Reachability achability Analysis alysis with th Consid - PowerPoint PPT Presentation

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 Manoj 2 and Guoyong Shi 1 1 School of Microelectronics, Shanghai Jiao Tong University 2 School of Electrical and Electronic Engineering, Nanyang Technological University Monday, March 25, 2013 1

Outline • SRAM failure analysis • SRAM nonlinear dynamics • Verification by reachability analysis • Experimental results • Summary Monday, March 25, 2013 2

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* V 2 (V dd , V dd ) State equilibrium Instate equilibrium State-variable space * W. Dong and et.al. ICCAD , 2008 (0,0) V 1 State equilibrium Monday, March 25, 2013 3

Write Failure Analysis Separatrix • Initial state (v1,v2) = (vdd, 0) Vdd • Target state (v1, v2) = (0, vdd) B V2 A WL Write vdd Failure M1 M2 0 Vdd V1 BR BL v1 “ 0 ” “ 0 ” v2 M5 “ 1 ” M6 Threshold voltage variation • 𝑊 M3 M4 𝑢ℎ6 causes difficulty to move 𝑊 𝑢ℎ4 state point to the target state. discharging charging Monday, March 25, 2013 4

Read Failure Analysis Vdd • Internal state is aimed to maintain regardless Read B perturbation during read Failure operation. V2 A WL vdd 0 Vdd M1 M2 V1 BR BL v1 • Mismatch between M4 “ 1 ” “ 0 ” v2 M5 “ 1 ” and M6 M6 M3 M4 • Mismatch among M1-4 charging Monday, March 25, 2013 5

Hold Failure Analysis Hold failure happens when the SRAM fails to • retain the stored data. WL Vdd vdd Hold M1 M2 V2 Failure BR BL v1 “ 0 ” v2 M5 “ 1 ” M6 M3 M4 Perturbation current 0 Vdd V1 Threshold variations in M1-4 affect the • position of seperatrix. Monday, March 25, 2013 6

Previous Work Parameter space • Statistical method: sampling Accept within the region for region searching* 2 . 𝑐 𝑊 Sampling 𝑢ℎ Failure points Parameter space region Accept 𝑏 0 𝑊 region 𝑢ℎ • Search for points on the boundary 𝑐 𝑊 of failure region in the parameter 𝑢ℎ space* 1 . Failure • Confined in 2-D space, i.e. only region two parameters considered. 𝑏 0 𝑊 * 1 W. Dong and et.al. ICCAD , 2008 𝑢ℎ * 2 D. E. Khalil and et.al. IEEE Tran. on VLSI, Dec 2008 Monday, March 25, 2013 7

Motivation State-variable space • Threshold variation is separatrix modeled as ad-hoc current source at input 𝐽 𝑒 + ∆𝐽 𝑒 = 1 2 𝑙 𝑋 2 𝑀 𝑊 𝑕𝑡 − 𝑊 𝑢ℎ + ∆𝑊 𝑢ℎ ∆𝐽 𝑒 ≈ −𝑙 𝑋 𝑀 (𝑊 𝑕𝑡 − 𝑊 𝑢ℎ )∆𝑊 𝑢ℎ trajectory v dd • Fast verification of SRAM Δ I d1 Δ I d2 nonlinear dynamics by reachability analysis • Variations from multiple Δ I d5 Δ I d6 Δ I d3 sources considered at the Δ I d4 I d3 same time Monday, March 25, 2013 8

SRAM Nonlinear Dynamics • One nominal point on the trajectory: , 𝑨 𝑈 𝑢 = 𝑦 𝑈 , 𝑣 𝑈 . 𝑦 = 𝑔 𝑨 𝑢 • One operating point in the neighborhood of the nominal point: 2 (𝑨 − 𝑨 ∗ ) 𝑈 ∙ 𝜖 2 𝑔 𝑦 = 𝑔 𝑨 ∗ + 𝜖𝑔 𝜖𝑨 | 𝑨=𝑨 ∗ 𝑨 − 𝑨 ∗ + 1 𝜖𝑨 2 | 𝑨=𝜊 ∙ 𝑨 − 𝑨 ∗ , 1 st order 2 nd order Taylor term Residue mean value 𝜊 ∈ 𝑨 ∗ + 𝛽 𝑨 − 𝑨 ∗ 0 ≤ 𝛽 ≤ 1 . theorem Neighbor point 𝑨 trajectory Δ𝑨 𝑨 ∗ Nominal point State-variable space Monday, March 25, 2013 9

SRAM Nonlinear Dynamics ( cnt’d ) 2 (𝑨 − 𝑨 ∗ ) 𝑈 ∙ 𝜖 2 𝑔 𝑦 = 𝑔 𝑨 ∗ + 𝜖𝑔 𝜖𝑨 | 𝑨=𝑨 ∗ 𝑨 − 𝑨 ∗ + 1 𝜖𝑨 2 | 𝑨=𝜊 ∙ 𝑨 − 𝑨 ∗ Linearization error L 𝑦 = 𝑔 𝑦 ∗ , 𝑣 ∗ + 𝜖𝑔 𝜖𝑦 | 𝑦=𝑦 ∗ 𝑦 − 𝑦 ∗ + 𝜖𝑔 𝜖𝑣 | 𝑣=𝑣 ∗ 𝑣 − 𝑣 ∗ + 𝑀 𝐵 = 𝜖𝑔 𝜖𝑦 | 𝑦=𝑦 ∗ 𝑦 = 𝑔 𝑦 ∗ , 𝑣 ∗ + 𝐵 𝑦 − 𝑦 ∗ + 𝐶 𝑣 − 𝑣 ∗ + 𝑀 𝐶 = 𝜖𝑔 𝜖𝑣 | 𝑣=𝑣 ∗ Nonlinear dynamics of nominal point Nonlinear dynamics for nominal point 𝑦 ∗ = 𝑔 𝑦 ∗ , 𝑣 ∗ (𝑦 − 𝑦 ∗ ) = 𝐵 𝑦 − 𝑦 ∗ + 𝐶 𝑣 − 𝑣 ∗ + 𝑀 Linear dynamics for distance from nominal point Monday, March 25, 2013 10

