1 Introduction

Friction and wear are interfacial phenomena commonly observed not only in daily life but also in many important fields of industry. Their effects are significantly enlarged due to the large surface-to-volume ratio when the system size reduces to micrometer scale, and have become bottlenecks that hinder the development of reliable and energy-efficient microelectromechanical systems (MEMS). Structural superlubricity (SSL) offers a unique solution to this challenge since its first realization on microscale^[1-3] based on earlier theoretical predications and experimental attempts^[4-6]. It refers to a state of nearly vanishing friction and wear due to systematic cancellation of lateral forces acting on each atom, when two solid surfaces are in incommensurate contact. One of the most promising application fields of microscale SSL is MEMS^[7-10]. The control of friction and minimization of wear are important aims within the MEMS regime, especially in devices where relative motion of contact-pairs is involved^[11]. Ultralow friction and wear can be achieved in superlubric contacts without adding any lubricants. This feature not only provides a friction reduction scheme to the existing MEMS, but also inspires disruptive designs of novel nano- and micro-devices. Since the theoretical prediction of superlubricity in 1980s to 1990s^[4-5], superlubricity was only achieved on nanoscale under strict conditions in early experiments^[6]. Meanwhile, pioneering works on the retraction and oscillation motion of van der Waals (vdW) layered materials attracted intensive research interest of using vdW materials for the moving parts in MEMS/NEMS for their ultralow friction^{[7-9, 12-13]}. The gigahertz oscillator proposed also became the first concept of nano-device that utilizes the ultralow friction of vdW materials^[9]. The first observation of microscale SSL in graphite^{[1, 10]} bridged the two parallel paths of fundamental studies on superlubricity and micro/nanoscale mechanical devices based on vdW materials. SSL technologies were made possible. From then, a lot of fundamental researches on microscale SSL have been conducted on graphite mesas^{[2-3, 14-16]}, as well as the heterojunctions between a graphite mesa and another material^[17-21]. While on the other hand, SSL based micro-devices with versatile functions were designed or conceptually proved, such as SSL generator^[22] and SSL oscillator^[23]. Some of the milestones in the history of microscale SSL and SSL MEMS/NEMS are summarized in Fig. 1 and discussed in Supplementary Materials.

An important challenge for the implementation of superlubric MEMS device is the difficulty in controlling the movement of a superlubric contact. In a superlubric contact, the slider has two in-plane translational degrees of freedom (x and y), and one in-plane rotational degree of freedom (θ). The slider can usually move in a large domain in the (x, y, θ) phase space. Meanwhile, the interfacial friction is ultralow, since it is superlubric. Thus external perturbations and imperfections of the structure itself can have significant disturbance on the movement of the slider^{[19, 25-28]}. Therefore, for the functioning of superlubric device, it is essential to guide the motion of the slider in a controllable way and maintain the contact in the low-friction superlubric state. Mechanical supports such as clamps are often used to constrain the motion of the deformable part in conventional MEMS. However, this is usually not feasible in superlubric contacts. Instead, methods with alternative physical principles have to be developed. Experiments have shown that the interfacial energy of the superlubric contact-pair itself can be utilized to drive the motion of the slider^{[10, 14, 29]}. Recently, a rotational instability was reported on graphite mesas, where the interfacial energy constrains/drives the rotation of the slider before/after a critical displacement^[27]. In the constraining regime, the interfacial energy self-aligns the sliding contact. Meanwhile, electrostatic forces have been widely used in MEMS for its simplicity, high energy density, and compatibility with fabrication process. Parallel-plate and comb-drive are commonly used for electrostatic actuators^[30]. It would be interesting to explore the possibility of using these two principles for controlled movements of superlubric MEMS.

