An Analytical Model for Negative Bias Temperature...
Transcript of An Analytical Model for Negative Bias Temperature...
An Analytical Model for Negative Bias Temperature
Instability (NBTI)Sanjay Kumar, Chris Kim, Sachin Sapatnekar
University of MinnesotaICCAD 2006
22
Outline
NBTI OverviewReaction-Diffusion (R-D) ModelOur Analytical NBTI ModelFrequency IndependenceDelay Estimation using NBTI Model
33
An Overview of NBTIVdd
G
S
0
-Vdd
Vdd
0
VG = 0 VG = Vdd
D S
VB
VS = VddVG = 0
VD
B
G
oxide
Negative Bias Temperature Instability
Stress StressRelaxation
SiH + h+ → Si+ + H
Si HSi HSi H
H2
Substrate PolyGate Oxide
H + H → H2
44
NBTI Effect
25-30% degradation in PMOS Vth
Effect increases with technology scaling
Around 10% delay degradationEffect worsens if thermal nitrides used instead of plasma nitrides in gate-oxide
Up to 25% delay degradation reported
0.20
0.22
0.24
0.26
0.28
0.30
time (s)
PMOS Vth versus time for a 65nm PMOS transistor
10 103 109105 1070
V th
(V)
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Outline
NBTI OverviewReaction-Diffusion (R-D) ModelOur Analytical NBTI ModelFrequency IndependenceDelay Estimation using NBTI Model
66
Reaction Diffusion (R-D) Model
Si
Si
Si H
SiSi
SiSi H
Reverse Reaction Rate = kr
H
Forward Reaction Rate = kf
SiSi
SiSi H
HH
Initial Si-H concentration = N0
Si HSi H
Si
Substrate PolyOxide
Si H
H
H
H
Diffusion of H2 into oxide
H
H H2SiH + h+ Si+ + Hkf
kr
77
dx
dND
dtdN HIT 2= 2
2 22
dx
dND
dtdN HH =
R-D model solved to obtain analytical equations for a stress phase followed by a relaxation phase
Numerical solution thenceforth
NBTI Modeling: R-D modelReaction-Diffusion (R-D) model to determine the number of interface traps. [Alam-IEDM’03]
Diffusion Phase
[ ] HrITfIT NNkNNk
dtdN
00 −−=
Reaction Phase Rate of diffusion of hydrogen
SiH + h+ Si+ + Hkf
kr
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Outline
NBTI OverviewReaction-Diffusion (R-D) ModelOur Analytical NBTI ModelFrequency IndependenceDelay Estimation using NBTI Model
99
Approach
Use R-D modelMechanism is diffusion limitedTrack the profile of H2 diffusion
Model shown for the special case of square waveforms
Equal periods of stress and recovery
Si HSi HSi H
H2
Substrate PolyOxide
Vdd
First Stress Phase
First Relax Phase
Second Stress Phase
Second Relax Phase
t0 2t0 3t0 4t00
x(t)
NH2(t) xd(t)
1010
First Stress Phase
xd(t1)
NH2(t) N0H2
x(t)
xd(t2)NH2(t) N0
H2
x(t)
t
NIT(t)
NIT →t1/6
NIT = Number of H atoms
= ½ Number of H2 molecules
= ½ Area of the triangle
NIT(t) found by solving the diffusion equation
kxN(x)N HH −= 022
NH2 is a linear function in x
Dt)t(xd 2=
61
)( CttNIT =
)t(xN)t(N dHIT0
2∝
t0 2t0 3t0 4t00
1111
First Relaxation PhaseAnnealing of traps due to
re-formation of bonds
Hydrogen continues to diffuse into the oxide
Si-H bond re-formation highest close to the interface
NIT = Number of traps at time t0– Number of traps annealed
xd(t0)
NH2(t)N0
H2
x(t)
xd(t+t0)
NH2(t) N∆H2
x(t))(2)( 00 ttDttxd +=+
),(1)()(0
00 ttf
tNttN ITIT +
=+
NIT(t)
Stress Relax
t
xd(2t0)NH2(t)
N∆H2
x(t)
t0 2t0 3t0 4t00
200 )2(2)2( txDtttx effd +=+
Second Stress PhaseExisting front diffuses beyond x(2t0)
New front begins at x=0 for time > 2t0
Combine into single “effective” frontNH2(t) xd(2t0)N∆H2
x(t)
xeff(2t0)
NH2(t) N0H2
x(t)
xd(t+2t0)NH2(t) N0
H2
x(t)
Boundary Conditions:
Equate area at time 2t0 and solve for xeff(2t0)
Diffusion continues beyond xeff(2t0) for time > 2t0
61
6
00 3
2)2(⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛+=+
ttCttNIT
t2t0
NIT(t)
t0
t0 2t0 3t0 4t00
1313
Comparison with Experimental Data
time (s)
NIT
DC
AC
Comparison of our model with experimental data from Chakravarthi-IRPS’04.
