objectives –scalingcourses.ece.ubc.ca/300/handouts/eece300-2011-scaling.pdf · 2011-09-15 ·...
TRANSCRIPT
Scaling Laws
• Motivation for miniaturization
S li f f• Scaling of forces
• Scaling in nature
• Scaling of mechanical, electrical, and fluidic systems
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Learning Objectives – Scaling
By the end of this section, students should be able to:able to:
• explain how the surface‐to‐volume ratio chan es as si e is decreasedchanges as size is decreased.
• explain how forces scale with size.
• explain which forces dominate at the micro‐and nanoscales.
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pick‐and‐place robotspick and place robots
pick and place machinep p
van der Waals forcesvan der Waals forces
F ~ L2
1/ ~vdw grF F L−
Fvdw L
adhesion force dominates the gravitational force at low L
The heterogeneous self-assembly process.
Stauth S A , Parviz B A PNAS 2006;103:13922-13927
©2006 by National Academy of Sciences
Details of the self-assembly process for a single microcomponent.
Stauth S A , Parviz B A PNAS 2006;103:13922-13927
©2006 by National Academy of Sciences
High yield (≈97%) self-assembly of 100-μm, circular, single-crystal silicon elements onto a flexible plastic template containing 10,000 binding sites.
Stauth S A , Parviz B A PNAS 2006;103:13922-13927©2006 by National Academy of Sciences
Further Reading
• S. Vogel, Life's Devices: The Physical World of Animals and Plants: Princeton University Press, 1988.
• W S N Trimmer "Micro Robots and Micro Mechanical Systems "W. S. N. Trimmer, Micro Robots and Micro Mechanical Systems, Sensors and Actuators, vol. 19, pp. 267‐287, 1988.
• M. Wautelet, "Scaling laws in the macro‐, micro‐ and nanoworlds," European Journal of Physics, vol. 22, pp. 601, 2001.
• Acknowledgment: part of this material is based on notes by G. F. Spencer, Texas State University San Marcos
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Scaling
• Fabrication processes are developed to create electronic and micro
Why is scaling important at the micro and nano scale?
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. p p g
• These changes are predictable since the forces are generally well understood.
• We need to develop new intuition of microscale phenomenamicroscale phenomena.
• These changes dictate how the device must be built and what forces it will use to operate. This issue is called scaling.
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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
• PerformanceI i i h i i d i d– Integration with circuitry can reduce noise and improve sensitivity
– Yield and reliability may be improved, fewer defects per chip
106 d f t / 3 → 1 d f t f 106 3• 106 defects/cm3 → 1 defect for every 106 μm3
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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, L0.j y 0
L0 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 LS = S ∙L0This gives a new, smaller size for the object as LS S L0• We assume that all 3 dimensions of the object are scaled by the same
amount, isomorphic scaling.
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Animal sizes
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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 highheat. (On the flip side, mice and other small animals, which have a high surface‐to‐volume ratio, often struggle to retain sufficient heat.)
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Walking on water
Surface tension or capillary forces scale with perimeter of wetted area ~ SWalking 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
i i ) h f f h ld b h f f f ior swimming) on the surface of water, held up by the force of surface tension on some hydrophobic (nonwettable) foot:
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Scaling in nature: velocity
Body mass (kg)
Comparison of theoretical predictions with the speeds, stroke frequencies, and force
v=speed, Mb=body mass, f=frequency (stride, flapping)
EECE 300 ‐ 2011 10Bejan A , Marden J H. J Exp Biol 2006;209:238-248
Comparison of theoretical predictions with the speeds, stroke frequencies, and force outputs of a wide variety of animals.
Scaling laws in natureVelocity as a function of body length –``natural law of speed``
T H hi ``Mi h i d th iT. Hayashi, ``Micromechanism and their characteristics``, Micro Electro Mechanical Systems, 1994, MEMS '94, Proceedings, IEEE Workshop.
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Swimming and flyingRe vs. L
Swimming:Mass of muscles ~ s3
Drag force FD ~ s2
L h
Re vs. L
Larger creatures have greater swimming speed.
21F c Avρ=
2D DF c Avρ=
Flying:Mass of muscles ~ s3
Weight ~ s3
Drag force FD ~ s2
Lift force FL ~ s2
Larger creatures have faster flight but
21L LF c Avρ=
Larger creatures have faster flight but more power to keep weight aloft.
T. Hayashi, "Micromechanisms and their
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2L LF c Avρ
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T. Hayashi, Micromechanisms and their characteristics," presented at Micro Electro Mechanical Systems, 1994, MEMS '94, Proceedings, IEEE Workshop on, 1994.
Scaling in nature: force output
EECE 300 ‐ 2011 13Bejan A , Marden J H. J Exp Biol 2006;209:238-248
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 × velocitymass ∝ length3
achievable velocity ∝ length of appendage (length of organism)
If an animal can throw a projectile of mass proportionate to its own, thenproportionate to its own, then
momentum ∝ length4 (!!!)
