1. 1 pt. for a correct cross-section

With perfect selectivity

PIT mask:

DIMPLE is 30um on a side

P1_P2_VIA is 16um on a side

ANCHOR2 is 6um on a side

  1. 1 pt. for a correct cross-section
  1. 1 pt. for a correct cross-section

Assuming a 175% overetch, we etch the full 1.5um of deposited POLY2. The nitride at the center of the pit gets (1.5um)(1.75)/5=0.525um etched away during the overetch. This does not fully breach the 0.6um of nitride.

  1. 1 pt. for stating that the nitride is not etched

The nitride is not breached. Assuming a more aggressive overetch or a lower selectivity, the nitride could potentially be etched away. It should noted that in Fouladi et al. the nitride breach actually etches 3um into the silicon substrate! (Figure 2).

If you made assumptions about the other overetches that gave you different answers (like P1P2via etching into ox1, and ANCHOR2 etching into nitride), give yourself full credit as long as your results are consistent with your assumptions. When I wrote this problem I thought that nitride was 0.5um thick (-1 for me!) and chose the selectivity such that nitride would be breached. -ksjp

  1. 1 pt showing that the polysilicon in the structure will get etched away
  1. 1 pt. for using oxide as the structural material

1 pt. for using POLY0 at the base of the cantilever as the piezoresistive sensing element

1 pt. for etching the substrate to release the cantilever

1 pt. for relating force to strain at the base of a cantilever

1 pt. for estimating the minimum detectable force using RMS noise

The layout and cross-section of the basic structure are shown below. The only two masks are POLY0 to define the piezoresistor and PIT to define where the silicon substrate is exposed and etched away by the TMAH etch.

Some simplifying assumptions:

  • The poly0 is at bottom surface at the base of the cantilever
  • The entire structure has the Young’s Modulus of oxide

The maximum strain will occur at the surface of the beam where z=a/2

Radius of curvature can be related to the moment on the beam by

The moment along the beam is given by

For a pure point force at the tip of the beam, . At the base of the beam, x=0

Assuming we have the piezoresistor in a wheatstone bridge configuration with excitation voltage

For and beam width ,

For the minimum detectable signal, we will have noise from the damping of the cantilever and the voltage across the resistor

For

For

The voltage noise dominates for large resistor values. We can bring the resistor value down so that our smallest source of noise is the damping of the cantilever. This happens when

  1. 1 pt. for finding the variance

Going from -40C to +85C, increases by over 54%

  1. 1 pt. for correct reasoning

is greatly reduced close to absolute zero. This lowers the noise floor of the experiment, allowing detection of very small signals.

  1. 1 pt. for an

The power spectral density is given by

The power dissipated through the resistor will be

We know that vn= 4nV/rt-Hz for R=1k, so for f=10kHz, vn=400nV. The noise equivalent acceleration will be the noise voltage divided by the sensitivity, so 400nV/(1mv/’g’) = 0.4 m’g’ or

  1. 1 pt.

Power spectral density is always . There is no dependence on any other physical parameter besides temperature.

  1. 1 pt. for each part (5pts. total)
  2. They are consistent. Equipartition, used in part a, tells you the noise on every energy storage element such as a capacitor or a spring. Fluctuation-dissipation tells you that there is constant noise power across all frequencies and this presents itself in elements like resistors and dampers. Integrating the noise density that you get from fluctuation-dissipation across all frequencies will give you equipartition.
  3. 1 pt. for each correct answer (5 pts. total)
  4. Release, the drying is performed at the critical point of CO2, so that the structure can dry without crossing a phase boundary.
  5. Both, dimples reduce device-substrate overlapping surface area and effective length of suspended structures
  6. Release, the XeF2 etch is gaseous. There is no liquid
  7. Both, SAM’s make surfaces hydrophobic, reducing stiction
  8. Release, same reason as XeF2. It is a gas phase etch.