1 Introduction

Recently, nanoscale structures have been applied to an increasing range of products, including nanosensors [1] and nanofluidic devices [2]. Many nanofabrication methods have thus been developed to meet growing demands for greater accuracy. Focused ion beam (FIB) fabrication [3] is a nanotechnology method that uses strong electrostatic fields to focus, accelerate, and control ions or particles to perform subtractive and additive material processes with sub-10 nm resolution [4] on various materials. However, FIB fabrication requires application of high voltages to establish the required electrostatic fields [5], leading to significant energy consumption. Electron beam lithography (EBL) [6] controls a finely focused electron beam to etch thin films of electron-sensitive materials to fabricate patterns on the scale of a few nanometers [7]. However, similar to FIB fabrication, nanoscale fabrication using EBL requires a significant power supply [8] and is limited by processable materials. Nanoimprinting lithography (NIL) is a low-cost method [9] that enables manufacturing of structures on the sub-10 nm scale [10]. The main limitation of NIL is that it can only replicate the specific pattern defined using a mold on the substrate, thus meaning that the method lacks flexibility. Scanning probe lithography (SPL) [11, 12] provides another low-cost and flexible nanoscale fabrication method.

The SPL nanofabrication technique uses the sharp-tipped probe available in scanning probe microscopes (SPMs) including the scanning tunneling microscope (STM) [13] and the atomic force microscope (AFM) [14]. The STM operates by inducing a tiny electric current between its sharp probe and the surface of a sample via the electron tunneling effect. In contrast, the AFM works by scanning across the sample surface using a tiny probe tip, which is often composed of a sharp cantilever, and subsequently generates a detailed map of the sample surface by measuring the force of the interaction between the tip and the surface. SPL enables precise fabrication to be performed on the nanoscale using the capabilities of these instruments. Several types of SPL technology have been developed. Thermal SPL technology [15] uses the heated tip of the probe in an SPM to soften or melt materials and thus achieves fast and reliable nanoindentation with sub-10 nm resolution [16]. Unfortunately, this method is only suitable for patterning of highly thermosensitive materials with high resolution and requires high-temperature-resistant probe materials. In oxidation SPL [17], an electronic bias or localized heat is applied to the probe tip to generate a controllable oxidation process on the sample surface, enabling the creation of stable patterns on the scale of a few nanometers [18]. Because the process relies on oxidation of the sample, oxidation SPL encounters challenges when processing materials that are difficult to oxidize. The dip-pen SPL [19] technique uses materials that are wrapped in or injected like ink into the probe tip, enabling direct deposition of nanopatterns [20] on the sample surface through controlled contact between tip and substrate. However, it is essential to note that various inks require specific processing conditions, including optimal humidity and temperature levels, to ensure consistent and continuous ink transport [21]. Inadequate processing conditions can lead to disrupted or irregular ink deposition, which compromises the quality and integrity of the patterned features. An alternative approach, mechanical SPL [22], has been developed to avoid the potential issues encountered in the methods described above. Furthermore, in comparison with additive processing, mechanical SPL techniques offer distinct advantages such as enhanced surface quality and low cost.

