Statistical Electro-Thermal Framework
4.1 Accuracy and Efficiency
To verify the simulator, the Monte Carlo (MC) method is also implemented by 105 samples as reference golden solutions which consider the same issues such as electro-thermal coupling,
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Power Source Layer of Die Interconnect Layer
C4/CBGA Package and PCB Board
Die
20nm
0.5mm 0.06mm
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Fig. 4.1: (a) The floorplan of test chip. (b) The geometry setting of test chip.
Table 4.1: Accuracy and efficiency compared with the Monte Carlo method.
Inter-die Intra-die Our Proposed Method† Monte Carlo‡ Speedup
/ Total / Total max. mean max. std. runtime (s) sampling runtime (s)‡ (X) Variations Variations error error Phase 1 Phase 2 knots
40% 60% 0.33% 1.70% 3.23 1.04 6736 326.49 313.93
50% 50% 0.35% 1.88% 3.27 1.04 6465 313.82 301.75
60% 40% 0.36% 1.84% 3.40 1.04 6422 311.47 299.49
† Our proposed method is compared with the golden solution constructed by Monte Carlo method using 105samples.
‡ To show the efficiency, Monte Carlo method here is simulated till achieving the same accuracy of standard deviation as our proposed method. The runtime here does not include the time of input parser which is only performed once in Monte Carlo simulation.
spatially intra-die variations, and inter-die variations. The proposed electro-thermal simulator takes 10 random variables to expand process variations and uses Smolyak sparse grid formula with q=11. Hence, the stochastic thermal profile over the test chip is interpolated by 21 individ-ual sampling points. The results with three different ratios of inter-die variations and intra-die variations to the total variations in a reasonable region are shown in Table 4.1.
Compared with the golden solution, the proposed simulator is extremely accurate and can be finished in seconds for the test chip. For example, in the case of inter-die variations being 50% of total variations, the proposed simulator can achieve the maximum errors of 0.35% and 1.88% in spatial mean and spatial standard deviation of temperature distribution, respectively.
The execution time is only 3.27 seconds and 1.04 seconds in Phase 1 and Phase 2, respectively.
The similar results can be found in the rest two cases.
Since each operation in Phase 1 of the proposed simulator is irrelevant to design pattern, they only need to be pre-performed once while applying the proposed simulator to the optimal thermal-aware design procedure. Therefore, to show the efficiency of the proposed simulator, the runtime of Phase 2 is compared with the execution time through the Monte Carlo simulation fulfilling the same accuracy of standard deviation as ours. Table 4.1 shows that the proposed simulator is orders of magnitude faster than the Monte Carlo analysis under the same accuracy level. the same Since each sampling point is independent, the parallel programming technique can be easily applied to further enhance the speedup.
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Fig. 4.2: The temperature profile at the top surface of the die. (a) The mean temperature dis-tribution without considering electro-thermal coupling. (b) The mean temperature disdis-tribution with considering electro-thermal coupling.
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Fig. 4.3: The temperature profile at the top surface of the die. (a) The spatial standard devi-ations without considering electro-thermal coupling. (b) The spatial standard devidevi-ations with considering electro-thermal coupling.
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Fig. 4.4: Distribution of the temperature using Monte Carlo (MC) simulation, with and without electro-thermal coupling, and the proposed method at the location of the hottest mean tempera-ture. (a) Probability density function (PDF). (b) Cumulative distribution function (CDF).
4.2 Without vs. With Including the Effect of Electro-Thermal Coupling
Fig. 4.2 and Fig. 4.3 show the spatial mean and spatial standard deviations of the temperature distribution at the top surface of the test chip, respectively. Fig. 4.2(a) and Fig. 4.3(a) are the results without considering electro-thermal coupling. Fig. 4.2(a) and Fig. 4.3(b) are the results with considering electro-thermal coupling. These two figures reveal the dramatic differences of the spatial mean and spatial standard deviation profiles between the results without considering electro-thermal coupling and the results considering electro-thermal coupling. As we can see, the difference of spatial mean profile can reach 6.54%, and the difference of spatial standard deviation profile is over 25.01%.
According to [8], the temperature profile of each location on the chip can be approximated as a log-normal distribution. The probability density function (PDF) and cumulative distribution function (CDF) of the temperature distribution at an arbitrary location on the chip are plotted in Fig. 4.4(a) and Fig. 4.4(b), respectively. The blue solid line marked in triangles is the result obtained from the Monte Carlo simulation with considering electro-thermal coupling. The red dash line marked in circles is the result acquired from the Monte Carlo simulation without considering electro-thermal coupling. The black solid line is an approximation using log-normal distribution and its mean and variance are obtained by the proposed simulator. Fig. 4.4 shows that the proposed method can provide accurate estimations of PDF and CDF for the thermal profile, and the simulation results without considering electro-thermal coupling are unreliable.
The similar result also happens in the statistical analysis of total leakage power. The PDFs and CDFs of the total leakage power of the test chip by the Monte Carlo simulation are shown in Fig. 4.5. Obviously, the statistical leakage power analysis without electro-thermal coupling is not reliable.
From the above discussion, it shows that the statistical thermal or leakage power analysis method without considering electro-thermal coupling can lead the simulation results into an unreliable region and provide a dubitable confidence interval. To give the correct and reliable analysis results for designers, it is necessary to take electro-thermal coupling into consideration for not only leakage power analysis but also thermal analysis, and the proposed electro-thermal
Fig. 4.5: PDFs and CDFs of the total leakage power using MC simulation with and without considering electro-thermal coupling.
simulator can accurately and efficiently achieve these.