Micro Electro-Mechanical Systems

Basic notion of kinematics ................................................................................................3

Basic notions of dynamics .................................................................................................4

One degree freedom oscillator ........................................................................................4

Basic notions of electrostatics ............................................................................................5

Parallel plate capacitors ................................................................................................6

Interdigitated “comb-finger” capacitors.............................................................................6

Coupled electro-mechanical problem ......................................................................................6

Parallel plate connected with an elastic spring .......................................................................6

Comb-finger connected with an elastic spring ........................................................................7

Capacitive accelerometers ...................................................................................................7

Capacitive gyroscopes ........................................................................................................9

Piezoelectricity ................................................................................................................9

Energy harvester .......................................................................................................... 10

Resonant accelerometers .................................................................................................. 10

A-Resonant accelerometer .............................................................................................. 10

B-Resonant accelerometer .............................................................................................. 11

Electrostatic stiffness variation ..................................................................................... 11

Momentum of inertia variation ...................................................................................... 11

Geometrical stiffness variation ...................................................................................... 11

Torsional resonator ....................................................................................................... 12

Thermo-mechanical problem .............................................................................................. 12

Electro-thermo mechanical actuators ................................................................................... 13

Fracture and fatigue ........................................................................................................ 13

Weibull approach ......................................................................................................... 14

Fatigue ...................................................................................................................... 15

Residual stresses ............................................................................................................. 15

Introduction

MEMS are made up of components between 1 to 100 µm (as the human hair diameter) in size. They usually

consists of a central unit that processes data, the microprocessor and several components that interact with

the outside such as microsensors or microactuators to exercise a control action.

Fabrication process 1

On a silicon wafer substrate, generally monocrystalline thanks to the Czochralski method , an oxide

2 3

sacrificial layer is deposited (with the desired pattern) , followed by a mechanical/structural layer . Both

of them are generally in silicon, respectively thin polysilicon and thick epitaxial polysilicon. The structural

layer is patterned by photolitography and etching. The end of the process consists in sacrificial oxide

removal and contact metallization deposition. Many processes start with the thermal oxidation of the

substrate in order to protect and electrically insulate the substrate from the upper device.

Contact printing occurs when the mask is directly posed on the photoresist, giving the possibility of

contamination. On the other hand, proximity printing can cause diffraction; actually projection printing is

the most used technique in order to avoid both the problems and even thanks to reduced costs (bigger masks

can be produced). Etching process consists of

three steps: mass transport

of reactants to the surface;

reaction between reactants

and the films to be etched at

the surface; mass transport

of reaction products. The

advantage is the high

selectivity, because it is

based on a chemical process;

the disadvantage is related to the poor process

control. Plasma etching has largely replaced wet etching because of the possibility to have directionality

(thanks to ionic components). Reactive Ion Etching is also used, in order to obtain nearly vertical sidewalls.

The complexity of the structure depends on the number of repeated steps, and generally is defined by the

number of used masks.

1 Raw polycristalline silicon is produced starting from selected sands and by reduction and refining in a

reaction furnace. A pure silicon seed crystal is placed into the molten bath, pulled out slowly as it is

rotated (homogeneous nucleation). The product is a monocrystalline silicon ingot, which is cut, lapped,

polished and cleaned.

2 Through dry or wet oxidation process, at 800 – 1200 °C.

3 Through CVD (epitaxy, electrodeposition) or PVD (evaporation, sputtering) processes.

A micromachined silicon wafer cap protects the

mechanical element and ensures the right damping to

the MEMS. Wafers are aligned to each other and

bonded by: direct fusion (1000 °C), anodic bonding

(Pyrex glass, 500 °C, positive voltage application),

glue layer (glass frit or gold, thermo-compression

process).

Packaging is applied to a unique box made of MEMS dies and ASIC (Application Specific Integrated Circuit).

It is possible to use: plastic, low cost, access for light, but possibility to break due to encapsulation process;

ceramic materials, durable, well sealed, higher costs; metals, solution for harsh environments, can be well

sealed. Hermeticity is very important.