Mechanical SPL enables fabrication of patterns on sub-10 nm scales [23] using a fundamental mechanical process that involves using the sharp tip of the cantilever as a cutting tool to perform indentation, scratching, and other actions on the sample surface. Because mechanical SPL is based on the most basic mechanical removal methods, it successfully circumvents the specific material and environmental requirements associated with the other SPL methods, thereby expanding its applicability to a broader range of operations. Jung et al. [22] not only conducted static scratching using the microcantilever in an AFM, but also introduced an innovative dynamic plowing lithography method, in which external oscillations were applied to the microcantilever while lateral oscillations were applied to the samples to mitigate the slip-stick motion, which affects the machined depth, along with the torsion of the microcantilever, which can affect the process resolution. Since then, application of vibration between the microcantilever and the samples to be processed has garnered increasing attention because of its numerous merits. Reducing the friction between the tip and the samples causes tip wear to be minimized [24, 25] and also significantly enhances the processing efficiency [26, 27]. In current AFM devices, cantilevers can be conveniently subjected to vibration using their built-in tap** mode [28], which usually drives the microcantilever into resonance via a piezo-actuator. Numerous studies have been performed using resonant microcantilevers in AFMs. Heyde at el. [29] drove a microcantilever using a dither piezo element operating in proximity to its natural frequency to perform nanolithography on polymethylmethacrylate and demonstrated that the amount of machining could be controlled using the input voltage applied to the dither piezo. Cappella et al. [30] performed a comparison study between nanolithography using a resonant microcantilever and using static indentation, thus revealing that nanolithography using the resonant microcantilever offers advantages in terms of processing speed and in enabling easier elimination of undesirable border walls around the cut grooves. ** mode, along static plowing lithography. Their study showed that the machining force when using dynamic plowing lithography was approximately half that required by static plowing lithography to achieve the same groove depths. Yang et al. [32] used the generation of self-excited oscillations within the microcantilever, rather than relying on external excitation. Their research substantiated the technique’s ability to achieve greater machined depths when using the self-excited microcantilever. Additionally, they proposed a machining depth control strategy based on manipulation of the phase shift in the feedback loop. However, detection of the microcantilever’s vibrations in the AFM is typically achieved through an optical lever system; consequently, the way in which the microcantilever tip moves during processing remains uncertain. Because the optical lever system only measures the deflection angle at one specific point, the same output waveform can correspond to multiple distinct vibration modes of the microcantilever. Varying vibration modes can cause the tip to perform machining using different mechanisms, which can potentially result in a divergent range of implications for the machining process. Therefore, it is essential to determine the mechanism behind the formation of the machined grooves. In this paper, research is conducted to elucidate the mechanism involved in nanoscale machined groove formation by the mechanical SPL.

In this paper, mechanical nanolithography is considered to be performed by using a self-excited microcantilever in an AFM as the machining tool. Two prospective machining modes are proposed: the first is the tap** mode, in which the microcantilever tip makes periodic contact with the sample, thus leading to formation of the machined grooves being achieved by the impact of the tip; the second is the indentation mode, in which the tip is fixed on the sample and sustained pressure is applied to penetrate the surface; this caused the machined grooves to be formed by the tip’s pressing and rubbing action.

To conduct the experiment, the redesigned microcantilever is used as the new machining tool and is equipped with a diamond abrasive grain on its tip. Harnessing from the inherent nonlinear dam** within the system, the steady-state self-excited oscillations in the machining tool and the amplitude control required are generated via phase modulation in the feedback loop [33]. The machining modes are verified through observation of the vibrational profiles of the self-excited machining tool during the nanolithography process. This observation is accomplished by strategic placement of a set of equidistant points along the machining tool that subsequently allows the shapes of the vibrational profiles to be approximated by evaluating the displacement of each point. Consequently, under the application of lower pressing loads, mechanical nanolithography is performed in tap** mode, where the diamond abrasive grain on the tip periodically taps the sample; in contrast, when the pressing load surpasses a threshold, the tip transitions into a simply supported end, and the indentation mode then takes over, which leads to continuous and sustained pressing and rubbing of the diamond abrasive grain against the sample. In addition, the manipulation of amplitude magnitude, limited by nonlinear dam**, through phase modulation is verified in both modes. Subsequently, investigations of the machined depths under various deflection amplitudes for the machining tool tip in tap** mode and under distinct deflection angle amplitudes for the tip in indentation mode are conducted. The findings indicate a strong correlation between the magnitude of the machining tool’s amplitude and the machined depth in tap** mode. The machined depths are strongly correlated with the deflection angle of the microcantilever tip in indentation mode. These results demonstrate that machining depth control through phase modulation is effective for both modes.

2 Methods and analysis

2.1 AFM application and generation of self-excited oscillations

Fig. 1
figure 1

a Schematic diagram of mechanical nanolithography system using the AFM. The microcantilever deflection angle is detected using the optical lever and is converted into a voltage signal. This signal is then processed by the filter that is designed on demand and the self-excited circuit. The piezo-actuator applies the displacement excitation required to generate self-excited oscillations in the microcantilever. The sample is placed on a tube scanner that is capable of three-dimensional movement. b Schematic of the phase shifter, which consists of an operational amplifier, two resistors with the fixed value R, a capacitor with a fixed value C, and a variable resistor \(R_0\)

