CDW Verification for BTI Modeling

Given the compression capabilities of a CDW, it is important to explore its usability for our target goal, namely to execute the atomistic BTI/RTN simulation over strides of𝑉𝑔𝑠 workload. We will be altering a version of the atomistic BTI model [265, 120] and will reformulate as follows: Starting from the premise of first order kinetics, the capture probability of a gate stack defect follows the ODE of Equation 3.7. The general solution returns𝑃𝑐, after𝑛 𝑉𝑔𝑠

pulses of specific 𝑓 and𝛼have been applied (equivalent duration Δ𝑡=𝑛/𝑓).

Parameters𝑎and𝑏are functions of 𝑓 and𝛼[265, 120].

𝑑𝑃_𝑐

𝑑𝑡 =𝑎𝑃_𝑐+𝑏 ⇒ 𝑃_𝑐= (︂

𝑃_𝑐0+𝑏 𝑎

)︂

𝑒^𝑎𝑛− 𝑏

𝑎 (3.7)

CDW VERIFICATION FOR BTI MODELING 43

0.2 0 0.6 0.4

1 0.8

0 2

4 6

8 10

−0.02 0 0.02 0.04 0.06

εα (p.u.) log₁₀ {ε_f (Hz)}

µ{es(t)} (V)

(a)

0.2 0 0.6 0.4

1 0.8

0 2

4 6

8 10

0 0.2 0.4 0.6 0.8

ε α (p.u.) log₁₀ {ε

f (Hz)}

σ{es(t)} (V)

(b)

0.2 0 0.6 0.4

1 0.8

0 2

4 6

8 10

0 100 200 300

ε α (p.u.) log10 {ε

f (Hz)}

Number of CDW Points (p.u.)

(c)

Figure 3.4: Accuracy assessment (signal integrity) of thewavapproxtool that produces CDW representations, using a 1µs bit stream. It should be noted that SPICE requires 17089 points to represent that signal and a VCD representation uses 521 points [234].

Each iteration of the general solution of Equation 3.7 is used as initial condition for the next model evaluation. The wavapprox tool, presented previously, identifies regions of similar𝑓 and𝛼in the𝑉_𝑔𝑠(i.e. a CDW representation is produced). Then, at each (𝑓, 𝛼,Δ𝑡) point, a single iteration of Equation 3.7 is performed. A CDW representation reduces aggressively the number of model iterations required for the calculation of𝑃_𝑐. That way, time-dependent BTI simulation becomes computationally feasible over long time spans (see Figure 3.2). Given the compression capabilities of the CDW representation (see Figure 3.4c), the number of model iterations is aggressively reduced, thus

decreasing the CPU time required for simulation.

In Section 3.3, we have assessed the accuracy of thewavapproxfrom a signal integrity point of view. Given the proposed model of Equation 3.7, we will evaluate the accuracy of a CDW approximation in terms of𝑃𝑐 evaluation. For our experiments we use 100 random bit streams with an average frequency (𝑓𝑎𝑣𝑔) of 1 GHz and a duration of 1µs. Given each signal, we define an error metric for BTI evaluation, according to Equation 3.8, where𝑃_𝑐,SPICE is the capture probability for a defect at the end of the signal based on SPICE evaluation (i.e. using Equation 3.1). 𝑃_𝑐,CDWis a similar capture probability, using a CDW file (for a specific pair of𝜀_𝑓 and𝜀_𝛼) and the model of Equation 3.7. We use 5 different cases, ranging from “fast” (𝜏𝑐𝐻 = 10⁻¹¹s) to “slow” (𝜏𝑐𝐻 = 10⁻³s) defects. That way, we can calculate the mean, standard deviation and maximum of𝑒𝑝across the 100 signals for various values of (𝜀𝑓, 𝜀𝛼) and for the five different defects (Figures 3.5a – 3.5f).

𝑒𝑝=𝑃𝑐,SPICE−𝑃𝑐,CDW (3.8)

A direct correlation exists between how “fast” a defect is and the average frequency of the𝑉𝑔𝑠bit stream (𝑓𝑎𝑣𝑔). More specifically, the defects that are significantly “slower” in comparison to the average frequency of the bit stream are accurately modeled, regardless of the compression imposed bywavapprox (Figures 3.5a through 3.5d). Faster defects (Figures 3.5e and 3.5f) have time constants comparable to the bit stream’s period (1/𝑓𝑎𝑣𝑔). The proposed model is unable to follow the𝑃𝑐 of these defects, irrespective of𝜀𝑓 and𝜀𝛼 values.

The relation between𝑓_𝑎𝑣𝑔 and the defect time constants has been identified earlier [120], however no comprehensive study has been performed for the accuracy limitations of the AC BTI/RTN model [265, 120]. Given that a version of this model, enhanced with workload memory, is the kernel of our simulation methodology, we should exhaustively explore its accuracy limitations for each model iteration. More specifically, it is important to identifyfor which defects and under which conditions is the accuracy of our model acceptable.

CDW VERIFICATION FOR BTI MODELING 45

0

5

10 0

0.5 5.28521 5.2854 5.2856 5.2858

x 10⁻⁶

log{ε_f (Hz)}

τ_cH=1.0e−003, τ_cL=1.0e+007, τ_eH=1.0e−001, τ_eL=1.0e−002

εα (p.u.) µ{ePc} (p.u.)

(a)

0

5

10 0

0.5 1.27761 1.2776 1.2776

x 10⁻⁴

log{ε f (Hz)}

τcH=1.0e−003, τ

cL=1.0e+007, τ

eH=1.0e−001, τ eL=1.0e−002

ε α (p.u.) max{ePc} (p.u.)

