Review of Electromechanical Concepts

Review of Electromechanical Concepts

Part I Electrical properties p of semiconductors Free charge carriers in silicon Energy bands of insulators, semiconductors, and metals Charge carrier concentration in silicon Conductivity/resistivity of silicon Fundamental concepts of crystal lattice Crystal planes Crystal directions Miller indices

Free Charge Carriers in Silicon T=0K T>0K Electrons of Si (14) atoms: 1s 2 2s 2 2p 6 3s 2 3p 2 Each Si atom form covalent bonds (3s 2 3p 2 ) with 4 adjacent Si atoms. Generation of free charge carriers When T>0K, some 3s 2 3p 2 electron could obtain enough energy to escape from its origin position and become free charge carriers in Si, which leaves a hole and form an electron/hole pair (EHP). Recombination The hole can be otherwise occupied by another free electron. The generation and recombination will be in equilibrium at a given temperature. Both thfree electrons and dholes are responsible for the conduction of semiconductors. If T increases, there will be more free charge carriers (temperature sensing).

Energy Bands T=0K Conduction band (high energy) and valence band (low energy). Free electrons have high energy and contribute the conductivity of semiconductor. They are associated with the conduction band. Otherwise, holes and other immobile electrons are associated with valence band. No intermediate energy level exists between the two bands (band gap). Insulators, semiconductor and conductors have T>0K different band gaps, thus different concentration of free electrons (and holes).

Free Charge Carrier Concentration in Silicon Intrinsic material Pure silicon No impurities incorporated in the crystal n 0 : electron concentration (cm -3 ) at equilibrium p 0 : hole concentration (cm -3 ) at equilibrium T=0K n n0 =p 0 = n i n i 1.5 10 10 cm 3 at room temperature n 2 * * 2 2 4π m nmpk T 2 4 e 2 h 4 2 i = 3 E g kt T>0K m n *, m p *: effective mass of electrons and holes K, T: Boltzmann s constant and absolute temperature E g : band gap h: Planck s constant

Free Charge Carrier Concentration in Silicon boron ExIntrinsic material Impurities incorporated in the crystal (doping) Acceptor (boron: 1s 2 2s 2 2p 1 ) (N a- ) Providing an extra hole n 0 <p 0 phosphorous p Donor (phosphorous: h 1s 2 2s 2 2p 6 3s 2 3p 3 )(N + d+ ) Providing an extra electron n 0 >p 0 Doping can be done controllably, so N a- and N + d are usually known. At equilibrium, n 0 p 0 = n i 2 n + 0 + N d+ = p 0 + N - a T>0K n i2 /p 0 + N d+ = p 0 + N a -

Doping Methods Diffusion Source type Temperature and time Drive condition Highest concentration at wafer surface Ion implantation Ion energy Heating coil Dosage Si Dopant source Activation Can form buried layers

Resistivity and Conductivity I = V / R = V S J = σ E Carrier mobility: μ = Average velocity of carrier/e : μ n and μ p σ = σ n + σ p = qnμ n + qpμ p ρ = 1/σ = 1/(qnμ μ n + qpμ p ) Sheet resistivity ρ s : R = ρl/wt = ρ/t L/w = ρ s L/w ρ s = ρ/t ρ (ohm/ )

Fundamental Concepts of Crystal Lattice Crystal lattice Periodic arrangement of atoms in a crystal Unit cell Representative volume regularly repeated throughout the crystal lattice. Lattice constant b a

Fundamental Concepts of Crystal Lattice Cubic lattice Simple cubic Body-centered cubic Face-centered cubic Diamond lattice (for Si and Ge)

Fundamental Concepts of Crystal Lattice Crystal planes Find the intercepts of the plane with the crystal axes (2,4,1) Take the reciprocals of the three integers (1/2,1/4,1) Round to the smallest set of integers h, k, l (2,1,4) Lable the plance (hkl) (Miller indices) (214)

Fundamental Concepts of Crystal Lattice Use {hkl} to represent equivalent planes (100), (010), (001) can be combined into {100}

Fundamental Concepts of Crystal Lattice Crystal direction Find the projections on the crystal axes of the vector point to that direction Round to the smallest set of integers h, k, l Lable the direction with [hkl] and equivalent directions with <hkl> [100], [010], and [001] can be combined into <100>

Different Crystal Planes and Direction for Silicon

Wet Bulk Micromachining z y x θ = 54.75 o (111) (100) {111} [100] <110>

Part II Stress and strain in mechanical structures Internal force analysis Normal stress and strain Shear stress and strain Characterization of stress and strain Mechanical properties of materials

Internal Force Analysis Newton s laws of motion Newton s First Law of Motion (The Law of Inertia): every object in a state of uniform motion tens to remain in the state of motion unless an external force is applied to it. Newton s Second Law of Motion: the relationship between an object s mass m, its acceleration a, and the applied force F is F=ma. Acceleration and forces are vectors. The direction of the force vector is the same as the direction of the acceleration vector. Newton s Third Law of Motion: for every action there is an equal and opposite reaction.