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Figure 1a shows a schematic diagram of the mechanical nanolithography system using the AFM device, where the optical lever can detect the deflection angle of the microcantilever tip and convert it into a voltage signal. This signal is then processed using the filter that is designed on demand and the self-excited circuit. The resulting output signal is then applied to the piezo-actuator to induce a proportional displacement that is fed back into the microcantilever to generate self-excited oscillations. The tube scanner is able to move the sample in three dimensions. In this case, control of the pressing load applied to the sharp tip is accomplished by manipulating the vertical position of the sample along the z-axis. To achieve both self-excited oscillation and amplitude control, the modulation of the phase difference between the deflection angle signal and the feedback signal is taken into account. The conventional phase shifter shown in Fig. 1b, which consists of an operational amplifier, two resistors with a fixed value R, a capacitor with the fixed value C, and the variable resistor \(R_0\), is considered. The phase frequency characteristic is described using the transfer function \(G(j\omega ) = \arctan \Big (-\frac{2\omega R_0C}{1-\omega ^2R_0^2C^2}\Big )\), where j is the imaginary unit, \(\omega \) is the angular frequency of the input signal \(V_\textrm{in}\), C is the capacitance of the capacitor, and \(R_0\) is the resistance of the variable resistor. Because the value and C is fixed, the phase difference between \(V_\textrm{in}\) with the angular frequency of \(\omega \) and the output \(V_\textrm{out}\) depends on \(R_0\) only.

Fig. 2
figure 2

Two likely vibrational profiles for the microcantilever. a Tap** mode, where the diamond abrasive grain vibrates along with the microcantilever tip and makes periodic contact with the sample. b Indentation mode, where the diamond abrasive grain makes continuous contact with the sample and rubs its surface along with the vibration of the microcantilever

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2.2 Vibrational profiles

Given that the self-excited oscillation is only generated in the fundamental mode in this research, two plausible machining modes that correspond to the two vibrational modes of the microcantilever are proposed: the tap** mode and the indentation mode, as illustrated in Fig. 2a and b, respectively. In tap** mode, the sharp tip taps the sample periodically with the vibration of the microcantilever. In this mode, formation of the machined grooves is attributed to the impact of the sharp tip. In indentation mode, the tip maintains continuous contact with the sample. In this mode, the microcantilever’s vibration causes the sharp tip to exert pressure on and rub against the sample, which then results in the formation of the machined grooves.

3 Experiments on observation of the vibrational profiles and amplitude control

3.1 Redesign of the microcantilever

To enhance the efficiency of nanolithography, a microcantilever with high stiffness is used. However, we encountered an issue with use of conventional microcantilever designs, where the self-excited oscillations resulted in excessively small amplitudes that limited the depth of the machined grooves that could be produced during the nanolithography process. To address this problem, the microcantilever was redesigned to have a trapezoidal shape. The machining tool used in this research is a stepped beam that is combined with the microcantilever and the base, which is glued onto a glass surface, as shown in Fig. 3a. To reduce tool wear, the diamond abrasive grain is attached to the microcantilever. The diamond is affixed at the tip of the machining tool. Additionally, to enhance the accuracy and effectiveness of the nanolithography process, the diamond abrasive grain is formed into a triangular pyramid shape, as shown in Fig. 3b, using FIB technology with an ion beam. A schematic diagram of the machining tool is shown in Fig. 3c, where the length, width, and thickness of the microcantilever are \(l_\textrm{m}\), \(a_\textrm{m}\), and b, respectively. For the base, the length of the portion that extends beyond the glass is \(l_\textrm{b}\), and the width and thickness are \(a_\textrm{b}\) and b, respectively. Table 1 offers a comprehensive summary of the specific dimensions and the physical properties of the trapezoidal beam used in our study.

Fig. 3
figure 3

a Top view of the machining tool. The microcantilever is attached to a base, which is glued to the glass. b Enlarged view of the microcantilever tip. The tip is equipped with a diamond abrasive grain that has been formed into a triangular pyramid shape by FIB processing. c Schematic diagram of the machining tool with specific dimensions

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Fig. 4
figure 4

Analytical mode of the microcantilever. w(x) and \(\theta (x)\) denote the deflection and deflection angle of the microcantilever at x, respectively. F is the pressing load on the on the microcantilever tip and \(l=l_\textrm{m}+l_\textrm{b}\) is the full beam length

