip-surface distances.
Table 2 demonstrates the effect of elastic membrane substrate on the coupled frequencies. The membrane inertia is neglected in both the force gradient method and the perturbation method. It implies that although the coupled frequency depends on the membrane stiffness, but no additional mode except the beam ones occurs. Nevertheless, the membrane inertia is considered in the beam-spring-membrane model and the analytical method. Several additional modes except the beam ones occur for small interacting distance D0 between the tip and the sample. For , the frequencies determined by the force gradient method and the perturbation method are significantly different from those by the beam-spring-membrane model and the analytical method. In the conventional measuring methodology, the topography of sample is observed based on the dynamic behavior of beam. Due to the additional modes, it likely results in the error of measuring the topography for small interacting distance D0. Further, the coupled frequencies determined by the analytical method are shown in Figure 4. Mode A denotes the closest one to the first natural frequency of an independent cantilever; Mode is B next. It is observed from Figure 4(a) that there is multimodes phenomenon for the probe-elastic membrane model. Figure 4(b) demonstrates that, for mode A, the amplitude of beam oscillation dominates. For mode B, the amplitude of membrane oscillation dominates.

(a)

(b)
Figure 5 demonstrates the effect of the rigidity ratio on the frequency shifts of modes A and B. When the interacting distance , the frequency shift is small. When the interacting distance , the frequency shift is obvious. There are negligible differences among the frequency shifts for the different rigidity ratio, . The smaller the interacting distance is, significantly the larger the frequency shift is. For the interacting distance , there is great difference among the frequency shifts for the different rigidity ratio, .

(a) Mode A

(b) Mode B
Further, the characteristic of the associated amplitudes of mode A is investigated and listed in Tables 3 and 4. Table 3 demonstrates the effects of the interacting distance and the rigidity ratio, , on the amplitudes of beam and membrane for mode A. The smaller the interacting distance is, the larger the amplitude of membrane is. The error of measuring the topography on the membrane is increased due to the amplitude of membrane. If the interacting distance is about 10.5 nm, the corresponding amplitude of membrane is about 0.125 nm. Table 4 demonstrates the effects of the interacting distance and the interacting position on the amplitudes of beam and membrane for mode A. It is observed that, for =0.1, the amplitude of membrane is very small. It means that the beam vibration dominates. The coupled frequency shift is close to that of the system with rigid substrate. If and =0.5, the corresponding amplitude of membrane is about 0.033 nm.
Figure 6 demonstrates the effects of the interacting distance and the interacting position on the frequency of mode A. When the interacting distance , the frequency shift is small. When the interacting distance , the frequency shift is obvious. There are negligible difference among the frequency shifts for different measuring positions, . The smaller the interacting distance is, significantly the larger the frequency shift is. For the interacting distance , there are great differences among the frequency shifts for different measuring positions, .

Figure 7 demonstrates the effect of the interacting distance D0 on the first five natural frequencies of the coupled probe-membrane system. When the interaction distance D0 is large enough, the coupled natural frequencies approach those of the independent probe or membrane without the coupled effect. It is because the van der Waals force decreases with the interaction distance D0. If the interaction distance D0 is decreased less than 6 nm, all the natural frequencies become smaller significantly.

5. Conclusion
In this study, the analytical solution, the force gradient method, the probe-spring-membrane model, and the generalized perturbation method are presented. The comparison of their results is made. Considering a rigid substrate, both the analytical method and the perturbation method are accurate. However, for an elastic substrate the errors of frequency shift determined by using the force gradient method and the perturbation method are satisfactory only for larger tip-surface distances. Several trends about the frequency shift and the induced error of measurement are obtained as follows:(1)Because the membrane inertia is considered, several additional modes except the beam ones occur for small interacting distance D0(2)For larger frequency shift sensitivity and small amplitude of membrane, smaller error of measure will be obtained(3)The smaller the interacting distance is, significantly the larger the frequency shift and the amplitude of membrane are(4)When the interacting distance is about 10 nm, the frequency shift is obvious. The corresponding amplitude of membrane is less than 0.15 nm. If is decreased, the amplitude of membrane is significantly increased. It will increase the error of measurement
Nomenclature
| : | Cross-sectional area of beam |
| : | Hamaker constant |
| : | Tip-membrane distance |
| : | Dimensionless tip-membrane distance, |
| : | Young’s modulus |
| : | Interacting force between beam and membrane |
| : | Thickness of membrane |
| : | Height of tip |
| : | Dimensionless height of tip, |
| : | Lengths of beam |
| : | Lengths of membrane in the x- and y-directions, respectively |
| : | Tip mass of beam |
| : | Mass of the membrane per unit area |
| : | Aspect ratios, , |
| : | Ratio of masses of membrane and beam, |
| : | Ratio of beam bending rigidity and membrane tension, |
| : | Tip radius |
| : | Tip radius, |
| : | Coordinate of beam |
| : | Time variable |
| : | Transverse displacements of beam and membrane |
| : | Coupled amplitude of beam and membrane, |
| : | Dimensionless coupled amplitude of beam and membrane, |
| : | Dimensionless transverse displacements of beam and plate, |
| : | Principal frame coordinates of membrane |
| : | Interacting position |
| : | Beam flexibility coefficient, |
| : | Dimensionless tip mass of beam, |
| : | Tensile force per unit thickness of membrane |
| : | Dimensionless principal beam coordinate, |
| : | Natural frequency |
| : | Dimensionless natural frequency, |
| : | Dimensionless principal frame coordinates of membrane, |
| : | Dimensionless interacting position, /, / |
| : | Mass density |
| : | Dimensionless time, |
| : | Beam |
| : | Membrane. |
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The support of the Ministry of Science and Technology of Taiwan is gratefully acknowledged (Grant no. MOST 106-2221-E-168-005).