SRAM cell의 통계적인 read static noise margin modeling

Abstract

Static noise margin (SNM) is an evaluation metric of SRAM cell stability. SNM is defined as maximum noise voltage while ensuring correct operation. SRAM cell has three operations: hold, write and read. In conventional 6-transistor SRAM cell, analysis of read stability is most important because of the conflicting problem during read operation. Therefore, read SNM (RSNM) is widely used as the evaluation metric of SRAM cell stability. Over decades, the scale of transistor has decreases to increase efficiency of area, yield and power. However, as the scale of the transistor decreases, process variations (PVs) have been increased. To consider PVs, variation-aware analysis has been studied in various circuit analysis. For SRAM, variation-aware analysis becomes more important as the trend of lowering the operating voltage for lower power design. As the operating voltage decreases, the stability of SRAM cell decreases and become more sensitive to PVs. Therefore, variation-aware RSNM analysis has been important to estimate RSNM distribution. There are two methods for variation-aware analysis: worst case analysis and statistical analysis. However, worst case analysis is not suitable because of over-design problem. For effective design of SRAM cell, statistical analysis needs to be used. In statistical analysis, Monte-Carlo (MC) is the most widely adopted method. While it provides reliable results, many samples need to be calculated by TCAD simulation, which results in high computational complexity. To address this problem, several pseudo MC methods have been proposed as an approach to reduce the required number of samples. The work in [6] used importance sampling to improve the extraction frequency of the RSNM value corresponding to extreme values. Also, in [7], a powerful pseudo MC method, Quasi Monte-Carlo (QMC), was proposed. However, these methods cannot estimate the required number of samples and still need many samples. In addition, these methods cannot provide design feedback for improvements because they cannot offer analytical equation for RSNM. Also, several analytical methods have been proposed to predict the influence of the PVs. In case of planar MOSFET SRAM cell, the work in [8], [9] offered an analytical equation for RSNM of SRAM cell. However, these methods did not consider PVs or only considered threshold voltage Vth variation. To reflect accurate PV effects, important physical parameter variations like length L and width W of the transistor have to be considered. Also, it is hard to reflect all phenomenon caused by short channel lengths in the equation. Therefore, these methods showed inaccurate results. In case of FinFET SRAM cell, the work in [10] discussed the dependencies on the physical parameter variation using TCAD simulation with no special methods. The work in [11] offered an analytical equation focused on aging model that does not consider PVs. We propose a method for statistical RSNM modeling for 6-transistor SRAM cell. The proposed method overcomes the disadvantages of MC-based method and analytical method. First, we derive the analytical equation form for RSNM high (RSNMH). In case of planar MOSFET, the equation form considers Vth, L and W of the transistor. For FinFET, the equation form considers Vth, L, fin height and fin width of the transistor. These analytical equation forms also consider Vth roll off and drain induced barrier lowering effect. In contrast to MC-based methods, the proposed method needs only a few samples with TCAD simulation. These simulation results are used to fit coefficients. Phenomenon caused by short channel length is taken into account in this process. Then, the distribution of RSNM is obtained using the relationship between RSNMH and RSNM [12]. Also, based on the obtained analytical equation, the proposed method offers design feedback for improvements. In experiment I, we analyzed the dependencies on parameter variations based on the obtained analytical equation for planar MOFET and FinFET SRAM cell. In experiment II, the accuracy of the proposed method was evaluated based on goodness of-fit tests with respect to MC simulation results. Also, the accuracy of the proposed method was compared with benchmark method results. In experiment III, we tested the accuracy of design feedback that provides direction of design improvement. As goodness-of-fit tests of experiment II, we used Kolmogorov-Smirnov (KS) test for cumulative distribution function and Chi-square test for probability density function. As benchmark methods for planar MOSFET, we used QMC and an analytical model [9]. In case of FinFET, only QMC was used because no model could be found to provide analytical equation considering parameter variations. As the results of planar MOSFET, the proposed method achieved about 89% (KS) and 99% Chi-square improvements compared with QMC for the same number of samples, and 92% (KS) and 98% Chi-square improvements compared with an analytical model [9]. For FinFET, the proposed method achieved about 99% (KS and Chi-square) improvements compared with QMC for the same number of samples.