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Table 1 Example of a lengthy table which is set to full textwidth
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To quantify the pressing load acting on the sample surface, the machining tool is modeled as shown in Fig. 4. l denotes the full length of the tool. F is the shear force acting on the tool tip, which is the opposite force of the pressing load on the sample surface, the relationships between the deflection w and the F of the new machining tool tip, and between the deflection angle \(\theta \) and F of the tip are derived. Considering the relationships [34]

$$\begin{aligned}&\theta =\int \frac{(l-x)F}{EI(x)}\textrm{d}x, \end{aligned}$$
(1)
$$\begin{aligned}&w=\int \int \frac{(l-x)F}{EI(x)}\textrm{d}x\textrm{d}x, \end{aligned}$$
(2)

where \(l=l_\textrm{m}+l_\textrm{b}\). In this specific scenario, the beam shape is not uniform, which leads to I varying with x that

$$\begin{aligned} I(x)=\left\{ \begin{array}{lr} a_\textrm{b}b^3/12; \quad 0\le x\le l_\textrm{b},\\ a_\textrm{m}b^3/12; \quad l_\textrm{b}<x\le l. \end{array} \right. \end{aligned}$$
(3)

Considering the boundary conditions: \(w(0)=0\) and \(\theta (0)=0\) and substituting the values given in Table 1 into Eqs. (1), (2), and (3) at \(x=l\) yield:

$$\begin{aligned} F=\textrm{k}_{\uptheta }\theta (l), \quad F=k_ww(l), \end{aligned}$$
(4)

where \(\textrm{k}_\uptheta =3.5\times 10^{-3}\) N/\(^\circ \) and \(k_w=2.5\times 10^2\) N/m.

3.2 Experimental setup

Fig. 5
figure 5

a Schematic diagram of the system used for observation. The velocity signal of the machining tool is detected with high precision using laser Doppler velocimeter I and converted into a corresponding voltage signal. This voltage signal is then transmitted to both the self-excited circuit and the low-pass filter, and the conditioned voltage signal is subsequently applied to the piezo-actuator housed within the cantilever holder. Ultimately, the piezo-actuator generates the proportional displacement excitation and then feeds it back to the machining tool that is fixed on the cantilever holder. Laser Doppler velocimeter II is used to observe the machining tool’s vibrational profile. Note that the laser beams are incident vertically and return along their original path, and are shown at an angle of incidence in the figure for ease of understanding. The \(x-y-z\)-axes stage can move the silicon sample in three dimensions. The output signal from laser Doppler velocimeter I is amplified, integrated, and phase-shifted using the self-excited circuit (b), which consists of a voltage follower, an integrator, three inverted amplifiers, and two phase shifters corresponding to Fig. 1b. c and d show photographic overviews of the experimental setup and the contents of the cantilever holder, respectively

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The experimental system used to observe the machining tool’s vibrational profiles is illustrated in Fig. 5a. The machining tool’s velocity is detected using laser Doppler velocimeter I (Ono Sokki Co., Ltd.: Laser Doppler Vibrometer LV-1800) and converted into a voltage signal. The signal is then filtered using a bandpass filter to ensure the fundamental-mode oscillations in the machining tool. The output signal is then integrated, phase-shifted, and amplified using the self-excited circuit. The self-excited circuit, which is a combination of a voltage follower, an integrator, two phase shifters corresponding to the circuit in Fig. 1, and three inverted amplifiers, is shown in Fig. 5b with specific component values. Phase shifter I is employed for the purpose of mitigating the phase distortion introduced by the filter. The phase difference between the velocity signal and output signal of this circuit and the output signal magnitude can be modulated by changing the values of variable resistors \(R_0\) and \(R_1\), respectively. The piezo-actuator is responsible for producing the proportional displacement excitation, which is then fed back to the machining tool that is fixed on the cantilever holder (which has the same specifications as that in the AFM device described in Sect. 3.2) to generate the self-excited oscillations. The silicon sample can be moved in three dimensions using the \(x-y-z-\)axes stage. Laser Doppler velocimeter II is used to observe the shapes of the vibrational profiles. Figure 5c and d shows photographs of an overview of the actual experimental setup and the contents within the cantilever holder, respectively.

