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⁾.

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)

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

where f(x) and g(x) are the original and distorted sampling functions, respectively. In the MC simulation with N samples, the estimated failure probability (P$_{f}$$^{IS}$) from importance sampling method is

(5)

$$P_{f}^{I S}=\frac{1}{N} \sum_{i=1}^{N} I\left(x_{i}\right) \cdot w\left(x_{i}\right), \quad w\left(x_{i}\right)=\frac{f\left(x_{i}\right)}{g\left(x_{i}\right)},$$

where I(x) is an indicator function, which indicates pass or failure according to x like as follows:

(6)

$$I\left(x_{i}\right)=\left\{\begin{array}{ll} 0, & \text { pass } \\ 1, & \text { fail } \end{array}\right.$$

There are two representative importance sampling methods. One is increasing-sigma method ⁽⁹⁾ and the other is mean-shift method ⁽¹⁷⁾. In the increasing-sigma method, g(x) is generated by increasing the standard deviation of f(x). In case of f(x) = N(${\mu}$, ${\sigma}$$^{2}$) and g(x) = N(${\mu}$, (k${\sigma}$$^{2}$) where k is an increasing-sigma coefficient, w(x) becomes

(7)

$$w(x)=\prod_{j=1}^{M} k_{j} \exp \left(\frac{-\left(x_{j}-\mu_{j}\right)^{2}}{2 \sigma_{j}^{2}} \times\left(1-\frac{1}{k_{j}^{2}}\right)\right),$$

where M is the number of distorted variables. In the mean-shift method, g(x) is generated by shifting a mean of f(x) as much as a mean-shift coefficient s. In case of f(x) = N(${\mu}$, ${\sigma}$$^{2}$) and g(x) = N(${\mu}$+s, ${\sigma}$$^{2}$), w(x) becomes

(8)

$$w(x)=\exp \left(-\sum_{j=1}^{M} \frac{s_{j}\left(2 x_{j}-2 \mu_{j}-s_{j}\right)}{2 \sigma_{j}^{2}}\right)$$

By applying (7) or (8) to (5), P$_{f}$ $^{IS}$ can be estimated. Because the importance sampling method does not use any assumption, it can be used when the distribution cannot be modeled as a known distribution. However, its efficiency is significantly degraded if variation parameters such as threshold voltage, line edge roughness, and oxide thickness are not controllable ⁽¹⁰⁾.

IV. TAIL FITTING YIELD ESTIMATION METHOD (TWO-POINT)

Read failure occurs when ΔV is close to 0 V or negative, and this mostly occurs in the tail region. Thus, the tail fitting method finds the ΔV distribution generated only from the saturation region because this region determines the read yield. A Gaussian distribution is characterized by two parameters, μ and σ. By using two points in the tail region of the ΔV distribution instead of the full ΔV distribution, μ$_{\mathrm{∆V}}$ and σ$_{\mathrm{∆V}}$ can be determined accurately to estimate the read yield. Fig. 4 shows how the Gaussian distribution from the saturation region can be obtained by connecting the two points (P$_{1}$ and P$_{2}$) and then extrapolating. The procedure of the tail fitting method is as follows.

Fig. 5. An illustration to show the error rates caused by the tail sample uncertainty according to P$_{1}$ when P$_{2}$ is fixed to a reliable point of 0.25 V.

Fig. 6. Error rate between real and estimated read yield according to P$_{1}$ with different number of MC simulations.

1. Obtain ΔV distribution from MC = N simulations.

2. Sort the ΔV distribution in ascending order. We define a function Sort$_{\mathrm{∆V}}$(i), which indicates the i$_{\mathrm{th}}$ smallest value among N samples of ΔV.

3. Select proper P$_{1}$ and P$_{2}$ for Sort$_{\mathrm{∆V}}$(P$_{1}$) and Sort$_{\mathrm{∆V}}$(P$_{2}$), respectively.

4. Determine μ$_{\mathrm{∆V}}$ and σ$_{\mathrm{∆V}}$ by solving the following simultaneous equations.

Φ$^{-1}$(P$_{2}$/N) is set to be 0.5σ apart from Φ$^{-1}$(P$_{1}$/N) instead of two neighboring points for robust fitting as shown in Fig. 4. Then, P$_{2}$ is expressed as

Therefore, by selecting proper P$_{1}$ and solving the simultaneous equations of (9) and (10), μ$_{\mathrm{∆V}}$ and σ$_{\mathrm{∆V}}$ in the tail region can be determined.

5. Finally, calculate the read yield using equation (3).

V. SELECTION OF P1 AND P2 IN TAIL FITTING METHOD

Fig. 7. Error rate between real and estimated read yield according to j$_{\mathrm{MAX}}$ in multiple-point tail fitting method.