This work shows that controlled movement of a superlubric MEMS is possible by utilizing interfacial energy and electrostatic force. When the shape of the contact-pair is designed appropriately, the interfacial energy of the contact-pair provides restoring and constraining forces to the slider. Extra driving and constraining forces are enabled by the electric fields induced by the electrodes under the slider. These two principles are demonstrated on the design of a superlubric oscillator. With the application of the design concepts, the oscillator shows a well-defined resonance frequency in the lateral translational mode, which is crucial for its practical use as a resonator. The unfavorable movements in other degrees of freedom (including rotation and translation) caused by external perturbations can be constrained. The design principles presented here should be applicable in general to superlubric devices and help the improvements of existing superlubric devices and the development of new ones. In the following part of this paper, the basic concept of superlubric MEMS are first described in Section 2. The two principles of controlling the movements of the slider in a superlubric MEMS device, i.e., the minimization of interfacial energy and the minimization of electrostatic energy, are detailed in Section 3 and Section 4, respectively. Conclusions are given in Section 5.

2 Concept of Superlubric MEMS

Before discussing the principles of controlled movement in superlubric MEMS, in this section, the basic concept of a superlubric MEMS and how it works are shown, by taking the superlubric oscillator as an example and comparing it with a conventional MEMS resonator.

Fig. 2(a) shows the simplified working principle of a conventional MEMS cantilever resonator actuated by electrostatic method. The oscillating part is typically a mechanically supported elastic body generating restoring force. The support anchor offers the constraints to the elastic beam while limits the available displacement to a certain extent at the same time. When the elastic response is in the linear regime, the oscillator is harmonic and has a well-defined resonance frequency. In this case, this oscillator acts as a resonator. Different from the conventional design, the design of a superlubric resonator is proposed in Fig. 2(b). The oscillating part is a slider which slides on a substrate, and the contact between the slider and substrate is superlubric. The substrate is made of an insulator, thus acting as an insulation layer at the same time. Thanks to superlubricity, the friction and wear of this sliding interface can be virtually zero. Interfacial energy provides both restoring and constraining forces to the slider, while driving forces are provided electrostatically, as will be detailed later in this paper. Compared with the elastic beam, the superlubric oscillator does not depend on the clamped anchor to generate restoring force, and the slider can move in a large range. As a result, the oscillation amplitude of the superlubric slider can be large compared with the elastic beam whose oscillation amplitude is essentially limited by the small amount of elastic deformation that the material can sustain.

3 Interfacial Energy 3.1 Principles

The effect of interfacial energy in a general superlubric contact can be understood by the kinematic model shown in Fig. 3. The model consists of a stationary substrate and a rigid but movable slider. The contact between the slider and the substrate is superlubric. The movable slider has three degrees of freedom, i.e., two translational ones, x and y, and the rotation in the xoy plane, θ. In a general case, the slider and substrate can be made from two different materials, whose specific surface energies are γ₁ and γ₂, respectively. Considering that the contact between the slider and substrate has a variable contact area A, the excess total interfacial energy is

$ {U_s} = {U_0} + ({\gamma _{{\rm{ int }}}} - {\gamma _1} - {\gamma _2})A = {U_0} - \varGamma A $

(1)

where U₀ is a constant reference value, γ_int is the specific interfacial energy of the slider-substrate contact interface, and Γ=γ₁+γ₂-γ_int is defined as the cleavage energy of this contact^{[14, 29]}. Eq. (1) implies that the system always tends to acquire the largest contact area as possible, in order to minimize the total interfacial energy.

As a result, the generalized forces induced by interfacial energy are

$ {Q_i} = - \frac{{\partial {U_s}}}{{\partial {q_i}}} = \varGamma \frac{{\partial A}}{{\partial {q_i}}} $

(2)

where q_i=x, y, θ are the generalized coordinates. The generalized forces Q_i corresponding to the generalized coordinates q_i either constrain or drive the displacement/rotation of the slider, depending on the geometry and loading cases.

