Improved sensitivity for subsurface imaging by contact resonance atomic force microscopy using Fano peaks

Recent developments in the semiconductor,^1–3 precision manufacturing,⁴ precision optics,^5,6 and biomedical industries^7–9 have led to the creation of many novel subsurface inspection methods. Techniques such as focused ion beam (FIB),¹⁰ industrial computed tomography (CT) imaging,¹¹ and ultrasonic imaging¹² can not only test the quality of industrial products but also find applications in probing the internal structure of biological cells.¹³ Among the available measurement methods, ultrasonic subsurface microscopy is advantageous owing to its non-destructive nature, low cost, and high efficiency. However, the detection capability of this system is limited by ultrasonic diffraction, which only allows subsurface detection at the level of several micrometers.¹⁴ Measurement methods based on atomic force microscopy (AFM) offer several advantages, including high spatial resolution that allows for the recognition of morphological changes in single atomic layers and the ability to cover a wide range of samples.^15,16 Contact resonance AFM (CR-AFM) combines the advantages of ultrasonic microscopy and AFM as a subsurface measurement technique. CR-AFM applies ultrasonic vibrational excitation to either the sample^17,18 or the probe.^17,19,20 Operating in AFM contact mode, the excitation frequency of either the probe cantilever or the sample is selected from its contact resonance frequency. This method detects changes in the subsurface morphology or material of the sample by analyzing the frequency,²¹ amplitude,^17,22 and phase¹⁹ of the contact resonance peaks between the probe and the sample surface during the scanning process.¹⁷ 

The selection of the probe excitation frequency is key to the CR-AFM subsurface measurement process. It has been shown that the choice of different contact resonance frequencies, each representing a specific order of vibration modes, can directly affect the results of subsurface measurements.^23,24 In addition, the shape of the contact resonance peak can be either symmetric or asymmetric.²⁵ This difference in symmetry results in divergent trends in amplitude development as the probe travels across different subsurface features.²⁴ Variations in the signal-to-noise ratio (SNR) were also observed when experiments were performed utilizing different types of resonant peaks.^17,24 However, the experimental performance evaluation of these different symmetry peaks and the study of their applicability in CR-AFM for subsurface imaging have not been fully discussed and thoroughly addressed.

Resonance is usually considered the enhancement of a system’s response to an external excitation at a particular frequency. In such a system, the vibration states between the two resonant objects can affect each other.²⁶ In general, there are two resonances located close to the eigenfrequencies ω₁ and ω₂ of two oscillators. One of the resonances demonstrates the standard enhancement of the amplitude near its eigenfrequency, while the other is an unusual sharp resonance with a smaller amplitude, showing the coupling effect of the two oscillators.²⁶ The first resonance is characterized by a symmetric profile, described by a Lorentzian function and is named Lorentzian resonance.²⁷ The second resonance, characterized by an asymmetric profile,²⁸ is called Fano resonance and exhibits total suppression of the amplitude of the forced oscillators at the eigenfrequency of the second oscillator ω₂.

Unlike in the case of symmetric Lorentzian curves, which have long been used as a general representation of spectral dependence, the first observation of asymmetric line shapes can be traced back to a discovery made by Wood in 1902.²⁹ He noted the presence of unexpected narrow bright and dark bands in the spectrum of an optical reflection grating illuminated by a light source with a slowly varying wavelength. Later, in 1961, Fano^30,31 showed that quantum interference between a continuum of states (inelastic scattering of electrons by helium atoms) and discrete states (resonant scattering via autoionization) results in a new resonance behavior characterized by an asymmetric shape. Since then known as Fano resonance, a manifestation of the interference between a localized wave and propagating states has demonstrated its universal behavior in physics, independent of the system configuration. Examining Fano’s microdynamic model reveals its similarity to tip–sample interaction pairs. The probe cantilever can be represented as a discrete state, a specific vibrational mode with a known modal, while the bulk sample can be represented as a continuous state with an infinite number of vibrational modes.

