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SRAM M Dynami namic c Sta tability bility Verification ification by Reachability achability Analysis alysis with th Consid nsideration eration of f Th Threshold eshold Voltage ltage Variatio riation Yang Song 1 , Hao Yu* 2 , Sai

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SLIDE 1

SRAM M Dynami namic c Sta tability bility Verification ification by Reachability achability Analysis alysis with th Consid nsideration eration

  • f

f Th Threshold eshold Voltage ltage Variatio riation

Monday, March 25, 2013 1

Yang Song1, Hao Yu*2, Sai Manoj2 and Guoyong Shi1

1School of Microelectronics, Shanghai Jiao Tong University 2School of Electrical and Electronic Engineering, Nanyang Technological University

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SLIDE 2

Outline

  • SRAM failure analysis
  • SRAM nonlinear dynamics
  • Verification by reachability analysis
  • Experimental results
  • Summary

2 Monday, March 25, 2013

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SLIDE 3

SRAM Failure Analysis

  • Becomes difficult as technology scales down

– Process variations, mismatch among transistors cause failures

  • Nonlinear dynamics of SRAM
  • Physical mechanisms of failures

– Separatrix: boundary separating two stable regions in state- variable space*

3 V2 (0,0) (Vdd, Vdd)

State equilibrium State equilibrium

V1

Instate equilibrium

State-variable space * W. Dong and et.al. ICCAD, 2008

Monday, March 25, 2013

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SLIDE 4

Write Failure Analysis

4

  • Initial state (v1,v2) = (vdd, 0)
  • Target state (v1, v2) = (0, vdd)

π‘Š

π‘’β„Ž4

V2 Vdd Vdd V1

Write Failure

Separatrix B A

WL BL BR v1 v2 vdd M1 M2 M3 M4 M6 M5 β€œ1” β€œ0” β€œ0”

discharging charging π‘Š

π‘’β„Ž6

  • Threshold voltage variation

causes difficulty to move state point to the target state.

Monday, March 25, 2013

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SLIDE 5

Read Failure Analysis

5

Vdd Vdd V1 V2

Read Failure

A B

WL BL BR v1 v2 vdd M1 M2 M3 M4 M6

M5

β€œ0” β€œ1” β€œ1” charging

  • Internal state is aimed to

maintain regardless perturbation during read

  • peration.
  • Mismatch between M4

and M6

  • Mismatch among M1-4

Monday, March 25, 2013

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SLIDE 6

Hold Failure Analysis

6

Vdd V1 V2 Vdd Hold Failure Perturbation current WL BL BR v1 v2 vdd M1 M2 M3 M4 M6 M5 β€œ1” β€œ0”

  • Hold failure happens when the SRAM fails to

retain the stored data.

  • Threshold variations in M1-4 affect the

position of seperatrix.

Monday, March 25, 2013

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SLIDE 7

Previous Work

7

π‘Š

π‘’β„Ž 𝑏

π‘Š

π‘’β„Ž 𝑐

Accept region Failure region

Parameter space

  • Search for points on the boundary
  • f failure region in the parameter

space*1.

  • Confined in 2-D space, i.e. only

two parameters considered.

*1 W. Dong and et.al. ICCAD, 2008 π‘Š

π‘’β„Ž 𝑏

π‘Š

π‘’β„Ž 𝑐

Accept region Failure region

Parameter space

  • Statistical method: sampling

within the region for searching*2.

