Digital Control of Active Network Microstructures on Silicon Wafers

This chapter presents a promising digital control of active microstructures developed and tested on silicon chips by current division and thus independent Joule heating powers, especially for planar submillimeter two-dimensional (2-D) grid microstructures built on silicon wafers by surface microfabrication. Current division on such 2-D grid networks with 2 2, 3 3, and n n loops was modeled and analyzed theoretically by employing Kirchhoff’s voltage law (KVL) and Kirchhoff’s current law (KCL), which demonstrated the feasibility of active control of the networks by Joule heating effect. Furthermore, in situ testing of a typical 2-D microstructure with 2 2 loops by different DC sources was carried out, and the thermomechanical deformation due to Joule heating was recorded. As a result, active control of the current division has been proven to be a reliable and efficient approach to achieving the digital actuation of 2-D microstructures on silicon chips. Digital control of such microstructural networks on silicon chips envisions great potential applications in active reconfigurable buses for microrobots and flexible electronics.


Introduction
Silicon-based microelectromechanical systems (MEMS) devices, including sensors, actuators, and generators, show wide applications in microrobots [1], medical devices [2,3], and flexible electronics [4]. Such miniaturized systems, on one hand, offer great potentials for improving the abilities of micromanipulation and functioning in some extreme conditions, such as limited working space and large displacements; On the other hand, however, the strong requirement of precise and effective control of these kinds of devices is not easy. Therefore, microstructures allowing for reliable and efficient actuation and large displacement are worth investigating.
Thermal microactuators have been proven to be able to realize large displacements more efficiently, and a variety of materials, for example, ceramics [5][6][7], polymers [8], composites [1,9,10], and metals [11][12][13][14], are available for building such active microstructures. Moreover, the geometries of these microstructures heavily depend on the selected materials, among which electrothermal bilayer beams have obvious advantages on large displacements, low costs, and high compatibility with mature microfabrication processes. Many efforts have been paid to develop electrothermal microstructures based on different applications and requirements. A safety and arming device composed of two V-shape electrothermal actuators was built, and the design of a cascaded V-beam amplification and two mechanical sliders enabled large deformation. As a result, a large planar displacement of 231.78 μm was achieved by applying a voltage of 15 V [15]. A typical Ushape electrothermal actuator made of a single material allowing for planar bending due to the thermal mismatch between the cold and hot arms was developed. Since the Joule heating power on the narrower arm was smaller than the wider one, the thermal expansion on the narrower (or hot) arm is larger than that of the wider (or cold) arm [16]. Another representative study is related to high frequency, low power, electrothermal bimorph actuators with shape memory effects, and the significant thermal mismatch and shape memory effect contributed to very large out-of-plane deformation [17].
However, previous research on electrothermal actuators usually focused on development beams with simple geometries, such as bridges, V-shape, U-shape, etc. The current flows through the beams were the same, which limited their ability of reconfiguration. To accomplish diverse reconfiguration of such electrothermal microstructures, active current division across planar (or 2-D) bilayer microstructures offers a promising approach to digital control of microstructures for distributed thermal balance and thus various deformations.
Our group has been endeavoring to design, fabrication, and characterization of 2-D multilayered microstructure consisting of beams for out-of-plane deflection, vertical deployment [18][19][20][21][22][23][24], and twisting under electrothermal actuation [9]. These microstructures can be created on whole wafers and tested separately by cutting the silicon wafer into chip-scale pieces [18]. Active parts of such microstructures are released by selective etching on the silicon substrates, while the anchored parts are protected from being etched. Different Joule heating powers and balanced temperatures on the grid networks could be obtained. However, instead of qualitative analysis, it is more important and meaningful to quantify the effect of electrothermal effect of bilayer 2-D microstructures, which will lay a solid foundation for modeling and predicting their potentials for large out-of-plane displacement.
The rest of this chapter is organized as follows. In Section 2, theoretical analysis of current divisions in some typical 2-D grid microstructures is firstly carried out, followed by the quantitative analysis of the equivalent resistance and Joule heating power. After that, fabrication and characterization of electrothermal and thermomechanical performances are presented experimentally in Section 3. The results are then shown and discussed in Section 4. Finally, the chapter is concluded in Section 5.

