Micro Electro Mechanical System Design - James J. Allen Part 6 ppsx

magnetic fields for actuationis to consider the energy density of an electric, Uelectric, and a magnetic, Umagnetic,field for a region of space at the appropriate operational condition F

One way to make this assessment of electric vs magnetic fields for actuation

is to consider the energy density of an electric, Uelectric, and a magnetic, Umagnetic,field for a region of space at the appropriate operational condition (Figure 4.11).Equation 4.25 and Equation 4.26 define the electric and magnetic field density,respectively, where ε is the permittivity and µ is the permeability of the region

that contains the electric field, E, and the magnetic field, B For purposes of this

assessment, the free space permittivity, ε0 = 8.84 × 10–12 F/M, and the free spacepermeability, µ0 = 1.26 × 105 H/M will be used The maximum value of the

electric field, E, and magnetic field, B, will be limited by the maximum obtainable

operational values

The maximum obtainable electric field is at the point just before electrostaticbreakdown This breakdown occurs when the electrons or ions in an electric fieldare accelerated to a sufficient energy level so that, when they collide with othermolecules, more ions or electrons are produced, resulting in an avalanche break-down of the insulating medium; high current flow is produced For air at standardtemperature and pressure, the electric field at electrostatic breakdown in macro-scopic scale gaps between electrodes (i.e., > ~10 µm) is Emax = 3 × 106V /M.

(4.25)

(4.26)

The maximum obtainable magnetic field energy density is limited by thesaturation of the magnetic field flux density in magnetic materials In materials,the spin of an electron at the atomic level will produce magnetic effects In many

FIGURE 4.11 Electric and magnetic fields in a region of space.

materials, these atomic level magnetic effects are canceled out due to their randomorientation However, in ferromagnetic materials, adjacent atoms have a tendency

to align to form a magnetic domain in which their magnetic effects collectivelyadd up Each magnetic domain can be from a few microns to a millimeter in size[17], depending upon the material and its processing and magnetic history How-ever, the domains are randomly oriented and the specimen exhibits no net externalmagnetic field If an external magnetic field is applied, the magnetic domainswill have a tendency to align with the magnetic field

Figure 4.12 shows a plot of the magnetic flux density, B, vs the magnetic field intensity, H, for a ferromagnetic material The magnetic field intensity, H,

is a measure of the tendency of moving charge to produce flux density (Equation

4.27) Figure 4.12 shows that, as H is increased, the magnetic flux density, B,

increases to a maximum in which all the magnetic domains are aligned For

magnetic iron materials, the saturated magnetic flux, Bsat, is approximately 1 to

2 T A Bsat of 1 T will be used for this assessment of magnetic field density

(4.27)

Using the limiting values of Emax and Bsat discussed earlier to calculate theelectric and magnetic field densities will yield the values shown next Theseresults indicate that the magnetic field energy density is 10,000 times greater thanthe electric field energy density This calculation explains why electromagneticactuation is dominant in the macroworld

FIGURE 4.12 An example a magnetization curve.

Irreversible growth

Reversible growth

H = Bµ

However, for MEMS scale actuators, the electrode spacing or gaps can befabricated as close as 1 µm MEM researchers [1,2,19] have noticed that the

electric field, E, can be raised significantly above the breakdown electric field,

Emax discussed earlier for macroscale gaps This increased breakdown electricfield for small gap sizes is predicted by Paschen’s law [18], which was developed

over 100 years ago This law predicts that the electric field at breakdown, Emax,

is a function of the electrode separation (d) – pressure (p) product Figure 4.13 illustrates the basic functional dependence of Paschen’s law, Emax = f(p,d) Figure

4.13 shows that the separation-pressure product decreases to a minimum, which

is the macroscopic breakdown electric field,

However, as the separation-pressure product is decreased further, the down electric field starts to increase This increase in the electric field requiredfor breakdown is because the gap is small and there are few molecules forionization to occur As the electrode separation becomes smaller, a fewer number

break-of collisions occur between an electron or ion with a gas molecule because themfp (mean free path) between collisions is becoming a greater fraction of theelectrode separation distance Decreasing the gas pressure also results in fewercollisions because decreasing the number of molecules increases the mfp lengthbetween collisions This means that fewer collisions occur in a given electrodeseparation distance The effect causes the breakdown electric field to increase

FIGURE 4.13 Paschen’s law: breakdown electric field, Emax (V/M), vs the electrode

separation — pressure product (M-atm).