To distinguish the effects of symmetric and asymmetric peaks on measurement results using the CR-AFM method, this study investigates the mechanism behind the different symmetries. Experiments were carried out to show the subsurface sensitivities of different resonant modes utilizing probes with different cantilever stiffnesses. Section II presents a series of experiments comparing the performance of the two types of resonant peaks, along with an introduction to the Fano peak and its relationship with subsurface features. Section III describes the construction of a theoretical model, and several proposals related to subsurface detection capability are made accordingly. This study explains why higher-quality subsurface images can be obtained using Fano peaks by simulating probe dynamics and evaluating the quality factor for both peaks. In addition, the pros and cons of different contact resonant measurement modes are discussed, and suggestions are provided on how to leverage the properties of these contact resonant modes to obtain more credible measurement results.

First, the subsurface measurement results are analyzed to outline the characteristics of these peaks and their interactions with different subsurface features.

A typical experimental CR-AFM system is shown in Fig. 1. The probe is scanned in contact mode, while a high-frequency actuation signal is provided by a signal generator to the piezoelectric sheet at the base of the AFM cantilever. This signal is converted into a time-varying vibrational interaction between the tip and the sample surface. As the surface contact stiffness of the sample varies, the resonant frequency of the probe–sample contact pair shifts from the drive signal, resulting in changes in the vibration phase and amplitude. The deflection signal from the vibrating cantilever is processed using a lock-in amplifier, and the amplitude/phase information is extracted to synthesize an image of pixels representing the variation in vibration amplitude across the surface.

The major difference between CR-AFM and conventional AFM is the contact resonance between the probe and the sample. This modification provides CR-AFMs with additional subsurface imaging capabilities. Therefore, to obtain a more resolved image of the subsurface features, resonance peaks with either a high-quality factor (Q) to achieve low dispersion or high sensitivity to subsurface variations are selected.

In general, there are symmetric peaks at all orders of modes of the probe and sample, as well as many smaller asymmetric peaks that exhibit asymmetry in the vicinity of these symmetric peaks. A primary comparison of the frequency spectra around these peaks with different symmetries is shown in the thumbnail plot in Fig. 1. These two types of peaks are referred to as Lorentzian and Fano peaks, respectively, as mentioned in Sec. I. However, despite being a common experimental phenomenon, there has been little research on Fano peaks and the mechanism behind them in the field of CR-AFM. Furthermore, their capabilities in micromechanics, such as cantilever–tip–sample triplets and promising performance in terms of extremely high-quality factors, have not been studied.

Experiments were carried out using a modified commercial AFM platform (Bruker Icon, USA). Two types of probes were utilized during the experiments, and their detailed specifications are listed in Table I. The probes were actuated from a standard probe base using a piezoelectric ceramic piece. The outputs were read from a signal-access module (Bruker SAM 6, USA) and fed back to the AFM controller. The experiments were carried out in an environment with a temperature of 25 °C and a humidity of ∼30%. The cantilever amplitude scale in the manuscript is derived from a feature of Bruker AFM that converts laser feedback position signals into amplitude, i.e., thermal deflection. Such a scale is derived from free space vibration modes and is only valid for vibration below the fundamental modes. The scale cannot reflect the actual vibration amplitude of the cantilever in the contact state or in the higher order modes. To avoid ambiguity, the amplitude scale for the measurement results is given in arbitrary units in the manuscript.

A part of spectrum of the cantilever (Rfespa-75 in Table I) contacted with a bulk of PMMA sample is shown in Fig. 2. The bandwidth of the phase lock loop utilized for frequency measurement was set to 0.4 kHz to capture the full spectrum. Apart from the aforementioned characteristic resonance peaks, there are some other spurious peaks in the full-frequency spectrum originating from various mechanisms. Modeling these peaks requires consideration of additional interactions or noise sources in the entire system. Various studies have considered elastic suspension,³² thermal noise,³³ and other systematic nonlinearities.^34,35 The effect of these mechanisms on the typical Fano peaks remains the subject of further research and is not included in this article for clarity.