Theoretical analysis of current divisions
In this section, Kirchhoff's Voltage Law (KVL) and Kirchhoff's Current Law (KCL) are employed appropriately to figure out the feasible current division in several representative 2-D grid structures. Note that the scale of the 2-D structures does not change the relative division ratios between these beams. So, we consider more general structures, instead of only microstructures, in this section. It also hints that such cross-scale research on the current distribution is applicable to multiple surroundings and uses. For the sake of simplicity, all the grid structures presented in this chapter are composed of bilayer beams with the same materials and dimensions. Furthermore, these beams are incorporated into grid networks with different geometries. Two representative 2-D grid structures with the 2 Â 2 loops and 3 Â 3 loops are shown in Figure 1, where the candidate input ports are marked as red dots. The 2 Â 2 grid structure consists of 12 beams for the current division, while the 3 Â 3 one consists of 24 beams. Besides that, it is worth noting that grid structures with n Â n loops can be similarly proven to be made of 2n(n + 1) beams.
Let us start with studying the current division through 2 Â 2 grid structures by assuming that a constant voltage V is applied to any two outer red nodes, as seen in Figure 1. To ensure the stable connection between the voltage source and the red input nodes, another two supporting beams with fixed ends anchored on silicon wafers are designed. Hence, it can be derived simply that there are six independent cases in total for voltage inputs, as illustrated in Figure 2. The resistances of the beams in grid structures, as well as those of the two supporting beams, are assumed to be the same R. To acquire the current distribution through these beams of all the cases, KCL and KVL are adopted to the nodes and loops, respectively. Taking Case 1 as an example, as presented in Figure 3, nine independent KCL equations at the nodes N i (i = 1, 2, … ,9) and five independent KVL ones at the loops C i (i = 1,2, … ,5) are established, as shown in Eq. [1].
Specifically, the KCL equations could be written as while the KVL equations are formulated as As a result, the 14 unknowns I i (i = 1, 2, … , 14) can be uniquely solved by the derived 14 independent equations, that is, which demonstrates the determination of current division factors for all the beams in Case 1. Similarly, the current division factors in Cases 2-6 can be obtained uniquely, as seen in Figure 2.
Further investigation on the current distribution across planar structures with 3 Â 3 loops was taken, and the division factors and directions of the current flows had been proved to be unique. Generally, there are 12 independent cases for voltage inputs, as seen in Figure 4. Particularly, current distributions of two representative geometries with 3 Â 3 loops, Case 3 and Case 6, are solved and illustrated in Figure 5. A very interesting phenomenon observed in Case 3 of the 3 Â 3 loops structure is that the current division factors are symmetric about the central axis marked as a green dashed line in Figure 5. Further exploration reveals that the directions of current flows symmetrical about the green line are opposite, and the magnitudes of the currents closer to the input ports tend to be larger. In addition, though the distribution regularity of Case 6 is a little more complicated than that of  the Case 3, a very interesting symmetric current distribution about the diagonal of the geometry is found. Similarly, current flows through more complex geometries with n Â n loops like Case 6 are shown in Figures 6 and 7. It is worth noting that the corresponding current division factors in Figure 6 are represented by E i (E 2 =2E 1 ,E n =E n-1 +E n-2 when n ≥ 3 and n is odd, E n =(E n-1 +E n-2 )/2 when n ≥ 3 and n is even). Generally, there are (2n À 2) different currents through the n Â n grid network, enabling digital control of diverse and regional Joule heating powers across such a 2-D network.
Based on the current divisions of 2 Â 2 loops structures with diverse voltage inputs' cases presented and discussed in this section, the effect of Joule heating on such conductive grid structures is available to be quantified and evaluated.
Assuming that the resistances of a single beam in the 2 Â 2 loops, as well as each of the supporting beams, are the same, R, and the external voltages applied are V, The equivalent resistances of each case of the 2 Â 2 grid structures shown in Figure 2 can be solved using KCL and KVL laws, which was presented in the previous research. The Joule heating power of the grid structures and supporting beams can be then determined and listed in Table 1, where R i represents the resistance in case i, while the P i represents the heating power when a voltage of V applied. It can be seen clearly from Table 1 that equivalent resistances are approximately three times over the resistance of a single beam, and thus the expected Joule heating power is about one-third of the power when a single beam is applied by a voltage of V.
Similarly, the equivalent resistances of the 3 Â 3 loops structure in Case 3 and Case 6 can be derived to be 293/56ÁR and 27/7ÁR, respectively.

