I. INTRODUCTION

Reconfigurable logic (RL) circuits such as field-programmable gate arrays (FPGAs) have received much attention due to their advantages: design flexibility and fast development^[1-7]. However, conventional RL circuits consisting of CMOS routing blocks (RBs) have shown some limitations in terms of chip density, energy efficiency and performance. To overcome these limitations, some alternative routing switches including nanoelectromechanical (NEM) memory switches have been intensively researched to replace CMOS RBs^[3-7]. It is feasible because NEM memory switches can be integrated in back-end-of-line (BEOL) metal interconnection layers. Previously, we have successfully proved that CMOS-NEM RL circuits have lower leakage power, signal delay and chip area than conventional CMOS-only RL ones^[5-7].

On the other hand, NEM memory switches suffer from reliability problems due to their operating mechanisms. Fig. 1(a) shows the structure of a conventional NEM memory switch, which acts as a one-to-two multiplexer. It consists of two fixed metal electrodes called Selection Line 1 (SL1), 2 (SL2) and a movable cantilever beam attached to the bit line (BL). The data signal path and logic functions are determined by the position of the cantilever beam. In the initial state, the cantilever beam is not stuck to either SL1 or SL2. Thus, the data signal is not transmitted. Then, the positive or negative voltage ($V_{SL1}$ or $V_{SL2}$) is applied to either SL1 or SL2. It makes the cantilever beam attach to either SL1 or SL2, which means data signal is transmitted. For nonvolatile switching, generally, the contact area between the cantilever beam and SLs is designed to be large^[5]. However, large contact area makes stress more concentrated around the anchor. It means that the movable beam becomes weaker than as the number of switching cycle increases, which is problematic in terms of reliability^[7,8]. Thus, for reliability improvement, the maximum stress around the beam anchor should be alleviated.

In this manuscript, we propose that the reliability of NEM memory switches can be improved by optimizing the dimension of SL as shown in Fig. 1(b). The validity of the proposed method is verified by finite-element-method (FEM) simulation. Fig. 2 shows the key process steps of NEM memory switches.

II. RESULTS AND DISCUSSION

The quasi-static electromechanical behavior of NEM memory switches is simulated by using three-dimensional (3D) FEM simulation^[12]. Table 1 and Table 2 summarize the beam design parameters material properties used in this work. $L_{SL}$ varies from 2.5 μm to 30 μm for optimization. For fair comparison, our proposed NEM memory switch has the identical design parameters except for $L_{SL}$. In order to evaluate stress level, von Mises stress analysis is introduced which is commonly used for the evaluation of metal damage^[13-16]. In our work, von Mises stress profile analysis is essential to understand the spatial variations of the stress depending on the beam bending degree determined by the applied voltage. Fig. 3(a) and Fig. 3(b) show the beam displacement in the beam width direction ($L_{\Delta}$) as a function of $L_{SL}$ It is observed that less portion of the movable beam becomes in contact with SLs as $L_{SL}$ decreases. It means more stress is concentrated around the anchor in the pull-in state as $L_{SL}$ increases. To clarify it, Fig. 3(c) and Fig. 3(d) show the stress contours over the beam cross-section at the anchor. The stress increases from the middle part of the beam to the edge, which means that beam fracture occurs from the edge to the center reflecting our experimental results.

A 1D parallel-plate model is introduced to calculate minimal $L_{SL}$ for nonvolatile switching^[17]. Fig. 4 shows the relationship between beam restoring force ($F_{r}$) and van der Waals adhesion force ($F_{ad}$) with the variation of $L_{SL}$. Each force component is defined as

$F_{r}$ is constant because it is determined only by the structure and material of the cantilever beam. On the other hand, $F_{ad}$ is linearly dependent on $L_{contact}$. The value of $L_{SL}$ when $F_{r}$ is equal to $F_{ad}$ corresponds to the minimal $L_{SL}$ for nonvolatile switching. According to our calculation, the minimal $L_{SL}$ for nonvolatile switching is 4.5 μm.

Fig. 5(a) shows the trade-off between the pull-in voltage ($V_{PI}$) and maximum stress ($\sigma_{max}$) as a function of $L_{SL}$. The data was extracted from FEM simulation. It is observed that $V_{PI}$ increases as $L_{SL}$ decreases. It is easily explained by the 1D parallel-plate model where $V_{PI}$ is derived as

where $k$ is the beam spring constant, $g_{0}$ is the air gap between the cantilever beam and the SL, $\varepsilon_{0}$ is the vacuum permittivity and $A$ is the area overlapped between the beam and SLs. $A$ is calculated as $L_{SL}$ * t. Thus, $V_{PI}$ increases continuously as $L_{SL}$ decreases. On the other hand, $\sigma_{max}$ is maintained at ~289 MPa as long as $L_{SL}$ ranges between 17.5 μm and 30 μm. It is because even if $L_{SL}$ decreases, beam contact length ($L_{cont}$) is constant until $L_{SL}$ reaches $L_{cont}$. $L_{cont}$ is defined as the length between the beam tip and the position where the beam displacement is $g_{0}$. Beam noncontact length ($L_{V}$) is defined as $L_{beam}$ - $L_{cont}$. On the contrary, if $L_{SL}$ decreases from 17.5 μm to 4.5 μm, $\sigma_{max}$ becomes steadily lower down to 123.2 MPa maintaining nonvolatile property. In this region, because both $L_{cont}$ is equal to $L_{SL}$, the decrease of $L_{SL}$ makes $V_{PI}$ larger but $\sigma_{max}$ smaller. Finally, if $L_{SL}$ becomes smaller than 4.5 μm, NEM memory switches fail to operate losing nonvolatile property. Fig. 5(a) shows a good example of $L_{SL}$ optimization If 20-% increase of $V_{PI}$ is allowed, $\sigma_{max}$ can be made > 40-% lower when $L_{SL}$ is 7.5 μm.

III. CONCLUSIONS

In this paper, the $L_{SL}$ of NEM memory switches is optimized by FEM simulation considering mechanical reliability and operating voltage. According to the simulation results, as $L_{SL}$ decreases, $V_{PI}$ increases while $\sigma _{max}$ concentrated at the anchor of a movable beam decreases. It is found that the selection of optimized $L_{SL}$ is helpful to make NEM memory switches more reliable with tolerable $V_{PI}$ increase.