The observation is accomplished via scanning a set of equidistant measurement points which is set strategically along the machining tool to detect the velocity signals of these points using laser Doppler velocimeter II (Polytec Ltd.: MSA-500 Micro System Analyzer) in the steady state of the oscillations. These velocity signals are subsequently integrated into displacement signals. The vibrational profiles are then approximated by evaluating the amplitude and phase of the amplitude signal at each point. In this work, 17 measurement points were placed as shown in Fig. 6a. Subsequently, the observation is performed by raising the sample along the z-axis incrementally to apply pressing loads on the machining tool’s tip; these loads are calculated using Eq. (4). \(R_1\) is set to have a value of 10 k\({\Omega }\), and at each specified pressing load, the displacement data for the individual points under the varying phase shifts, i.e., the distinct values of \(R_0\), are acquired. The excitation voltage is taken as the reference, and the onset time for measuring the displacement at each measurement point corresponds to the moment when the excitation voltage starts rising from zero.

Fig. 6
figure 6

a Positions of the measurement points on the machining tool. b Displacements at each measurement point at fractions of 0, 1/6, 1/3, 1/2, 2/3, and 5/6 of a time period, respectively, when the pressing load is \(125\,\upmu \)N and \(R_0=75\) \(\Omega \). Please watch the animation (Online resource 1) to provide a better understanding of the process. c Displacements of each measurement point at fractions of 0, 1/6, 1/3, 1/2, 2/3, and 5/6 of a time period when the pressing load is \(375\,\upmu \)N and the \(R_0=75\) \(\Omega \). Please watch the animations (Online resource 2) to provide a better understanding of the process

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3.3 Results and analysis

Linear feedback is employed for self-excited oscillations; however, the oscillations tend to reach steady states which prompts consideration of the role played by inherent nonlinear dam** within the system. From the nonlinear resonance profiles of the machining tool, when the pressing load is low, nanolithography is performed in tap** mode, but as the pressing load intensifies, the process then transitions to operation in indentation mode. Illustrative examples of vibrational profiles for both the tap** mode and the indentation mode are provided. As examples, the vibrational profiles are shown in Fig. 6b and c which show the amplitude of the machining tool at each measurement point at fractions of 0, 1/6, 1/3, 1/2, 2/3, and 5/6 of a time period, respectively. The pressing loads for the tap** and indentation modes are given as 125 \(\upmu \)N and 375 \(\upmu \)N, respectively, and \(R_0\) is 75 \(\Omega \). Please watch the animations (Online resource 1 for Fig. 6b and Online resource 2 for Fig. 6c) to provide a better understanding of the process.

Fig. 7
figure 7

a Amplitudes of the machining tool tip which is represented by the rightmost measurement point under pressing loads of 125 \(\upmu \)N, 250 \(\upmu \)N, 375 \(\upmu \)N, 500 \(\upmu \)N, and 625 \(\upmu \)N. b Displacements of the machining tool tip with respect to time under a 125 \(\upmu \)N pressing load with \(R_0=75\) \(\Omega \) and a 375 \(\upmu \)N pressing load with \(R_0=75\) \(\Omega \). c and d Displacements of the machining tool at the rightmost measurement point and rightmost second measurement point with respect to time under a 125 \(\upmu \)N pressing load with \(R_0=75\) \(\Omega \) and a 375 \(\upmu \)N pressing load with \(R_0=75\) \(\Omega \), respectively