3.2 Design of Slider Shape

As a demonstration, in this subsection, the general principles described above is applied to the design of slider shape in a superlubric oscillator. Four candidate designs, including one widely considered in previous studies, are investigated. We will show that, through appropriately designing the shapes of the slider and substrate, it is possible to develop a superlubric resonator with a well-defined resonance frequency on the order of 1 MHz, and the unfavorable rotation of the slider can be constrained for oscillating amplitude up to 1 μm.

3.2.1 Linear restoring force

For practical use as a resonator, an oscillator should have a well-defined resonance frequency, which requires a linear relation between the restoring force and displacement. The restoring force in a superlubric oscillator is provided by the retraction force induced by interfacial energy. Following Ref. [10], the retraction force is defined as

$ {F_{{\rm{ret}}}} = - {F_x} = - \frac{{\varGamma \partial A}}{{\partial x}} $

(3)

In previous works, rectangular sliders were used, as shown in Fig. 4(a). The retraction force is a step function as shown in Fig. 4(e)^{[10, 29]}. In this case, the F_ret-x relation is non-linear, and the frequency of the oscillator depends on its oscillating amplitude. To solve this problem, a structure with rhombus shaped slider and a rectangular substrate is designed (denoted as the 2^nd design), as shown in Fig. 4(b). In this case, the retraction force is linear for -a/2 < x < a/2, as shown in Fig. 4(f). The resonance frequency is given as (detailed in Supplementary Materials)

$ f = \frac{1}{\pi }\sqrt {\frac{\varGamma }{{\rho {\kern 1pt} {a^2}t}}} $

(4)

where ρ is the mass density of the slider, t is the slider thickness, and a is the slider length shown in Fig. 4(b). Despite the linear F_ret-x relation, it will be shown later that this structure cannot sustain a translation-only movement. The slider rotates spontaneously when it starts to move. It has been indicated that the rotational stability is sensitively dependent on slider shape^[27]. Therefore, two different designs (denoted as the 3^rd and 4^th designs) are proposed, as shown in Figs. 4(c) and 4(d). For the 3^rd design, the retraction force is step-like at x=0, but linear for 0 < |x| < (a/2)(1-2h/b). The F_ret-x response of this design is non-linear, however, the step size is dependent on the width of the bar-shaped segments on the slider, h, as ΔF_ret=4Γ h. When h is small, the F_ret-x curve can be regarded as approximately linear. For the 4^th design, the F_ret-x response is linear in the range of -(a-h)/2 < x < (a-h)/2, as shown in Fig. 4(h). The resonance frequency is the same as the 1^st design, as given by Eq. (4).

For the 2^nd and 4^th designs, the resonance frequency can be estimated by Eq. (4). For a graphite slider-substrate, the cleavage energy Γ is on the order of 0.2 J/m^{2[14, 29, 31]}. With the typical values of geometry parameters a=5 μm and t=100 nm, the frequency is estimated to be on the order of 1 MHz.

3.2.2 Rotational stability

Other than the requirement of a linear F_ret-x relation, the slider has to be rotationally stable. A recent research showed that the interfacial energy can drive the slider to suddenly rotate at a certain critical displacement δ_cr, and the rotation usually leads to irreversible destruction of the superlubric interface^[27]. Optical microscopic snapshots of an experiment showing such an instability is presented in Fig. 5. Such rotation must not happen for a reasonable range of displacement where the superlubric oscillator is designed to work in. Following the algorithms developed in Ref. [27], assuming that the y-direction translation of the slider is constrained, i.e., only x-translation and rotation θ is permitted, the critical displacement δ_cr is found dependent on the rotation center (x_C, y_C). A δ_cr-contour on the (x_C, y_C) plane is drawn for each of the 4 designs described above, as shown in Figs. 6(a)-(d). Higher δ_cr indicates that the slider can move without any rotation for a larger displacement in the x-direction. For the 1^st, 3^rd, and 4^th designs, it is found that δ_cr is always larger than 0.3a, which is 1.5 μm for a=5 μm, regardless of the rotation center. This result implies that the interfacial energy can effectively constrain the rotation of the slider when its oscillating amplitude is smaller than 1.5 μm. On the other hand, the situation is completely different for the 2^nd design. Non-zero δ_cr is found only when the rotation center is located exactly on the centerline of the slider, i.e., y_C=0. Otherwise, the critical displacement is zero, which implies that the slider will rotate immediately as it starts to move.

