*Sophomore "Engineering Enthusiasm" Contest*

**March 17, 2006**

You have 60 minutes to complete the contest. You may use all materials at your disposal (within the prescribed limits). Stay at your seat until you are ready to hand in your work.

May the student with the most engineering enthusiasm win!

__Problem Statement__:

In the modern world, many devices are made with multi-functioning characteristics. Among them are micro-electromechanical devices, called MEMs. Let us assume that a cantilever-beam, MEMs-type structure is made entirely of silicon, after being etched using photolithographic processes. Using chemical doping techniques, the bottom layer (2) of the silicon will have different mechanical properties than that of the top layer (1).

A very small electrical current is being passed through the bottom layer, hence the resistive drawing. Assume no current is going through the top layer (perfect insulator). For the bottom layer, the current causes a power loss equal to *P* = *i*^{2}*R*, where * i* is the current and *R* is the resistance. For MEM-type structures, power levels are usually below 50 mW (*P* = 50).

As power is increased, it is found that the temperature increase of each layer above ambient increases proportionately. Due to the fairly high thermal conductivity of silicon and the high aspect ratio of the layers, assume the temperatures of both layers to be the same at all times.

A material′s coefficient of thermal expansion (CTE) is a description for how much the material will elongate (actually, "strain") for each degree Celsius increase in temperature. It is found that doping the bottom layer with cyanide nitrates allows CTE, *a*, of the bottom layer to increase. For every 1% weight concentration of doping, *D*, the CTE of the silicon increases by 0.003%. (Assume uniform concentration of the dopant throughout the thickness of layer 2 and no dopant for layer 1. Also, note that *D* = 2 for 2% doping, for instance.)

The curvature of the cantilever beam is given by the equation:

where *k* = curvature (mm^{-1}),

*a*_{i} = CTE of layer *i* (x 10^{-6}/^{o}C),

*E*_{i} = elastic modulus of layer *i* (GPa)*,*

*h _{i}* = thickness of layer

D*T* = temperature change of the structure from that of the bonding temperature (^{o}C).

__Notes__:

Â· Curvature is positive for upward bending; negative for downward bending.

Â· Deflection, *d*, is the amount of vertical deflection at the end of the beam and is positive for upward deflection and negative for downward deflection.

Â· Deflection is related to curvature since curvature can be approximated as the second derivative of deflection with respect to beam length.

Â· Assume the curvature of the beam to be constant with length.

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__Technical Information__:

Â· The thickness of both layers (*h*_{1} = *h*_{2}) of silicon is 25 mm.

Â· The length, *L*, is 150 mm.

Â· The elastic modulus, *E*, is 160 GPa for both layers (doped and undoped).

Â· For undoped silicon, the CTE (*a*) is 3 x 10^{-6}/^{o}C.

Â· The two layers are chemically bonded together at *T _{bond}* = 150

Â· *P _{min}* (minimum power input) = 1 mW.

Â· *D _{min}* (minimum dopant level) = 1%.

Â· *T*_{2} at minimum power input = 22 ^{o}C.

Â· Ambient temperature = 20 ^{o}C.

Â· Mechanical problems (delamination between layers) tend to occur when curvature magnitude exceeds 5 x 10^{-4}/mm.

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__Objectives:__

**Just turn in this answer sheet; your work will be ignored.*

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A. **[10 pts]** If curvature increases by 10%, deflection increases by ______ %.

B. **[10 pts]** At a power level of 1 mW, *T _{2}* is _______

C. **[20 pts]** At a dopant level of 20%, what is the deflection for *P* = 1 mW and 7 mW? (each answer equally weighted) ANSWERS: ___________ , ___________

D. **[30 pts]** At what power would deflection cause likely mechanical problems for a dopant level 20% (*D* = 20)? ANSWER: __________

E. **[20 pts]** At what power would deflection be zero (completely flat)? ANSWER: __________

F. **[10 pts]** Assuming no more than a 20% dopant level, should we expect delamination problems for this structure at high, yet reasonable, power levels? ANSWER: circle __YES__ or __NO__