Motivation for miniaturization Scaling of forces Scaling in nature Scaling of mechanical, electrical, and fluidic systems EECE

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1 Scaling Laws Motivation for miniaturization Scaling of forces Scaling in nature Scaling of mechanical, electrical, and fluidic systems EECE Learning Objectives Scaling By the end of this section, students should be able to: explain how the surface to volume ratio changes as size is decreased. explain how forces scale with size. explain which forces dominate at the microand nanoscales. EECE

2 pick and place place robots pick and place machine van derwaals forces F ~ 2 vdw L F / F ~ L vdw gr 1 adhesion force dominates the gravitational force at low L

3 The heterogeneous self-assembly process. Stauth S A, Parviz B A PNAS 2006;103: by National Academy of Sciences Details of the self-assembly process for a single microcomponent. Stauth S A, Parviz B A PNAS 2006;103: by National Academy of Sciences

4 High yield ( 97%) self-assembly of 100-μm, circular, single-crystal silicon elements onto a flexible plastic template containing 10,000 binding sites by National Academy of Sciences Stauth S A, Parviz B A PNAS 2006;103:

Further Reading S. Vogel, Life's Devices: The Physical World of Animals and Plants: Princeton University Press, 1988. W. S. N.

22, pp. 601, 2001. Acknowledgment: part of this material is based on notes by G. F.

Fabrication processes are developed to create electronic and micro electromechanical devices at ever decreasing sizes. MEMS are often 1000x smaller than macro counterparts.

5 Further Reading S. Vogel, Life's Devices: The Physical World of Animals and Plants: Princeton University Press, W. S. N. Trimmer, "Micro Robots andmicro Mechanical Systems, " Sensors and Actuators, vol. 19, pp , M. Wautelet, "Scaling laws in the macro, micro and nanoworlds," European Journal of Physics, vol. 22, pp. 601, Acknowledgment: part of this material is based on notes by G. F. Spencer, Texas State University San Marcos EECE Scaling Why is scaling important at the micro and nano scale? Fabrication processes are developed to create electronic and micro electromechanical devices at ever decreasing sizes. MEMS are often 1000x smaller than macro counterparts. As these sizes shrink, different physical forces become more important or less important depending on their nature. These changes are predictable since the forces are generally well understood. We need to develop new intuition of microscale phenomena. These changes dictate how the device must be built and what forces it will use to operate. This issue is called scaling. EECE

6 Why Miniaturize? Motivation Batch fabrication, lower cost per device Less energy, less material consumed, disposable Arrays of sensors possible, minimally invasive Similar size scale as individual cells Can take advantage of different scaling laws (e.g., electrostatic forces) Breakdown of macroscale laws of physics Performance Integration with ihcircuitry i can reduce noise and improve sensitivity Yield and reliability may be improved, fewer defects per chip 10 6 df defects/cm t/ 3 1 defect df tfor every 10 6 μm 3 EECE Geometrical Scaling In many instances, the behavior of a physical system can be predicted using geometric scaling from larger to smaller sizes. Assume an object is defined by some initial characteristic size, L 0. L 0 LS Next this object is reduced in size by a multiplicative factor S, where the scale factor is in the range 0 S 1. This gives a new, smaller size for the object as L S = S LL 0 We assume that all 3 dimensions of the object are scaled by the same amount, isomorphic scaling. EECE

7 Animal sizes EECE Why are there no mice in the arctic? The larger an animal is, the less surface area it has relative to overall body mass and the harder it is for the creature to rid its body of excess heat. (On the flip side, mice and other small animals, which have a high surface to volume ratio, often struggle to retain sufficient heat.) EECE

Walking on water Surface tension or capillary forces scale with perimeter of wetted area ~ S Walking on water is a feat some creatures such as water striders can do.

8 Walking on water Surface tension or capillary forces scale with perimeter of wetted area ~ S Walking on water is a feat some creatures such as water striders can do. Make a dimensionless index to examine the practicality of standing or walking (not floating or swimming) i on the surface of water, hld held up by the force of surface tension on some hydrophobic (nonwettable) foot: EECE Scaling in nature: velocity Body mass (kg) v=speed, M b =body mass, f=frequency (stride, flapping) Comparison of theoretical predictions with the speeds, stroke frequencies, and force outputs of a wide variety of animals. Bejan A, Marden J H. J Exp Biol 2006;209: EECE

