1. Physical Limits of Computing Dr. Mike Frank CIS 6930, Sec
Physical Limits of Computing Dr. Mike Frank CIS 6930, Sec. #3753X Spring 2002
Lecture #25 Limits on Adiabatics: Friction, Leakage, & Clock/Power Supplies Fri., Mar. 15

2. Administrivia & Overview
Don’t forget to keep up with homework!
We are 8 out of 14 weeks into the course.
You should have earned ~57 points by now.
Course outline:
Part I&II, Background, Fundamental Limits - done
Part III, Future of Semiconductor Technology - done
Part IV, Potential Future Computing Technologies - done
Part V, Classical Reversible Computing
Adiabatic electronics & CMOS logic families, - Mon. & Wed
Limits of adiabatics: Friction,Leakage,Power supplies. TODAY
RevComp theory I: Emulating Irreversible Machines - Fri. 3/15
RevComp theory II: Bounds on Space-Time Overheads - Mon. 3/18
(plus ~7 more lectures…)
Part VI, Quantum Computing
Part VII, Cosmological Limits, Wrap-Up
First we presented the fundamental limits, because we need to keep these in mind as bounds on what any possible future computing technology is going to be able to do.
Next, we will survey the future of semiconductor technology, because this has near-term importance and will help to ground our later studies on things like nanoelectronics, molecular electronics, and other alternative technologies, as well as our discussions of reversible semiconductor circuits in part V and some semiconductor-based approaches to quantum computing in Part VI.

3. The system’s energy is “in here”
Structured Systems
A structured system is defined as a system about whose state we have some knowledge.
Some of its physical information is known.
 Its entropy is not at a maximum (by defn.).
 It is not at equilibrium (by defn.).
For states with a given energy E,
we say the system’s energy is distributed among those states, in proportion to their probability.
All states of the abstract system having energy E
The system’s energy is “in here”
States w. prob. > 0

4. Desired Trajectories
Any structured system we build to serve some purpose has some desired trajectory, or set of trajectories, through its configuration space that we would ideally like it to follow at all times.
Think of any given state as having a specific “desirability” at any given time.
Time
Config- uration
Desired trajectories

5. Energy that has departed from desired trajectories
Energy Losses
Energy dissipation can be viewed as a departure of part of the system’s energy away from the system’s desired trajectory.
E.g., 1 of 106 electrons leaks out of a DRAM cell = system’s energy has departed from desired trajectory (all 106 stay) by a small amount
Time
Config- uration
Energy that has departed from desired trajectories

6. Limits of Adiabatics I: Friction

7. Generalized Friction
Any force leading to departure from desired trajectory that obeys the adiabatic principle:
I.e., force strength (& total energy loss) is proportional to velocity along trajectory at low velocities
Examples:
Ordinary sliding friction
Fluid viscosity
Electrical resistance
Forces causing electromagnetic radiative losses
Forces causing losses in inelastic collisions

8. Ways to Quantify Friction
Normal friction measures referring to length, mass, etc. may not apply to all processes.
For a given mechanism executing a specified process (i.e., following a specified desired trajectory or -ies) over a time t:
Energy coefficient: cE = Elost·t = Elost/q
Energy dissipated from traj. per unit of “quickness”
Note quickness q = 1/t has units like Hz
Entropy coefficient: cS = Smade·t = Smade/q
New entropy generated per unit of quickness
Note that cE = cS·T at temperature T.
What matters!

9. Energy Coefficient in Electronics
For charging capacitive load C by voltage V through effective resistance R: cE = Elostt = (CV2RC/t)t = C2V2R
If the resistances are voltage-controlled switches with gain factor k controlled by the same voltage V, then effective R  1/kV cE = C2V/k
In constant-field-scaled CMOS, k  1/dox  , C  , and V  , so cE  3/ = 4; Elost = cE/t  4/ =  (like CV2 energy)

10. Degree of Reversibility of CMOS
What is the Q of a min-size CMOS transistor?
Q = Efree/Ediss
= Efree/(cE/t) = ½CV2/(C2V2R/t) = ½(t/RC) = ½ s (s = slowdown factor)
Note: Using transistors wider than minimum-size (larger C, smaller R) wouldn’t change RC or Q, and would increase overall dissipation by increasing cE.

