Characteristic time of transition from write error to retention error in voltage-controlled magnetoresistive random-access memory ¹¹1 This work was partly supported by JSPS KAKENHI Grant Number JP19H01108, and JP20K12003.

Keeping information for enough time to ensure the reliability for computing is a key policy of the modern computing systems based on so-called the von Neumann architecture. Hierarchical memory system with cache, main memory, and storage has been successfully realized an effective and reliable computing [1, 2]. However, the performance of the von Neumann architecture is limited not only by the performance of the central processing unit (CPU) but by the data transfer rate between the memory and the CPU through the system bus, which is known as the von Neumann bottleneck.

In-memory computing is a promising approach to alleviate the von Neumann bottleneck, where resistive memory devices are used for both processing and memory [3, 2]. In-memory computing generally requires fast, high-density, low-power, scalable resistive memory devices, such as resistive random access memory (ReRAM) [4, 5], phase-change memory (PCM) [6, 7, 8, 9], ferroelectric RAM (FeRAM) [10], and magetoresistive RAM (MRAM) [11, 12, 13, 14, 15, 16, 17, 18]. A crosspoint array of these resistive memory devices provides a hardware accelerator for the matrix-vector multiplication (MVM).

Recently much attention has been focused on the in-memory computing because it provides a powerful MVM tool for machine learning tasks such as image recognition, object detection, voice recognition, and time-series data analysis [7, 19, 20]. Because the final outputs of these machine learning tasks are provided as a probability and its algorithms are error tolerant [21, 22], the accuracy required for the memory is not very high to obtain a practical result. T. Hirtzlin et al. showed that up to 0.1% bit error rate can be tolerated with little impact on recognition performance of a standard binary neural network [23].

Voltage-control (VC) MRAM is a promising candidate for the key element of the fast and low-power in-memory computing because of the fast ($\sim$ 0.1 ns) and low-power ($\sim$ 1 fJ) writing [24, 25, 26, 27, 28, 29, 30, 31, 32]. The VC MRAM uses the voltage control of magnetic anisotropy (VCMA) effect [24, 26, 33, 34, 35] to induce the precession of magnetization for writing, and little Joule heating is generated.

The main causes of bit errors in VC MRAM are write error and retention error. The write error in VC MRAM is a failure in magnetization switching due to thermal agitation field during the precession and relaxation. The write error rate (WER) of the VC MRAM is of the order of $10^{-6}$ [29] which is low enough to obtain practical recognition accuracy. The retention error in VC MRAM is an accidental switching of magnetization that occurs while retaining the data. The origin of the retention error is also thermal agitation field. The retention error rate (RER) of the VC MRAM can be calculated by solving the Fokker-Planck equation as shown in Ref. [37]. As the size of the memory cell decreases, the data retention time decreases, which causes a transition from the write-error-dominant region to the retention-error-dominant region at a certain operating time. For developing the fast and low-power in-memory computing it is important to know the characteristic time of the transition.

In this paper we define the characteristic time, $t_{\rm c}$, of the transition from the write-error-dominant region to the retention-error-dominant region and analyze the dependence of the characteristic time on the effective anisotropy constant, $K_{0}$. We obtained the approximate expression of the characteristic and show that $t_{\rm c}$ increases with increase of $K_{0}$ for large value of $K_{0}$.

Figure 1(a) schematically shows the magnetic tunnel junction nanopillar which is the basic element of the VC MRAM. The nonmagnetic insulating layer is sandwiched by the two ferromagnetic layers called the free layer and the reference layer. The size of the MTJ nanopillar is assumed to be so small that the magnetization dynamics can be described by the macrospin model. We denote the direction of the magnetization in the free layer by the unit vector ${\bm{m}}=(m_{x},\ m_{y},\ m_{z})$. Application of the voltage, $V$, pulse reduces the magnetic anisotropy through the VCMA effect and induces the precession of magnetization around the static external field, ${\bm{H}}_{\rm ext}$, applied in the positive $x$ direction. The magnetization unit vector in the reference layer, $\bm{p}$, is fixed to the positive $z$ direction, i.e. $\bm{p}=(0,0,1)$. The information stored in the VC MRAM is read as the change of the tunnel resistance due to the tunnel magnetoresistnace effect [38, 39, 40, 41].

