An optimized harmonic probe with tailored resonant mode for multifrequency atomic force microscopy

Abstract

For simultaneously measuring specimen’s surface morphology and material properties, multifrequency atomic force microscopy is often employed. In this kind of atomic force microscopy, if the probe’s higher-order resonance frequencies match the integer multiples of its fundamental frequency, the probe’s responses at such harmonic frequencies will be enhanced. Meanwhile, an enlarged effective slope during vibration at the probe’s tip results in an improved probe sensitivity. Moreover, increasing the probe’s natural frequency leads to a fast scanning speed. In this study, we propose to design cantilever probes that satisfy the aforementioned requirements via a structural optimization technique. A cantilever probe is represented by a three-layer symmetrical geometric model, and its width profile is continuously varied through the optimization procedure. Thereafter, an optimized design of probe considering the fifth harmonic is prepared by focused ion beam milling. Both simulation and experiment results show that the prepared probe agrees well with design requirements.

Introduction

Given its increasing applications in biology, physics, and material science, tapping-mode atomic force microscopy (TM-AFM) ¹ is receiving increasing attention. In contrast to traditional contact-mode AFM, TM-AFM avoids lateral friction in measurement, and it results in minimal damage to the specimen. In a TM-AFM measurement, a vibration signal excites the probe at its fundamental resonant frequency, so that the tip of probe touches the specimen surface periodically. During the vibration, the optical lever method^2,3 is used to detect the probe’s deflection. The topography of the specimen is attained by scanning the desired area. Apart from traditional topography measurement, TM-AFM has also been used for material property characterization. Periodic tapping results in a periodic pulse-like nonlinear interaction force. Higher harmonic components extracted from the response of the probe contain rich information of the material properties of samples.^4–7 Therefore, the detection of higher harmonic signals that are sensitive to the variations of local mechanical properties provides a way for material discrimination. ⁸ Moreover, harmonic amplitude images have been proven to possess higher spatial resolution than topography when biological samples are scanned. ⁹

In TM-AFM, the dynamic behavior of the key sensing component, that is, the cantilever probe, considerably influences the resulting image. ¹⁰ The harmonic signals decay fast at off-resonance frequencies, so that they are usually quite weak for reliable detection. ¹¹ Therefore, effective signal enhancement methods should be developed. On the basis of the theory of vibration, a common approach is to make a higher harmonic frequency match one of the probe’s resonance frequency, thereby greatly strengthening the frequency response of the probe at this harmonic. Several research groups have made efforts by selectively altering the mass and stiffness distribution of the cantilever to achieve the above requirements. Sahin et al. ¹² cut a notch at the region where mechanical stress is large along the cantilever in the third mode to make frequency of the 16th harmonic be equal to its third-order resonance frequency. Li et al. ¹³ tailored the second and third resonance frequencies to make them match the fundamental frequency’s integer multiples by attaching a concentrated mass. Zhang et al.^11,14 tuned the frequency ratio between the higher-order and first-order frequencies by cutting one or two rectangular slots at a specific position. These frequency ratio tuning methods use intuitive regular cuttings or concentrated mass, and the aforementioned works cannot be readily extended to a general probe design case. Our previous work proposed two types of harmonic probes through structural optimization, and the final design had a variable width and a step cross section.^15,16 Sriramshankar et al. ¹⁷ designed and fabricated harmonic probes with exchangeable tip, and their proposed probe structure comprised a series of beams and blocks.

Despite the progresses, almost all the previous studies on harmonic probe designs have only focused on tuning frequency ratio; however, other objectives are also valuable for practical applications, for instance, natural frequency and its resonant mode shape, which have significant influence on measurement performance. Improving the resonant frequency of the probe enhances the scanning speed and signal-to-noise ratio. ¹⁸ Tuning an “effective slope” of the fundamental resonant mode shape at probe’s free end can improve the sensitivity of measurement because a large effective slope produces a large reflection angle of laser beam in the optical lever.^19,20

