Mechanics of Microelectromechanical Systems - N.Lobontiu and E.Garcia Part 14 pdf

The length scales toitself, namely to the area A scales to the volume V scales to whereasthe moment of inertia scales to One important feature in scaling properties is the surface-to-vol

The maximum capacitive sensitive can be found from Eq (4.36) as:

Similarly, the minimum capacitive sensing is:

The linear dimension variation can be written as:

By substituting Eqs (6.42) into Eqs (6.40) and (6.41), the ratio of themaximum to the minimum sensitivity becomes:

Figure 6 28 is a plot corresponding to Eq (6.43)

Figure 6.28 Ratio of maximum to minimum capacitive sensitivity ratios as a function of the

tolerance fraction

It can be seen that the sensitivity ratio varies non-linearly from a value of 1for p = 0 (this is the ideal case, when the dimensions are perfect) to a value ofapproximately 2.75, when p = 0.25

Although not a microfabrication defect in itself, stiction – the

phenomenon of adhesion of thin-film structures such as microcantilevers,microbridges or membranes, especially during the wet etching of sacrificiallayers – constitutes a source of shape damaging and even mechanical failure

in MEMS As shown by Mastrangelo and Hsu [15], the magnitude of theforces that are developed through stiction are significant, such that attempts

to normally operate microfilm structures that have adhered to the substratecan terminally damage the microdevice The mechanisms that generate suchhigh levels of forces are the capillary phenomena at the liquid- solid interface

in very small interstices, as well as the solid-solid adhesion which isestablished after contact However, further discussion and quantitativeanalysis of this phenomenon is beyond the scope of this book

Another source of errors in MEMS design is the precision of modelingthe mechanical behavior of a microcomponent Simplifying assumptions areoften used to keep the modeling process tractable while preserving thecertitude of its prediction accuracy, Examples can be cited wheremodifications of the basic assumptions used in modeling have to be applied,such as considering the shearing effects for relatively-short beams or usingthe large deformation theory, when it warranted by reality It is also veryimportant in MEMS that move through elastic deformation of theircomponents, such as the ones discussed in this book, to separate between themembers that can be considered rigid and the ones that cannot Otherwise,errors – sometimes significant – can be introduced in thestiffness/compliance of the microdevice of interest An example will beanalyzed next, highlighting errors that are produced through modelingassumptions in boundary conditions

It has been shown previously that the displacements at a certain point on

a free-fixed chain, as the one considered here, can be calculated bycombining the various compliances of the flexible links in terms of thespecific configuration of the chain In this case, the deflection and slopecan be expressed as:

expressed in the form:

Figure 6.29 Microcantilever with anchor: (a) Three-dimensional sketch; (b) Loading and

boundary conditions

where the stiffnesses are:

If one takes and then Eqs (6.48), (6.49) and (6.50) reduce tothe known equations for a simple cantilever, namely:

The following ratios can be formulated and calculated by means of Eqs.(6.48) through (6.53):

and Fig 6.30 is the plot of in terms of h and for the case where

and Obviously, the relative errors increasewhen the height h increases and the thickness decreases

Figure 6.30 Stiffness error when considering the leg compliance in a microcantilever

Other examples of modeling errors have been presented in Chapter 5 whenanalyzing the out-of-the-plane stiffness of microsuspensions in connection tothe regular, in-plane stiffness and motion of microdevices

The theory of similitude enables comparing the behavior of systems that

are similar by extrapolation of the available data (either experimental ornumerical) characterizing one system to another system, in order to predictthe response of the latter In many cases data can be acquired by using alaboratory model which suitably produces experimental data By applying

properties and to predict the behavior of a similar system that is built at adifferent scale The theory of similitude is implemented by means of the

dimensional analysis, which is the analytical tool that takes into

consideration the dimensions of the pertinent amounts defining a givenphenomenon The dimensional analysis, as shown by Murphy [16] or Taylor[17], produces qualitative relationships, and in combination withexperimental/numerical data, yields quantitative results that lead to accuratepredictions The dimensional analysis is based on two axioms, namely:

Two quantities are numerically equal only when they arequalitatively similar (have the same dimensions) For instance, aquantity that is measured in length units can only be equal to anotherquantity that is also being measured in length units

The magnitude ratio of two similar quantities is independent of themeasurement units when the same units are used for both quantities.The width-to-thickness ratio of one microcantilever’s cross-section isthe same, regardless whether the measuring units are meters orinches

The scaling laws, as mentioned by Spearing [4], can be fundamental (or

quasi-fundamental), which are basically obtained from the theory of

similitude and the dimensional analysis, and which consider that materialproperties and physical quantities such as density, elastic constants, orthermo-electric properties are constant Another category of scaling laws are

the mechanism-dependent ones, which take note that under a threshold value

of approximately certain material and physical properties are nolonger constant, and their values will vary according to the mechanisms thatdominate their behavior A third branch of scaling laws, as also mentioned by

