Sample case-study

Title
Resonant beam pressure sensor

Principal source
"Resonant beam pressure sensor fabricated with silicon fusion bonding," K. Petersen, et al., Proceedings of Transducers 91, San FRancisco, CA, June 24-27, 1991, pp. 177-180.

Purpose of the device
It is a pressure sensor. Its specific purpose is to measure pressure with an accuracy of 0.01% of absolute pressure. The applications are air pressures in aerospace systems and in engine control systems.

Principle of operation
It consists of a resonant beam on top of which piezo resistors (red) are formed. Underneath the beam, there is a diffused electrode (orange) that actuates the beam in resonance. There is also a pit underneath the beam as shown. Below the pit is a square disphragm. When the pressure is applied from bottom throught the pressure access hole, the diaphragm and the portion of the device above it experience change in tension. This changes the resonant frequency of the beam. This is detected via the piezo resistors. The change in the resonance frequency is correlated to the pressure under the diaphragm.
The beam is kept in resonance by a simple analog maintainer circuit while the change in resonance frequency is monitored by a microprocessor. These are outside the packaged sensor.

Microfabrication

1. This process begins with an n-type silicon wafer.
   
   
2. A shallow pit is etched as shown. One can see this as a rectangle in the top view.
   
   
3. p-type diffusion is performed to define electrodes in the shallow pit. These electrodes actuate the resonant beam that will be formed later. Diffused electrodes are connected to the top of the wafer as shown so that electrical connection can be easily made to them.
   
   
4. A second wafer is bonded and then ground to 6 microns. This allows for "released" mechanical structures in the portion of the top wafer above the shallow pit in the first wafer. This is where a resonant beam will be formed later.
   
   
5. A passivation layer (blue) is formed for electrical isolation. Piezo resistors (red) are created by ion implantation.
   
   
6. Contact holes are etched through the passivation layer and top silicon wafer to enable electrical contact with the diffused electrodes.
   
   
7. Bond pads and electrical interconnections are formed using metallization. Not see in the figure are electrical connection made to the piezo resistors. These will be seen another cross-section as they go perpendicular to the plane of this image.
   
   
8. The bottom wafer is then etched back from below to leave a square diaphragm. The square shape an can be seen if we view from top.
   
   
9. Two slots are etched in the top wafer to form the resonant beam. The piezoresistors are on top of this including the metallic interconnects. This is a top-view of the relevant portion of the wafer. The actuating electrode underneath the beam can also be seen.
   
   
10. A glass wafer with a pressure access hole is bonded to the bottom.
    
    

Modeling challenges
This device involves five energy domains.

1. Elasto-static and elasto-dynamic (diaphragm deflection and beam vibration)
2. Electrostatic (beam actuation)
3. Electrical (power supply to actuating electrodes and resistance measurement in piezoresistors
4. Fluidic (squeezed film effects under the beam)
5. Thermal (as a secondary effect in piezoresistors)

Modeling tasks
1. Modeling the piezoresistor behavior as a lumped model
2. Dynamic modeling of the beam to gets its requency response. This can be done first as a lumped 1 dof model and later as a distributed beam model. In both cases, squeezed film effects and external pressure effects are to be included.
3. Modeling the diaphragm deformation to see how the pressure on it changes the natural frequency of the beam. 3-D finite element analysis could be used to do this.
4. Secondary influence of temperature raise on the mechanical behavior (due to thermal loads) and electrical resistance can be studied. Again both as a lumped model as well as distributed model.
5. System level model of the whole device so that the influence of the ambient pressure and temperature can be studied.

Required background and governing equations
The following aspects need to be understood in order to model the device.

1. Piezeoresistive effect
2. 3-D elastic deformation
3. Simulation of dynamic behavior of beams
4. Electrostatic forces on elastic beams
5. Squeezed film effects
6. Change in electrical resistance due to temperature raise

Other references

1. Kovacs, G.A., "Resonant pressure sensors," Micromachined Transduers, WCB-McGraw-Hill, 1998, Boston, pp. 260-261.

Model analysis using the finite element modeling

In order to compute the first natural frequency of the resonant beam, a solid model of the device was built in I-DEAS and its model analysis was performed. The solid model was constructed in accordance with the dimensions given in the principal reference for this case-study. The following material properties were assumed.

• Young's modulus = 160 GPA
• Poisson's ratio = 0.29
• Density = 2300 kg/m^3