The conclusions from the δ_cr-contours are further confirmed by fully dynamic simulations considering all the 3 degrees of freedom of the slider (i.e., x, y, θ). The simulation was conducted by solving the Lagrange's equations with the potential energy given by Eq. (1). Two runs were carried out for each design. In the first run, the initial conditions (denoted as #1) are defined as x(0)=0.3 μm, x^'(0)=0, θ(0)=θ^'(0)=0, y(0)=y^'(0)=0. This set of initial conditions corresponds to the scenario that the slider is displaced for 0.3 μm in x-direction and then released free. In the second run (denoted as #2), a perturbation is applied as θ(0)=0.1°, and the rest of the initial conditions are the same as #1. The results of the two runs are plotted in Figs. 6(e)-(l) as the black and red curves, respectively. For the 1^st, 3^rd, and 4^th designs, the black and red curves are almost identical, implying that a small perturbation does not influence the system response. Especially, Figs. 6(i), 6(k), and 6(l) show that θ(t) stays at θ(t)≈0, suggesting that the system is rotationally stable. However, the black and red curves are significantly distinct with each other for the 2^nd design, suggesting that the slider is very sensitive to perturbations and a small perturbation in the initial value of θ leads to completely different system response.

Combing the results of F_ret-x curves and rotational stability, the 4^th design is the only one among the 4 designs that meets the design targets for a superlubric resonator.

4 Electrostatically Actuated Superlubric Resonator 4.1 Design and Working Principle

Actuation by electrostatic force is the most widely used method to drive the mechanical motion in MEMS, because of its simple principles, high energy density, and excellent processability. Based on the defined superlubric contact-pair design with linear restoring force and rotational stability discussed above, an electrostatically actuated superlubric resonator is designed in this section. As shown in Fig. 7, the resonator consists of a substrate, three actuation electrodes (in blue), an insulation layer (in purple), and a slider (in red). The superlubric contact-pair is formed by the slider and the insulation layer, and has a linear restoring force characteristic. Three actuation electrodes are symmetrically distributed below the slider. The middle electrode is grounded, and the left and right ends are an input end (V_i) and an output end (V_o), respectively. It can be seen in Figs. 7(a) and 7(d) that the designed slider covers the grounded electrode entirely, and overlaps with the input and output electrodes by a small area. Therefore, capacitors are formed between the slider and three electrodes, denoted as C₁, C₂, and C₃, respectively. When an external excitation signal is applied on the left, the generated electrostatic force in the horizontal direction will vibrate the slider, and the resulting capacitance variation between the slider and electrodes can be detected by the output end.

In the designed superlubric resonator, electrostatic force works as a driving force, and simultaneously a position constraining force to prevent the slider from shifting in y-direction due to external perturbations. To derive the driving and constraining forces, an equivalent circuit of the designed superlubric resonator is shown in Fig. 8. Combining with the energy principle, the electrostatic driving force F_x in the x-direction, and the electrostatic constraining force in the y-direction are given as

$ {{F_x} = - \frac{{\partial {\kern 1pt} {U_e}}}{{\partial x}} = \frac{{V_i^2}}{2}\frac{{\partial {\kern 1pt} {C_a}}}{{\partial x}}} $

(5)

$ {{F_y} = - \frac{{\partial {\kern 1pt} {U_e}}}{{\partial y}} = \frac{{V_i^2}}{2}\frac{{\partial {\kern 1pt} {C_a}}}{{\partial y}}} $

(6)

where U_e is the total electrostatic energy, and C_a is the total system capacitance, which is, according to the equivalent circuit, written as

