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1. (Department of EE, Incheon National University, Incheon 22012, Korea)

Distribution, multiple-point, non-volatile computing system, resistive device, spin-transfer-torque magnetoresistive random access memory (STT-MRAM), tail fitting, yield estimation

## I. INTRODUCTION

Because of technology shrinks, process variations such as random dopant fluctuation and line edge roughness among others (1), (2), have become an issue as they cause significant variations in circuit characteristics (3). As a result, the read yield in non-volatile memories such as spin-transfer-torque magnetoresistive random access memory (STT-MRAM), phase-change RAM, and resistive RAM has decreased and a large error correction code or redundancy has become an essential component. With an increase in variations and decrease in supply voltages, Monte Carlo (MC) simulations have become indispensable in correctly estimating the memory yield and optimizing performance, area, and power (4). Thus, MC simulations have been widely used in various memories to analyze statistical properties (5)-(8). In general, the normal fitting and importance sampling yield estimation methods are used to estimate the read yield, considering the effect of process variations in deep submicron technology nodes (4), (5), (9). However, the normal fitting method cannot avoid a significant error unless a distribution follows the Gaussian distribution (10). The importance sampling method can overcome this problem because it does not assume a Gaussian distribution. However, the efficiency is considerably degraded when industry-compatible model parameters are used because most variables affected by process variations cannot be controlled as the conditions are fixed based on the silicon results and not allowed to modify with company confidential (10).

As an extension of our previous work (11), this study shows that the output voltage distribution of the sensing circuit used in STT-MRAM is not a Gaussian distribution but is composed of two Gaussian distributions. Deep analysis of two-point tail fitting method is also presented for the first time in the literature. A novel multiple-point tail fitting yield estimation method that can estimate the read yield with higher accuracy is proposed.

The remainder of this paper is organized as follows. In Section II, the distribution analysis of the output voltage difference (ΔV) in the sensing circuit is studied. The challenges of existing yield estimation methods are presented in Section III. Two-point tail fitting yield estimation and selection of P$_{1}$ and P$_{2}$ in tail fitting method are described in Sections IV and V, respectively. The proposed multiple-point tail fitting method is presented in Section VI. Section VII presents the simulation results and comparison. Finally, Section VIII concludes the paper.

Fig. 1. (a) Conventional sensing circuit in STT-MRAM. (b) I-V curves with single corner simulation showing the generations of V$_{\mathrm{DATA0}}$, V$_{\mathrm{DATA1}}$, and V$_{\mathrm{REF}}$.

Fig. 2. (a) Distributions of ΔV$_{0}$ and ΔV$_{1}$ with MC = 300,000 simulations. (b) I-V curves with MC = 1,000 simulations showing large variations in V$_{\mathrm{DATA0}}$ and V$_{\mathrm{DATA1}}$.

Fig. 3. Normal probability plots of ΔV$_{0}$ and ΔV$_{1}$ distributions (MC = 300,000) showing that the distributions of ΔV are composed of two Gaussian distributions; one is from the linear region and the other from the saturation region.