Two types of samples were used to compare the effects of Lorentzian and Fano peaks on subsurface morphology. The first sample, shown in Fig. 3, was made of a silicon grating covered with a graphene oxide (GO) film. The silicon grating was fabricated by photolithography with a gate width of 2 mm and a period of 4 mm, as shown schematically in Fig. 3(a). The silicon grating was then covered with a GO film with a thickness of 317 nm. The GO film and the top of the sample were firmly bonded together by van der Waals forces. A cross-sectional scanning electron microscopy (SEM) image of the specimen used in this experiment is shown in Fig. 3(b). The second sample is shown in Fig. 4. In this sample, Au particles of ∼50 nm in diameter were spun onto a silicon substrate, which was further coated with a 163 nm thick polymethyl methacrylate (PMMA) film, as shown in the schematic in Fig. 4(a). A cross-sectional SEM image of the prepared specimen is shown in Fig. 4(b).

The Otespa-R4 (Table I) probe was used for these measurements. The selected resonance frequencies of the Fano and Lorentzian peaks were 1763 and 1990 kHz, respectively (Fig. 5). The probe was maintained in contact with the sample surface at a nominal load of 20 nN. The lateral speed of the scanner was set to 20 mm/s to achieve a 1 Hz line rate on a 20 × 10 mm² scan region, resulting in a 512 × 256 pixel rectangular image of the measurement results.

A comparison of the subsurface measurement results using either the Lorentzian or Fano peaks under the same experimental conditions is shown in Figs. 6(b) and 6(c). The subsurface structures are clearly visible in both figures, whereas in the surface result plot shown in Fig. 6(a), the subsurface structures are barely discernible. Furthermore, measurements using the Fano peak can distinguish the boundaries of the grating structure with higher contrast. In contrast, the Lorentzian peak yields an almost binary plot with a worse lateral resolution and blurred patterns. This comparison suggests that the shift in the central frequency of the Fano peak, which intuitively originates from changes in the probe–sample contact stiffness, results in a more balanced and stable change in amplitude compared to the Lorentzian peak.

In another experiment, the Rfespa-75 (Table I) probe was used to obtain the subsurface tomography of a sample. The selected resonance frequencies of the Fano and Lorentzian peaks were 1438 and 2430 kHz, respectively, as shown in Fig. 7. The peak at f_F was asymmetric, indicating that it was a Fano peak, whereas the peak at f_L was symmetric, indicating that it was a Lorentzian peak. The imaging setup was the same as that in the first experiment.

The measured surface topography results are shown in Fig. 8, and the corresponding subsurface measurement results utilizing the Lorentzian and Fano peaks are shown in Figs. 9(a) and 9(c), respectively. Figures 9(b) and 9(d) show the results of the experiments performed on the magnified regions of interest (ROI) from Figs. 9(a) and 9(c). The results obtained using Fano peaks were better resolved with a relatively lower noise level. Evidently, the utilization of asymmetric Fano contact resonant peaks results in superior transversal resolution and contrast compared to its Lorentzian counterpart, despite having a lower amplitude.

To validate the above findings, a smaller area with a denser distribution of Au particles from Fig. 4 was selected for a more detailed comparison of the Lorentzian and Fano peaks. The ROI was set to 1.6 × 1.6 mm², and the line rate was maintained at 1 Hz, yielding a scanning speed of 1.6 mm/s. The pixel rate and load on the probe were identical. The subplots in Fig. 10(a) show a peak-to-valley (PV) value of ∼60 nm across the measured sample surface. Figures 10(b) and 10(c) display the subsurface measurements using the Fano resonant peak and Lorentzian peak, respectively, selected from the frequency spectrum in Fig. 11. The two peaks are adjacent to each other to avoid parasitic parameter variations.