$ {C_a} = \frac{{(\frac{{{C_3} \cdot {C_5}}}{{{C_3} + {C_5}}} + {C_2}) \cdot {C_1}}}{{\frac{{{C_3} \cdot {C_5}}}{{{C_3} + {C_5}}} + {C_2} + {C_1}}} + {C_4} $

(7)

where C₄ and C₅ are the capacitances between the left and central electrodes, and between the central and right electrodes, respectively. The capacitances C₁, C₂, and C₃ are further dependent on the overlapping areas as

$ {C_i} = \frac{{{ \epsilon _0}{ \epsilon _r}{A_i}}}{d} $

(8)

where i=1, 2, 3, and A_i is the corresponding overlapping area, ∊₀ is the permittivity of vacuum, ∊_r is the relative permittivity of the medium, and d is the thickness of the insulation layer.

4.2 Electrostatic Driving and Constraining Forces

To quantitatively demonstrate the driving and constraining effects, the driving and constraining forces are calculated for a typical geometry as shown in Fig. 9(a), with a 50 V input voltage applied to the left electrode. When the slider (outlined in red) moves in the x- or y-direction, as shown in Fig. 9(b), the total electrostatic energy and corresponding electrostatic forces are calculated. In x-direction, the electrostatic energy reaches its minimum at x≈-2 μm, as shown in Fig. 9(c). This result implies that the slider tends to move left, when released from the center. At the position of x=0, the electrostatic force that drives this movement is on the order of 1 μN, as shown in Fig. 9(d). To estimate the frictional force, it is first noticed that the friction force for a 3 μm square mesa is typically 40-80 nN, and can be reduced to below 30 nN with proper treatment^[15]. Assuming that the friction force of the specific design shown in Fig. 9 is on the same order, it can be concluded that the electrostatic driving force is indeed large enough to overcome the friction force between the slider and the substrate. It is noted that the electrostatic force here is nonlinearly dependent on x. In fact, with appropriate design, linear-spring-like restoring force is also possible, which will be detailed in a future publication. Meanwhile, in the y-direction, the electrostatic energy minimum is located at x=0, as shown in Fig. 9(e). Thus the slider tends to stay on the middle in y-direction. Any y-displacement due to external perturbations will be constrained and the constraining force is on the order of 1 μN as well, as shown in Fig. 9(f). The geometric parameters used for calculation are a=6 μm, b=3 μm, h=0.5 μm, a=6 μm, c=0.5 μm, g=1.5 μm, and d=20 nm. As a proof-of-concept estimation, ∊_r=1 is assumed for all the capacitances. The thickness of the electrodes is assumed to be 50 nm when calculating C₄ and C₅.

In a real device, energy dissipation is inevitable. However, thanks to the ultralow friction of a superlubric interface, the energy dissipation can be very low in a superlubric oscillator whose quality factor can be as high as 1.3×10⁷, as shown in Ref. [23]. Meanwhile, the electrostatic actuation discussed in this section can also supply energy to the system and maintain a sustainable oscillation. In a real device, the non-zero friction at the sliding interface also has effect on the rotational stability discussed in Section 3.2.2. When only friction and interfacial energy are considered, the friction at the interface induces an extra torque that resist any rotation of the slider. Therefore, the friction generally benefits the stability against rotation in this regard. On the other hand, if taking electrostatic actuation into consideration as well, having larger friction means a larger voltage is needed to drive the motion. The rotational stability should be dependent on the total energy (sum of interfacial energy and electrostatic energy) as well as the frictional resistance torque. In this case, the stability is dependent on the specific geometry and design, and a systematic study is left for further research.