## II. DISTRIBUTION ANALYSIS OF ΔV IN SENSING CIRCUIT

The operation of the conventional sensing circuit in STT-MRAM (Fig. 1(a)) can be explained by using the I-V curves shown in Fig. 1(b). Each I-V curve represents the relationship between the current through each transistor (clamp NMOS or load PMOS) and the drain voltage of each transistor. Because the crossing point of the I-V curves is the same as the operating point at a steady state, V$_{\mathrm{DATA}}$ and V$_{\mathrm{REF}}$ can be estimated. In the reference voltage generator, reference cells with low resistance (R$_{\mathrm{L}}$) and high resistance (R$_{\mathrm{H}}$) magnetic tunnel junctions (MTJs) are connected; thus, the I-V curve of NC$_{\mathrm{R}}$ is in the middle of the I-V curves of NC$_{\mathrm{D}}$ when R$_{\mathrm{DATA}}$ = R$_{\mathrm{L}}$ and NC$_{\mathrm{D}}$ when R$_{\mathrm{DATA}}$ = R$_{\mathrm{H}}$. The crossing point of the I-V curves between NC$_{\mathrm{R}}$ and PL$_{\mathrm{R}}$, which is a diode-connected configuration, becomes V$_{\mathrm{REF}}$. Because V$_{\mathrm{REF}}$ is used as the gate voltage of PL$_{\mathrm{D}}$, the I-V curve of PL$_{\mathrm{D}}$ passes through the crossing point between the I-V curves of NC$_{\mathrm{R}}$ and PL$_{\mathrm{R}}$. At the intersection of the I-V curves between NC$_{\mathrm{D}}$ when R$_{\mathrm{DATA}}$ = R$_{\mathrm{L}}$ (NC$_{\mathrm{D}}$ when R$_{\mathrm{DATA}}$ = R$_{\mathrm{H}}$) and PL$_{\mathrm{D}}$, V$_{\mathrm{DATA0}}$ (V$_{\mathrm{DATA1}}$) is also generated. The operating point of V$_{\mathrm{DATA0}}$ (V$_{\mathrm{DATA1}}$) is generated at the boundary between the linear and saturation regions in the NC$_{\mathrm{D}}$ (PL$_{\mathrm{D}}$). To take the process variation into account, 300,000 MC simulations were performed. Fig. 2(a) shows the distributions of the output voltage differences (ΔV$_{0}$ = V$_{\mathrm{REF}}$ - V$_{\mathrm{DATA0}}$, ΔV$_{1}$ = V$_{\mathrm{DATA1}}$ - V$_{\mathrm{REF}}$). It is seen clearly that both ΔV$_{0}$ and ΔV$_{1}$ distributions have large variations. Further, there are tail regions in the left side of the distribution, which implies that it is not a Gaussian distribution. Because of the process variation, V$_{\mathrm{DATA0}}$ and V$_{\mathrm{DATA1}}$ move toward the linear and saturation regions, as shown in Fig. 2(b). The variations in ΔV$_{0}$ and ΔV$_{1}$ are small if they move toward the linear region. However, if they move toward the saturation region, the variations in ΔV$_{0}$ and ΔV$_{1}$ become large, resulting in a tail region, which dominantly degrades the read yield. In contrast, the variation in V$_{\mathrm{REF}}$ is relatively small because the slope of the diode-connected I-V curve is steep. Because the variations in V$_{\mathrm{DATA0}}$ and V$_{\mathrm{DATA1}}$ are generated from the linear and saturation regions, the ΔV distribution is non-Gaussian. Furthermore, the tail region is generated from the saturation region; hence, the ΔV distribution has a large tail region.

In the normal fitting method, this tail region is not considered. Because the read yield is primarily determined by the tail region, the estimated read yield from the normal fitting method is overestimated with a significant error. Fig. 3 shows the normal probability plots of ΔV$_{0}$ and ΔV$_{1}$ distributions. A straight line in the normal probability plot implies that the distribution can be modeled as a Gaussian distribution. Fig. 3 clearly shows that the ΔV distribution is composed of two straight lines, representing two Gaussian distributions. One is generated from the linear region and the other is from the saturation region. Thus, by using the Gaussian characteristic in the tail region, the accurate read yield can be estimated with a moderate number of simulations. This method is called the tail fitting method.

## III. CHALLENGES OF EXISTING YIELD ESTIMATION METHODS

In this section, several existing yield estimation methods and their challenges are briefly described. For further details, refer to (4) and (10).

### 1. Standard MC Yield Estimation Method

This method performs many MC simulations until the appropriate number of failures occur. By comparing the total number of simulations with the number of failures, the yield can be simply estimated. However, the required number of simulations exponentially increases as the read yield increases; thus, the standard MC yield estimation method is impractical in most cases. For a given confidence interval ${\varepsilon}$ and acceptable error criteria d, the required number of simulations becomes (4)

##### (1)
$$N=\frac{\left(\Phi^{-1}(1-\varepsilon)\right)^{2}}{d^{2}} \cdot \frac{1-P_{f}}{P_{f}},$$

where Φ() is the cumulative distribution function (CDF) of the Gaussian distribution, Φ$^{-1}$() is the inverse of Φ(), and P$_{f}$ is the failure probability. With 95% confidence and 5% acceptable error criteria, the required number of simulations is approximately equal to

##### (2)
$$N \approx \frac{384}{P_{f}}$$

For example, to estimate the read yield in an STT-MRAM having a real read yield of 6-sigma with 95% confidence and 5% acceptable error criteria, the required number of simulations should be greater than 3.89${\times}$10$^{11}$, which is impractical.