The subsurface image obtained using the Fano peak was less affected by surface topography and showed better contrast between the Au particles and the sample substrate, with sharper boundaries. In particular, for the two labeled regions, we observe the following:

• Region Ⅰ represents a shallow pit on the surface. The Fano result is only slightly affected, as depicted in Fig. 10(b), whereas the Lorentzian result is more significantly affected in the same region.

• Region Ⅱ represents a small bump on the surface. The Fano result is almost unaffected, whereas the Lorentzian result fails to recognize the subsurface particles in the vicinity.

The Fano peaks also provide a better definition of the subsurface features. The profiles of the three recognized particles, denoted by lines 1, 2, and 3 in Figs. 10(b) and 10(c), are plotted in Figs. 10(d) and 10(e), respectively. The results from the Fano peak show better repeatability and are more compatible with the properties of the sample itself; the Au particles supposedly have diameters of ∼50 nm, which is very close to the true value. Contrarily, the recognized particle diameters under the Lorentzian peak are exaggerated by a factor of ∼1.5 and suffer from much worse conformity. This discrepancy may be due to the larger amplitude of the Lorentzian peaks, which exerts a larger force on the sample surface. When the contact load increases, the accumulated deformation on the sample surface may reach the subsurface features, causing the dislocation of the surface to shift laterally from the exact subsurface features.³⁶ 

These findings indicate that Fano resonant peaks are, to some extent, better suited for sensing small, isolated features below the surface. In Sec. III, we explain why Fano peaks show such superiority from a theoretical perspective based on dynamical modeling and modal analysis.

Based on the aforementioned experimental phenomena, a more detailed analysis is presented in this section, starting with an examination of the generation of the Fano peaks. Similar to optoelectronics and other fields,³⁷ it is believed that Fano peaks in the case of contact resonance are generated by the coupling of two vibrational states. The classical spring–mass model was introduced to analyze the probe–sample contact resonance. The measurement advantages of the Fano peak were also analyzed from the perspective of the Q-value. Furthermore, although beyond the scope of this study, the selection of the stiffness of the cantilever and sample surface is important for obtaining a better-resolved subsurface image. The criteria for the selection of the experimental design can be found in the  Appendix and are not discussed in this section.

For the coupled cantilever–tip–sample surface system, the model starts with a simplified schematic of the tip–sample interaction pair, as shown in Fig. 12(a), and a simplified system of coupled point masses, dampers, and springs is proposed in Fig. 12(b). Unlike traditional tip–sample models, to characterize the coupling of the probe to the sample vibrations, a pair of spring–damper systems (k_cs, γ_cs) is added to describe the tip–sample contact, while another pair of spring–damper systems (k_s, γ_s) is added between the thin layer of the sample surface and the sample substrate. The probe and sample surfaces are then represented by mass blocks m_c and m_s, respectively.

Notice that without the interaction pair $ksc,csc$⁠, the system shown in Fig. 12 will degenerate into an ordinary two-degree-of-freedom (DOF) dynamical system. This kind of 2-DOF system model, as well as its minor modifications, lacks an embedded mechanism to demonstrate the presence of asymmetry in the frequency domain, as proven in many previous studies. In addition, in the derivation of the above equation, the pair of spring–damper systems (k_s, γ_s) characterizes the vibration of the surface layer of the sample. It can be interpreted as follows: When features such as cavities or heterogeneous particles are present on the subsurface of the sample, it is equivalent to the surface of these structures being covered with a thin film. Thus, the coupling of the probe and sample vibrations can be explicitly demonstrated, thereby elucidating the appearance of the Fano peak.

In classical mechanics, Fano resonances can be analogized as disturbances between two successive states of vibration.³⁷ In the case of contact resonance between the tip and sample surface, the vibrations of the cantilever and sample surface also interfere with each other. As a result, the presence of a Lorentzian peak in the amplitude spectrum of the probe implies that there is a free-vibration mode of the cantilever near this frequency. However, the appearance of the Fano peak implies that there is a vibration mode of the sample surface near that frequency, with a frequency shift originating from interference with the vibrational mode of the cantilever.

