Nonlinear oxide devices and

brain-like analog computing

Suhas Kumar

2

Thanks for the funding and user facilities

The voids of computing

3

Arithmetic heavy

Data heavyNonlinear dynamics

Floating point arithmetic

Boolean logic

Weather prediction

Gene sequencing(+ other NP-class)

Real-time regex matching

Image processing

Present digital computers

Major limitations of digital computers:

• End of Moore’s law

• Von Neumann architecture• Boltzmann tyranny• Boolean logic• Turing limit

The post-Moore’s law physics-driven computer

4

Materials discovery

(new behaviors)

Functional devices

(new functions)

Interacting devices

(new algorithms)

Outline of the talk

1. Neuronic devices – information processing or communication

2. Synaptic devices – information storage

3. The fundamental connection between synaptic and neuronic devices

4. Synaptic + neuronic NP-hard accelerators

5

6

Neuronic devices

Mott Memristors

7

Volatile resistive switching can emulate neuron-like spiking!

NbO2 Mott insulator

Nature Materials 12, 114 (2013)

T

R

Devices and static behavior

8

NbO2 devices

Diameter ~70 nm

Local activity

?

Kumar et al., Nature Communications, 8, 658 (2017)

The cause of NDR in NbO2

9Kumar, Nature Comms. 8, 658 (2017)

0.0 0.5 1.00.0

0.5

1.0

1.5

Cu

rre

nt (m

A)

Voltage (V)

NDR-1

NDR-2

Crystalline

Thermal anomaly in NbO2

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𝜎𝑡ℎ𝑀𝑒𝑡𝑎𝑙 < 𝜎𝑡ℎ

𝐼𝑛𝑠𝑢𝑙𝑎𝑡𝑜𝑟

Counter Wiedemann Franz postulate

Lee et al., Science 355, 371 (2017)

Kumar et al., Nature Communications 8, 658 (2017)

Model for the electrical behavior

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𝑖𝑚 = 𝛼e−Ea2kBT

kBT

β 𝑣𝑚

2

1 +β 𝑣𝑚/t

kBT− 1 e

β 𝑣𝑚/t

kBT +𝛼e−EakBT

2t𝑣𝑚 … Equation (1)

dT

dt=𝑖𝑚𝑣𝑚

Cth−T−Tamb

CthRth T… Equation (2)

Rth T = 1.4 × 106 for T ≤ TC2 × 106 for T > TC

… Equation (3)

Modified 3D Poole-

Frenkel Equation

Temperature

dynamics

Mott transition in Rth

Kumar et al., Nature, 548, 318 (2017)

Kumar et al., Nature Communications, 8, 658 (2017)

Temperature-controlled instability (Chua Corsage)

12

Kumar et al, Nature Communications, (2018)

Kumar et al., Nature Communications, 8, 658 (2017)

Mannan, Int. J. Bifurcation Chaos, 27, 1730011 (2017)

NDR + Local activity

Mott transition + Local activity(Chua Corsage)

New type of current-voltage behavior!

Theorem: Chua corsage leads to chaos in dynamics

Modeling the dynamical behavior

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𝑑𝑇

𝑑𝑡=𝑣𝑚2

𝑅 𝑇 𝐶𝑡ℎ−𝑇−𝑇𝑎𝑚𝑏

𝑅𝑡ℎ𝐶𝑡ℎ

𝑑𝑣𝑚

𝑑𝑡=𝑣𝑖𝑛

𝑅𝑠𝐶− 𝑣𝑚

1

𝑅 𝑇 𝐶+1

𝑅𝑆𝐶

For chaos, where is the third state variable or oscillating force?

η T = 𝑇𝑘𝐵𝐶𝑡ℎ

12 4π

𝑅𝑡ℎ𝐶𝑡ℎcos2π𝑡

𝑅𝑡ℎ𝐶𝑡ℎ

What does it take to produce chaos?

1. Local activity

2. A minimum of three state variables; OR two state variables + a coupled dynamic signal

Kumar et al., Nature, 548, 318 (2017)

0 5 10

0.0

0.5

1.0

Cu

rre

nt

(mA

)

Time (s)

+η(𝑇)

Fluctuation-dissipation theorem of local activity

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Local Activity:

The biology-based theory that now allows us to predict dynamics of electronic devices using their static behavior.

Local activity leads to chaos

Kumar et al., Nature Communications, (2018)

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What is chaos?

– Extreme sensitivity to variations in initial conditions

– In between perfectly ordered and completely random behavior

– Very difficult to predict (but not impossible)

– Why is edge of chaos behavior favored in computing systems?

– Units communicate/cooperate with to compute

– But no global synchronization

– Applies to neurons in a brain

Thermal fluctuations

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η T = 𝑇𝑘𝐵𝐶𝑡ℎ

12 4π

𝑅𝑡ℎ𝐶𝑡ℎcos2π𝑡

𝑅𝑡ℎ𝐶𝑡ℎ

0 5 10

0.0

0.5

1.0

Curr

ent (m

A)

Time (s)0 5 10

Time (s)

a

b cNo fluctuations With fluctuations

400 800 1200-2

0

2

dT

/dt (K

/ps)

Tave

(K)

0

2

dTstatic

/dt

T

|T

| (K

)

Simulations

0 5 10

0.0

0.5

1.0

Cu

rre

nt

(mA

)

Time (s)

dWith fluctuations, large device

Double pendulum with a large driving force!

