Interconnect Design Sachin S. Sapatnekar University of Minnesota - PowerPoint PPT Presentation

Electromigration-aware Interconnect Design Sachin S. Sapatnekar University of Minnesota Acknowledgments Vivek Mishra (PhD 16), Palkesh Jain (PhD 17) Vidya Chhabria (PhD student) ISPD 2019 University of Minnesota

I P D S 2 0 0 2 ISPD 2019 2 University of Minnesota

Outline • Overview of electromigration • EM modeling • The weakest- link model (and why it’s problematic) ISPD 2019 3 University of Minnesota

Interconnect aging • Electromigration (EM) Metal 2 [Li, IRPS ’09] via Metal 1 void Cross-section TEM image − + ISPD 2019 4 University of Minnesota

Traditional view of EM +FinFETs,GAAFETs + I/O drivers [Jain, TVLSI June 16] ISPD 2019 5 University of Minnesota

Self heating • Joule heating in wires • Multigate FETs make things worse – Larger degrees of self-heating, worse paths to the ambient Bulk FinFET SOI FinFET GAAFET [Chhabria, ISQED 19] ISPD 2019 6 University of Minnesota

Which interconnects? • Power grids Cu e - – Largely unidirectional current Vac e - • Signal interconnects – Bidirectional current flow – Recovery effects seen Power Network Signals Cell Y A DC Cell AC Cell-Internal ISPD 2019 7 University of Minnesota

Outline • Overview of electromigration • EM modeling • The weakest- link model (and why it’s problematic) ISPD 2019 9 University of Minnesota

Black’s law • Black’s law ≈ – Predicts mean time to failure Lognormal • TTF follows a lognormal distribution – For a fail fraction FF , defects in parts per million (DPPM) – Constraint on t z → Constraint on t 50 → Constraint on j AVG – Joule heating → Constraint on j RMS • Circuit-level EM constraint: – For each wire, stay within j RMS , max , j AVG,max ISPD 2019 10 University of Minnesota

Physics of mortality and the Blech criterion Blech criterion Tensile stress Compressive stress at cathode ( σ) at anode ( – σ) At steady state, F electron wind = F back-stress If: At steady state, σ < σ critcal then: wire is immortal! (voids never form) σ < σ critcal ⟹ 𝑲 × 𝑴 < 𝑳 𝟐 (Blech criterion) Atomic diffusion creates stress gradient that causes F back-stress σ critcal : Critical stress needed for void formation ISPD 2019 11 University of Minnesota

Physics-based EM analysis • Korhonen model – Void nucleation 𝜖𝜏 𝜖𝑢 = 𝜖 𝜖𝑦 𝜆 F back−stress + F electron wind Stress at a blocking boundary (cathode) Stress evolution along the wire [Korhonen, JAP 1993] ISPD 2019 12 University of Minnesota

EM mortality: Issues with classical approach Blech criterion Black’s equation For potential if: J × L < K 1 mortal wires : Wire immortal to EM else: wire is TTF = K 2 J n 𝐟𝐲𝐪 K 3 potentially mortal T J : Current density Steady state L : Wire length Empirical model, approach for issues for Cu K 1 : Constant mortality [Lloyd, MER ’07] ISPD 2019 13 University of Minnesota

EM mortality: Classical vs. filtering approach Blech criterion Black’s equation Filtering approach For potential if: J × L < K 1 mortal wires : Wire immortal to EM Transient else: wire is TTF = K 2 J n 𝐟𝐲𝐪 K 3 state approach potentially mortal T for mortality Steady state Physics-based, Empirical model, approach for applicable for Cu issues for Cu mortality [Lloyd, MR ’07] ISPD 2019 14 University of Minnesota

EM mortality: Mechanical stress evolution Potentially mortal by Blech criterion σ steady state Compressive stress Tensile stress at anode ( – σ ) Stress ( σ ) at cathode (MPa) at cathode ( σ ) σ critical 𝝉( t lifetime ) 𝝉 < 𝝉 critical throughout the lifetime. EM-safe! Cu atoms t lifetime Time (years) Atomic diffusion creates stress gradient that causes F back-stress 1. Practical EM mortality: relative Blech criterion presumes steady state to the product lifetime between F electron wind and F back-stress 2. Transient stress evolution instead of steady state ISPD 2019 15 University of Minnesota

EM mortality: Modeling transient stress Wire length, L EM equation 𝝐𝝉 𝝐𝒖 = 𝝐 𝝐𝒚 𝝀 F back−stress + F electron wind 𝝐𝝉 Blech criterion assumes • 𝝐𝒖 = 𝟏 Stress at cathode, 𝝉 ( t ), 2 options: 1. Semi-infinite (SI) : 𝝉 ( t ) = 𝜷 𝟐 𝑲 𝒖 Efficient, but overestimates stress 2. Finite ( F) : ∞ 𝟑 𝒖 𝜷 𝟒 −𝒏 n 𝟐 𝒇 𝑴 𝟑 L =75 µ m 𝝉 ( t ) = 𝑲 𝑴 𝜷 𝟑 𝟑 − ෍ 𝒏 𝒐 𝟑 𝒐=𝟏 Inefficient, but accurate prediction Extension to interconnect trees using [Park, IRPS10] ISPD 2019 16 University of Minnesota