M U

Ionization occurs

with decreasing separation-pressure product up to a maximum, , for cale electrode spacings The electric field for small electrode separation distances

micros-in vacuum have been reported [20] to be

Using this new value for Emax will change the comparison of the electric andmagnetic field energy density calculation of Equation 4.29 as shown next Thisresults in a more favorable but neutral comparison of the energy density of electricand magnetic fields However, the literature indicates that, for MEMS applica-tions, electrostatics predominates This is due to the added fabrication and assem-bly complexity of fabricating MEMS scale permanent magnets, coils of wire,and the associated resistive power losses with their use

(4.29)

In another simple comparison of electric and magnetic fields, it can be seen

that the magnetic field energy density, Umagnetic, does not change with size scaling

because Bsat and µ are material properties that do not change appreciably withscaling to the microdomain However, assuming that the applied voltage remains

constant up to the limit of Emaxat electrostatic breakdown shows that the electric

field energy density, Uelectric, varies with scale as shown in Equation 4.30 Thisgives electrostatic actuation increasing importance as devices are scaled to themicrodomain

(4.30)

4.1.6 OPTICAL SYSTEM SCALING

Optical MEMS applications and research is an extremely active area, with MEMSdevices developed for use in optical display, switching, and modulation applica-tions These MEMS scale optical devices [23,24] include LEDs, diffraction grat-ings, mirrors, sensors, and waveguides Their operation can depend upon opticalabsorption or reflection for functionality

1

12

0 2ε

Optical absorption-based devices are governed by Beer’s law (Equation 4.31),which can be seen to scale unfavorably to MEMS size because absorption depends

on path length This has spurred the development of folded optical path devices[22] to overcome this disadvantage, but this is ultimately limited by the reflectivitylosses incurred with a large number of path folds

L = distance into the medium

Optical reflection-based MEMS devices are used for optical switching, play, and modulation devices MEMS optical devices that have a displacementrange from small fractions of a micron to several microns can be made Thiscorresponds to the visible light spectrum up to the near infrared wavelengths(Figure 4.1) Because electrostatic actuation is frequently used in MEMS devices,very precise submicron displacement accuracy is attainable Also, very thin low-stress optical reflective coatings are possible These attributes make a MEMSoptical element very attractive

dis-4.1.7 CHEMICAL AND BIOLOGICAL SYSTEM CONCENTRATION

Miniaturization of fluidic sensing devices with MEMS technology has mademiniature chemical and biological diagnostic and analytical devices possible[25,26] To assess the effect that reduction in scale will have on these devices,the concentration of chemical or biological substances and how it is quantifiedmust be studied

Before the concentration of a chemical solution can be defined, a few

pre-liminary definitions will be stated A mole (mol) is a quantity of material that contains an Avogadro’s number (N A = 6.02 × 1023) of molecules The mass ingrams of a mole of material is the molecular weight of the chemical substance

in grams The is known as the gram molecular weight (MW) and has units of

grams per mole Example 4.5 illustrates how the MW is calculated for salt

Example 4.5

Problem: Calculate the gram molecular weight (MW) of common table salt (i.e.,sodium chloride, NaCl) The atomic mass of sodium (Na) = 23.00 The atomicmass of chlorine (Cl) = 35.45 The molecular weight of NaCl = 58.45 The grammolecular weight of NaCl is MW = 58.45 g/mol

Solution : The concentration, C, of a chemical in a solution is known as the molarity of the solution A 1-molar solution (i.e., 1 M) is 1 mol of a chemical

A=εCL ∝S

dissolved in 1 liter of solution For example, a 1-M solution of NaCl consists of

58.45 g of NaCl dissolved in a liter of solution This relationship is expressed inEquation 4.32

(4.32)