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Notably, the origin of the z-axis shown in the figures does not mean the positions of individual points in the natural state of the machining tool because the displacements are integrated from the velocity signal, where the information of deflection caused by the pressing load is not involved. However, the origin of the z-axis here is implied to be the position of the maximum velocity. The occurrence of negative amplitude of the rightmost measurement point is attributed to intermolecular repulsive forces, causing the machining tool tip to reach its maximum velocity before impacting the sample. As shown in Fig. 6b, the machining tool’s tip, which is indicated by the rightmost measurement point, impacts the sample periodically in synchronization with the oscillation. In contrast, Fig. 6c illustrates a scenario in which the tip of the machining tool remains fixed on the sample surface. The amplitudes of the microcantilever tip are shown in Fig. 7a. The graph’s horizontal axis represents the applied pressing load and the vertical axis represents the amplitude. When the pressing load ranges up to 250 \(\upmu \)N, the displacement of the microcantilever tip is obvious, and it shows the characteristics of the tap** mode. When the pressing load exceeds or is equal to 375 \(\upmu \)N, the displacement then becomes inconspicuous and remains nearly constant, regardless of the magnitude of the applied pressing load. Under these conditions, the machining tool works in the indentation mode. However, in the range of pressing loads greater than 375 \(\upmu \)N, the displacement of the rightmost measurement point should be zero in the proposed indentation mode. A plausible explanation for the existence of the deflection is proposed, in that the measurement point cannot be positioned exactly at the location of the diamond abrasive grain. Therefore, the minuscule vibrations can be discerned. Two pieces of evidence were presented, firstly, in contrast with scenarios where the pressing load remains below 250 \(\upmu \)N, when the pressing load surpasses 375 \(\upmu \)N, the displacement–time graph of the rightmost measurement point demonstrates a pronounced symmetry. The displacements of the rightmost measurement point in Fig. 6b and c are plotted with respect to time in Fig. 7b. When the pressing load is 125 \(\upmu \)N, the displacement with respect to time shows a strong asymmetry, where the downward displacement is suppressed significantly. In contrast, the displacement waveform shows good symmetry when the pressing load is 375 \(\upmu \)N. Secondly, when the pressing load is below 250 \(\upmu \)N, the displacement waveforms are in phase at the rightmost first measurement point and the rightmost second measurement point. Conversely, their waveforms become out of phase when the pressing load surpasses 375 \(\upmu \)N. Figure 7c and d shows the displacements of the rightmost and rightmost second measurement points under a 125 \(\upmu \)N pressing load with \(R_0=75\) \(\Omega \) and a 375 \(\upmu \)N pressing load with \(R_0=\)75 \(\Omega \), respectively, as an example.

When the pressing load is less than or equal to 250 \(\upmu \)N, nanolithography is conducted in tap** mode, and the displacement amplitude of the machining tool tip shows clear variations in response to changes in \(R_0\), i.e., to the phase shift, as shown in Fig. 7a. Conversely, when the pressing load exceeds or is equal to 375 \(\upmu \)N, the nanolithography process transitions into the indentation mode, and the displacement amplitude of the machining tool tip cannot demonstrate the phase shift role adequately. Therefore, the deflection angle variation of the machining tool tip is taken into account. Here, the deflection angle variation of the machining tool tip is approximated using the displacements at the rightmost and rightmost second measurement points in Fig. 6a. As shown in Fig. 8a, the circles denote the measurement point positions, and the deflection angle variation \(\mathrm{\theta }_n\) is defined approximately by the relationship \(\mathrm{\theta }_n\approx -\arctan {[(w_2-w_1)/d]}\), where \(w_1\) and \(w_2\) denote the displacements of the rightmost and rightmost second measurement points, respectively, and \(d=115\) \(\upmu \)m. Because the static positions of the measurement points cannot be detected, \(\mathrm{\theta }_n\) in this context represents only the temporal variations in the magnitude of the deflection of the machining tool tip. The magnitudes of the amplitude of the machining tool tip for various values of \(R_0\) are shown in Fig. 8b. The magnitude is closely associated with the value of \(R_0\) when referring to the phase shifts.

Fig. 8
figure 8

a Analytical mode of the machining tool tip, where the circles denote the measurement point positions. The first point from the right represents the measurement point on the machining tool’s tip. The deflection angle variation \(\theta _n\) is efined as \(\uptheta _n\approx -\arctan {[(w_2-w_1)/d]}\), where \(w_1\) and \(w_2\) denote the displacements of the first and second measurement points from the right, respectively. d (\(=115 \upmu \)m) is the interval between the measurement points. b The relationship between the calculated \(\theta _n\) and the value of \(R_0\)

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4 Experiments on machining depth control

4.1 Experimental setup

The nanolithography process is performed on a silicon sample using an AFM device (Seiko Instrument Inc.: SPA300HA system) that operates based on the working principle illustrated in Fig. 1a. The machining tool and the filter used in this experiment are the same as those that are used in Sect. 3. The self-excited circuit used in Sect. 3 is modified by considering the distinctions between the operational principle of the laser Doppler velocimeter and the principle of the optical lever, inverted amplifier III and the integrator are removed. The pressing load acting on the diamond abrasive is calculated using Eq. (4). The deflection angle of the machining tool tip is detected via the optical lever and the sensitivity is 14.25 V\(/\ ^{\circ }\). The nanolithography process is performed in tap** mode (250 \(\upmu \)N) and in indentation mode (375 \(\upmu \)N). With the objective of achieving discernible differences in the machining grooves, R is deliberately set to have values of 100 \(\Omega \), 120 \(\Omega \), 145 \(\Omega \), and 175 \(\Omega \). The value of \(R_1\) remains at 10 k\(\Omega \).