Selecting P$_{1}$ has a trade-off between tail sample uncertainty and tail region reliability. If P$_{1}$ is selected in the deep tail region (e.g., P$_{1}$ is 1-3), Sort$_{\mathrm{∆V}}$(P$_{1}$) can deviate from the Gaussian distribution of the tail region because the number of samples below P$_{1}$ is very few. This is known as the tail sample uncertainty problem. Fig. 5 clearly illustrates the tail sample uncertainty problem according to P$_{1}$ when P$_{2}$ is fixed to a reliable point of 0.25 V. This distribution is the same as the ΔV$_{1}$ distribution (having real read yield of 4.58σ) in Fig. 4. Only the difference is the number of MC simulations. When P$_{1}$ is 1, the estimated read yield becomes 5.21σ having the error rate of 13.8% due to the tail sample uncertainty, where the error rate is defined as

When P$_{1}$ of 5 is used, the error rate reduces to 5.2% because the tail sample uncertainty is alleviated. If the P$_{1}$ is selected in the region far from the tail, Sort$_{\mathrm{∆V}}$(P$_{1}$) can also deviate from the Gaussian distribution of the tail region because it is no longer in the tail region. This problem is referred to as the tail region reliability problem. Thus, a small P$_{1}$ increases the tail sample uncertainty and large P$_{1}$ decreases the tail region reliability. Therefore, P$_{1}$ should be carefully selected. To check the influence of the tail sample uncertainty, Gaussian distributions are generated with a given μ$_{\mathrm{∆V}}$ and σ$_{\mathrm{∆V}}$. Then, μ$_{\mathrm{∆V}}$ and σ$_{\mathrm{∆V}}$ are determined using the tail fitting method according to P$_{1}$ and the read yield is calculated. Fig. 6 shows the error rate between the real and the estimated read yield according to P$_{1}$ with different number of MC simulations. It is seen that the error rate decreases with increasing P$_{1}$ because of the mitigation of the tail sample uncertainty. The error rate also decreases with the number of simulations because the tail sample uncertainty decreases with decreasing probability of sample occurrence. Note that the Fig. 6 is only related to the tail sample uncertainty. To estimate the read yield correctly, not only the tail sample uncertainty but also the tail region reliability must be considered. And considering the tail region reliability is much more important than considering the tail sample uncertainty, because if a ΔV distribution does not show any tail region (the Gaussion distribution from the saturation region), it is the same as the normal fitting method causing a significant error. To have the tail region in the ΔV distribution, the number of MC simulations should be increased. And the best way to check if the ΔV distribution contains the tail region is to display the normal probability plot and observe the distribution. Because of the trade-off between the tail sample uncertainty and the tail region reliability, the smallest P$_{1}$ satisfying the target error rate of 5% should be selected. Note that even though the target error rate of 5% is set in this paper empirically based on whether the optimal voltage/size/time can be found or not, it can be changed according to applications. For example, if 10% error rate is acceptable in an application, the target error rate of 10% can be used. MC = 1,000 simulations cannot satisfy the target error rate in the tail region (P$_{1}$ < 50), but MC = 100,000 simulations can satisfy the target error rate in the deep tail region (P$_{1}$ < 10). However, this requires considerable a simulation running time. Therefore, P$_{1}$ of 10 is selected for MC = 10,000 simulations. It is worth noting here that even though the simulation running time depends on the number of nets in the netlist, the general sensing circuit simulation containing bit-line and source-line paths takes 10-20 (100-200) minutes in case of 10,000 (100,000) MC simulations.

Fig. 7. Error rate between real and estimated read yield according to P$_{2}$ when P$_{1}$ is 10 with MC = 10,000 simulations.

There is a trade-off between tail sample uncertainty and tail region reliability when selecting P$_{2}$. Fig. 7 shows the normalized error rate between the real and the estimated read yield according to P$_{2}$ when P$_{1}$ is 10 with MC = 10,000 simulations. If P$_{2}$ is close to P$_{1}$, the tail sample uncertainty problem exacerbates the error rate. If P$_{2}$ is very far from P$_{1}$, the tail region reliability problem can degrade the error rate. Thus, P$_{2}$ should be selected as close as possible to P$_{1}$ with minimized tail sample uncertainty. Therefore, P$_{2}$ of P$_{1}$ + 0.5σ is selected.

VI. MULTIPLE-POINT TAIL FITTING METHOD

Mathematically, μ$_{\mathrm{∆V}}$ and σ$_{\mathrm{∆V}}$ can be extracted by using the two-point tail fitting method. However, in practice, there is the tail sample uncertainty problem because of the limited number of simulations. Therefore, the extrapolation of the Gaussian distribution found by connecting the two points can have a significant error even if the effect of the tail sample uncertainty is small. Moreover, this is exacerbated as the number of simulations decreases. To address this, the multiple-point tail fitting yield estimation method is proposed. This method alleviates the tail sample uncertainty in the Sort$_{\mathrm{∆V}}$(P$_{1}$) by averaging the multiple values around P$_{1}$. Multiple P$_{1}$ is expressed as

where P$_{1}$ is a default value (e.g., 10 in MC = 10,000 simulation), and j is 1, 2, 3, ... , j$_{\mathrm{MAX}}$. Multiple P$_{2}$ is also expressed as

Then, Sort$_{\mathrm{∆V}}$$^{\mathrm{Multi\_P1}}$ and Sort$_{\mathrm{∆V}}$$^{\mathrm{Multi\_P2}}$ are defined to substitute the Sort$^{\mathrm{∆V}}$(P$_{1}$) and Sort$^{\mathrm{∆V}}$(P$_{2}$) in (9) and (10) as

Finally, by solving the following simultaneous equations, a more accurate tail fitting can be achieved.

Fig. 8. Error rate between real and estimated read yield according to j$_{\mathrm{MAX}}$ in multiple-point tail fitting method.

Fig. 9. Error rate between real and estimated read yield from normal fitting, importance sampling, tail fitting, and multiple-point tail fitting yield estimation methods according to the number of MC simulations.

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⁾.