11. Lower Bounds on Friction?
No general (technology-independent) lower bounds on friction coefficients for interesting types of processes (e.g. computation) are currently known.
Clever engineering may eventually reduce the friction in desired processes to values as small as is desired.
Some ways:
Reduce number of moving parts (or particles)
Isolate “moving parts” of system from unwanted interactions w. environment

12. Entropy coefficients of some reversible logic gate operations
From Frank, “Ultimate theoretical models of nanocomputers” (Nanotechnology, 1998):
SCRL, circa 1997: ~1 b/Hz
Optimistic reversible CMOS: ~10 b/kHz
Merkle’s “quantum FET:” ~1.2 b/GHz
Nanomechanical rod logic: ~.07 b/GHz
Superconducting PQ gate: ~25 b/THz
Helical logic: ~.01 b/THz
How low can you go? We don’t really know!

13. Is Adiabatic Limit Achievable?
Even if there is some lower bound on cS, it seems we can have Smade 0 as t  .
What factors may prevent this?
Any lower bound >0 on the number of irreversible bit-operations performed. (Each has Smade  1.)
Fortunately, the lower bound can always be made 0.
Any lower bound on the rate of energy leakage, even when system is completely stopped.
Any upper bound on the Q of the clocking & synchronization system.
The system dissipates Efree/Q on every cycle.
No technology-independent upper bounds on Q known

14. Limits of Adiabatics II: Leakage

15. Some Synonyms
Leakage of energy or (equivalently) probability mass out of a desired configuration or trajectory.
Occurrence of errors in the desired analog or digital state of a system. (Motion away from desired states.)
Decay of structure of a structured system. (The state departs from desired state.)
Leakage = Error = Decay

16. Perfect Mechanisms? If a structured system is perfectly closed,
I.e. non-interacting with other systems, at all!
And if its internal interactions are perfectly known,
Then, and only then, is its (von Neumann) entropy going to be a constant.
Otherwise, its entropy will continuously increase as we lose track of its state.
In this case, no mechanism is perfect, in that some of its energy (i.e. some probability mass) is always leaking away from the desired trajector(y/ies) at some nonzero base rate, even when the rate of system’s progress along its trajectory is zero.

17. Energy that has leaked from desired configuration
Leakage Limits
Claim: No real, structured system can have absolutely zero rate of energy leakage out of its desired trajectories, even if not moving.
However: No general, technology-independent lower bound on leakage rates is known (other than zero.)
Engineering advances might make leakage as small as desired.
Time
Config- uration
Energy that has leaked from desired configuration

18. Quantifying Leakage For a given structured system:
Leakage power: Pleak = dEleak / dt
Spontaneous entropy generation rate: Sleak = dSleak / dt
Again, note Pleak = Sleak · T at temperature T. • •

19. Ways to Decrease Leakage
Have high potential-energy barriers
slows down thermally excited leakage exponentially
Have thick potential-energy barriers
slows down quantum tunneling exponentially
Example: Older generations of CMOS.
Mechanical (clockwork) systems have high potential energy barriers, for their size:
Decay may require atoms to diffuse out of tightly-bonded spots.
Mechanisms that avoid making/breaking contacts (e.g. buckled logic) avoid losses due to stiction.

20. Minimum Losses w. Leakage
Etot = Eadia + Eleak
Eleak = Pleak·tr
Eadia = cE / tr

21. Minimum Loss Derivation

22. Leakage in CMOS
See transparencies.

23. Limits of Adiabatics III: Clock/Power Supplies
See transparencies.