Internal Force Analysis Assumption: neglecting the gravitational force on the bar F=0 F 0 To balance the external force F, the wall must provide a force F, pointing toward left. At any cross-section, there must be a set of two Fs. The two internal Fs are force/counteract force: equal magnitude, but in opposite direction. They are not canceling each other (since they are on different surfaces). They are caused by the fact that atoms or molecules try to restore their own equilibrium positions. o s

Internal Force Analysis Assumption: neglecting the gravitational force on the bar F=0 F 0 To balance the external force F, the wall must provide a force F and a moment M=FL. At any cross-section, there must be a set of two Fs and a set of two Ms (=FL ) The two internal Fs and Ms are force/counteract force: equal magnitude, but in opposite direction. They are caused by the fact that atoms or molecules try to restore their own equilibrium positions.

Stress and Strain Normal stress and strain External forces perpendicular to structure surface (elongation, compression) Shear stress and strain External forces parallel with structure surface (grinding, shifting) In most load conditions, both stresses and strains exist.

Normal Stress and Strain Normal stress: σ = F / A (Pa) Normal strain: ε = ΔL /L(%) Hook s law: σ = E ε (for small ε) (microscopic view) Recall F=kx E: modulus of elasticity y( (Young s modulus) )(Pa) Intrinsic property of material Tensile stress(pull) Compressive stress(push) σ, ε > 0 σ, ε < 0

Poisson s Ratio The stretching/compression in one direction (x) will cause the shrinkage/expansion in two other direction (y,z). Poisson s s ratio:ν ν = ε ε y / ε x = ε ε z / ε x Intrinsic property of material Tensile stress(pull) Compressive stress(push) σ, ε x > 0, ε y, ε z < 0 σ, ε < 0, ε y, ε z > 0

Shear Stress and Strain Shear stress: τ = F / A (Pa) Shear strain: γ = Δx / L (radians) Hook s law: τ = G γ (for small ε) (microscopic view) G: shear modulus of elasticity (Pa) Intrinsic property of material a G = E / [2(1+ν)] Negative shear stress Positive shear stress z z y y x x Δx, γ, σ < 0 Δx, γ, σ > 0 See Chapter 2 for more description

Vector Relation Between Stress and Strain σ zz Stiffness matrix τ zx τ yz σ yy τ xy σ xx z Compliance matrix x y T 1-6 : σ xx, σ yy, σ zz, τ xy, τ yz, τ zx, S 1-6 : ε xx, ε yy, ε zz, ν xy, ν yz, ν zx

Characterization of Tensile Stress and Strain Pulling test (destruction test) Most mechanical structures experience pulling and pushing at the same time. Many materials exhibit weaker properties when subject to tension (pulling), other than compression. Easier to break Internal defects (cracks) more easily to grow and expand.

Characterization of Tensile Stress and Strain Stress-strain curve (similar to I-V curve) Structure usually can restore its original shape after the removal of load, if stress is always smaller than σ y. A stress greater than ultimate strength would cause the structure to crack or break. Ultimate strength Plastic deformation Slope:E (elastic deformation) Yield strain Ultimate strain

Mechanical Properties of Materials Brittle material v.s. ductile material Strong material v.s. tough material Hard material v.s. soft material Resilient material

Mechanical Properties of Materials Brittle material v.s. ductile material (Strain) Brittle material (Si, SiO, SiN) Small ultimate strain Little plastic deformation before fracture Ductile material (Au, Cu, Al) Large ultimate strain Large plastic deformation before fracture Fracture Plastic deformation Ultimate strain Ultimate strain

Mechanical Properties of Materials Strong material v.s. tough material Strong material (stress) High yield stress value and ultimate stress value (high yield strength and ultimate strength) Not easy to yield or damage under large force Tough material (stress and strain) High ultimate stress value High ultimate strain Not easy to yield or damage under large impact (energy) Si is stronger than steel, but steel is tougher. Si Si Steel Steel

Mechanical Properties of Materials Hard material v.s. soft material (surface) Capability to resist surface scratch Not closely related to the stress-strain curve Usually, hard materials are more brittle. High carbon steel v.s. low carbon steel Resilient material High yield strain (such as rubber) Large elastic region Si Steel Rubber Yield strain of rubber

Mechanical Properties of Materials Mechanical properties of materials are affected by a large number of factors, such as Material itself Crystal direction Deposition condition Sample size and shape The mechanical properties of thin film structures are usually different from their bulk forms. The mechanical strength of materials degrade after subject to alternating stresses for a long time (fatigue). Think about the timing belt of your car. Microscopic cracks will grow and expand. Even a stress much smaller than the ultimate strength can cause damage and failure Cautions should be taken when referencing material property data from other sources.