5 Conclusions

In this paper, the concept of constraining rotation of a superlubric slider by interfacial energy reported in Ref. [27] is generalized to controlled movement of superlubric slider of an arbitrary shape. The concept of controlled movement of superlubric MEMS is proposed through minimizing interfacial energy and minimizing electrostatic energy. Detailed studies are carried out for superlubric oscillators, which show that interfacial energy and electrostatic force are effective to provide driving, restoring, and constraining forces to the slider. The interfacial energy induced retraction force is investigated for four different slider shape designs, along with analyses of the rotational stability. When the shape of the contact-pair is designed appropriately, interfacial energy provides retraction force to the slider in x-direction and at the same time constrains the unfavorable movements due to external perturbations in the other degrees of freedom. It is also shown that with proper designs, linear-spring-like restoring force can be realized. Thanks to this linear restoring force, the slider has a well-defined resonance frequency which is an important feature for the practical use of a resonator and was not explored before for superlubric oscillators to our best knowledge. By applying voltage to the electrodes distributed under the slider, actuation of the superlubric resonator by electrostatic forces is feasible. Quantitative calculations show that tunable driving force and extra constraining force are enabled by the capacitive forces between the slider and the electrodes. In a broader view, the results provide a method to drive, control, and constrain the motion of the contact-pair in superlubric MEMS. It should be applicable to microscale superlubric devices in general, and guide future application research on structural superlubricity.

Supplementary Materials

Supplementary materials contain derivation of retraction forces and resonance frequency.

1 History of Microscale SSL and SSL MEMS/NEMS

Fig. 1 in the main text summarizes some of the milestones in the history of microscale SSL and SSL MEMS/NEMS. The works listed there are briefly reviewed in this section. Multi-walled carbon nanotubes are considered as candidate for movable parts in NEMS. A pioneering work on this subject is the observation of spontaneous retraction of telescoped carbon nanotube wall against its outer shell^[7]. When the inner tube is telescoped and then released, it spontaneously retracts back to minimize surface energy. Nano-devices were proposed based on this structure, such as gigahertz mechanical oscillator^[9], nanoscale linear bearing^{[7, 12]}, nanoscale rheostat^[8], nano-switch^[13], etc. The friction between the inner and outer tubes was estimated to be ultralow^[32]. Due to similar van der Waals interactions between the layers, the ultralow friction between the walls of multi-walled carbon nanotubes should be also observed in stacked graphene layers (graphite). Thus similar self-retraction motion was observed in graphite mesas in 2008^[10], and later proved to be an outcome of microscale superlubricity at the graphite-graphite contact^[1]. The concept of superlubricity was theoretically proposed in 1980s to 1990s^[4-5]. In an incommensurate contact, the lateral forces acting on each atom cancel out systematically, resulting in vanishing friction. In the early attempts, superlubricity was only experimentally achieved on nanoscale scanning probe systems^[6]. The microscale superlubricity in graphite mesas bridged the two parallel paths of fundamental researches on SSL and van der Waals mechanical micro/nano-devices, and made SSL technologies possible. From then, a lot of fundamental researches on microscale SSL have been conducted on graphite mesas^{[2-3, 14-16]}, as well as the heterojunctions between a graphite mesa and another material^[17-21]. Two recent reviews on SSL across length scales and SSL in crystalline materials are recommended to the readers^{[3, 18]}. While on the other hand, SSL based micro-devices with versatile functions were designed or conceptually proved, such as SSL generator^[22] and SSL oscillator^[23].

2 Derivations of Retraction Forces and Resonance Frequency

As mentioned in the main text, the retraction force F_ret is related to the overlapping area of the slider and substrate A as

$ {F_{{\rm{ret}}}} = - {F_x} = - \frac{{\varGamma \partial A}}{{\partial x}} $

(S1)

The derivations of F_ret-x relations for the 4 designs shown in Fig. 3 are detailed here. Considering the symmetry of the slider and substrate shapes, F_ret(x) should be an odd function for all the 4 designs. Thus, only the F_ret-x relations for x>0 are considered here.

The 1^st design

The retraction force for the 1^st design has been derived in detail in Ref. [10] and is omitted here.