For chemical detection, the number of molecules, N, in a given sample volume, V, may be important to quantify This relationship between number of molecules in a given concentration of solution, C, and volume of solution, V, is:

(4.33)

Figure 4.14 shows the relationship between concentration, C, and sample

volume, V, as expressed by the preceding equation The boundary for less than one molecule, N1, of chemical or biological substance in a given sample volume

is shown; this is an absolute minimum sample volume for analysis The number

of molecules required for detection, N D , is some amount greater than N1 (i.e.,

N D > N1) The required sample volume for analysis would be at the intersection

of the N Dboundary with the concentration of the analyte available for analysis.Petersen et al [26] have shown that the typical concentrations of chemicaland biological material available for a few types of analyses are as shown inTable 4.2

The miniaturization of chemical and biological systems has a few tal limits:

fundamen-• The trade-off between sample volume, V, and the detection limit, N D,

for a given concentration of analyte, C, is illustrated in Figure 4.14.

• Further miniaturization may require increasing the concentration ofanalyte or increasing the sample volume

• The use of small sample volumes requires increasingly sensitive tors, which may be limited by other scaling issues (i.e., electrical,fluidic, etc.)

detec-• The physical size limitation of biological sensing devices is limited bythe size of the biological entity A cell is approximately 10 to 100 µm,whereas DNA has a width of only ~2 nm but is very long

gram gram

mole

mole liter liter

A

4.2 COMPUTATIONAL ISSUES OF SCALE

The computational aspects of the scale of MEMS devices need to be consideredbecause much of modern engineering design depends upon numerical simulation

to achieve success Due to fabrication challenges, long fabrication times, andexperimental measurement difficulties, MEMS applications rely more upon sim-ulation than their macroworld counterparts do Therefore, time would be well spent

in assessing the unique issues encountered in simulation of MEMS scale devices.Engineering calculations are almost exclusively performed on digital com-puters in which the numbers representing the input data (i.e., mechanical andelectrical properties, lengths, etc.) and the variables to be calculated are repre-sented by a fixed number of digits Due to this digital representation of numbers,

FIGURE 4.14 Concentration vs sample volume.

TABLE 4.2 Typical Analyte Concentrations for Various Types of Analyses

Uses

Concentration (moles/liter)

Clinical chemistry assays 10 –10 –10 –4

ND

the quantity known as machine accuracy,εm, is the smallest floating point numberthat can be represented on a given computer The machine accuracy is a function

of the design of the particular computer Two types of errors arise in the lations performed on digital computers [38]:

calcu-• Truncation error arises because numbers can only be represented to a

finite accuracy (i.e., machine accuracy) on a digital computer

• Round-off error arises in calculations, such as the solution of equations,

due to the finite accuracy of the computer Round-off error accumulateswith increasing amounts of calculation If the calculations are per-formed so that the errors accumulate in a random fashion, the totalround-off error would be on the order of , where N is the number

of calculations performed However, if the round-off errors accumulate

preferentially in one direction, the total error will be of the order Nεm.The topics of truncation and round-off error arise in regular macroscaleengineering simulation; however, a unique aspect of computation for MEMS scalesimulation needs to be addressed:

• Convenient units scale of numbers for MEMS simulation The system

of units typically used in engineering simulations (e.g., MKS) usesunits of measure of quantities typically encountered for macroscaledevices For example, the MKS system of unit length measure ismeters However, MEMS devices are on a size scale of microns (i.e.,0.000001 m)

• Numerically appropriate scale of unit for MEMS simulation Numerical

simulations such as finite element analysis (FEM) [39,40] typicallyinvolve the solution of a large system of equations (e.g., 1,000 →1,000,000) This system of equations will become ill conditioned whenthe quantities involved in the equations vary widely in magnitude Alarge ill-conditioned system of equations can produce inaccurate results

or may even be unsolvable For example, ill conditioning can arisewhen a very small number is subtracted from a very large number; thiswill make the result unobservable due to the truncation and round-offerrors of digital computation