Part III Bending analysis Bending equation Bending moment of inertia Boundary condition Two most useful deformations to generate large movement in Torsion analysis MEMS Torsion moment of inertia Basic analysis Intrinsic stress in materials Origin of intrinsic stress Effects of intrinsic i i stress on material properties Compensation of intrinsic stress Mechanical behavior at resonance (dynamic behavior) Resonant frequency of mechanical structures Quality factor Active tuning of resonant frequency

Bending Analysis y x y F Consider the 2D case Beam bending equation for small bending 2 d y EI = dx M ( x ) 2 x Solving the bending equation to obtain y=f(x) Calculate the bending moment of inertia (I) Determine the boundary condition Find the loading condition (force and torque on the beam) (M(x))

Neutral Axis y x y F Neutral axis Neutral axis: σ = ε = 0 Not necessarily is the symmetric axis Dependent on the shape of the cross section and composition of the beam The position of neutral axis can be determined by calculation. 1 2 If E 1 < E 2

Stress Distribution σ y There is only normal stress in purely bent beams. There is no stress on the neutral axis. Both tensile and compressive stresses exist. The normal stress increases linearly in y direction. The top and bottom surfaces encounter highest stress condition.

Bending Moment of Inertia (I) Bending moment of inertia (I) y The beam consists of single material. I = y' A A number to determine the bending resistance of the beam Material at outer shell contribute more than the material inside. 2 da Rectangular beam: I = wt 3 /12 Circular beam: I = πr 4 /4 Adding and subtraction for complex feature calculation l

Boundary Conditions For most micro beams in MEMS, one end of the beams is anchored on substrate and has no degree of freedom (DOF). For the other end, there are three possible boundary conditions. For free B.C., the two linear motions and the angular rotation are related, so there are only 2DOFs, either 2LDOFs, or 1LDOF and 1ADOF. Fixed B.C. Bending profile will be affected by the boundary condition as the beam is becoming more or less difficult to bend.

Boundary Conditions: Examples (b), (f) are clamped beams (c) and (h) can be considered as two guided beams (a),(d), (d) (e), and (f) are cantilever beams, with one free end.

F

Force Constant (Stiffness) Force constant: k = F/ d (max. displacement) Example d = Fl 3 /(3EI) I = wt 3 /12 k = F/d = 3EI/l 3 = Ewt 3 /(4l 3 ) (In the above, the lateral movement will be small). See Appendix B for more formulas for beam calculation.

Force Constant: Example For one guided beam d = Fl 3 /(12EI) k = F/d = 12EI/l 3 = Ewt 3 /l 3 Left case k =2(Ewt 3 /l 3 ) Right case k = 4(Ewt 3 /l 3 )

Force Constant: Example Zigzagged beams First-order estimation: treat it like a straight beam with equal length Accurate calculation will require finite element method.

Torsion Beams Torsion beams are used to obtain large rotational angles. Bending at such large angles usually will break the Si structure.

Torsion Analysis In pure torsion, all the internal forces are within (or parallel to the cross-sectional sectional surfaces): shear force. Shear stress (τ) and shear strain (θ = τ/g). Torsional moment of inertia J= r 2 da = πr 04 /2 (for cylinder) J = 2.25a 4 for square cross-section beam with the length of each side being 2a. Φ = d/r 0 = Lθ/r 0 = TL/JG (rotation angle at the free end)

Intrinsic Stress in Thin-film Materials Tensile stress Compress stress Thin-film structures with intrinsic tensile stress is pre-streched and tends to shrink after releasing from substrate. t Thin-film structures with intrinsic compressive stress is presqueezed and tends to expand after releasing from substrate. t

Intrinsic Stress in Thin-film Materials Tensile stress Compress stress Thin-film membrane with intrinsic tensile stress remain flat after releasing from substrate. Too much stress causes cracks and fracture. Thin-film structures with intrinsic compressive stress tends to buckle after releasing from substrate.

Intrinsic Stress Gradient Tensile stress gradient Compress stress gradient Thin-film structures with intrinsic stress gradient tends to shrink/expand and bend after releasing from substrate. Think about the beam bending seen before.

Examples Zero stress and stress gradient material is single crystal silicon (SCS). SCS thin-film are used in making micro mirrors.

Minimizing the Effect of Intrinsic Stress and Stress Gradient Optimize the deposition condition to minimize the intrinsic stress and stress gradient. Use stress compensation by adding layers with contradictory t stress condition. Tensile stress

Dynamic Behavior at Resonance Resonant frequency: f r Can also have resonance at a frequency of multiple f r (higher order modes). Quality factor: Q = f r /FWHM (FWHM full width at half maximum) Q = (total stored instantaneous energy)/(energy loss per cycle) Mechanical resonators needs high Q to enhance the frequency selectivity. Micromachined resonators to replace quartz f r = f(1/l) Smaller structure has higher resonant frequency Related to force loading conditions We can drive the device at f r to obtain large movement with a small driving force. Think about the swing