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)

55

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

88

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