4.2 Results and analysis

Fig. 9
figure 9

Temporal dynamics of the deflection angle in tap** mode (250 \(\upmu \)N) (a) and in indentation mode (375 \(\upmu \)N) (b) when \(R_0\) is 100 \(\Omega \). The dashed lines represent the deflection angle of the machining tool tip when the tip has already been subjected to the pressing load. \(\theta _\textrm{b}\) represents the deflection angle before the nanolithography process. \(\theta _\textrm{a}\) denotes the deflection angle after the nanolithography process

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Fig. 10
figure 10

a Relationship between the amplitude of the machining tool’s tip deflection and the value of \(R_0\), i.e., the phase shift. b Variation in the deflection angle of the machining tool tip with respect to \(R_0\), i.e., the phase shift

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In the mechanical nanolithography process, when \(R_0\) is 100 \(\Omega \), the time histories of the deflection angles of the machining tool tip in tap** mode (250 \(\upmu \)N) and in indentation mode (375 \(\upmu \)N) are shown in Fig. 9a and b, respectively. \(\theta _\textrm{b}\) represents the deflection angle prior to mechanical nanolithography due to the application of the pressing loads. In Fig. 9a, \(\theta _\textrm{b}\) depicts the deflection angle under a pressing load of 250 \(\upmu \)N, while \(\theta _\textrm{b}\) in Fig. 9b corresponds to the deflection angle characteristic under a pressing load of 375 \(\upmu \)N. Additionally, \(\theta _\textrm{a}\) represents the deflection angle after mechanical nanolithography processing. Notably, the deflection angle decreases after the mechanical nanolithography process is complete, indicating that the tip penetrates the sample surface. In Fig. 9a, the deflection angle waveform also shows significant asymmetry. In contrast, the time history shown in Fig. 9b is symmetrical relative to \(\theta _\textrm{a}\). This scenario is similar to the previous case presented in Sect. 3. The frequencies of the first-order self-excited oscillation in tap** mode (250 \(\upmu \)N) and indentation mode (375 \(\upmu \)N) are 13.5 kHz and 13.7 kHz, respectively. For comparison, under the same conditions used in the experiment for observation of the vibrational profiles, the corresponding frequencies were 13.8 kHz and 13.9 kHz, respectively. Therefore, we can conclude that the nanolithography process is conducted in tap** mode under the 250 \(\upmu \)N pressing load; as the deflection angle approaches the approximate value of \(\theta _\textrm{a}\), the original negative directional vibration then undergoes a transition toward the positive direction. We attribute this behavior to the occurrence of tip–surface impacts. Following impact, the tip comes to a halt at the surface while the remaining portion of the machining tool continues its oscillation in the negative direction, thus leading to an angular change in the positive direction. Subsequently, when the remaining parts reach the lowest point in their vibration and then initiate oscillation in the positive direction, the tip still remains in contact with the surface, and this contact results in the variation in the deflection angle in the negative direction. Finally, as the tip lifts off from the surface during the oscillation, the deflection angle then begins to transition toward the positive direction. Under the 375 \(\upmu \)N pressing load, the indentation mode takes over, and the tip thus maintains constant contact with the sample surface, causing the deflection angle to oscillate around \(\theta _\textrm{a}\) as its center.

To quantify the effect of the phase shift, which is represented by the value of \(R_0\) in the tap** mode, the upper deflection angle of the machining tool tip \(\theta _\textrm{a}\) is converted into the deflection using Eq. (4). The relationship between the deflection amplitude and the value of \(R_0\), which represents the phase shift, is illustrated in Fig. 10a. In addition, in the indentation mode, the variation in the deflection angle with respect to the value of \(R_0\) is depicted in Fig. 10b. As the value of \(R_0\) increases, the amplitudes of the physical quantities depicted in both models show decreasing trends.