This work focuses on designing a high-performance harmonic probe. The probe has the characteristic that a required harmonic of excitation frequency is equal to a probe’s selected higher-order resonance frequency. In this research, only flexural vibration mode is considered; harmonic probe with torsional mode can be referred to Sriramshankar and Jayanth’s ²¹ work. In addition, the probe’s fundamental resonance mode shape is tuned such that it has a large effective tip slope. Mode shape tuning is addressed as an “inverse mode shape” problem, and geometry of the structure is designed for prescribed mode shapes. ²² Moreover, probe’s fundamental frequency is optimized to be increased. Increasing the probe’s resonant frequency and tuning its resonant mode shape to enlarge the effective slope at free end have conflicting requirements of the probe. Intuitively integrating a probe with certain special structural features to satisfy all these demands, along with the frequency ratio constraints, is nearly an impossible task. Therefore, in this study, a systematic design method through structural optimization is developed. The design problem is formulated as an optimization problem, and it automatically evaluates and modifies the probe design until it arrives at an optimum. ²³

Optimization model and solution

A three-layer geometric model of a cantilever probe that has been discussed in previous research ¹⁶ is adopted in this study, as shown in Figure 1. In this model, the cantilever, whose total thickness is $h$ , comprises three layers that are symmetric to the neutral plane. The middle layer, whose thickness is $t$ , is not subject to optimization and intrinsically maintains the integrity of probe. Design variable in the optimization is the top/bottom width profile $w(x)$ . The probe’s triangular shape area which is reserved for laser beam reflection is defined as a non-design region. In the finite element analysis (FEA), the probe is modeled using Bernoulli–Euler beam elements. ¹⁶ The method of FEA is referred to our previous study. ²³ In this study, only flexural eigenmodes are considered for design, and torsional modes used in torsional AFM are not considered here.

Figure 1.

Geometry model of a three-layer cantilever probe.

To design a high-performance harmonic probe, three optimization objectives are considered. First, as a harmonic probe, a probe’s selected resonance frequency ${\omega }_k,k=2,3,\dots$ , should match the desired harmonics of the excitation frequency, that is, ${\omega }_k=m{\omega }_1$ , where $m$ is a given integer. Second, the fundamental resonant mode shape ${\mathit{u}}_1$ should possess a large slope at probe’s free end. For this purpose, we set a specific defined target resonant mode shape ${\mathit{u}}^{*}$ . It has a large effective slope at free end compared with the original mode shape, and its slope can be adjusted with requirement on measurement sensitivity. Note that it is impractical to have a target mode whose shape is drastically different from that of the objective mode, because it is limited by the predetermined middle layer of the cantilever. The target mode shape is commonly a vector, which is built with discrete points. The optimization procedure is to tailor the probe’s actual resonant mode shape ${\mathit{u}}_1$ to be more similar to the target mode shape ${\mathit{u}}^{*}$ . Similarity between the two vectors is described quantitatively using the modal assurance criterion (MAC). ²⁴ Third, the probe’s fundamental resonant frequency $f_1$ is maximized to increase the scanning speed. Now, the optimization is considered as a multi-objective problem

$$\begin{array}{c}\multicolumn{1}{c}{\mathit{max}J={\lambda }_1+\alpha ·\mathbf{MAC}({\mathit{u}}_1,\mathit{u}*)}\\ \multicolumn{1}{c}{\mathit{s}.\mathit{t}.\mathbf{K}{\mathit{u}}_k={\lambda }_k\mathbf{M}{\mathit{u}}_k}\\ \multicolumn{1}{c}{{\lambda }_{\mathrm{k}}=m^2{\lambda }_1}\end{array}$$

(1)

where ${\lambda }_k({\lambda }_k={{\omega }_k}^2)$ and ${\mathit{u}}_k$ are the kth eigenvalue and eigenvector of natural flexural vibration; $\alpha$ is a user-specified weighting parameter; $\mathbf{K}{\mathit{u}}_k={\lambda }_k\mathbf{M}{\mathit{u}}_k$ is the natural flexural vibration equation for FEA. The $\mathbf{MAC}({\mathit{u}}_1,{\mathit{u}}^{*})$ is defined as

$$\mathbf{MAC}({\mathit{u}}_1,{\mathit{u}}^{*})=\frac{{\left({{\mathit{u}}_1}^T{\mathit{u}}^{*}\right)}^2}{\left({{\mathit{u}}_1}^T{\mathit{u}}_1\right)\left({{\mathit{u}}^{*}}^T{\mathit{u}}^{*}\right)}$$

(2)

