An Analytical Model for Negative Bias Temperature Instability NBTI

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An Analytical Model for Negative Bias
Temperature Instability (NBTI)

• Sanjay Kumar, Chris Kim, Sachin Sapatnekar
• University of Minnesota
• ICCAD 2006

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Outline

• NBTI Overview
• Reaction-Diffusion (R-D) Model
• Our Analytical NBTI Model
• Frequency Independence
• Delay Estimation using NBTI Model

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An Overview of NBTI
Negative Bias Temperature Instability
Stress
Stress
Relaxation
VG Vdd
VG 0
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NBTI Effect

• 25-30 degradation in PMOS Vth
• Effect increases with technology scaling
• Around 10 delay degradation
• Effect worsens if thermal nitrides used instead
  of plasma nitrides in gate-oxide
• Up to 25 delay degradation reported

Vth (V)
103
105
109
107
0
10
time (s)
PMOS Vth versus time for a 65nm PMOS transistor
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Outline

• NBTI Overview
• Reaction-Diffusion (R-D) Model
• Our Analytical NBTI Model
• Frequency Independence
• Delay Estimation using NBTI Model

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Reaction Diffusion (R-D) Model
H
Diffusion of H2 into oxide
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NBTI Modeling R-D model

• Reaction-Diffusion (R-D) model to determine the
  number of interface traps. Alam-IEDM03

Reaction Phase
Diffusion Phase
Rate of diffusion of hydrogen

• R-D model solved to obtain analytical equations
  for a stress phase followed by a relaxation phase
• Numerical solution thenceforth

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Outline

• NBTI Overview
• Reaction-Diffusion (R-D) Model
• Our Analytical NBTI Model
• Frequency Independence
• Delay Estimation using NBTI Model

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Approach

• Use R-D model
• Mechanism is diffusion limited
• Track the profile of H2 diffusion
• Model shown for the special case of square
  waveforms
• Equal periods of stress and recovery

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First Stress Phase
t0
2t0
3t0
4t0
0
NH2 is a linear function in x
NIT Number of H atoms ½ Number of H2
molecules ½ Area of the triangle
NIT(t) found by solving the diffusion equation
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First Relaxation Phase
t0
2t0
3t0
4t0
0
Annealing of traps due to re-formation of bonds
Si-H bond re-formation highest close to the
interface
Hydrogen continues to diffuse into the oxide
NIT Number of traps at time t0 Number of
traps annealed
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Second Stress Phase
t0
2t0
3t0
4t0
0
Existing front diffuses beyond x(2t0) New front
begins at x0 for time gt 2t0 Combine into single
effective front
NH2(t)
Boundary Conditions Equate area at time 2t0 and
solve for xeff(2t0)
Diffusion continues beyond xeff(2t0) for time gt
2t0
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Comparison with Experimental Data
Comparison of our model with experimental data
from Chakravarthi-IRPS04.
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Threshold Voltage Degradation
Vth degradation larger for static NBTI stress
(DC) as compared with dynamic NBTI (AC)
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sk Notation
Stress
Stress
Relax
Relax
Can obtain closed form expression using sk
notation
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sk Notation
For DC, sk is simply k1/6
For AC, sk is given by
sk values computable for any arbitrary waveform
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Outline

• NBTI Overview
• Reaction-Diffusion (R-D) Model
• Our Analytical NBTI Model
• Frequency Independence
• Delay Estimation using NBTI Model

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Frequency Independence
freq f1
T1
n1 cycles
freq f2
n2 cycles
T2
Number of interface traps for both cases
same Trap generation independent of frequency
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Frequency Independence Plots
DC
DC
freq f
AC freq f
?Vth (mV)
time (s)
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Frequency Independence Plots
DC
DC
freq f
freq 0.1f
freq f
? Vth (mV)
freq 0.1f
time (s)
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Frequency Independence Plots
DC
freq f
freq 0.1f
freq 0.01f
? Vth (mV)
time (s)
Vth degradation same for all three cases
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Outline

• NBTI Overview
• Reaction-Diffusion (R-D) Model
• Our Analytical NBTI Model
• Frequency Independence
• Delay Estimation using NBTI Model

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Issues

• Estimate the delay degradation after a time
  period equal to 10 years of operation, i.e.,
  (3X108 s)
• f1GHz implies 1017 cycles
• Need fast-forwarding
• NBTI effect is temporal
• Requires exact nature of stress and relaxation to
  determine NIT
• Impossible to determine temporal input activity
• Need to use statistical inputs

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Signal Probability and Activity Factor

• Signal Probability (SP)
• Probability that the signal is high (or low)
• Activity Factor (AF)
• Probability that the signal switches

Clock
Signal
AF 0.6
SP 0.4
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NBTI Activity Factor (AF) Independence
DC
?Vth (mV)
f
0.1f
0.01f
1Hz
time (s)

• Three square waveforms with same signal
  probability (SP) of 0.5
• 1X, 0.1X and 0.01X activity factor (AF) values
• Same amount of Vth degradation
• Trap generation is AF independent

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NBTI Signal Probability (SP) Dependence
? Vth (mV)
time (s)

• Four waveforms with same frequency
• SP values are 0.25, 0.5, 0.75, 1.00
• ?Vth values differ significantly
• NBTI effect is SP dependent

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SPAF Method

• Converting a random waveform to an equivalent
  deterministic periodic waveform
• Dont care about AFs
• Maintain same SP

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Validity of SPAF Method
Generate a random waveform for 10000
cycles Estimate number of traps
Determine SP for each sample
Build periodic waveforms with same SP
value Estimate number of traps
Compare sk values
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Circuit Delay Estimation

• Simulations on ISCAS85 benchmarks 65nm PTM
  technology
• Clock frequency 1GHz

• Estimate Vth of each transistor after 10 years
  using a Vth SP look-up table
• Calculate new arrival times

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Results
9 degradation in delay of circuits after 10
years of operation
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Conclusion

• NBTI growing threat to reliability
• Need accurate estimation of its effect
• NBTI Modeling
• Analytical model for NBTI presented
• Circuit delay characterized due to temporal NBTI
  stress and relaxation
• 9 increase in delay estimated
• Model can be used for NBTI-aware design