### 2. Normal Fitting Yield Estimation Method

A commonly used method for reducing the required number of simulations is the normal fitting method. It models the entire ΔV distribution as a Gaussian distribution and extrapolates to long tails to estimate the read yield. Because the ΔV distribution is fully characterized by its mean (μ$_{\mathrm{∆V}}$) and standard deviation (σ$_{\mathrm{∆V}}$), the read yield can be easily obtained with a moderate number of simulations. The read yield can be expressed as

##### (3)
$$\text { read yield }=\frac{\mu_{\Delta \mathrm{V}}}{\sqrt{\sigma_{\Delta \mathrm{V}}^{2}+\sigma_{\mathrm{SA}_{-} \mathrm{oS}}^{2}}}$$

where σ$_{\mathrm{SA\_OS}}$ is the standard deviation of offset voltage in the second stage sense amplifier (12), (13). However, this method is effective only when the ΔV distribution is Gaussian in entire region, otherwise, a significant error is obtained. It is important to note that the estimated read yield is generally overestimated. For example, in an STT-MRAM with real read yield of 5-sigma, the estimated read yield using the normal fitting method could be greater than 8-sigma.

### 3. Importance Sampling Yield Estimation Method

The importance sampling method is a well-known variance reduction technique for rare event applications. It executes the MC simulation under an intentionally distorted process variation condition to produce more samples around the failure region. Then, by transforming the simulation results mathematically, the accurate read yield can be estimated with a smaller number of simulations (14). The concept of the IS method is based on

Fig. 4. Tail fitting illustration using two points (MC = 300k).

##### (4)
$E_{f(x)}[\theta (x)]=E_{g(x)}[\theta (x)\cdot \frac{f(x)}{g(x)}],$

##### (19)
μ$_{\mathrm{∆V}}$ + Φ$^{-1}$(P$_{2}$/N)${\cdot}$σ$_{\mathrm{∆V}}$ = Sort$_{\mathrm{∆V}}$$^{\mathrm{Multi\_P2}}$

## VII. SIMULATION RESULTS AND COMPARISON

To verify the effectiveness of the proposed method, HSPICE MC simulations were performed using industry-compatible 45-nm model parameters. A MTJ model is based on (15). R$_{\mathrm{L}}$ of 3 kΩ and R$_{\mathrm{H}}$ of 7.5 kΩ were assumed from the TMR of 150%, and a critical switching current (I$_{\mathrm{C}}$) of 100 μA was used with a current pulse width of 10 ns. A standard deviation of 4% was also assumed for the variation in the MTJ resistance and I$_{\mathrm{C}}$ caused by the process variation (16). A low supply voltage of 1.0 V (~90% of typical core voltage of 1.1 V) was used. The simulations were executed between -45 $^{\circ}$C and 90 $^{\circ}$C so that the result of the read yield includes wide temperature variation effects. The σ$_{\mathrm{SA\_OS}}$ of 20 mV was used to calculate the read yield (12). Fig. 8 shows the error rate between the real and the estimated read yield according to j$_{\mathrm{MAX}}$ in the multiple-point tail fitting method. Because of the decrease in the tail sample uncertainty, the error rate initially decreases with increasing j$_{\mathrm{MAX}}$ till j$_{\mathrm{MAX}}$ = 17. However, above 17, the error rate starts increasing with j$_{\mathrm{MAX}}$ because the tail sample uncertainty increases again owing to the deeper tail samples. Thus, j$_{\mathrm{MAX}}$ of 17 is selected. The constant of 0.025 is used in (13) and (14) as an example to verify the effectiveness of the multiple-point tail fitting method. Although optimal j$_{\mathrm{MAX}}$ varies according to the constant value, it is not sensitive to the error rate because the optimal j$_{\mathrm{MAX}}$ increases (decreases) if the constant decreases (increases). Fig. 9 shows the error rates obtained from many MC simulations for the normal fitting, importance sampling, tail fitting, and multiple-point tail fitting methods. The commonly used normal fitting method cannot estimate the yield accurately because of the non-Gaussian distribution of ΔV. The importance sampling method cannot be an efficient way for variance reduction when industry-compatible model parameters are used because most variables affected by process variation are not controllable (10). Because of the Gaussian characteristic of the ΔV distribution in the tail region, the tail fitting methods show considerably less error rates compared with existing methods. The proposed multiple-point tail fitting method clearly shows an accuracy improvement of 1.9x compared with the two-point tail fitting method by mitigating the tail sample uncertainty.