MAC value is a rational number ranges from 0 to 1, and a value 1 means the two vectors are the same. Lagrangian of the optimization problem is written as

$$\mathrm{L}={\lambda }_1+\alpha ·\mathbf{MAC}({\mathit{u}}_1,{\mathit{u}}^{*})-\beta ({\lambda }_{\mathrm{k}}-m^2{\lambda }_1{)}^2$$

(3)

Gradient-based method is adopted to obtain the optimal solution. The Lagrangian $L$ is differentiated with respect to every element width $w_i$ , namely, the design variables. And, we obtained

$$L\prime =\lambda {\prime }_1+\alpha ·\frac{2\left({\mathit{u}}^{*T}{\mathit{u}}^{*}\right)\left({\mathit{u}}_1^T{\mathit{u}}^{*}\right)[\left({\mathit{u}}_1^T{\mathit{u}}_1\right)\left({\mathit{u}}^{*T}{\mathit{u}\prime }_1\right)-\left({\mathit{u}}_1^T{\mathit{u}}^{*}\right)\left({\mathit{u}}_1^T{\mathit{u}\prime }_1\right)]}{{\left[\left({\mathit{u}}_1^T{\mathit{u}}_1\right)\left({\mathit{u}}^{*T}{\mathit{u}}^{*}\right)\right]}^2}-2\beta (\lambda {\prime }_k-m^2\lambda {\prime }_1)({\lambda }_{\mathrm{k}}-m^2{\lambda }_1$$

(4)

Derivatives of eigenvalue and eigenmode, namely, $\lambda {\prime }_k$ and ${\mathit{u}\prime }_{\mathrm{k}}$ , can be calculated by Nelson’s ²⁵ method. The optimization algorithm can be found in our previous research. ²³

Example of design a harmonic probe

We start the optimization with a commercial AFM silicon probe (240AC-NA; Opus). Its length, width, and thickness are 240, 40, and 2.6 µm, respectively. The density of silicon material is 2330 kg/m³ and its Young’s modulus is 169 GPa. ²⁶ The attached tip is modeled as a lumped mass positioned at the triangular area with 4 µm to the edge. The region reserved for laser beam reflection is measured via scanning electron microscopy (SEM) to obtain its size. Thereafter, it is accounted for in the FEA by accordingly changing the width of beam elements. The nominal first resonance frequency of the probe is 70 kHz, and the frequency measured by an AFM (MultiMode 8; Bruker) is 71.9 kHz.

In practical material property mapping, the harmonic order selection is chosen on the requirement of sensitivity. Here, we select the fifth harmonic in this example, and our objective is to match the second resonance frequency of the probe to its fifth harmonic, that is, ${\omega }_2=5{\omega }_1$ . As a multi-objective optimization problem, the selection of the frequency ratio may not be arbitrary. The target frequency ratio should not be significantly different from the original case. Before optimization, the frequency ratio between the second- and first-order resonances is 6.1, which is measured by AFM. The middle layer has the thickness of 1.6 µm in this example. Optimization results with 400 iterations are shown in Figure 2. Weighting parameter here is $1\times {10}^{13}$ . Width profile is shown in Figure 2(a), and convergence history of the optimization objective is shown in Figure 2(b). The gray area represents the region to be removed. Convergence history of fundamental frequency and MAC value are presented in Figure 2(c) and (d), respectively. The fundamental frequency is changed from 72.6 to 66.07 kHz, while its MAC value is increased to 0.998 (original value is 0.962). With the sacrifice of fundamental frequency, the optimized probe’s resonant mode shape obtains more similarity with the target one. Convergence history of frequency ratio between the second- and first-order resonances is shown in Figure 2(e). The final frequency ratio is 4.98.

Figure 2.

Optimization results: (a) optimized probe with its width profile, (b) convergence history of objective function, (c) convergence history of fundamental frequency, (d) convergence history of MAC value, and (e) convergence history of frequency ratio.