9 Scaling laws in nature Velocity as a function of body length ``natural law of speed`` T. Hayashi, ``Micromechanism i and their characteristics``, Micro Electro Mechanical Systems, 1994, MEMS '94, Proceedings, IEEE Workshop. EECE Swimming and flying Swimming: Mass of muscles ~ s 3 Drag force F D ~ s 2 Larger creatures have greater swimming speed. 1 FD = cd ρ Av 2 Flying: Mass of muscles ~ s 3 Weight ~ s 3 Drag force F D ~ s 2 Lift force F L ~ s 2 Larger creatures have faster flight but more power to keep weight aloft. 1 F L = c L ρ Av 2 EECE Re vs. L T. Hayashi, "Micromechanisms and their characteristics," presented at Micro Electro Mechanical Systems, 1994, MEMS '94, Proceedings, IEEE Workshop on,

Scaling in nature: force output Bejan A, Marden J H.

Throwing projectiles: Chimps and humans throw projectiles (rocks) as one of

momentum = mass velocity mass length 3 achievable velocity length of appendage

10 Scaling in nature: force output Bejan A, Marden J H. J Exp Biol 2006;209: EECE When animals attack! Throwing projectiles: Chimps and humans throw projectiles (rocks) as one of various aggressive tactics. Smaller animals, even manually dexterous ones do not. momentum = mass velocity mass length 3 achievable velocity length of appendage (length of organism) If an animal can throw a projectile of mass proportionate to its own, then momentum length 4 (!!!) EECE

11 Surface area In some cases, an important quantity is given by the ratio of the surface area to volume of the object (for example, heat transfer and fluid dynamics problems). If a quantity depends on the surface area per unit length, then it scales as EECE Scaling laws in Mechanics For elements with typical linear dimension L, assume all linear dimensions vary proportionally to L How do the following quantities scale with L? Area Volume Mass gravitation force pressure EECE

12 Scaling of Mechanical Components Two important mechanical properties for MEMS are mass and stiffness. These will determine the device inertia and the degree of its response to applied forces. Example: spring loaded with a mass A spring (spring constant k) with a mass m attached. From Hooke s law, the restoring force when the spring is stretched or compressed by Δx from its equilibrium length is F = kδx = ma (where a is the acceleration from Newton s 2 nd law) The spring constant k is a measure of the stiffness of the spring. The mass m measures the inertia (the resistance to changes in its own motion). If this spring is oscillating, the resonant frequency is ω = k m EECE Stiffness Hooke s law can be used to provide a definition of the stiffness of a mechanical component. Using the spring force law, k = F/Δx The spring constant k has units of [k] = force/length = newtons/meter. force k = deflection from equilibrium i There are several modes of motion or oscillation for this cantilever. EECE

13 Axial loading on a cantilever Forces applied along the axis provide an axial load to either lengthen or shorten the cantilever. F ΔL Hooke s law relation for stress and strain: AΔL A L L 0 0 F = E = E Δ L= kδx 0 0 From this relation, the stiffness coefficient is 2 A S 0 k = il~ axial E kaxial E S L S 0 EECE Transverse load on a cantilever Forces applied at the end of the cantilever will give a bending load. F The stiffness coefficient is given by: k bending I = E L 3 where I = moment of inertia around the end of the bar. 3 4 wt S kbending ~ E E S 3 3 L S = EECE

14 Scaling resonant frequency l w Cantilever spring, fixed free t F x F = k x l=length, w=width, t=thickness E=Young modulus 3 3 wt S S k = E ~ = S 3 3 4l S Interpretation: aswe scale down the dimensions, the springconstant will decrease. i clicker: What will happen to the resonant frequency? A. f 0 remains the same B. f 0 increases C. f 0 decreases EECE Scaling in electronics A ρ L EECE

15 Parallel plate capacitor A capacitor stores electrical charge on its plates. It contains energy that depends on the amount of stored charge or on the electric field between the two plates. + Q Q E area A plate lt separation d Given a parallel plate capacitor with plate area A and plate separation d, the capacitance is given as where ε is the permittivity of the gap insulator material EECE Non isomorphic scaling How these components come together in an electronic or MEMS device depends on how the actual circuit/device components are scaled. The actual scaling may not be isomorphic. In designing the device, the nature of the actual scaling must be accounted for in the design of each component. For example, if a device contains both R and C, then there will be an associated RC time constant that governs the device behavior. How might this scale as the device is shrunk? One example of shrinking some part of a device is the interconnects, the wires that connect individual components in an IC. The C is the parasitic capacitance along the wire length. In shrinking these, the lengths are decreased, but the major shrink is the wire widths or areas and the film thicknesses. How do these shrinking interconnect dimensions and thinner films affect the RC time constant, and hence the speed at which these devices operate? EECE

16 Interconnects Resistance L Material Resistivity [Ω m] Silver (Ag) 1.6x10 8 Copper (Cu) 17x10 1.7x10 8 Gold (Au) 2.2x10 8 Aluminium (Al) 2.7x10 8 Tungsten (W) 5.5x10 88 H W EECE Interconnects capacitance parallel plate capacitance: L W H d di Dielectric Substrate EECE

If either d or H are reduced, RC increases as ~ 1/S Example: Intel 0.