Takes hundreds of transistors to emulate!

Kumar et al., Nature, 548, 318 (2017)

Chaotic oscillations

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0 10 20Time (s)

1.02 V 1.03 V 1.04 V

0.95 1.00 1.05 1.100.01

0.1

1

(

s-1,x1

07)

Voltage (V)

b

c

0 10 200

150

300

Curr

en

t (

A)

Time (s)0 10 20

Time (s)

a

Are temperature fluctuations the only route to chaos?

Kumar et al., Nature, 548, 318 (2017)

Controlled chaos in a single

electronic device!

Nonlinearity and bifurcations

18A different route to chaos in nonlinear devices.

Bifurcations due to instabilities The missing pieces of device models:

1. Minimization of internal energy or ΔH

2. Spontaneous symmetry breaking during nonlinearity

3. Amplification and coupling of ambient thermal fluctuations in nanoscale devices

Kumar et al., Nature Communications, (2018)

Kumar et al., Advanced Materials, 28, 2772 (2016)

Ridley, Proc. Phys. Soc., 82, 954 (1963)

)𝑗U. 𝐴 = 𝑗L. 𝐴 − 𝑥. 𝐴 + 𝑗H. (𝑥. 𝐴

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Synaptic devices

Ion migration in Hf and Ta oxides

20O K-edge

Pristine/Virgin

HfOx+Hf device

Material stack:

Pt/HfOx/Hf/Pt

I570eV I570eV

Operated

Non-volatile switching and neuronic devices

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The origin of nonvolatile storage in ReRAM

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The origin of non-volatile storage

Fundamental requirement for S-NDR non-volatility

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𝑖 = 𝐺𝑣

𝐺 = 𝜓𝑇𝜉d𝑇

d𝑡=𝑖𝑣

𝐶𝑡ℎ−𝑇 − 𝑇amb𝐶𝑡ℎ𝑅th

𝑇 = 𝑇𝑎𝑚𝑏 + 𝑅𝑡ℎ𝑣2𝜓𝑇𝜉

d𝑇

d𝑖= 𝑅th𝑣

2𝜓𝜉𝑇𝜉−1d𝑇

d𝑖+ 2𝑅th𝑣𝜓𝑇

𝜉d𝑣

d𝑖

𝑅th𝑣2𝜓𝜉𝑇𝜉−1 = 1

𝑇 = 𝑇NDR =𝜉 𝑇amb𝜉 − 1

ξ >1

Steady state: d𝑇

d𝑡= 0

Estimate only this material parameter and one can predict most of the nonlinear and information storage behaviors.

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A brain-like computer using synaptic and neuronic oxide memristors

The traveling salesman problem

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Objective:

Find the shortest path

Constraints:

1. Visit every city once2. Visit every city no more than once3. Do not visit more than one city in a given stop

“Hard” problems

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It is non-deterministic polynomial (NP) complete.

Other NP-complete/hard problems:

Gene sequencing/traveling salesman

Sudoku

Vehicle routing

Open shop scheduling

Bandwidth allocation

NFL scheduling256 games20k variables50k constraints

Takes ~3 months on a 1000-core system to solve!

1M100

10

10M

#co

nstr

ain

ts#vars

Hopfield network

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E = −1

2 𝑖 𝑗𝑠𝑖,𝑗

𝑘 𝑙𝑠𝑘,𝑙𝑤 )𝑖,𝑗 ,(𝑘,𝑙 +

𝑖 𝑗𝑠𝑖,𝑗θ

Synapses + neurons

Energy function:

Kumar et al., Nature, 548, 318 (2017)

US Patent App. 15/141,410

Recursive feedback neural network

An incredibly compact transistorlessall-analogue implementation of a

Hopfield network

Example of one solution with chaos

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The traveling salesman problem

Kumar et al., Nature, 548, 318 (2017)

US Patent App. 15/141,410 (2017)

1 mm

Statistics of many solutions with and without chaos

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Literally annealing the system into its solution!

Kumar et al., Nature, 548, 318 (2017)

We only want better solutions quickly.

High precision prohibitive slow downs

Room temperature analogue of quantum

adiabatic annealing

Mem-HNN outperforms digital hardware and quantum annealing

10,000x better Solutions/sec/Watt !!

Benchmarking on a suite of 60 node max-cut problems

Analog Digital Quantum

Mem-HNN GPU D-wave 2000Q

Clock frequency 1 GHz 1.5 GHz

Time-to-solution 0.3 µs 10 µs 104 s

Power 792 mW <250 W 25,000 W

Solutions/s/Watts 4.6 x 106 >400 4 x 10−9

Major Caveat: Circuit level analysis only – no scalable architecture analyzed yet

Reason for optimism: NP-hard problems are compute-intensive, not data-intensive

arXiv:1903.11194: Kumar et. al. Harnessing Intrinsic Noise in Memristor Hopfield Neural Networks

for Combinatorial Optimization

Summary

–Analog, brain-like computing can outperform any digital alternative in solving certain classes of intractable problems

–But we need new device physics to do that

– Controlled chaos

– Nano-size coupling with ambient thermal fluctuations

– Enthalpy minimization and symmetry breaking

– Super-linearity and non-volatility (the connection between neurons and synapses)

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