Spearing [4], form the extrinsic or indirect category, where restrictions

imposed by the peculiarities of a specific microfabrication technology, inrelationship with the given set of geometric shapes that can be obtained bythat microtechnology, affect scaling properties Addressed will be here onlythe fundamental scaling laws

Among the features that determine the scaling trends of MEMS, the

length is a key parameter because the length directly reflects the dimensional

differences between small- and large-scale, and therefore carries over theamount whose scale-dependence is being studied As a consequence, thelength – denoted by 1 in the following – is the paramount basic variable, andall the derivative amounts of interest are related to it The length scales toitself, namely to the area A scales to the volume V scales to whereasthe moment of inertia scales to

One important feature in scaling properties is the surface-to-volume ratio

(SVR) of a body, which is:

R, this ratio becomes:

and Fig 6.31 plots this ratio when the radius ranges from to

and it can be seen that for small geometric features, this ratio becomes verylarge

Figure 6.31 Surface-to-volume ratio for a sphere

It is interesting to mention that when the cube and the sphere have the samevolume, the surface-to-volume ratio of the sphere is approximately 1.21larger than the cube’s ratio, whereas when the cube and the sphere have thesame area, the of the sphere is approximately 1.34 larger than the ofthe cube In other words, forces that are proportional to the area (such asexternal friction or superficial tension) become more important than forcesthat are proportional to the volume (such as gravity, for instance) at smallscale, and the forces that are proportional to surface-to-volume are alwayslarger for sphere-like microcomponents than those of equivalent cube-likeones

Another amount which is of interest in qualifying the specific resistance

of a mechanical component is the strength-to-weight ratio, which is defined

as the maximum load over the mass For a strut that is compressed by anaxial load F, the strength-to-weight ratio, SWR, is:

When the yield stress and the density are constant, SWR scales (isproportional) to the surface-to-volume ratio SVR, which was shownpreviously to scale to

Stiffness is another feature which is of particular interest to this book, as

the main objective here was to qualify MEMS quasi-statically and therefore

to define/determine relationships between load anddisplacement/deformation by means of stiffness

By taking into account that E is considered constant and scales to itfollows that scales to scales to and scales to Thetorsional stiffness is similar to the direct rotary bending stiffness andconsequently scales to Because of the linear relationships between forceand deflection, on one hand, and moment and slope (rotation angle), on theother hand, which are:

the force will also scale to as does, and scales to similar to

Trimmer [18] and [19] introduced the so-called vertical Trimmer bracket

symbolism as a way of determining the scaling proprieties of amounts thatresult from forces/moments (which will be discussed shortly) and otheramounts, whose scalability is known The load (composed of forces and/ormoments), depending on the specific character of the actuation – as detailed

in Chapter 4 –, can scale with different powers (exponents) of 1, and thefollowing formalism can be used to illustrate this connection:

which simply indicates that one specific load can scale to either of the

defined as:

combines to F in a way that produces another amount X, in the form:

then the scale definition of the new amount X is:

Equation (6.65) shows that when the force scales with for instance, thederivative amount X will scale to

A direct application of the Trimmer symbolism is calculatinglinear/angular displacements by utilizing corresponding forces/moments andstiffnesses

Example 6.12

Establish the scaling properties of linear and angular displacements of amicrocomponent by utilizing the vertical Trimmer bracket symbolism.Solution:

A linear displacement is calculated as the ratio between the generatingforce and the corresponding stiffness and it scales as:

Similarly, a rotary displacement which is produced by a moment is the ratiobetween the moment and the corresponding stiffness and it scales as:

Eventually, a rotary displacement which is produced by a force is the ratio ofthe force to the corresponding cross stiffness and scales as:

The static work can also be scale-evaluated by using the verticalTrimmer bracket method, as shown in the next example

Similarly, the rotary work produced by a moment is proportional to theproduct between the moment and the resulting rotation angle, and it scales as:

where ml, m2, , mn represent the (potential) scaling laws of the moment.Analyzed will be next the scaling of the various forces introduced inChapter 4

The thermal force that is developed by a fixed-free bar under a

temperature increase of has been evaluated in Chapter 4 as:

and therefore, the force scales with the cross-sectional area, whenYoung’s modulus E and the coefficient of thermal expansion are constant

Forces such as piezoelectric or produced by shape-memory alloys also do

scale to and this can simply be shown by considering again a fixed-freebar whose expansion/contraction needs to be prevented by an axial force,which is proportional to the product between the maximum stress (which is aconstant material-dependent feature) and the cross-sectional area As a result,the respective force scales with A and consequently to