17 Interconnect scaling To fit more devices, reduce the width of the interconnects. This allows a denser network of wires between the high density devices underneath the interconnect layer. The width shrink cancels out. If either d or H are reduced, RC increases as ~ 1/S Example: Intel 0.25 micron Process 5 metal layers Ti/Al Cu/Ti/TiN Polysilicon dielectric i EECE Digital Integrated Circuits, 2nd Edition, Jan M. Rabaey, et al 27 Power dissipation How does the power dissipated per unit area scale? A. L 2 B. L 1 C. L 0 D. L 1 E. L 2 EECE

Example: IBM air gap technology Present microelectronics is constrained by the transmission of information: wires (interconnects) get much slower when scale shrinks ( R ) + interference (C) 2007:

The constraint motivates specific technological solutions: 1 st generation: Cu interconnects, low k dielectrics 2 nd generation: IBM air gap (vacuum embedded in the insulation that surrounds the

18 Example: IBM air gap technology Present microelectronics is constrained by the transmission of information: wires (interconnects) get much slower when scale shrinks ( R ) + interference (C) 2007: ~10km of wiring into a space smaller than a postage stamp! The constraint motivates specific technological solutions: 1 st generation: Cu interconnects, low k dielectrics 2 nd generation: IBM air gap (vacuum embedded in the insulation that surrounds the interconnects), Intel (nanopore technology), STMicroelectronics IEEE Spectrum, Jan EECE Energy stored in a capacitor How does stored energy scale with capacitor size? 1. for constant charge density A. L 1 B. L 0 C. L 1 D. L 2 E. L 3 2. for constant applied voltage A. L 2 B. L 1 C. L 0 D. L 1 E. L 2 EECE

19 Electrostatic Forces Calculate the electrostatic force exerted between the plates of a parallel plate capacitor. EECE Paschen Curve Two regimes in breakdown voltage V b vs. spacing d curve: Breakdown voltage In an air capacitor the maximum E field is given by the electrical breakdown of the air in its gap. In microscale devices, the breakdown E field can be much greater (as the gap d approaches the mean free path λ of insulator molecules, fewer molecules are around to be ionized). E bd ~100MV/m bd ~200V EECE ~2μm@ 1atm Distance*pressure (meter*atm) 32

Electrostatic Forces EECE 300 2011 33 Physical Forces & Scaling One concern

used at the larger sizes are still appropriate (or even operable) at the

Everyday common electric motors use electromagnetic (EM) forces for

Alternating electric currents in the motor windings create magnetic forces

20 Electrostatic Forces EECE Physical Forces & Scaling One concern when devices are scaled is whether the interactions or forces which were used at the larger sizes are still appropriate (or even operable) at the smaller sizes. A common example is that of motors. Everyday common electric motors use electromagnetic (EM) forces for operation. Alternating electric currents in the motor windings create magnetic forces to cause motion of the rotor. Magnetic actuated motor Electrostatic rotary motor At small sizes, these forces may no longer be appropriate to use in microdevices. Why? EECE

21 Magnetic forces For example, take the case of an electric motor. At macroscale, it relies on magnetic fields to generate forces, involving a combination of coils and permanent magnets. However, at microscale, most of the micromotors use electrostatic instead of magnetic fields. EECE Conclusions Scaling modifies the relative importance of various physical effects dominant physical quantities between different scales change: gravitational, inertial forces become less effective electrostatic forces, surface tension forces become more important However, intensive parameters (material properties) generally impose the scaling invariants (J max, E max, T max ) EECE

Outline. 4 Mechanical Sensors Introduction General Mechanical properties Piezoresistivity Piezoresistive Sensors Capacitive sensors Applications

Outline. 4 Mechanical Sensors Introduction General Mechanical properties Piezoresistivity Piezoresistive Sensors Capacitive sensors Applications Sensor devices Outline 4 Mechanical Sensors Introduction General Mechanical properties Piezoresistivity Piezoresistive Sensors Capacitive sensors Applications Introduction Two Major classes of mechanical