Fig. 11
figure 11

Machined grooves in tap** mode (250 \(\upmu \)N) (a) and in indentation mode (375 \(\upmu \)N) (b). a Hole Nos. 1–4 are the machined grooves corresponding to values of \(R_0 = 100\) \(\Omega \), 120 \(\Omega \), 145 \(\Omega \), and 175 \(\Omega \), respectively. b Hole Nos. 5–8 are the machined grooves corresponding to \(R_0=100\) \(\Omega \), 120 \(\Omega \), 145 \(\Omega \), and 175 \(\Omega \), respectively. The cross sections of the machined grooves are measured four times in the directions along lines I–IV, as shown in the left upper corner of (a). The cross sections of machined holes No. 1 (c) and No. 5 (d) from direction I are shown as examples

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The results of the mechanical nanolithography process are shown in Fig. 11a and b. The images were generated by the AFM within an area of 12 \(\upmu \)m \(\times \) 12 \(\upmu \)m. Hole Nos. 1–4 in Fig. 11a are the machined grooves in ta** mode (250 \(\upmu \)N) corresponding to \(R_0\) settings of 100 \(\Omega \), 120 \(\Omega \), 145 \(\Omega \), and 175 \(\Omega \), respectively. Hole Nos. 5–8 in Fig. 11b represent the machined grooves in indentation mode (375 \(\upmu \)N) corresponding to \(R_0\) settings of 110 \(\Omega \), 120 \(\Omega \), 145 \(\Omega \), and 175 \(\Omega \), respectively. Cross-sectional images of each hole passing through its deepest point are plotted in four distinct directions corresponding to the lines I-IV shown in the left upper corner of Fig. 11a. Figure 11c and d shows the cross-sectional plots of hole No. 1 in Fig. 11a and hole No. 5 in Fig. 11b from direction I, respectively. The machining depth is defined by the difference between the average height of the sample surface in these four measurements and the height of the deepest point in the hole. Note that the regions in which the height greatly exceeds the sample’s surface height are not taken into account when the average surface height is calculated. One example is the area between 1000 and 2000 nm along the horizontal axis in Fig. 11d, which is considered to be formed by the machining scraps.

Fig. 12
figure 12

b Relationship between the machining tool and groove in tap** mode (250 \(\upmu \)N) and \(R_0=90\) \(\Omega \) conditions. b Relationship between the deflection angle of the machining tool and groove depth in indentation mode (375 \(\upmu \)N) and \(R_0=90\) \(\Omega \) conditions

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The relationship between the amplitudes of the machining tool tip deflections and the depths of the machined grooves is shown in Fig. 12a. A positive correlation is observed between the amplitudes and the depths, i.e., an increase in amplitude leads to a corresponding increase in the depth. Additionally, the relationship between the amplitude of the machining tool tip deflection angle and the depth of the machined grooves is shown in Fig. 12b, illustrating the obvious correlation between the depths and the amplitudes. This implies that greater amplitudes will lead to deeper machined holes.

5 Conclusions

In this research, the mechanism of formation of machined grooves in a mechanical nanolithography process using a self-excited microcantilever in an AFM was investigated. Two likely machining modes are proposed. The first is the tap** mode, where the microcantilever tip makes periodic contact with the sample, leading to the formation of the machined grooves being achieved by the impact of the tip; the second is the indentation mode, where the tip is fixed on the sample and sustained pressure is applied to penetrate the surface, causing the machined grooves to be formed by the rubbing action of the tip. In addition, the machining depth control process based on use of modulation of the magnitude of the microcantilever amplitude is analyzed experimentally for both modes. Greater microcantilever tip deflection amplitudes lead to deeper machined grooves in the tap** mode. Similarly, in indentation mode, an increase in the amplitude of the deflection angle of the microcantilever tip results in an increase in the depth of the machined grooves.

Mechanical nanolithography experiments were performed using a redesigned microcantilever. Generation of steady-state self-excited oscillations in the new machining tool and amplitude control was realized by applying phase modulation in the feedback loop due to the nonlinear dam** inherent within the system. The machining modes were verified by observing the vibrational profiles of the machining tool during the nanolithography process. As a result, the nanolithography process was shown to operate initially in tap** mode, but as the pressing load intensified, the process then transitioned into the indentation mode. Following this observation, machining depth control was investigated experimentally. In the tap** mode, the machined depth is positively correlated with the magnitude of the machining tool deflection amplitude. Additionally, a positive correlation was also observed between the machined depth and the magnitude of the deflection angle of the machining tool in the indentation mode. The experimental results aligned well with the theoretical predictions.