ESE370:
Circuit-Level
Modeling, Design, and Optimization
for Digital Systems
Day 37: December 9, 2011
Repeaters in Wiring
Uncorrelated Noise Sources
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Penn ESE370 Fall2011 -- DeHon
Previously
• Unbuffered wire delay scales as L2
– 0.5 Rwire Cwire
– 0.5 L2 Ru Cu
• Correlated Noise
– Crosstalk
– Inductance
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Penn ESE370 Fall2011 -- DeHon
Today
• Interconnect Buffering
• Uncorrelated Noise sources
– Ionizing particles, thermal, shot
• Final
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Penn ESE370 Fall2011 -- DeHon
Delay of Wire
•
•
•
•
Long Wire: 1mm
Rwire = 60K W (for the 1mm)
Cwire = 0.16 pF (for the 1mm)
Driven by inverter
– R0 = 25K W
– C0 = 0.01 fF
– Assume velocity saturated, sized Wp=Wn=1
• Loaded by identical inverter
Penn ESE370 Fall2011 -- DeHon
4
Should be able to do these calculations on final.
Formulate Delay
Delay of inverter driving wire?
(

Rbuf  Cself  Cwire  Cload  0.5Rwire  Cwire  Rwire  Cload
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Penn ESE370 Fall2011 -- DeHon
Calculate Delay
• Cload = 2 C0
• Rbuf = R0
• Cself = g 2 C0 = 2 C0
(

Rbuf  Cself  Cwire  Cload  0.5Rwire  Cwire  Rwire  Cload
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Penn ESE370 Fall2011 -- DeHon
Should be able to do these calculations on final.
Buffer Middle
• Delay if add buffer to middle of wire?
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Penn ESE370 Fall2011 -- DeHon
Formulate and Calculate
Delay




 Rwire Cwire  Rwire
Cwire
2 Rbuf  Cself 
 Cload   0.5

 Cload 



 2
2
2 
2


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Penn ESE370 Fall2011 -- DeHon
N Buffers
• Delay for N buffers?




 Rwire Cwire  Rwire
Cwire
N Rbuf  Cself 
 Cload   0.5

 Cload 



 N
N
N  N


(

N  Rbuf  Cself  Cload  Rbuf
 Rwire  Cwire 
 Cwire  0.5
  Rwire  Cload


N
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Penn ESE370 Fall2011 -- DeHon
Minimize Delay
• How minimize delay?
• Differentiate &

R

C
wire
wire

Solve for N:
N  0.5
R  C C
self
load
 buf
(
(

N  Rbuf  Cself  Cload  Rbuf
(





 Rwire  Cwire 
 Cwire  0.5
  Rwire  Cload


N

  Rbuf  Cself  Cload  Rbuf  Cwire  Rwire  Cload
2 0.5Rwire  Cwire
Penn ESE370 Fall2011 -- DeHon
Equalizes delay in buffer and wire
10
Calculate: Delay at Optimum
Stages for Example
•
•
•
•
Rwire = 60K W (for the 1mm)
Cwire = 0.16 pF (for the 1mm)
Rbuf=R0 = 25K W
Cself=Cload=2(C0 = 0.01 fF)=0.02fF
(

2 0.5Rwire  Cwire  Rbuf  Cself  Cload  Rbuf  Cwire  Rwire  Cload
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Penn ESE370 Fall2011 -- DeHon
Segment Length
• Rwire = L×Runit
• Cwire = L×Cunit

Rwire  Cwire

N  0.5
R  C C
self
load
 buf
(


Ru  Cu

N  L 0.5
R  C C
self
load
 buf
(





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Penn ESE370 Fall2011 -- DeHon





Optimal Segment Length
• Delay scales linearly with distance once
optimally buffered
*
seg
L

(
R  C C
L
buf
self
load

  2

N
Ru  Cu


Ru  Cu

N  L 0.5
R  C C
self
load
 buf
Penn ESE370 Fall2011 -- DeHon
(










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Buffer Size?
• How big should buffer be?
– Rbuf = R0/W
– Cload = 2 W C0 (assuming velocity saturation)
– Cself = g 2 W C0
(

