Analytical Modeling of SRAM Dynamic Stability Bin Zhang, Ari Arapostathis, Sani Nassif and Michael O

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Analytical Modeling of SRAM Dynamic
StabilityBin Zhang, Ari Arapostathis, Sani
Nassif and Michael Orshansky University of
Texas at Austin IBM Austin Research Lab
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Outline

• Motivation
• System modeling setup
• Dynamic state space analysis
• Criterion of dynamic stability of SRAM
• Transient Behavior of SRAM under Noise
• Dynamic noise margin of SRAM
• Experimental Results

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Motivation

• Verification of SRAM stability is an essential
  design task
• Smaller design windows due to power supply
  scaling
• Identify the maximum possible unintended
  violations
• Static vs Dynamic
• Growing number and amplitude of dynamic noise
  sources
• Single Event Upset (SEU)
• Cross-coupling Noise
• Power supply noise
• Use non-linear system theory to analyze the
  stability of cross-coupled inverter system of an
  SRAM in the presence of transient noise

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Prior Work

• Static Noise Margin (SNM) analysis posits a
  constant noise signal at circuit nodes Seevinck
  et al 1987 Agarwal et al 2006
• Mirror-and-Maximum-Square Approach
• Neglects time-dependent aspect of noise, e.g.
  pulse duration. Gives pessimistic noise tolerance
  for short noise pulse

Static Noise Margin
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Prior Work

• Simulation-based techniques can capture the
  transient behaviors of SRAM and noise, but
  provide little insight into problem and requires
  iterative search to find critical noise pulses
• Device level Fu et al 1984
• Transistor level Hazucha et al 2000
• Mixed mode Mayaram et al 1993
• Breaking the feedback loop analysis ignores the
  strong feedback effect in SRAM Shepard 98
• No analytical approach exists for dynamic noise
  margin of SRAM

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Inverter-Level SRAM Cell Modeling
Standby mode
in(t), Noise Current
in(t)
In
V1(t)
V2(t)
pw
0
t
C
C
V2(t0)0
State Vector V(t)(V1(t), V2(t))
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State Space Trajectory

• Case 1 Short noise pulse does not result in a
  state flip

Noise Duration
Recovery
Noise Duration
Recovery
Noise Duration
Recovery
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State Space Trajectory
Case 2 Long pulse results in state flip
Noise Duration
Failed Recovery
Failed Recovery
Noise Duration
Noise Duration
FailedRecovery
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SRAM Cell Coupled Nonlinear System with Feedback
Solvable?
driving current of the inverter
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Linear Gate I-V Modeling

• To enable analytical solutions, need to use a
  linear and separable inverter I-V model
• Ignore short-circuit current
• Piece-wise linear model (Horowitz84)
• Assumptions justified by accurate waveforms
  (transient behavior!) predicted for logic gates

(Cutoff) (Saturation) (Linear)
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Dynamic State Space Analysis

• Equilibrium states

Time-dependent

Equilibrium states evolve with time
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Dynamic State Space Analysis
1.6

V2 (V)
0.8
0.0
0.0
0.8
1.6
V1 (V)
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Dynamic State Space Analysis
1.6

V2 (V)
0.8
0.0
0.0
0.8
1.6
V1 (V)
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Dynamic State Space AnalysisRecovery Stage

• Equilibrium states have regions of attraction
• Any vector in attraction regions is pulled to the
  equilibrium state
• If noise pushes state vector beyond the boundary
  flip occurs!
• Boundary of attraction regions for symmetric
  system

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SRAM State Evolution under Noise

• The operation of SRAM can be approximately
  classified into two modes before the attraction
  boundary is reached
• Weak coupling V2ltVdsat
• Strong feedback V2gtVdsat

Strong Feedback
V1
Weak Coupling
V2
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SRAM Modeling Weak Coupling Mode
Linearized equations
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SRAM Modeling Strong Feedback Mode
Linearized cross-coupled equations
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Solving Transient SRAM Equations Critical Pulse
Width

• Critical pulse width, Tcrit, the time when the
  state trajectory crosses the attraction boundary.
  
• Tcrit can be solved from the linearized
  equations

Asymptotic behavior, for large noise amplitude,
a simple bound
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Dynamic Noise Margins Experimental Results

• Amplitude (In) vs pulse width (Tcrit) of the
  critical pulse
• Safe region is below the curves
• Analytical result gives larger and more realistic
  DNM than the bounds

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Critical Pulse Width Dependence on Supply Voltage

• Model allows stability-driven cell design
  exploration
• Dynamic noise margins increase for higher Vdd
• For small noise magnitudes, Tcrit is very
  sensitive to Vdd

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Critical Pulse Width Dependence on Cell Area
(Size)

• DNM can be increased by sizing up devices
• But transistor sizing becomes less effective for
  large noise amplitude

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Conclusions

• We present an analytical modeling of SRAM dynamic
  stability
• Model the transient current noise as a
  square-pulse
• Apply linear gate modeling
• Obtain a closed-form solution of critical pulse
  in terms of the noise amplitude.
• Excellent match is achieved compared to SPICE