We all have them in our phones: MEMS (Micro Electric Mechanical Systems). Inertial MEMS, for instance, are the tiny accelerometers and gyroscopes in our smart phones. They make playing games on the phone more fun and help us finding the next pizza restaurant. To do that, they measure our movements by recognizing rotation and acceleration. MEMS are really tiny, i.e. in the micro-meter (µm) size regime. You need special microscopes to see their set up in detail, e.g. a scanning electron microscope. However, people are trying to make them even smaller. Why? Well, there are many reasons and of course one of them is the cost. If the devices were smaller, the production cost would also decrease.

The MEMS accelerator and the gyroscope are two of the most important examples of the growing zoo of MEMS. Therefore, we will take them as examples in our discussion on what limits the shrinkage of MEMS? To give an overview, we will first consider some general aspects of dimension scaling. Then we will proceed to more specific aspects which limit the area shrinkage of an inertial MEMS.

The very beginning of nanotechnology is often associated with the famous lecture “There’s plenty of room at the bottom.” by Nobelist Richard Feynman. According to him, you can always decrease the size of a device by altering the setup design. So you can make a device smaller by using a completely different "machine". But, until new reliable designs for such "bottom-up" ideas have been developed, it is convenient to decrease the technology which is already available. Such top-down developments generally experience a limit, when the size of a device part reaches the scale of the physics, that defines the part’s important material properties. For example, a metal segment, where the electric conductivity is the important property, cannot be reduced without limitations, because it looses electric conductivity when nano-meter scale is reached. The metallic state is revoked by the emergence of a band gap due to the limited number of atoms.

Apart from such very fundamental limitations, we have to be aware of the quasi-fundamental scaling of physical properties also for bigger size regimes. For instance, the length **L** scales linearly, when a device is decreased in size. However, the surface area scales with **L ^{2}** and the volume with

**L**, from which results, that the area/volume ratio scales with

^{3}**L**. This can be important when considering heat flow. Decreased replica may therefore not have the same thermal properties as their larger analogs. Similar points can be made for mass (which scales with

^{-1}**L**), the mechanical strength or the resistance.

^{3}To understand the size limitations of MEMS in more detail, we have to have some basic knowledge of their working principle: Let's first consider the accelerator. They measure the acceleration **a** of the hosting device into any of the three spacial dimensions using silicon spring-mass systems. Acceleration results in a force **F _{acc} = ma (eq. 1)** acting on the proof mass

**m**. The springs allow a movement of the mass due to this force and the displacement can be registered by capacitor plates. So, when the mass moves, the distance between the different capacitor plates changes. This results in a potential difference which can be easily translated into digital data, which is our signal

**(S)**.

As accelerometers, also gyroscopes detect a force acting on a proof mass. However, gyroscopes use the Coriolis force, which acts on a moving and rotating mass: **F _{cor} = 2m Ω × v (eq. 2)**. Here,

**Ω**is the rotation vector and

**v**is the velocity vector of the mass. Just as in the case of the accelerometer, a spring-mass system is used, but this time, the mass is set in constant oscillation. To keep the mass in such motion, obviously a more sophisticated capacitor setup must be supplied, but we don't want to learn this set up right know.

If we want our MEMS to work properly, we need an appropriate signal to noise ratio **(S/N)**. Anything that significantly reduces **S/N** due to area shrinkage is therefore a potential limitation. Evidently, **S** should be high and **N** low. Our signal is dependent on the displacement of the proof mass. The displacement in turn is dependent on the force that acts on the mass and the force is dependent on the mass, see **eq. 1-2**. So, we need a big mass, but as we have seen before, the mass scales with **L ^{3}**. On the other hand, the noise must be small. The noise is affected by a lot of factors. One of the important ones is the

*mechanical thermal noise*. In fact, the smallest displacement registration of the proof mass is limited by mechanical thermal noise. So the proof mass limits

**S/N**from both sides: A small mass gives a small signal on the one hand and more noise on the other hand. So at a certain size, the

**S/N**becomes too small, because we have shrunk the proof mass too much.

Another limitation of the MEMS shrinkage is due to air friction. Although they are so small, the proof masses suffer from air friction and therefore their movement is damped. This is an important problem, because damping impairs the signal quality and so the sensitivity of the measurement. Reducing the gas pressure by near-vacuum packaging can only decrease the fraction to a certain extend, but not completely.

A further factor that limits area shrinkage is the rigidity of silicon. To compare stiffness values of different materials, the Young’s modulus can be used, which is a

measure for the materials elastic behaviour. The greater the Young’s modulus, the more stiff and rigid is the material. Do you think that a Quartz crystal is stiff? Quartz has a Young's parameter of around 70 GPa. Silicon (the material the proof masses are made of) has a Young's which is twice as big. So, it is a quite stiff material, which is actually good, because it means also that it is a stable material. Yet, this stability is size dependent. When silicon segments become too fine, they break easily and this contradicts the consumer requirement of a stable device. Therefore, we cannot just make it more small.

In summary, we have seen that size and shape of the proof mass (and other segments) are essential parameters that limit the shrinkage of inertial MEMS. The S/N ratio and the stability are significantly influenced by the size of the segments and that leads to an overall shrinkage limitation for the MEMS device.

**Author:** Philomena Schlexer

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