2 0.5Rwire  Cwire  Rbuf  Cself  Cload  Rbuf  Cwire  Rwire  Cload
R0
R0
2 0.5Rwire  Cwire 
 (1 g 2WC0 
 Cwire  Rwire  2WC0
W
W
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Penn ESE370 Fall2011 -- DeHon
Implication W
• Rwire = L×Runit
• Cwire = L×Cunit
•  W independent of Length
– Depends on technology
R0  Cwire
W 
Rwire  2C0
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Penn ESE370 Fall2011 -- DeHon
Delay at Optimum W
2 0.5Rwire  Cwire  R0  (1 g 2C0  2 R0  Cwire  Rwire  2C0
• With g=1, 1+g=2
• Same size as first term
4 R0  Cwire  Rwire  2C0
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Penn ESE370 Fall2011 -- DeHon
Ideas
• Wire delay linear once buffered
• Optimal buffering matches
– Buffer delay
– Delay on wire between buffers
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Penn ESE370 Fall2011 -- DeHon
Uncorrelated Noise Sources
Ionizing Particles
Thermal Noise
Shot Noise
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Penn ESE370 Fall2011 -- DeHon
Ionizing Particles
• Alpha Particles (He nucleus=He2+)
• Impact with mega-electron-volts of
energy (3—10MeV)
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Penn ESE370 Fall2010 -- DeHon
Ionizing Particles
• Alpha Particles (He nucleus=He2+)
• Can penetrate microns into Si
• Creating 2×106 electron-hole pairs
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Penn ESE370 Fall2010 -- DeHon
Ionizing Particles
• Alpha Particles (He nucleus=He2+)
• Can be generated by decay in
packaging materials
– Lead common one including some fraction
of radioactive isotopes 210Pb
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Penn ESE370 Fall2010 -- DeHon
Src: http://en.wikipedia.org/wiki/File:Wirebonding2.svg
Comparisons
• How many electrons in:
– Capacitor: 1fF charges to 1V
– e = 1.6×10-19 Coulombs
• Recall C0 = 0.01fF,
– typical load around 10-20C0
• How large a capacitor to withstand loss
of 2×106 electrons?
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Penn ESE370 Fall2010 -- DeHon
Disrupt
• Alpha particle will disrupt DRAM Cell
• Can disrupt undriven nodes
– Latch
– Dynamic node
– Memory bit
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Penn ESE370 Fall2010 -- DeHon
Ionizing Particles
• There are other particles with different
energies
– Neutrons from cosmic rays
• 10x energy of alpha particles
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Penn ESE370 Fall2010 -- DeHon
Particle Flux
• Differs with location
– Altitude
• Denver vs. Philadelphia
• Ground vs. aircraft at 30,000 feet
• Space (outside atmosphere)
– Near poles
• Changes upset rate seen by chips
– By orders of magnitude
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Penn ESE370 Fall2010 -- DeHon
SEU/bit Norm to 130nm
Scaling and Error Rates
Increasing Error Rates
10
2X bit/latch count increase per
generation
logic
cache
arrays
1
180
130
90
65
45
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Technology (nm)
Penn ESE370 Fall2010 -- DeHon
Source: Carter/Intel
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Thermal Noise
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Penn ESE370 Fall2010 -- DeHon
Thermal Background
• Except at absolute 0 (Temperature)
– Particles are moving around randomly
• Thermal bath means free energy
around
• Electron can be borrow the thermal
energy to hop over barrier
– Out of an energy well, bond cite
– …Out of a capacitor
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Penn ESE370 Fall2010 -- DeHon
Day 8
Doping with P
• End up with extra electrons
– Donor electrons
• Not tightly bound to atom
– Low energy to displace
– Easy for these electrons
to move
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Penn ESE370 Fall2010 -- DeHon
Day 8
Doped Band Gaps
• Addition of donor electrons makes more
metallic
– Easier to conduct
Semiconductor
0.045ev
1.1ev
Ec
ED
Ev
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Penn ESE370 Fall2010 -- DeHon
Day 8
Electron Conduction
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Penn ESE370 Fall2010 -- DeHon
Thermal Background
• Except at absolute 0 (Temperature)
– Particles are moving around randomly
• Thermal bath means free energy
around
• Electron can be borrow the thermal
energy to hop over barrier
• Are doing it all the time to give us our
semiconductors
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Penn ESE370 Fall2010 -- DeHon
Rising above
the Thermal Noise
• Must apply more energy than
background noise to
– Hold electron in place
– Move an electron from place to place
• Charge/discharge a node with some reliability
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Penn ESE370 Fall2010 -- DeHon
Probability of Noise Error
• Probability exponential in energy
– Not exactly this…but basic dependence
• To keep error rate sufficiently low
– Need energy of operation (of storage) to be
some multiple of kT
Perror  e
E
 