Fabrication of 2-D grid microstructures
To demonstrate the effect of electrothermal actuation of the 2-D structures, a series of ultrathin (or 2-D) microstructures consisting of grid beams and supporting beams that are mentioned in Section 2 are designed, fabricated, and tested.
Several previous research by our group can be referred to on a typical design and fabrication processes of such 2-D microstructures with two different materials. Specifically, aluminum and NiTi alloys (which are in the austenite phase in the range of testing temperatures) are selected as the bottom layer and top layer, respectively. What needs to be emphasized here is that although NiTi alloys show great potential for shape memory effects, this effect is not introduced intentionally in this research. It is the effect of digital Joule heating that we want to present in this chapter. Aluminum was chosen due to its significantly larger coefficient of thermal expansion than that of the NiTi alloys in the austenite phase.
As a result, a typical 2-D microstructure with the geometry presented in Figure 3 was imaged by the SEM after being selectively released from the silicon chip, as seen in Figure 7. It is worth noting that these two contact pads attached to the silicon chip were connected to gold wires with a diameter of 20 μm. The gold wire was used for electrical signal transfer from the logic printed circuit board after being fixed on the in situ imaging stage in SEM.

Results and discussion of in situ test of the microstructure
The experimental setup for in situ electrothermal actuation of the microstructure is illustrated in Figure 8. The stage of the SEM was tilted from zero degree to 45 degree for easier observation and measurement, and the configuration of the microstructure without heating was reimaged as shown in Figure 9. The in situ electrothermal testing of such a microstructure on the silicon chip started with the   video about this process was recorded simultaneously. It can be seen from Figure 10, as well as the video, that the microstructure could be digitally actuated by distributed currents for diverse and regional Joule heating. Therefore, the effect of active control of microstructures using digitally distributed currents is demonstrated. It is important to highlight that although the "digital currents" were inherently ensured, the thermomechanical reconfiguration of corresponding beams does not seem to be that "digital". It could be explained by the effect of scaling which could have a significant influence on the thermal conduction between the beams. The scaling effect is expected to be alleviated gradually with structural scale-up. Generally, such silicon chip-based microfabrication processes show great compatibility, effectivity, and efficiency in the development and validation of 2-D microstructures.

Conclusions
In conclusion, effective and efficient development of active control of 2-D microstructures based on silicon chips is presented in this chapter. Representative planar structures composed of grid beams are introduced to quantitatively analyze possible current distribution across the conductive geometry using KCL and KVL. Diverse current divisions of structures with different loops and voltage inputs have been proven to be available for digital control of electrothermal actuators. Besides that, the determination of equivalent resistances and resulting Joule heating powers have contributed to the evaluation of in situ experiments on the representative microstructure created on silicon chips by microfabrication. The process and critical steps of thermomechanical deformation of such a microstructure are shown to demonstrate the effect of digital control by Joule heating. The unsignificant digital deformation could be attributed to the scale effect of thermal conduction.
It is worth highlighting that much more various and precise current divisions can be obtained by superposition of different voltage inputs, which can be an attractive topic in the future. Another promising research is to investigate the scale effect on Joule heating in different current distributions.