Sampling points *2 D. E. Khalil and et.al. IEEE Tran. on VLSI, Dec 2008

Monday, March 25, 2013

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SLIDE 8

Motivation

8

vdd

Ξ”Id1 Ξ”Id2 Ξ”Id6 Ξ”Id3 Ξ”Id5 Ξ”Id4 Id3

  • Threshold

variation is modeled as ad-hoc current source at input trajectory separatrix

State-variable space 𝐽𝑒 + βˆ†π½π‘’ = 1 2 𝑙 𝑋 𝑀 π‘Š

𝑕𝑑 βˆ’ π‘Š π‘’β„Ž + βˆ†π‘Š π‘’β„Ž 2

βˆ†π½π‘’ β‰ˆ βˆ’π‘™ 𝑋 𝑀 (π‘Š

𝑕𝑑 βˆ’ π‘Š π‘’β„Ž)βˆ†π‘Š π‘’β„Ž

  • Fast

verification

  • f

SRAM nonlinear dynamics by reachability analysis

  • Variations

from multiple sources considered at the same time

Monday, March 25, 2013

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SLIDE 9

SRAM Nonlinear Dynamics

9

𝑦 = 𝑔 𝑨 𝑒 , π‘¨π‘ˆ 𝑒 = π‘¦π‘ˆ, π‘£π‘ˆ . 𝑦 = 𝑔 π‘¨βˆ— + πœ–π‘” πœ–π‘¨ |𝑨=π‘¨βˆ— 𝑨 βˆ’ π‘¨βˆ— + 1 2 (𝑨 βˆ’ π‘¨βˆ—)π‘ˆβˆ™ πœ–2𝑔 πœ–π‘¨2 |𝑨=𝜊 βˆ™ 𝑨 βˆ’ π‘¨βˆ— , 𝜊 ∈ π‘¨βˆ— + 𝛽 𝑨 βˆ’ π‘¨βˆ— 0 ≀ 𝛽 ≀ 1 .

π‘¨βˆ— 𝑨

State-variable space

Δ𝑨

  • One nominal point on the trajectory:
  • One operating point in the neighborhood of the nominal point:

1st order Taylor term 2nd order Residue Nominal point Neighbor point

mean value theorem trajectory

Monday, March 25, 2013

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SLIDE 10

SRAM Nonlinear Dynamics (cnt’d)

10

𝑦 = 𝑔 π‘¨βˆ— + πœ–π‘” πœ–π‘¨ |𝑨=π‘¨βˆ— 𝑨 βˆ’ π‘¨βˆ— + 1 2 (𝑨 βˆ’ π‘¨βˆ—)π‘ˆβˆ™ πœ–2𝑔 πœ–π‘¨2 |𝑨=𝜊 βˆ™ 𝑨 βˆ’ π‘¨βˆ— 𝑦 = 𝑔 π‘¦βˆ—, π‘£βˆ— + πœ–π‘” πœ–π‘¦ |𝑦=π‘¦βˆ— 𝑦 βˆ’ π‘¦βˆ— + πœ–π‘” πœ–π‘£ |𝑣=π‘£βˆ— 𝑣 βˆ’ π‘£βˆ— + 𝑀 𝑦 = 𝑔 π‘¦βˆ—, π‘£βˆ— + 𝐡 𝑦 βˆ’ π‘¦βˆ— + 𝐢 𝑣 βˆ’ π‘£βˆ— + 𝑀 Nonlinear dynamics of nominal point Linearization error L

π‘¦βˆ— = 𝑔 π‘¦βˆ—, π‘£βˆ— (𝑦 βˆ’ π‘¦βˆ—) = 𝐡 𝑦 βˆ’ π‘¦βˆ— + 𝐢 𝑣 βˆ’ π‘£βˆ— + 𝑀

Nonlinear dynamics for nominal point Linear dynamics for distance from nominal point

𝐡 = πœ–π‘” πœ–π‘¦ |𝑦=π‘¦βˆ— 𝐢 = πœ–π‘” πœ–π‘£ |𝑣=π‘£βˆ—

Monday, March 25, 2013

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SLIDE 11

Reachability Analysis: Zonotope

11

𝑨 = 𝑑 +

𝑗=1 𝑓

𝛾𝑗𝑕𝑗 , βˆ’1 ≀ 𝛾𝑗 ≀ 1

Zonoto tope pe Set of point nts in n-dimen ensi sion

  • nal

l polygo gon n with gener erato ator r gi

1

g

2

g

c

Initial Set Trajectory Approximated Reachable Set Final Set Unsafe region Unsafe Final Set center generator

Monday, March 25, 2013

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SLIDE 12

Reachability Analysis: Linear Multi-Step

12

𝐻 = 𝐡𝐻⨁𝐢𝑉⨁𝑀, 𝐻 = [𝑕1, 𝑕2, … , π‘•π‘œ]

Zonotope for noise current vector with given variation range Zonotope for state vector Calculation of generators Generator matrix

𝐻𝑙 = (𝐽 βˆ’ β„Žπ΅)βˆ’1π»π‘™βˆ’1β¨β„ŽπΆπ‘‰π‘™β¨β„Žπ‘€π‘™

Backward Euler with h time-step

Initial state solution Input solution Linearization error

𝑦 = π‘¦βˆ— +

𝑗=1 π‘œ

𝛾𝑗𝑕𝑗 , βˆ’1 ≀ 𝛾𝑗 ≀ 1 𝑣 = π‘£βˆ— +

𝑗=1 𝑛

π›½π‘—βˆ†π½π‘’