From a CAD layout perspective, the unit of length most appropriate for aMEMS scale device is a micron (i.e., 1 µm = 0.000001 m) This will allow theCAD design of the device to be done using reasonable multiples of a basic unit

of measure

From a numerical computation perspective, the system of units needed toexpress the basic quantities used in MEMS device simulation should be a numer-ically similar order of magnitude This will avoid the ill conditioning of thenumerical simulation problem A system of units for MEMS simulation has beenproposed [41] for finite element analysis Appendix C provides the conversion

Nεm

factors between the MKS system and the µMKS system, which will be used inthe design sections of this book Several different permutations of an appropriate

system of units are possible However, a consistent set of units must be used in

any simulation This will maintain dimensional consistency for material propertiesand simulation problem parameters such as loads and boundary conditions

4.3 FABRICATION ISSUES OF SCALE

To assess the fabrication issues unique for MEMS scale devices, it is necessary

to put MEMS fabrication processes and technologies in perspective with facturing processes for other size scales The size scales for manufacturing thatwill be discussed are large-scale construction, macroscale machining, MEMSfabrication, and integrated circuit (IC) and nanoscale manipulation These areindividually discussed next These four size groups provide a wide spectrum thatwill enable the evaluation of any fabrication issues due to scale

manu-• Large-scale construction (>15 m) The fabrication of things in this sizecategory includes civil structures, marine structures, and large aircraft.Manufacturing at this size scale involves a wide array of processes formaterials such as wood, metal, and composite materials

• Macroscale machining (2 mm to 15 m) Manufacturing at this scaleincludes a plethora of processes and materials In many cases, the man-ufacturing processes and materials have been under development andimprovement for an extended period These manufacturing processesare mature and quite flexible In most instances, more than one approach

to the manufacture of a given item is available Examples of itemsmanufactured in this category include automobile or aircraft engines,pumps, turbines, optical instruments, and household appliances

• MEMS scale fabrication (1 µm to 2 mm) MEMS fabrication includesthe processes and technologies discussed in Chapter 2 and Chapter 3

to produce devices that range in size from 1 µm to 2 mm This category

of manufacturing has been under development for 30 years and hasstarted to produce commercial devices within the last 10 years To alarge degree, the fabrication methods for MEMS are rooted in the ICinfrastructure As a result, the range of materials and the flexibility ofthe fabrication processes are more restrictive than in macroscalemachining Silicon-based materials are frequently used in surface andbulk micromachining LIGA uses electroplateable materials (e.g.,nickel, cooper, etc.) When LIGA molds are used with a hot embossing,plastic materials can be utilized to create devices

• IC and nanoscale manipulation (<1 µm) The size scale for thesefabrication technologies is 1 µm and below (i.e., <1 µm) IC fabricationtechnology has been under development and continuous improvementfor 40 years [29] and relies on leading edge photolithography, CVDdeposition, and etching techniques similar to those presented in Chap-

ter 2 The IC manufacture included in this category are state-of-the-artcapabilities that are rapidly approaching 0.1 µm feature sizes andbelow Nanoscale manipulation [32] is a recent demonstrated use ofsurface profiling tools [30,31] such as an atomic force microscope(AFM) and a scanning tunneling microscope (STM) These enable theindividual manipulation of molecules Nanoscale manipulation is alaboratory-based research capability as contrasted with IC manufac-ture, which is a mature large industrial capability.

The smallest feature that can be fabricated on a part is the feature size From

a design perspective, a more useful quantity to assess a fabrication capability is

the relative tolerance Relative tolerance is defined as the feature size divided by

part size; this provides a measure of the precision with which a fabrication processcan produce a part of any given size

Figure 4.15 shows a graph of the relative tolerance vs size over a considerablerange The four size categories defined earlier are noted in this figure, and thedata for this graph are extracted from a number of sources [2,27,28,30–35] Due

to the extended size range and large number of fabrication processes that exist,the data in this graph should be viewed as a broad statement of the fabricationprocesses in a given size range rather than as indicative of any specific fabricationprocess or capability Because of the large number and variety of macroscalefabrication processes, data were extracted [27,33] for some broad ranges ofprocesses (e.g., grinding, milling, etc.) within this category Figure 4.15 showsthat macroscale fabrication has the smallest relative tolerance or precision, withthe relative tolerance increasing as the size scale increases or decreases Thisshows that MEMS scale fabrication has about the same precision as that of large-scale fabrication (i.e., MEMS devices have about the same level of precision asone’s house!)