Basic notion of kinematics

One reference frame in motion w.r.t. another, considering P fixed: the components c and n change in time.

i ij

The velocity of the point P as seen from O’ is given by where the c derivative is the

velocity of O w.r.t. O’. So we obtain . We can also compute the relative

acceleration

One reference frame in motion w.r.t. another, considering P in motion: motion composition. 4

r = with respect to a non-inertial observer ; d = drag

term; c = Coriolis acceleration.

4 Inertial reference frame: all inertial frames are in a state of constant, rectilinear motion w.r.t. one

another, they’re not accelerating.

Basic notions of dynamics

so “apparent forces” can be defined

. Note that these equations are referred to a material

point, not to a rigid body. However, they can be interpreted as governing the motion of the centroid of the

body.

One degree freedom oscillator The material point is in dynamic equilibrium under the action of: linear

elastic, viscous damping, external and inertial forces. The equation of

motion is

The general solution is given by the sum of a general integral of the homogeneous equation plus a particular

solution of the whole equation.

Forced oscillations, considering an external sinusoidal force with amplitude A. Looking for a particular

solution we find .

It is like a “filtering effect”: forcing the system to higher frequencies then its characteristic frequency, the

system doesn’t respond. Forcing it to lower frequencies, the static response is obtained. To get resonance,

a frequency near to the characteristic one is needed.

The general solution in case of forced oscillations thus becomes

Case with r = 0 (no damping): Case with r ≠ 0 (damping):

When non-linear effects are considered we can have hard or soft spring effect: the characteristic frequency

of the system increases (or decreases) with the amplitude B of the external force.

Basic notions of electrostatics

Gauss’s law: the flux of the electric field through a closed surface S is equal to the sum of electric charges

contained inside S, divided by ε. Faraday’s law: the curl of the electric field is different from zero so the

field isn’t conservative, exception made for stationary condition, where a

potential function Φ such that can be defined.

The “electrostatic problem” is governed by suitable boundary conditions: surface charge density and

potential are assigned. Capacitance is the ability of a body to hold an electrical charge. C = Q/Φ

The electrostatic energy can be defined as the work necessary to bring

electric charges in the considered configuration, starting from infinite

distance.

The capacitor is a device in which electrostatic energy (equal to the work done to charge it) can be stored.

By considering the charging process as a sequence of displacements of infinitesimal charges dq at potential

difference Φ: The electrostatic force is the variation of

the electrostatic energy due to a virtual

displacement.

Parallel plate capacitors

The electric potential is a linear function of the distance between the plates.

Interdigitated “comb-finger” capacitors

A plate is placed symmetrically at distance d with respect to other two parallel plates. It

is assumed that the distance r is large enough and that the electric field of interest is

only at the sides of the central plate.

Usually, comb-finger capacitors are used as actuators while parallel plates as sensor

(more sensible to displacements, that are to the second power at the denominator).

Coupled electro-mechanical problem

Parallel plate connected with an elastic spring The static equilibrium of the plate

connected with the elastic spring is

guaranteed by the presence of

elastic and electrostatic forces.

The first derivative w.r.t. u is

The maximum of the plot is reached when u = g /3, and it is called the pull-in value: At increasing voltage,

0

the pull-in situation is unstable, the static equilibrium can no more be guaranteed (prevalence of the

electrostatic force over the elastic one) and the parallel plate goes suddenly toward the opposite plate,

closing the gap and creating a short circuit.

Let us now consider the dynamic equilibrium of the plate connected with the elastic spring without damping.

That is the equation of a non-linear 1dof oscillator. Using a Taylor

expansion of the electrostatic load, up to the first order, we obtain an

equation of motion in with the total stiffness is made up of two

contributions, one mechanical and the other coming from electrostatics. When the voltage increases, the

stiffness decreases: the eigenfrequency of the equivalent linear oscillator depends on the voltage (at the