The 2^nd design

For the 2^nd design, in the range of x < a/2 as shown in Fig. S1(a), the overlapping area is

$ A = \frac{1}{2}ab - \frac{{b{\kern 1pt} {x^2}}}{a} $

(S2)

Thus, the retraction force is

$ {F_{{\rm{ret}}}} = \frac{{2\varGamma bx}}{a} = kx $

(S3)

where k=2Γb/a is the effective stiffness of the linear-spring-like restoring force. Therefore, the resonance frequency of the resonator is

$ f = \frac{1}{{2\pi }}\sqrt {\frac{k}{m}} = \frac{1}{\pi }\sqrt {\frac{\varGamma }{{\rho {\kern 1pt} {\kern 1pt} {a^2}t}}} $

(S4)

While for x>a/2 as shown in Fig. S1(b), the overlapping area is

$ A = \frac{{b{{(a - x)}^2}}}{a} $

(S5)

Thus, the retraction force is

$ {F_{{\rm{ret}}}} = \frac{{2\varGamma b(a - x)}}{a} $

(S6)

The 3^rd design

For the 3^rd design, in the range of 0 < x < (a-c)/2=(a/2)(1-2h/b) as shown in Fig. S2(a), the overlapping area is

$ A = {A_0} - (\frac{{b{\kern 1pt} {x^2}}}{a} + 2hx) $

(S7)

where A₀=(b+2h)(a-c)/2+bc is the total area of the slider which is a constant. Thus, the retraction force is

$ {F_{{\rm{ ret }}}} = 2\varGamma (bx/a + h) $

(S8)

It can be seen that F_ret(0⁺)=2Γh. Since F_ret(x) is an odd function, thus F_ret(0^-)=-2Γh and the step at the discontinuous point is ΔF_ret=4Γh.

For (a-c)/2 < x < (a+c)/2 as shown in Fig.S2(b), the overlapping area is

$ A = {A_0} - \frac{{(b + 2h)(a - c)}}{4} - b(x - \frac{{a - c}}{2}) $

(S9)

Thus, the retraction force is

$ {F_{{\rm{ ret }}}} = \varGamma b $

(S10)

For x>(a+c)/2 as shown in Fig. S2(c), the overlapping area is

$ A = \frac{{b{{(a - x)}^2}}}{a} + 2h(a - x) $

(S11)

Thus, the retraction force is

$ {F_{{\rm{ret}}}} = 2\varGamma [\frac{{b(a - x)}}{a} + h] $

(S12)

The 4^th design

For the 4^th design, in the range of x < (a-h)/2 as shown in Fig. S3(a), the overlapping area is

$ A = A_0^\prime - \frac{{b{\kern 1pt} {\kern 1pt} {x^2}}}{a} $

(S13)

where ${A'_0} = \frac{{ab}}{2}\left[ {1 + {{\left( {\frac{h}{a}} \right)}^2}} \right]$ is the total area of the slider which is a constant. Thus, the retraction force is

$ {F_{{\rm{ret}}}} = \frac{{2\varGamma bx}}{a} = kx $

(S14)

which is exactly the same as Eq. (S3). Therefore, the 4^th design has the same linear-spring-like restoring force and the same resonance frequency as the 2^nd design.

For (a-h)/2 < x < (a+h)/2 as shown in Fig. S3(b), the overlapping area is

$ A = A_0^\prime - \frac{{b{{(a - h)}^2}}}{{4a}} - b(x - \frac{{a - h}}{2}) $

(S15)

Thus, the retraction force is

$ {F_{{\rm{ ret }}}} = \varGamma b $

(S16)

For x>(a+h)/2 as shown in Fig. S3(c), the overlapping area is

$ A = \frac{{b{{(a - x)}^2}}}{a} $

(S17)

Thus, the retraction force is

$ {F_{{\rm{ ret }}}} = \frac{{2\varGamma b(a - x)}}{a} $

(S18)