(b)

0

5

10 0

0.5 1 1 1.5 2 2.5 3

x 10⁻³

log{ε_f (Hz)}

τcH=1.0e−007, τ

cL=1.0e+007, τ

eH=1.0e−005, τ eL=1.0e−006

εα (p.u.) µ{ePc} (p.u.)

(c)

0

5

10 0

0.5 1 0.03 0.04 0.05 0.06

log{ε f (Hz)}

τ

cH=1.0e−007, τ

cL=1.0e+007, τ

eH=1.0e−005, τ eL=1.0e−006

ε α (p.u.) max{ePc} (p.u.)

(d)

0

5

10 0

0.5 0.41 0.5 0.6 0.7 0.8

log{ε f (Hz)}

τcH=1.0e−011, τ_cL=1.0e+007, τ_eH=1.0e−009, τ_eL=1.0e−010

εα (p.u.) µ{ePc} (p.u.)

(e)

0

5

10 0

0.5 0.99011 0.9901 0.9901 0.9901 0.9901

log{ε f (Hz)}

τ

cH=1.0e−011, τ

cL=1.0e+007, τ

eH=1.0e−009, τ eL=1.0e−010

ε α (p.u.) max{ePc} (p.u.)

(f)

Figure 3.5: Accuracy assessment of the atomistic BTI model of Equation 3.7 for various degrees of CDW-compression [234].

In order to formally conclude on the accuracy limitations of our approach, we first create a set of 400 different defects, with time constants (𝜏𝑐𝐻, 𝜏𝑐𝐿, 𝜏𝑒𝐻, 𝜏𝑒𝐿) uniformly distributed in the logarithmic scale. The defects are arranged in a two dimensional plane according to min{𝜏𝑐𝐻, 𝜏_𝑐𝐿} and min{𝜏𝑒𝐻, 𝜏_𝑒𝐿}.

We use three sets of bit streams with 𝑓𝑎𝑣𝑔 equal to 1 GHz, 100 MHz and 10 MHz for each set. Each set contains 100 signals, each one with a duration of 1 µs. Each signal is CDW-approximated with maximum compression (𝜀𝑓 →+∞

and𝜀𝛼→1) and𝑒𝑝 is calculated according to Equation 3.8. That way, we can calculate 𝜇{𝑒𝑝}, 𝜎{𝑒𝑝} and max{𝑒𝑝} for each defect and for each 𝑓𝑎𝑣𝑔 case.

The results are shown in Figures 3.6a through 3.6f. It is important to note that, these results represent the accuracy of the proposed modelfor a single iteration of Equation 3.7. However they provide useful insight on the accuracy limitations of the proposed model, when using it for an arbitrary number of iterations.

For all𝑓_𝑎𝑣𝑔values, we can always find a subset of defects that we are unable to model accurately. As 𝑓𝑎𝑣𝑔 is increasing, this subset is reduced and an acceptable accuracy of our model spreads to more defect time scales. For the case of𝑓𝑎𝑣𝑔= 1 GHz, our model can accurately simulate all defects that satisfy min{𝜏𝑐𝐻, 𝜏𝑐𝐿} ≥10⁻⁷ sand min{𝜏𝑒𝐻, 𝜏𝑒𝐿} ≥10⁻⁷s.

CDW VERIFICATION FOR BTI MODELING 47

−10

−5 5 0

10

−10 0

10 0

0.2 0.4 0.6 0.8 1

min{logτ_e (s)}

f_avg=1GHz

min{logτ c (s)}

µ{ep} (p.u.)

(a)

−10

−5 5 0

10

−10 0

10 0

0.2 0.4 0.6 0.8 1

min{logτ e (s)}

f_avg=1GHz

min{logτ c (s)}

max{ep} (p.u.)

(b)

−10

−5 0 5 10

−10 0

10 0

0.2 0.4 0.6 0.8 1

min{logτ_e (s)}

favg=100MHz

min{logτ_c (s)}

µ{ep} (p.u.)

(c)

−10

−5 0 5 10

−10 0

10 0

0.2 0.4 0.6 0.8 1

min{logτ e (s)}

favg=100MHz

min{logτ c (s)}

max{ep} (p.u.)

(d)

−10 0 −5

5 10

−10 0

10 0

0.2 0.4 0.6 0.8 1

min{logτ e (s)}

f_avg=100MHz

min{logτ_c (s)}

µ{ep} (p.u.)

(e)

−10 0 −5

5 10

−10 0

10 0

0.2 0.4 0.6 0.8 1

min{logτ e (s)}

f_avg=10MHz

min{logτ c (s)}

max{ep} (p.u.)

(f)

Figure 3.6: Errors from a single iteration of Equation 3.7 on bit streams of different 𝑓_𝑎𝑣𝑔. The error of the proposed model is restricted to small defect subsets, as𝑓_𝑎𝑣𝑔is increased [234].

−15 −10 −5 0 5 10 15 0.8

1 1.2 1.4 1.6

−20

−10 0 10 20

τ0 (s) V_dd (V)

τ (s)

(a) Scaling defect time constants for various bias (𝑉𝑑𝑑) conditions, according to Equation 3.9

−15 −10 −5 0 5 10 15

0 50 100 150 200

−20

−10 0 10 20

τ0 (s) T (^oC)

τ (s)

(b) Scaling defect time constants for various temperature (𝑇) conditions, according to Equation 3.10

Figure 3.7: Equations 3.9 and 3.10 enable full support of bias and temperature variations in the CDW-based BTI simulation framework [234].

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