## Electro Micro Metrology

All MEMS are subject to process variations that can drastically change performance behavior from that predicted by modeling and simulation of its prefabricated design. In the following figures, we describe process variations, their effects, and how EMM can characterize such MEMS.

The following illustrations provide a concise introduction to how EMM can be used to accurately measure the effective mass, damping, stiffness, and temperature of MEMS. In addition, an explanation of how EMM is able to quantify the uncertainty of its measurements is also provided. An introduction to process variations and how it significantly affects performance is given at the bottom of this page

Measure change in capacitance

The device chosen for this example has comb drives, flexures, and two asymmetric gaps. For this version of EMM, the required components are the comb drive and gaps. The type of flexure is irrelevant.

A 3-step process is illustrated above. Step1: Measure the initial capacitance before actuation. Step2: Apply enough voltage to close the left gap, and measure the capacitance. Step3: Apply enough voltage to close the right gap, and measure capacitance. These measurements of the change in capacitance that the comb drive must undergo to fully traverse the size of the gap are all that is necessary to measure the fabricated gap, force, displacement, and stiffness. These and other characteristics follow as described blow.

Determine the geometric difference between layout and fabrication

Once the capacitances due to gap-closure are measured, an accurate measurement of the change in gap geometry is known.

The parasitic capacitance, which is constant between each pair of measurements, is eliminated by taking the difference in capacitance. This enables EMM to be uniquely repeatable between labs and facilities.

Equation of motion

For the first time, the complete equation of motion is measurable in MEMS. This includes measurements of effective mass, displacement, comb drive force, stiffness, damping, etc. In addition to gap capacitance, the measurements of mass and damping require measuring resonance.

This shows a sequence of quantity extractions, where each quantity follows from a previous quantity. That is, once gap is known, the comb constant can be determined, then they are used to determine displacement, which is used to determine force, which is used to determine stiffness, which is used to determine mass, etc. Well-accepted definitions are used to link each quantity.

Note: velocity resonance in non-vacuum yields the same value as amplitude resonance in vacuum. Therefore, velocity resonance is used because achieving a good vacuum is often expensive, time-consuming, or impractical.

Uncertainty

The  uncertainty from a single measurement is possible with EMM, which is done by taking a multivariate Taylor expansion of the desired quantity about the uncertainty in the electrical measurands. The uncertainty of the electrical measurands are easily and conservatively determined by using the order of the most significant flickering digit from the electronic readings. Such flickering includes the totality of all sources of noise.

Accuracy

Since absolute mechanical quantities are expressed in terms of changes in electrical measurands, the accuracy of the mechanical quantities do NOT depend on the accuracy of the electrical measurands. However, the accuracy of mechanical quantities DOES depend on the precision of the electrical measurands. This is also why EMM is reliable and repeatable.

Temperature

For the first time, EMM enables MEMS to be used as accurate absolute thermometers. The technique used there is the reverse of what has been used to estimate the stiffness of atomic force microscope (AFM) cantilevers. When used for the AFM, the measurements of thermally-induced displacement and temperature of a nearby thermometer are used to estimate cantilever stiffness by the equipartition theorem. However, in this case, since EMM can more accurately measure the stiffness and displacement of MEMS, then temperature can be easily and accurately determined.

Validation

Two types of EMM methods for measuring geometry have been validated and published [see publications]. The figure shows how EMM compares to the scanning electron microscope (SEM). The MEMS under test is shown [Top-Left], where a large number of measurements are taken with the SEM [Bottom-Left]. Due to the coarse side walls, the actual width of a flexure is indeterminate. Even if an average value is determined, such a value could not be used to accurately predict for performance because flexure stiffness is not linear in width, it is cubic in width. More over, the courseness is not just along the length of the flexure, but it is also coarse into the plane. [Right] This plot compares EMM to the SEM, where the y-axis is the overcut in going from layout to fabrication, and the x-axis pertains to the amount of capacitance change in EMM. That is, the larger the actuation, the smaller the relative error for EMM. The important aspect of this plot is that all EMM measurements have a much smaller relative error than SEM.

The accuracy of EMM is  two-fold. (1) EMM is shown to be repeatable, where we disassembled the test-bed and shut down the equipment between tests. And (2) EMM is performance based, which means that its results, when plugged into the model, match the performance of the real device. Conversely, SEM is not repeatable, since each operator would get an entirely different set of width measurements each time. And SEM is not performance based, that is, it is not guaranteed to achieve a match between the model, experiment, and the highly-uncertain SEM results.

Test-bed

EMM may be perform-ed on-chip or off-chip using standard lab equipment, such as capacitance meter and voltage source.

[Left] A couple of test-beds are shown in the figure. Each test-bed is covered with a Faraday shield to reduce interference by external electromagnetic fields.

[Right] Electrical connectivity between the MEMS and capacitance meter. The capacitance meter and interface used is here is an ADI 7746 (\$150) with a 4-attofarad resolution.

# Introduction to process variations

Material properties and geometry vary from run to run and across the wafer. [Top-Left]  This plot shows the variation in residual stress versus fabrication run. [Top-Right] This plot shows the elasticity of polysilicon versus publication date of various research organizations and testing methods. [Bottom-Left] The variable geometry of a fabricated flexure is shown next to its experimentally-accurate model, where EMM determines the model parameters that are required such that a simulation of the model exactly matches the behavior of the real flexure. [Bottom-Right] Original layout geometry (dashed lines) superimposed onto its fabricated counterpart. Compared to the desired layout geometry, the fabricated geometry shows a decrease in capacitance between the gap and a decrease in flexure stiffness.

[Left] Image of a microstructure used to model the effect of overetch on effective stiffness, as shown in the adjacent plot. The minimum feature size is 2um. [Right] Plot of the modeled relative-error in stiffness versus overetch for the adjacent microstructure. It can be seen that a relatively small increase in overetch of 0.25um can greatly increase the structure's overall stiffness by 98%. If the variation in Young's modulus is also taken into account, then the increase in stiffness can be as large as 188%.  That is, small process variations in material properties and geometry significantly affect the behavior of MEMS.