Sequential mortal wire filtration ISPD 2019 17 University of Minnesota

Sequential mortal wire filtration M1 ⊃ M2 ⊃ M3 𝝉 ( t ) = 𝜷 𝟐 𝑲 𝒖 𝝉 ( t ) ∞ 𝟑 𝒖 𝜷 𝟒 −𝒏 n 𝟐 𝒇 𝑴 𝟑 = 𝑲 𝑴 𝜷 𝟑 𝟑 − ෍ 𝒏 𝒐 𝟑 𝒐=𝟏 ISPD 2019 18 University of Minnesota

IBMPG case study: PG2 mortal wire distribution Potential Mortal wires from the Blech criterion 1.2 1 Current density (MA/cm 2 ) 0.8 0.6 Potential 0.4 Mortal wires mortal wires 0.2 Blech criterion 0 20 40 60 80 Length ( µ m) ISPD 2019 19 University of Minnesota

IBMPG case study: PG2 mortal wire distribution Immortal wires filtered out using pessimistic Filter 2 (SI) 1.2 𝑛𝑏𝑦 𝑲 𝑻𝑱 1 Current density (MA/cm 2 ) 0.8 Product lifetime = 10 years 0.6 Temperature ( T ) = 105C Potential mortal wires Mortal wires 0.4 Filter 2 (SI) 0.2 Blech criterion 0 20 40 60 80 Length ( µ m) ISPD 2019 20 University of Minnesota

IBMPG case study: PG2 mortal wire distribution Immortal wires filtered out using pessimistic Filter 2 (SI) & accurate Filter 3 (F) 1.2 𝑛𝑏𝑦 𝑲 𝑻𝑱 1 Current density (MA/cm 2 ) 0.8 Product lifetime = 10 years 0.6 Temperature ( T ) = 105C Actual Mortal wires mortal wires 0.4 Filter 2 (SI) 0.2 Filter 3 (F) Blech criterion 0 20 40 60 80 Length ( µ m) ISPD 2019 21 University of Minnesota

What about lines with branches? Vias? Flux Divergence • – Current flow in neighboring wire affects EM flux – Use effective current for EM X Y 2J J Ta barrier J EM (Y) = 2J + J • The above is approximate – There’s a physics -based version for this too [Park, IRPS10] ISPD 2019 22 University of Minnesota

Outline • Overview of electromigration • EM modeling • The weakest- link model (and why it’s problematic) ISPD 2019 23 University of Minnesota

Circuit impact • Conventional way to overcome EM – Constraint on t z → Constraint on t 50 → Constraint on j AVG – Joule heating → Constraint on j RMS • Circuit-level EM constraint: – For each wire, stay within j RMS , max , j AVG,max • Weakest-link model ISPD 2019 24 University of Minnesota

Handling catastrophic errors • A simple analysis of an n-component system F i = probability of failure of the i th component – 1 – F i = probability that the i th component works – – n = number of components in the system (1 – F i ) n = probability that all n components work – – Probability of system failure = 1 – (1 – F i ) n • Implicit assumptions – All failures are catastrophic – All failures are equally serious – All failures are independent ISPD 2019 25 University of Minnesota

Interconnect redundancy • Several on-chip interconnect systems are built to be redundant Power grids Clock grids • A system fails when it’s key parameters fail – and NOT at first failure! ISPD 2019 University of Minnesota

Electromigration in power grids • Power grids are built to contain redundancies! 0 0 1 2 -3 2 ∆ R / R = 50% -13 -8 6 Worst ∆ V (mV) A better failure criterion: [Mishra, DAC13] ISPD 2019 University of Minnesota

Analyzing redundancy • Two component system: one of the two fails first [Jain, IRPS15] ISPD 2019 University of Minnesota

Analyzing redundancy • Two-component system: one of the two fails first • Post-failure: current goes through intact component [Jain, IRPS15] ISPD 2019 University of Minnesota

Reliability under changing stress Fail Fraction Two parallel leads – 𝐺 1 (𝑢) A single lead – 𝐺 2 (𝑢) Shifted CDF : 𝐺 2 (𝑢 − 𝜀 1 ) System CDF TTF 0 [Jain, IRPS15] ISPD 2019 University of Minnesota

Reliability under changing stress Fail Fraction CDF : 𝐺 1 (𝑢) Unshifted CDF : 𝐺 2 (𝑢) Shifted CDF : 𝐺 2 (𝑢 − 𝜀 1 ) System CDF TTF 0 [Jain, IRPS15] ISPD 2019 University of Minnesota

Reliability under changing stress Fail Fraction CDF : 𝐺 1 (𝑢) Unshifted CDF : 𝐺 2 (𝑢) Shifted CDF : 𝐺 2 (𝑢 − 𝜀 1 ) System CDF TTF 0 [Jain, IRPS15] ISPD 2019 University of Minnesota

System impact for a clock grid Vdd Y Circuit Delay A Vss time [Jain, IRPS15] ISPD 2019 University of Minnesota

System impact for a clock grid Vdd R1 Y Circuit Delay A R1 fails Vss time [Jain, IRPS15] ISPD 2019 University of Minnesota

System impact for a clock grid Vdd R2 R1 Y Circuit Delay A R1 fails R2 fails Vss time [Jain, IRPS15] ISPD 2019 University of Minnesota

System impact for a clock grid Vdd R2 R1 Y Circuit Delay A R1 fails R2 fails R3 fails R3 Vss time [Jain, IRPS15] ISPD 2019 University of Minnesota