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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).j ( p , y p )
If a quantity depends on the surface area per unit length, then it scales as
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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?
AreaArea
Volume
Mass
gravitation force
pressure
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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 massA 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 2nd law)F = ‐kΔx = ma (where a is the acceleration from Newton s 2nd 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 k
mω =
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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 Th i k h i f [k] f /l h /The spring constant k has units of [k] = force/length = newtons/meter.
force
d fl ti f ilib ik =
deflection from equilibrium
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There are several modes of motion or oscillation for this cantilever.
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Axial loading on a cantilever
Forces applied along the axis provide an axial load to either lengthen or shorten the cantilever.
F
ΔLΔL
0 0
0 0
A L AF E E L k x
L L
⎛ ⎞Δ= = Δ = Δ⎜ ⎟
⎝ ⎠
Hooke’s law relation for stress and strain:
From this relation, the stiffness coefficient is
0i l
Ak E=
2
~i l
Sk E S≈
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0axialk E
Laxialk E S
S
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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:
3bending
Ik E
L=
3 4
3 3~bending
wt Sk E E S
L S≈ =
where I = moment of inertia around the end of the bar.
L S
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Scaling – resonant frequencyFF
w
lt
F k x= ⋅
• Cantilever spring, fixed‐free
xk x
• l=length, w=width, t=thickness
• E=Young modulus 3 3w t S S⋅ ⋅3 3
~4
w t S Sk E S
l S
⋅ ⋅= =
Interpretation: as we scale down the dimensions the spring constant will decreaseInterpretation: as we scale down the dimensions, the spring constant will decrease.
i‐clicker: What will happen to the resonant frequency? A. f0 remains the same
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B. f0 increasesC. f0 decreases
Scaling in electronics
A ρ
L
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Parallel plate capacitorA 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.
area Al t ti d
+ Qplate separation d
Q
‐ Q
E
Given a parallel plate capacitor with plate area A and plate separation d, the
it i icapacitance is given as
where ε is the permittivity of the gap insulator material
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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 componentdesign 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 areasshrinking 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?
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Interconnects ‐ Resistance
L
WMaterial Resistivity
H
Material Resistivity [Ω‐m]
Silver (Ag) 1.6x10‐8
Copper (Cu) 1 7x10‐8Copper (Cu) 1.7x10
Gold (Au) 2.2x10‐8
Aluminium (Al) 2.7x10‐8
( ) 0 8
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Tungsten (W) 5.5x10‐8
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Interconnects ‐ capacitance
parallel plate capacitance:
L
W
H
Dielectric
Substrate
ddi
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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 Process5 metal layersTi/Al ‐ Cu/Ti/TiN
l ili di l iPolysilicon dielectric
EECE 300 ‐ 2011 “Digital Integrated Circuits”, 2nd Edition, Jan M. Rabaey, et al 27
Power dissipationHow does the power dissipated per unit area scale?
A. L‐2
B. L‐1
C. L0C. LD. L1
E. L2
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Example: IBM air‐gap technology
• Present microelectronics is constrained by the transmission of information: wires (interconnects)information: wires (interconnects) get much slower when scale shrinks ( R ) + interference (C)
• 2007: ~10km of wiring into a space• 2007: 10km of wiring into a space smaller than a postage stamp!
• The constraint motivates specific technological solutions:technological solutions: – 1st generation: Cu interconnects, low‐k
dielectrics– 2nd generation: IBM air gap (vacuum
IEEE Spectrum, Jan. 2008
embedded in the insulation that surrounds the interconnects), Intel (nanopore technology), STMicroelectronics
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Energy stored in a capacitor
How does stored energy scale with capacitor size?
A. L‐1
B. L0
1. for constant charge density
C. L1
D. L2
E. L3
2. for constant applied voltage
A. L‐2
B. L‐1
C. L0
D. L1
E. L2
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Electrostatic Forces
Calculate the electrostatic force exerted between the plates of a parallel plate capacitor.
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Paschen Curve Two regimes in breakdown voltage Vb vs. spacing d curve:
In an air capacitor the maximum E field is given by the
Breakdownlt
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
voltage greater (as the gap d approaches the mean free path λ of insulator molecules, fewer molecules are around to be ionized).
Ebd~100MV/mbd
~200V~200V
2 @
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Distance*pressure(meter*atm)
~2μm@1atm
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Electrostatic Forces
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Physical Forces & ScalingOne concern when devices are scaled is whether the interactions or forces which wereOne 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
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At small sizes, these forces may no longer be appropriate to use in microdevices. Why?
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Magnetic forcesFor 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 fieldselectrostatic instead of magnetic fields.
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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 importantmore important
• However, intensive parameters (material properties) generally impose the scaling invariants (Jmax, Emax, Tmax)
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