When Lorentzian peaks are used for measurement, changes in the central frequencies reflect changes in the contact resonance peak of the probe, indirectly characterizing changes in the sample surface stiffness and, thus, the subsurface morphology. However, when the Fano peak is used for subsurface morphology measurements, the change in the peak directly reflects changes in the intrinsic frequency of the sample near the peak, thereby revealing subsurface information. This is consistent with our previous experimental results in Fig. 10, which show that the Fano peak heavily relies on variations in the sample properties, whereas the Lorentzian peak reflects more coupled interference between the probe and the sample surface.

The geometric model shown in Fig. 12(b) was used for the probe vibration-frequency simulation. Finite element simulations were performed using COMSOL Multiphysics. When the equivalent mass and stiffness of the probe are constant and those of the sample are varied, the first-order contact resonance amplitude spectrum of the cantilever, according to Eq. (3), is shown in Fig. 13. The equivalent mass and stiffness of the probe are held constant at 0.07 mg and 26 N/m, respectively. As the equivalent stiffness and mass of the samples increase, the equivalent mass of the sample increases, indicating that defects on the subsurface of the sample become larger or that the subsurface material is softer, causing the equivalent stiffness of the sample surface to decrease. These changes lead to an overall leftward shift in the probe spectrum. Conversely, the frequency spectrum shifts to the left. Evidently, the Fano peak is more sensitive to the equivalent contact stiffness and mass changes of the sample than the Lorentzian peak. This sensitivity implies that the Fano peaks in the probe spectrum are more responsive to changes in the sample and may provide higher subsurface image contrast in the CR-AFM method.

Reflecting on the experiment in Fig. 9, the Q-values of the Lorentzian and Fano peaks in Fig. 10 are 12 and 433, respectively. The quality factor of the Fano peak is 35 times higher than that of the Lorentzian peak.

This means that as long as the criterion ω_c < 0.618ω_s is satisfied, Q_Fano is always higher than Q_Lorentzian.

The advantages of a higher Q-value for subsurface measurements are as follows: A higher Q-value means that less vibrational energy is lost in a cycle.^39,40 In the subsurface scanning process, a less energy loss for similar energy and excitation frequencies usually implies a greater penetration depth. Conversely, a high Q-value means that the system requires less energy input for the same penetration depth. In other words, the probe cantilever requires less amplitude for the same probing capability, which reduces the plastic deformation of the surface and minimizes damage from contact measurements. Second, a higher-quality factor results in a narrower full width at half maximum (FWHM) in the spectrum, enhancing frequency selectivity. This means that the measurement process is closer to a single-frequency measurement, suffering less dispersion.⁴¹ When mutual interference between frequencies is reduced, the sharpness of the image can be significantly improved, as confirmed by our experimental results.

CR-AFMs are promising and powerful tools for subsurface topography exploration and characterization. In this study, we investigated the subsurface detection capabilities of CR-AFMs in detail. We found that under certain circumstances, measurements using Fano peaks not only benefit from higher sensitivity to subsurface features but also effectively reduce surface damage owing to their generally smaller amplitude compared to Lorentzian peaks.

This study can be summarized as follows: First, a series of experiments on various samples was conducted, confirming the existence of anti-resonant Fano peaks and their superior capability for subsurface imaging applications. While these peaks are well-known in quantum photonics, they are rarely discussed in micromechanics, despite originating from idealized micromechanics models.

Second, we revisited the theory of Fano peaks and derived a kinematic model of the contacts between the probe and sample in CR-AFM. This helped explain the basics of symmetric and asymmetric peaks in the probe resonance spectrum. Concepts from quantum mechanics for identifying Lorentzian and Fano peaks were introduced and extended to the dynamic modeling of surface contact resonance to distinguish them.

Finally, based on experimental results and subsequent modeling work, we revealed why Fano peaks achieve better subsurface measurements than Lorentzian peaks from a quality factor point of view.