Similarly, electrostatic forces scale to In a transverse (plate-type)actuation for instance, the electrostatic force has been defined in Chapter 4

as:

and therefore for constant electric permittivity and constant electric field, theelectrostatic force scales proportionally to the plate area and therefore toFor longitudinal (comb-finger) actuation, the electrostatic force has beenfound to be:

and again for constant values of and E, the electrostatic force scales to

The attraction magnetic force developed between a permanent magnet

and a mating ferroelectric surface was found to be in Chapter 4:

which indicates that, being proportional to the square of the cross-section, themagnetic force will scale to

The Lorentz electromagnetic force that acts on a conductor of length lcarrying a current I when placed in an external magnetic field B was definedas:

in the case where the magnetic field B is perpendicular to the linearconductor The current is defined as:

where j is the current density and A is the cross-sectional area of theconductor In the case where the current density is a constant, the current I isproportional to A, and therefore scales to as suggested by Eq (6.76),which means that the electromagnetic force will scale to for constant B

Example 6.14

Determine the scaling with respect to length of the electromagnetic forceacting between two parallel conductors of lengths and (as shown in Fig.6.32) that are placed at a distance d in vacuum and carry the currents andrespectively

Figure 6.32 Electromagnetic force between two conductors

Solution:

The electromagnetic force that will act on the mobile conductor is:

It is known that the magnetic field generated by the left conductor is:

and therefore the attraction force of Eq (6.77) becomes:

It has been shown previously that the current is proportional to the sectional area for constant current densities, and therefore Eq (6.79)indicates that the Lorentz force is proportional to the square of the currentand therefore scales to

cross-Example 6.15

Establish the length-related scaling law of the force generated between amagnet and an electromagnet, as shown in Fig 4.38

Solution:

It has been shown that the magnet of Fig 4.38 (a) can be substituted by

an equivalent coil, as sketched in Fig 4.38 (b) and the corresponding forcegenerated between the two coils (the real one and the equivalent one) is re-written here for convenience:

where is the distance between the two coils and is the number ofwindings of the real coil If one carries a dimensional analysis in terms oflength, it is apparent that the force is proportional to (from strictly looking

at the relationship between and and to the square of the current(because of the product and therefore, because the current itself isproportional to as shown in the previous problem, the force will scale totimes which is

Two other possibilities that can occur and are connected to the currentdensity are analyzed by Trimmer [18] and the conclusions are brieflymentioned here In the case where there is a constant heat flow generated inthe conductor, the electromagnetic force scales to whereas in the situationwhere a constant temperature increase is applied to the conductor, theelectromagnetic force will scale to

Problem 6.1

A thermal bimorph, which is composed of two materials with

and having

temperature increase of Young’s modulii are only known

approximately, namely ranging in a interval and in

interval Calculate the minimum and the maximumcurvature radii that can be produced when p varies from – 0.2 to + 0.2

Answer:

Problem 6.2

A bent beam thermal actuator, as the one shown in Fig 4.3 is

microfabricated with the following nominal dimensions: cross-sectional

angle Young’s modulus is E = 160 GPa and the temperature increase

is The microfabrication process ensures that is realized with a

precision of around the nominal value Also the precision of

measuring temperature variations is within a tolerance of

Considering that and (where p is a tolerance fraction

that can vary between - 0.2 and + 0.2) determine the maximum and

minimum values of the output displacement

In a transverse (parallel-plate) electrostatic actuator, as the one pictured

in Fig 4.19, the microfabrication technology produces linear dimensions

within a tolerance field of such that the dimension for

instance is located anywhere un the interval Calculate the

percent relative error between the maximum and minimum initial force in

terms of the minimum initial force (x = 0), namely 100

Numerical application: p = 0.1

Answer:

Numerical percentage error: 82.58%

Problem 6.4

A microcantilever of the type shown in Fig 6.29 of Example 6.9 is

realized through wet etching and therefore the cross-section, instead of being

rectangular, is trapezoid with the dimensions and

(as shown in Fig 6.19) The microcantilever is used in a reading

AFM application, which gives a tip deflection of and a slope of 1° Find

the force producing these deformations and determine the error on this

force when the cross-section is considered rectangular (w x t) The length is

and Consider that the anchor leg is rigid and that

there are no other forces acting on the microcantilever,

Answer:

Problem 6.5

The vertical microcantilever in Fig 6.33 is microfabricated by DRIE in

order to be utilized in a vibration detection application The dimensions of

results through microfabrication Find the difference in the tip

deflection between the real microcantilever and the ideal one (with no

runout) when a force is applied Young’s modulus is E = 150

GPa

Figure 6.33 Microcantilever for vibration monitoring

Answer:

Problem 6.6

Determine the deflection at the midpoint of the microbridge shown in Fig

6.34 when considering the flexibility of all three segments Find the error in

deflection when only the flexibility of the middle (horizontal) link is taken