Thereafter, focused ion beam (FIB) milling (Helios NanoLab G3 CX; FEI) is used to fabricate the designed profile. Fabrication parameters are given: 30 kV for ion beam voltage and 2.5 nA for ion beam current. The prepared probe is observed via SEM, and its image is shown in Figure 3(a). Then, it is mounted on an AFM (MultiMode 8; Bruker) for resonance frequency measurement. The first resonance peak $f_1$ and second resonance peak $f_2$ are 66.33 and 332.00 kHz, respectively, as shown in Figure 3(b), and their frequency ratio is 5.01. The results show that the frequency ratio of the fabricated probe matches the design requirement. Small difference between the measurement results and theoretical model analysis is mainly attributed to the fabrication or measurement error and the non-uniform of material distribution. The designed probe can be batch fabricated by lithography process, and fabrication needs further study. Also, robust design optimization of AFM probe subjected to uncertainty in fabrication ²⁷ will also be studied in the future.

Figure 3.

(a) Fabricated probe with its SEM images and (b) measured frequency spectrum of prepared probe.

By setting $\alpha$ in equation (1) to zero, another harmonic probe is obtained through the optimization without considering mode shape tuning. All other optimization parameters are the same, and the result width profile is plotted in Figure 4(a). The optimized probes’ mode shapes are studied with ANSYS, which is a commercial FEA software. The cantilever probe’s finite element model is built by shell elements, and its attached tip is mimicked by a mass element. We plot the first resonant mode shape of the two optimized probes, as well as the original mode shape and the target one all in the same coordinate as in Figure 4(b). To compare their mode shape clearly, their deflection at free ends are normalized to the same value. From Figure 4(b), a larger “effective slope” can be clearly observed in the probe that obtained with mode shape tailoring. The tip slope in the first resonant mode of the probe without considering mode shape tailoring is clearly smaller. Assume that the amplitude of the free end is 3 µm, and tip slope of the original probe and the optimized probe with mode shape tailoring are −0.0168 and −0.0198, respectively. The later slope is 17.41% larger than the previous one. The results prove that our method is effective in designing a harmonic probe with a large tip slope at its fundamental resonant mode. The original probe and our optimized one as in Figure 3(a) are used to measure scratch morphology in a thin film. The results are shown in Figure 5. It can be seen that the morphology measured by optimized probe has clearer edge, which verifies improvement of the sensitivity.

Figure 4.

(a) Optimized harmonic probe without considering mode shape tuning and (b) first resonant mode of the two optimized probes, as well as the original mode shape and the target one.

Figure 5.

Scratch morphology in a thin film: (a) measured by the optimized probe and (b) measured by the original probe.

As a general method, we can also match the second resonance frequency of the probe to its fourth harmonic in the second example. The optimization problem is the same as before except that the frequency ratio constraint is changed to ${\omega }_2=4{\omega }_1$ . The middle layer has a thickness of 1.3 µm in this example. Optimized width profile is given in Figure 6(a); other optimization results are shown in Figure 6(b)–(d). The first resonance frequency is increased from 72.6 to 74.4 kHz as presented in Figure 6(b); meanwhile, MAC value with respect to the target mode is increased from 0.927 to 0.998 as shown in Figure 6(c). The optimized MAC value is closer to 1, and a larger similarity between the target mode shape and our optimized one can be observed in Figure 6(e). Assume that the amplitude of the free end is 3 µm, and that tip slope of the original probe and the optimized probe with mode shape tailoring are −0.0168 and −0.0218, respectively. The later slope is 29.31% larger than the previous one. Convergence history of the frequency ratio between the second- and first-order resonances is presented in Figure 6(d).

Figure 6.

Optimization results: (a) optimized probe with its width profile, (b) convergence history of the fundamental frequency, (c) MAC value variation with iteration, (d) convergence curve of frequency ratio, and (e) optimized probe’s first resonant mode shape compared with the original and the target mode shape.

Conclusion

In this study, structural optimization is adopted to design high-performance harmonic probes with tailored resonant mode. Apart from assigning a higher harmonic frequency to be equal to one of the probe’s higher resonance frequency, the designed probe also possesses large tip slope in working resonance mode. The fundamental resonant frequency is maximized during optimization to ensure scanning speed. The probe is modeled as a three-layer symmetrical cantilever beam, and its width profile is continuously varied through the optimization. Matching the second resonance frequency of the probe to its fifth harmonic is as a design example. The optimized probe is fabricated using a commercial probe and FIB technique. The resonant frequency is measured by AFM, and the optimized resonant mode shape is computed by FEA, both results demonstrate the effectiveness of the proposed structural optimization. Example of design the fourth harmonic is also given.