Due to the large variety and flexibility of macroscale fabrication processes,

a number of categories of precision or relative tolerance have been defined[27,33]; these are shown in Figure 4.16 andTable 4.3 Ultraprecision machining

is at the extreme level of precision and is reserved for only a few applicationsdue to the time and expense necessary Only a few instances, such as some largeoptical applications [36,37], require this level of precision Figure 4.16 showswhere these levels of precision lie relative to the MEMS-scale and nanoscalemanipulation

The fabrication issues of scale show that a MEMS designer is faced withfewer options and more restrictions than those faced by the macroworld designengineer MEMS scale fabrication imposes the following concerns for the designengineer; they will need to be addressed in the device design:

• Limited material set availability

• Fabrication process restrictions upon design

• Reduced level of precision in the fabricated device

4.4 MATERIAL ISSUES

As the size of a device is decreased, two general trends become evident:

• The granularity of the solid or fluid materials becomes increasingly

apparent This granularity can be expressed by quantities (see Table4.4) such as the grain size of a material or the mfp in a gas Does this

FIGURE 4.15 Manufacturing accuracy at various size scales.

FIGURE 4.16 Relative tolerance levels.

Large scale construction MEMS

IC fabrication and nano-scale manipulation

X X

m ng

lappin

g an

d polishing

grinding

violate the assumption of continuum mechanics frequently used in themacroworld to model engineering phenomena?

• New physical phenomena (e.g., Brownian motion, Paschen effect,

elec-tron tunneling current) become significant due to the reduced volume

or spacing in MEMS devices

The classical engineering models used to design and simulate macroworldphysics and devices are based upon continuum mechanics, which models thephysics of interest with a set of partial differential equations Table 4.5 shows

a sampling of the array of physical phenomena modeled by such equations.These equations involve partial derivatives of the variable of interest, such as

Large scale construction Cutting, forging, forming

processes, welding and fastening

Nanoscale manipulation Focused ion beam, scanning

tunneling microscope, atomic force microscope

<100 nm ~0.1 32

TABLE 4.4 Size Scale of Phenomena Relevant to MEMS

Mean free path of air @ STP 65 nm @ STP Lattice constant 5.431Å for silicon Material grain size 300–500 nm for polysilicon Magnetic domains 25 µm

stress, displacement, or temperature, and some parameters (i.e., modulus ofelasticity, heat transfer coefficients, speed of sound in a media) that model thedomain that the set of equations govern For these equations to be easily solved,the parameters must be known and the variable of interest smoothly varyingover the domain of interest (i.e., differentiable) If a material is discrete or

TABLE 4.5

Physical Phenomena Modeled by Continuum Mechanics

Three-dimensional heat flow

Three-dimensional wave equation

Elastic equations of equilibrium for

solid mechanics

Maxwell’s free space electromagnetic

equations

Navier–Stokes equations for

compressible fluid dynamics

u z

2 2

2 22

2 2

2 2

2 2

2 22

2 2

2 2

u z

t

J t

discontinuous (e.g., granular), it is more difficult to model the system with a

continuum mechanics approach

As one tries to design and model systems on smaller scales, a certain ularity of the physics is observed In Chapter 2, the material structures of crys-talline, polycrystalline, and amorphous were discussed (Figure 2.2 illustratesthese three material structures.) The spacing of atoms in crystalline and amor-phous materials is at the atomic scale (i.e., <1 nm) The size of the individualcrystals in a polycrystalline material are on the order of 100 to 500 nm, dependingupon the material processing used Many materials of engineering significanceare polycrystalline The physical parameters used to describe material behavior(e.g., Young’s modulus, speed of sound) in a continuum mechanics model arestatistical averages of the effects of the individual grains or molecules of materialwithin a large object (relative to the grain size)

gran-For example, for a macrodevice that is 2 cm wide with a 500 nm grain size,the statistically averaged property representing a parameter such as Young’smodulus is adequate However, a 2-µm wide microdevice contains only a fewgrains of material, and a statistically averaged approximation of a material prop-erty is not adequate Research has been ongoing to measure microscale effects[42]; develop theories that apply at the microscale [43,44]; and incorporate theseeffects into simulations of the microscale phenomena [45]