Reachability Analysis: Zonotope Unsafe Final Set Initial Set Trajectory Unsafe region Approximated Final Set Reachable Set Zonoto tope pe Set of point nts in n-dimen ensi sion onal l center c polygo gon n with gener erato ator r g i g g 2 1 𝑓 generator 𝑨 = 𝑑 + 𝛾 𝑗 𝑕 𝑗 , −1 ≤ 𝛾 𝑗 ≤ 1 𝑗=1 Monday, March 25, 2013 11

Reachability Analysis: Linear Multi-Step 𝑦 ∗ = 𝑔 𝑦 ∗ , 𝑣 ∗ Transient simulation 𝑜 (𝑦 − 𝑦 ∗ ) = 𝐵 𝑦 − 𝑦 ∗ + 𝐶 𝑣 − 𝑣 ∗ + 𝑀 𝑦 = 𝑦 ∗ + 𝛾 𝑗 𝑕 𝑗 , −1 ≤ 𝛾 𝑗 ≤ 1 𝑗=1 Calculation of generators Zonotope for state vector 𝐻 = 𝐵𝐻⨁𝐶𝑉⨁𝑀, 𝑛 𝑣 = 𝑣 ∗ + 𝑗 , −1 ≤ 𝛽 𝑗 ≤ 1 𝐻 = [𝑕 1 , 𝑕 2 , … , 𝑕 𝑜 ] 𝛽 𝑗 ∆𝐽 𝑒 𝑗=1 Generator matrix Zonotope for noise current vector with given variation range Backward Euler with h time-step 𝐻 𝑙 = (𝐽 − ℎ𝐵) −1 𝐻 𝑙−1 ⨁ℎ𝐶𝑉 𝑙 ⨁ℎ𝑀 𝑙 Initial state solution Input Linearization solution error Monday, March 25, 2013 12

Linearization Error 2 (𝑨 − 𝑨 ∗ ) 𝑈 ∙ 𝜖 2 𝑔 𝐾 𝑙 = 1 𝑙 𝜖𝑨 2 | 𝑨=𝜊 ∙ 𝑨 − 𝑨 ∗ 𝜖 2 𝑔 𝑙 2 𝑛𝑏𝑦|𝑨 − 𝑨 ∗ | 𝑈 ∙ max 1 𝜖𝑨 2 | 𝑨=𝜊 ∙ 𝑛𝑏𝑦|𝑨 − 𝑨 ∗ | 𝑀 𝑙 = Approximation Adjacent trajectory Linearization Reachable set Error 𝐾 𝑙 Linearization error • depicts the nonlinear Incorrect dynamics. adjacent • is approximated at each trajectory iteration step. Nominal trajectory Monday, March 25, 2013 13

Reachable Set Refinement • Reachable set is over- expanded without refinement. Two nominal trajectories • Zonotope is split into smaller ones confine linearization error in each set. Split Adjacent trajectory Over-expansion if 𝐉𝐈 ℎ𝑀 𝑙 ⊆ −𝜁, 𝜁 , Nominal ⇒ 𝑡𝑞𝑚𝑗𝑢 𝑡𝑓𝑢 trajectory • A new trajectory is created. Monday, March 25, 2013 14

Experimental Results • Implemented in Matlab 7.12 and C • Platform – Core i5 3.2GHz processor – 8GB memory • Simulation parameters of SRAM – Initial state 𝑤 1 ∈ 1.7,1.8 , 𝑤 2 ∈ 0,0.1 𝑋 – Variation range ∆𝐽 𝑒 = 𝜀𝑙 𝑊 𝑕𝑡 − 𝑊 𝑢ℎ 𝑊 𝑢ℎ , 𝜀 = 1%, 5%, 10% 𝑀 Monday, March 25, 2013 15

Write Operation • Write pulse is set as 70ns. • Write operation succeeds. • Write pulse is set as 50ns with relative threshold variation of 5% in each transistor. • Write operation fails. Monday, March 25, 2013 16

Write Operation • Write pulse is set as 60ns. Trajectory splits near • Write operation fails. separatrix after write pulse stops. separatrix Monday, March 25, 2013 17

Read Operation • Relative threshold variation in each transistor is set as 5%. • Read operation fails. • Relative threshold variation in each transistor is set as 1% with 50ns read pulse. Mismatch between • Read operation succeeds. MC & RA Monday, March 25, 2013 18

Hold Operation • 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 19

Performance Pulse(ns) Threshold Reachability Monte Speedup Variation Analysis (s) Carlo (s) 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 Monte- Carlo setup with 500 samples are considered Monday, March 25, 2013 20

Summary • 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 optimization Monday, March 25, 2013 21

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