𝑗 , βˆ’1 ≀ 𝛽𝑗 ≀ 1

π‘¦βˆ— = 𝑔 π‘¦βˆ—, π‘£βˆ— (𝑦 βˆ’ π‘¦βˆ—) = 𝐡 𝑦 βˆ’ π‘¦βˆ— + 𝐢 𝑣 βˆ’ π‘£βˆ— + 𝑀 Transient simulation

Monday, March 25, 2013

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SLIDE 13

Linearization Error

13

Reachable set Adjacent trajectory Incorrect adjacent trajectory Nominal trajectory

Linearization Error 𝐾𝑙

𝐾𝑙 = 1 2 (𝑨 βˆ’ π‘¨βˆ—)π‘ˆβˆ™ πœ–2𝑔

𝑙

πœ–π‘¨2 |𝑨=𝜊 βˆ™ 𝑨 βˆ’ π‘¨βˆ—

𝑀𝑙 =

1 2 𝑛𝑏𝑦|𝑨 βˆ’ π‘¨βˆ—|π‘ˆ βˆ™ max πœ–2𝑔𝑙 πœ–π‘¨2 |𝑨=𝜊 βˆ™ 𝑛𝑏𝑦|𝑨 βˆ’ π‘¨βˆ—|

Linearization error

  • depicts

the nonlinear dynamics.

  • is

approximated at each iteration step. Approximation

Monday, March 25, 2013

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SLIDE 14

Reachable Set Refinement

14

  • Reachable

set is

  • ver-

expanded without refinement.

  • Zonotope is split into smaller
  • nes

confine linearization error in each set.

Over-expansion Adjacent trajectory Nominal trajectory Two nominal trajectories Split

if π‰πˆ β„Žπ‘€π‘™ βŠ† βˆ’πœ, 𝜁 , β‡’ π‘‘π‘žπ‘šπ‘—π‘’ 𝑑𝑓𝑒

  • A new trajectory is created.

Monday, March 25, 2013

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SLIDE 15

Experimental Results

15

  • Implemented in Matlab 7.12 and C
  • Platform

– Core i5 3.2GHz processor – 8GB memory

  • Simulation parameters of SRAM

– Initial state 𝑀1 ∈ 1.7,1.8 , 𝑀2 ∈ 0,0.1 – Variation range βˆ†π½π‘’ = πœ€π‘™

𝑋 𝑀

π‘Š

𝑕𝑑 βˆ’ π‘Š π‘’β„Ž π‘Š π‘’β„Ž, πœ€ = 1%, 5%, 10%

Monday, March 25, 2013

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SLIDE 16

Write Operation

16

  • Write pulse is set as 50ns with

relative threshold variation of 5% in each transistor.

  • Write operation fails.
  • Write pulse is set as 70ns.
  • Write operation succeeds.

Monday, March 25, 2013

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SLIDE 17

Write Operation

17

  • Write pulse is set as 60ns.
  • Write operation fails.

Trajectory splits near separatrix after write pulse stops. separatrix

Monday, March 25, 2013

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SLIDE 18

Read Operation

18

  • Relative threshold variation in

each transistor is set as 1% with 50ns read pulse.

  • Read operation succeeds.
  • Relative threshold variation in

each transistor is set as 5%.

  • Read operation fails.

Mismatch between MC & RA

Monday, March 25, 2013

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SLIDE 19

Hold Operation

19

  • Injected noise current lasts 12.5ns.
  • Relative threshold variation in each transistor is set as 5%

315uA noise current 300uA noise current

Monday, March 25, 2013

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SLIDE 20

Performance

Pulse(ns) Threshold Variation Reachability Analysis (s) Monte Carlo (s) Speedup 50 1% 12.71 5635.57 443.38X 5% 13.13 5817.71 443.01X 10% 12.63 6078.68 481.24X 60 1% 52.70 6224.09 118.09X 5% 52.58 6535.35 124.29X 10% 52.68 6387.28 121.25X 70 1% 13.72 5931.76 432.32X 5% 14.43 6245.45 432.73X 10% 13.21 6348.54 480.45X

20

Monte- Carlo setup with 500 samples are considered

Monday, March 25, 2013

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SLIDE 21

Summary

21

  • Introduced SRAM failure mechanisms in the state

space.

  • Presented

reachability analysis for nonlinear continuous systems.

  • Proposed reachability-based verification for SRAMs

with consideration of threshold voltage variations.

  • Reachability verification for SRAMs achieved good

speed and precision, and can be extended for

  • ptimization

Monday, March 25, 2013

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SLIDE 22

Thank you!

Please send comments to haoyu@ntu.edu.sg http://www.ntucmosetgp.net