Thus, despite their smaller amplitudes, Fano peaks provide better-resolved subsurface information. Their smaller amplitudes reduce the likelihood of surface damage, and their anti-resonant nature makes them less sensitive to surface topography. In addition, their high Q-factor and narrower bandwidth make them more sensitive to changes in subsurface characteristics. Fano peaks, despite providing high-resolution subsurface information, have some disadvantages. Their high sensitivity limits their use for measuring large variations in samples, requiring more sophisticated measures, such as frequency-tracking UAFM, to cover a wide dynamic range. Fano peaks also operate at higher actuation frequencies than Lorentzian peaks, posing challenges in the selection of both the actuation system and signal processing equipment. Further investigation into subsurface detection capability is promising and will be confirmed and justified in more detail in future experiments.

This work was supported in part by the National Natural Science Foundation of China under Grant No. U22A20207, the National Natural Science Foundation of China under Grant No. 51975522, the National Key R&D Program of China under Grant No. 2022YFB34000102, and the Zhejiang Provincial Key R&D Program of China under Grant 2023C01056. The authors also gratefully thank Wei Wang for her assistance with the SEM.

The authors have no conflicts to disclose.

Yuyang Wang: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Methodology (equal); Resources (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Mingyu Duan: Conceptualization (equal); Software (equal); Validation (equal). Yuan-Liu Chen: Funding acquisition (equal); Project administration (equal); Resources (equal); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

The values of k_nL under free-vibration conditions are constant, and the first four roots of the above characteristic equation are listed in Table II.

More detailed research considering the probe tip shape and position relative to the cantilever-free end, as well as the contact angle against the sample surface, can be found in a previous study.⁴² For example, in this study, we utilized two types of probes, the properties of which are listed in Table I.

In addition, consider that the cantilever is at an oblique angle of α = 15° against the sample surface. The solutions for such a system can be found numerically using MATLAB and are listed in Table III.

The first four resonance frequencies f_n of a spring-coupled cantilever are shown as functions of k_cs/k_c in Fig. 14. Here, f_n is normalized to the first resonance frequency of the clamped–free cantilever.

As shown in the plot, the resonance frequency of each mode of the clamped–spring-coupled cantilever lies between the resonance frequency of the clamped–free cantilever mode and the resonance frequency of the clamped–pinned cantilever mode. Furthermore, it can be concluded from the plot that the first-order resonant modes have a much wider frequency-tuning range under minor changes in the sample surface than the others. In particular, f_n/f₀ varies by ∼5.751 times for k_cs/k_c from 1 to ∼10³, making it suitable for sensing similar materials of lower stiffness with large contrast. Higher resonant modes, although less sensitive to small variations in sample stiffness, can distinguish much larger changes in sample stiffness, i.e., k_cs/k_c from ∼10² to ∼10⁶, making them more capable of imaging hard materials. Combined with more advanced instruments offering a higher frequency resolution, the acquired images can present more detailed information about the materials between the k_cs/k_c upper and lower limits with a balanced weight, rather than extreme binary demonstrations limited to a relatively small stiffness range.

This method helped us select the stiffness of the probe and its vibration modes to obtain a clear characterization of the subsurface morphology. However, because the Fano vibration is a peak generated by the resonant coupling between the probe and sample, the above vibration modes do not include the Fano vibration. To compare the sensitivity of the Fano vibration with the modal vibration of the probe itself, it is preferable to select the first-order vibration with the widest effective range for a simulated comparison with the Fano peak. This guideline informed the parameter selection in our experimental system, as shown in Figs. 6–8.

In general, we consider the Fano peaks to have similar properties to other antisymmetric peaks in that vibrational minima occurs at the antisymmetric resonance point because of force balance. In our case, the contact region between the probe tip and the sample surface at such peaks should exhibit a very low deformation. As a result, the modal shape of Fano peaks shows a relatively small displacement near the probe tip base, which distinguishes such peaks greatly with general spurious peaks. We have also done a series of modal analyses using the commercial FEM software COMSOL, and a fundamental resonant mode simulation can show the difference clearly as in Fig. 16.