The optimal parameter values (P$_{1}$ = 10, P$_{2}$ = P$_{1}$ + 0.5σ, MC = 10k, j$_{\mathrm{MAX}}$ = 17) are only applicable in the specific scenario presented in this paper, where the STT-MRAM sensing circuit, target error rate = 5%, 45-nm model parameters, and R$_{\mathrm{L}}$ of 3 kΩ MTJ model with 150% TMR and 4% resistance variation. In the different appications and scenarios, however, if the combination of NMOS and PMOS is used to generate or amplify a voltage, then this voltage distribution will be composed of the two Gaussion distributions and it can be easily and accurately analyzed by using the proposed tail fitting or multiple-point tail fitting method.

## VIII. CONCLUSIONS

In this paper, by analyzing the output voltage distribution of the sensing circuit used in STT-MRAM, it is shown that the distribution does not follow the Gaussian distribution but is composed of two Gaussian distributions. The inaccuracy of the normal fitting method and the limitation of the two-point tail fitting method are also shown. Finally, the multiple-point tail fitting method is proposed to overcome the accuracy limitation of the two-point tail fitting method by alleviating the tail sample uncertainty. The simulation results confirm the effectiveness of the proposed method. Therefore, the proposed method can be utilized for fast and accurate design in STT-MRAM, such as (18)-(27).

### ACKNOWLEDGMENTS

This work was supported by Incheon National University Research Grant in 2019.

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## Author

##### Taehui Na

Taehui Na received the B.S. and Ph.D. degrees in Electrical & Electronic Engineering from Yonsei University, Seoul, Republic of Korea, in 2012 and 2017, respectively.

From 2017 to 2019, he was with Samsung Electronics Co., Ltd., Hwasung, Republic of Korea, where he worked on phase-change random access memory (PRAM) and high-performance NAND (ZNAND) core circuit designs. Since 2019, he has been a professor at Incheon National University, Incheon, Republic of Korea. His current research interests are focused on process-voltage-temperature variation tolerant and low-power circuit designs for memory, microcontroller unit, and neuromorphic SoC.

##### Seung H. Kang

Seung H. Kang is the Director of Engineering, Corporate R&D, Qualcomm Technologies, Inc., and leads an emerging memory technology group for mobile systems, including IOT and wearables. He received B.S. and M.S. degrees from Seoul National University, Korea, and the Ph.D. degree in Materials Science and Engineering from the University of California at Berkeley.

Dr. Kang worked at Lawrence Berkeley National Laboratory in the fields of SQUID sensors and VLSI interconnects. From 1998 till 2005, he worked at Lucent Technologies Bell Laboratories and led advanced device-reliability projects as a Distinguished Member of the Technical Staff. In 2006, he joined Qualcomm and has pioneered embedded STT-MRAM and spintronic devices for mobile systems. He has served on numerous technical committees. He has published over 70 papers and delivered more than 35 keynote and invited speeches at international conferences. He also holds over 350 patents granted globally. Dr. Kang currently serves as an IEEE Electron Device Society Distinguished Lecturer.

##### Seong-Ook Jung

Seong-Ook Jung received the B.S. and M.S. degrees in Electrical & Electronic Engineering from Yonsei University, Seoul, Republic of Korea, in 1987 and 1989, respectively. He received the Ph.D. degree in Electrical Engineering from the University of Illinois at Urbana-Champaign, Urbana, IL, in 2002.

From 1989 to 1998, he was with Samsung Electronics Co., Ltd., Hwasung, Republic of Korea, where he worked on specialty memories, e.g., video RAM, graphic RAM, and window RAM, and merged memory logic. From 2001 to 2003, he was with T-RAM Inc., Mountain View, CA, where he was the leader of the thyristor-based memory circuit design team. From 2003 to 2006, he was with Qualcomm, Inc., San Diego, CA, where he worked on high-performance low-power embedded memories, process-variation-tolerant circuit design, and low-power circuit techniques. Since 2006, he has been a professor at Yonsei University. His research interests include process-variation-tolerant circuit design, low-power circuit design, mixed-mode circuit design, and future generation memory and technology.

Dr. Jung is currently a board member of the IEEE SSCS Seoul Chapter.