**MEMS energy harvesters using piezoelectric materials are compact and have the potential for high-volume production. As the design and optimization of these devices is complex, numerical simulation is a necessary tool in order to obtain high performance. However, an uncritical use of simulation tools may lead to significant inaccuracies.**

The need to sense, control, and communicate is growing rapidly due to the deployment of a large number of wireless applications, often in the context of the *internet of things*, and harvesting of energy from ambient vibrations can reduce the need for battery replacement and disposal. By using a MEMS energy harvester (EH) based on piezoelectric materials, the potential for large-scale and low-cost production emerges. However, the design of such components is far from trivial. In the EH itself, the mechanical and electrical properties must jointly be optimized when tuning the device to the expected excitation and at the same time the constraints related to MEMS manufacturing must be taken into account. Furthermore, the output energy must be stored and the variations in frequency and output power must be managed, typically using a low-power ASIC.

Thus, the complete story is very complex and the intention of this newsletter is only to give an overview description of EH numerical simulations and point out a number of potential pitfalls that must be avoided. For the simulations, we mainly use COMSOL Multiphysics as it is capable of jointly describing the mechanical and piezoelectric effects. We base this description on the simplest non-trivial example, which is the cantilever EH shown in Fig. 1. The EH material is silicon and the EH length is 3 mm. The beam thickness is 10 µm and the thickness of the PZT, selected to be PZT-5H and colored blue in the figure, is 2 µm. With these dimensions, using a thin beam and a, relatively speaking, large proof mass, the first resonance is at 260 Hz. This is to be considered a relatively low frequency in this context.

This EH is straightforward to simulate. The EH is fixed at its base and is excited by inertial forces due to the acceleration. The PZT is assumed to be operated in the 31 mode, i.e., the bending of the PZT leads to a vertical electric field. The two terminals are placed at the top and the bottom of the PZT and connected by a load resistance. The output power can be studied in time, using a time-dependent excitation, or in frequency, using a harmonic excitation, and many kinds of parameter studies can be performed. However, an uncritical use of simulation software leads to incorrect results as will now be exemplified.

In all FEM simulations, a high-quality mesh is essential. As is clear from Fig. 1, there is a separation of length scales in this problem; the PZT is thin as it is typically limited by the deposition techniques to less than 5 µm, leading to a difference of a factor ~1000 between the PZT thickness and the EH length. This is illustrated in Fig. 2, which shows a very dense automatically generated mesh. Although the mechanical properties would be very well captured by this mesh, the resolution of the PZT is very low. In fact, there is only a single element in the vertical direction of the PZT and the vertical electric field is not resolved. This shows that manual inspection and adjustment of the mesh is necessary.

A further effect from the separation of length scales is shown in Fig. 3. In this case, no damping is modeled as only the mechanical properties are simulated. In the left figure, the EH is released at time zero and is allowed to “fall” and oscillate. This is achieved by simulating the time evolution, starting from a case with no deflection and adding gravitational forces (1 *g*) at the start of the simulation. It is seen that the system loses energy rapidly, corresponding to a very low quality factor. This is not a physically correct behavior and this is caused by the fact the deflection is so small that the default error tolerances are not sufficiently strict. In the right figure, the gravitational force has been increased to 50 *g* and the error tolerances have been reduced. The motion is now the expected vertical oscillation with a high quality factor. This illustrates the fact that the simulation error must be monitored.

Figure 3. The oscillating motion of the EH with a step external acceleration. (left) 1*g* acceleration and default error tolerances. (right) 50*g* acceleration and reduced error tolerances.

The fact that the PZT is thin also has further implications for the numerical simulations. As the rate of energy loss due to energy output is small, the quality factor is high. Thus, the resonance is very narrow with high amplitude and the deflection amplitude becomes large when using harmonic excitation, often to the point that the simulations fail to converge. Two possible ways to work around this problem is (i) to use time-domain simulations or (ii) to model the damping. In the first case, a reasonable choice is to use an excitation pulse with a limited extension in time. In this way, the power spectral density of the excitation is spread over a band of frequencies and the mechanical energy of the system is kept at a reasonable level. In the second case, the most important damping mechanism must be identified and modeled. Unfortunately, this is not always a simple task but in some situations squeeze film damping can be identified as the main energy loss. This typically occurs when the EH is oscillating towards a wall, has a considerable area, and when the EH/wall separation is small. In this case, the damping can be introduced in COMSOL Multiphysics or even estimated analytically (see, e.g, M. Bao and H. Yang, Sens. Actuators, 136, 3-27, 2007). As the air moves in and out of the volume between the EH and the wall, viscous effects lead to a pressure distribution on the EH as shown in Fig. 4. This opposes the motion and dissipates energy.

In summary, numerical FEM simulation is an important tool when studying and designing EHs, but a critical view on the results based on physical understanding is essential when interpreting the results.

Author(s):

Pontus Johannisson, Acreo