 kT 
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Penn ESE370 Fall2010 -- DeHon
Where are we today?
• How does kT compare to switching
10C0 at 1V?
– k=1.4×10-23 J/K
– T=300K (Room Temperature)
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Penn ESE370 Fall2010 -- DeHon
Where are we today?
• How does kT compare to switching
10C0 at 1V?
– k=1.4×10-23 J/K
– T=300K (Room Temperature)
• kT=4.2×10-21 J
• Eswitch=CV2 = 0.1fJ=10-16J
• Eswitch~=2×104 kT
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Penn ESE370 Fall2010 -- DeHon
Scaling
• What happens as we scale?
– 0.5V 4.5nm (versus 1V 45nm)
• Eswitch~=2×104 kT
• 45nm to 4.5nm  impact on Eswitch?
– reduce capacitance by 10x
– reduce voltage by 2x
– Eswitch~=500 kT
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Penn ESE370 Fall2010 -- DeHon
Shot Noise
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Penn ESE370 Fall2010 -- DeHon
Shot Noise
• Actual electron transport is probabilistic
• Current is a statement about average
rate of electron flow
• For large numbers of electrons
– Law of large numbers convergence to
mean
 s ~= Sqrt(N)
– Large N  sqrt(N)/N small
• Small percentage variation
Penn ESE370 Fall2010 -- DeHon
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Shot Noise
• For small number of electrons (N)
 s ~= Sqrt(N)
– sqrt(N)/N not so small
– Higher variation
– Noise in switching time
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Penn ESE370 Fall2010 -- DeHon
Electron Counts
• How many electrons (N)
– in 0.1fF, 1V switching event?
– in 0.01fF, 0.5V switching event?
s?
• How many s out to only get 50% of
electrons moving?
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Penn ESE370 Fall2010 -- DeHon
Will we see?
• Large chips, fast clock rates  many
events….samples far out on curve
From: http://en.wikipedia.org/wiki/File:Standard_deviation_diagram.svg
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Penn ESE534 Spring2010 -- DeHon
Gaussian Distrubution
Number Sigma
1 in How many
1
2
3
4
5
6
3.2
22
370
16K
1.7 M
510M
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Penn ESE370 Fall2010 -- DeHon
Idea
• Many sources cause upsets
– Ionizing particles, thermal, shot noise
• Tend to depend on charge
– Of node, of switching even
• Scaling decreases charge
– Lower voltage, lower capacitance
– Also increases susceptible nodes
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Penn ESE370 Fall2010 -- DeHon
Final
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Penn ESE370 Fall2011 -- DeHon
Final
• Everything
– New stuff: (see last year’s final)
•
•
•
•
Clocking and dynamic logic
Memories
Crosstalk
Transmission lines
– Midterm 1 and 2 questions all fair game
• Esp: Delay of circuit, optimize delay, Energy
• Noise Margins (interaction with noise effects?)
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Penn ESE370 Fall2011 -- DeHon
Admin
• Project Due at 11:59pm
– 20% per day late penalty
• Final
– Noon—2pm
– Dec. 20
– Moore 212
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Penn ESE370 Fall2010 -- DeHon