Vdd
DC Stress
0
Vdd
AC Stress
0
Vdd
1414
Threshold Voltage Degradation
0
2
4
6
8
10
12
14
0 1000 2000 3000 4000
time (s)
∆Vth
(mV)
DC
AC
Vth degradation larger for static NBTI stress (DC) as compared with dynamic NBTI (AC)
ITth NV ∝∆
1515
0.0
0.2
0.4
0.6
0.8
1.0
1.2
“sk” Notation
t0 2t0 3t0 4t00
)t(V)kt(Vs
th
thk
0
0
∆∆
=
Can obtain closed form expression using sk notation
Stress StressRelax Relax
s1 = 1
s2 = 0.66s4 = 0.89
s3 = 1.02
s0 = 0
)t(V)t(V
th
th
0∆∆
1616
“sk” Notation
0
1
2
3
4
5
6
AC
DC
sk
k (no. of half cycles)
For DC, sk is simply k1/6
For AC, sk is given by
sk values computable for any arbitrary waveform
( )⎪⎪
⎩
⎪⎪
⎨
⎧
>+
>+
==
=
−−
−
(relax) even
(stress) odd
k,kss
k,kskk
s
kk
kk
131
32
111100
21
61
61
1717
Outline
NBTI OverviewReaction-Diffusion (R-D) ModelOur Analytical NBTI ModelFrequency IndependenceDelay Estimation using NBTI Model
1818
Frequency Independence
Number of interface traps for both cases same
Trap generation independent of frequency
freq = f1
freq = f2
T1
T2
n1 cycles
n2 cycles
Vdd
Vdd
1919
Frequency Independence Plots
AC freq = f
DC
time (s)
∆Vth (mV)
0
3
6
9
12
15
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
DC
freq = f
2020
Frequency Independence Plots
DC
freq = ffreq = 0.1f
time (s)
∆ Vth (mV)
0
3
6
9
12
15
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
DC
freq = ffreq = 0.1f
2121
Frequency Independence Plots
0
3
6
9
12
15
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
DC
freq = ffreq = 0.1ffreq = 0.01f
time (s)
∆ Vth (mV)
Vth degradation same for all three cases
2222
Outline
NBTI OverviewReaction-Diffusion (R-D) ModelOur Analytical NBTI ModelFrequency IndependenceDelay Estimation using NBTI Model
2323
Issues
Estimate the delay degradation after a time period equal to 10 years of operation, i.e., (~3X108 s)
f=1GHz implies 1017 cyclesNeed “fast-forwarding”
NBTI effect is temporalRequires exact nature of stress and relaxation to determine NIT
Impossible to determine temporal input activityNeed to use statistical inputs
2424
Signal Probability and Activity Factor
Signal Probability (SP)
Probability that the signal is high (or low)
Activity Factor (AF)Probability that the signal switches
AF = 0.6 SP = 0.4
Clock
Signal
0 0 0 0
Vdd
2525
NBTI – Activity Factor (AF) Independence
Three square waveforms with same signal probability (SP) of 0.5
1X, 0.1X and 0.01X activity factor (AF) valuesSame amount of Vth degradation
Trap generation is AF independent
1Hz
0
3
6
9
12
15
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
DC
0.01f0.1f
f
∆Vth
(mV)
time (s)
2626
NBTI – Signal Probability (SP) Dependence
Four waveforms with same frequencySP values are 0.25, 0.5, 0.75, 1.00
∆Vth values differ significantlyNBTI effect is SP dependent
0
2
4
6
8
10
0 100 200 300 400 500 600 700 800 900 1000
SP=1.0SP=0.75SP=0.5SP=0.25
time (s)
∆V t
h(m
V)
2727
SPAF Method
Converting a random waveform to an “equivalent”deterministic periodic waveform
Don’t care about AFsMaintain same SP
(SP, AF)
Vdd
k
m k-m
m = k * SP
Vdd
2828
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
k
m k-m
m = k * SP
sk SP = 0.25
SP = 0.75
2929
Circuit Delay Estimation
Simulations on ISCAS85 benchmarks – 65nm PTM technologyClock frequency = 1GHz
0.5
0.5
0.5
0.5
0.5
0.25
0.25
0.375
0.375
SP= 0.25 SP = 0.375Estimate Vth of each transistor after 10 years using a Vth – SP look-up tableCalculate new arrival times
3030
Results
8.73Average8.7738683556C62888.6811391048C53159.0313401229C35408.74771709C26708.351064982C19089.12730669C13558.40671619C8808.92745684C4997.69966897C4329.598073C17
% IncreaseNBTI Delay (ps)Nominal Delay (ps)Benchmark
~9% degradation in delay of circuits after 10 years of operation
3131
Conclusion
NBTI – growing threat to reliabilityNeed accurate estimation of its effect
NBTI Modeling Analytical model for NBTI presentedCircuit delay characterized due to temporal NBTI stress and relaxation
9% increase in delay estimatedModel can be used for NBTI-aware design