The statistically averaged assumption also plays a role in the failure model

of materials The stress at which a material yields or fails is quantified by the

parameters, yield strength, S y , or failure strength, S u These parameters also havestatistics in their origin A material has a certain number of defects in the materialstructure (e.g., crystal lattice imperfections, corrosion products in the grain bound-aries) that give rise to locations at which a material will yield or ultimately fail.These defects are assumed to be statistically distributed throughout the material

The defect density of a material and statistical process control is frequently used

in the microelectronic community [46] in assessments and modeling of the yield

(i.e., percentage of good devices manufactured) of their processes A potential advantage of scaling devices down to densities approaching the defect density

of the material is that devices could be produced with a low defect rate

4.5 NEWLY RELEVANT PHYSICAL PHENOMENA

Several new phenomena are enabled or become relevant at the MEMS scale.The three briefly discussed next are examples of such phenomena, which gainimportance because of the size of a MEMS device or the small gaps used inMEMS devices

• Brownian noise Also called thermal noise or Johnson noise for

elec-trical systems, Brownian noise is a low-level noise present in elecelec-tricaland mechanical systems This thermal noise is present everywhere inthe environment and is due to such things as the vibrations of atoms

in the materials from which a device is made and the environment in

which the device operates This indicates that the thermal noise is afunction of temperature of these materials The mechanisms that couplethese thermal vibrations to the mechanical or electrical device of inter-est are the energy dissipation mechanisms (i.e., damping for mechan-ical devices, resistance for electrical devices) As a device is reduced

in size, these thermal noises or vibrations become significant forMEMS scale sensors A detailed discussion of Brownian noise is inthe chapter on MEMS sensors

• Paschen’s effect The phenomenon that the breakdown voltage in a gap

increases as the product of the pressure of the gas in the gap and gapspacing is reduced was discovered in 1889 [19] This phenomenon iseffective when the gap size is very small (<2 µm), which is typical ofMEMS devices This enables increased effectiveness of electrostaticactuation as discussed in detail in Section 4.1.5

• Electron tunneling current Quantum entities such as electrons can

“tunnel” across a very small gap (on the order of nanometers) due tothe uncertainty in the wave description of quantum mechanical entities.This especially appears to be strange due to the barrier of classicalphysics in which like charges repel This phenomenon can be used inMEMS devices as a very sensitive displacement transduction methodcapable of resolving displacements on the order of 0.01 nm A MEMScantilever can be fabricated with a tip suitable for tunneling that iselectrostatically brought within operating distance for this phenomenon

to be effective The tunneling phenomenon will be discussed in moredetail in the chapter on MEMS sensors

4.6 SUMMARY

A MEMS designer needs to be aware of a number of wide ranging issues andcannot rely solely on macroworld engineering experiences and training whenconsidering the implementation of a MEMS design System parameters willchange in relative importance as the system scale is reduced Table 4.6 showsfour quantities that can be directly or indirectly related to actuation forces (i.e.,gravity, surface tension, electrostatic, magnetic) in a device If these forces allscaled in the same manner, heuristic macroworld intuition would be valid; how-ever, these forces all scale differently

Gravity forces become increasingly small with reduced size, and surfacetension increases in importance Surface tension forces can be used for assembly

of devices; however, they can be a concern during MEMS fabrication releaseprocesses Also, the table shows that the electric and magnetic fields and theforces derived from them scale differently, with the magnetic field forces notdepending on scale Table 4.7 summarizes a number of scaling effects for mechan-ical, fluidic, and thermal systems The data in this table show that mechanicaland thermal time constants are reduced for MEMS systems, and regimes ofoperation for thermal and fluidic systems are different at MEMS scale The