Figure 1(b) shows the shape of the voltage pulse and the corresponding time evolution of the effective anisotropy constant, $K$, in our simulations. The initial direction of $\bm{m}$ at $t=0$ is set to the equilibrium direction with $m_{z}>0$, which minimizes the energy density

where $K_{0}$ is the effective anisotropy constant of the free layer at $V=0$, which consists of the crystal magnetic anisotropy and the demagnetizing energy density, $\mu_{0}$ is the permeability of vacuum, $M_{\rm s}$ is the saturation magnetization. For the first 5 ns, the initial state are thermalized by the thermal agitation field to obey the Boltzmann distribution at temperature, $T$. During the thermalization no voltage is applied, and the effective anisotropy constant is $K_{0}$. Then the voltage pulse with the magnitude of $V_{\rm p}$ is applied for the duration of $t_{\rm p}$. During the pulse the effective anisotropy constant is zero, and the magnetization precesses around the external magnetic field. The pulse width, $t_{\rm p}$, is set to a half of the precession period to switch the magnetization. After the pulse the magnetization is relaxed under the condition of $K=K_{0}$. The WER is calculated using the direction of magnetization at 5 ns after the end of the pulse. The contribution from the retention error is estimated by analyzing the error rate after the relaxation for the extra relaxation time, $t_{\rm r}$.

Temporal evolution of $\bm{m}$ is obtained by solving the Landau-Lifshitz-Gilbert (LLG) equation,

where $\gamma$ is the gyromagnetic constant and $\alpha$ is the Gilbert damping constant. The first and second terms on the right hand side represent the torque resulting from the effective field, $\bm{H}_{\rm eff}$, and the damping torque, x respectively. The effective field comprises the external field, anisotropy field, $\bm{H}_{\rm anis}$, and thermal agitation field, $\bm{H}_{\rm therm}$, as

The anisotropy field is defined as

where $\bm{e}_{z}$ is the unit vector in the positive $z$ direction. The thermal agitation field is determined by the fluctuation-dissipation theorem [42, 43, 44, 45, 46] and satisfies the following relations

where $\langle\ \rangle$ denotes the statistical average, the indices $i$, $j$ denote the $x$, $y$, and $z$ components of the thermal agitation field. $\delta_{i,j}$ represents Kronecker’s delta, and $\delta(t-t^{\prime})$ represents Dirac’s delta function. The coefficient $\xi$ is given by

where $k_{\rm B}$ is the Boltzmann constant, $T$ is temperature, and $\Omega$ is the volume of the free layer. The WER is obtained by counting the number of failure in $10^{7}$ trials of writing. The success or failure of each trial is determined by the sign of $m_{z}$ at $t=t_{\rm p}+10$ ns [see Fig. 1(b)]. We also calculate the error rate after $t=t_{\rm p}+10$ ns by performing the same simulation until $t=t_{\rm p}+t_{\rm r}+10$ ns.

The RER is the probability of the accidental switching of magnetization by thermal agitation during the period of data retention. We employ the theory of RER given by Kalmykov in Ref. [37], to save the simulation time, and to provide a theoretical analysis. Introducing the escape ratio, $\Gamma$, over the potential barrier which separate the two different equilibrium directions of the magnetization, the RER is given by the switching probability defined as

which is the solution of the master equation as

The inverse of $2\Gamma$ is called the relaxation time and is denoted by $\tau$, i.e., $\tau=1/(2\Gamma)$.

Kalmykov derived the expressions of $\tau$ for three different dissipation regions: the very low damping (VLD, $\alpha<0.001$) region, the intermediate-to-high damping (IHD, $\alpha\gg 1$) region, and the crossover ($0.01<\alpha<1$) region. For VC MRAM, the typical value of $\alpha$ is in the crossover region. The expression of $\tau$ in the crossover region is given by Eq. (18) in Ref. [37] as

where

and

Here, $\tau_{N}=\beta M_{\rm s}(1+\alpha^{2})/(2\gamma\alpha)$, $h=\beta\mu_{0}M_{\rm s}H_{\rm ext}/(2\sigma)$, $\sigma=\beta K$, and $\beta=\Omega/(k_{\rm B}T)$. The index of $S_{i}$ is the label of the equilibrium directions, e.g., ”1” for $m_{z}>0$ and ”2” for $m_{z}<0$. Since the system we consider has an inversion symmetry with respect to the $x-y$ plane, i.e., $z\to-z$, Eq. (10) can be expressed as $\tau=\tau_{\rm IHD}A(2\alpha S_{1})/A(\alpha S_{1})^{2}$.

The following parameters are assumed. The saturation magnetization is $M_{\rm s}$ = 0.955 MA/m [36], and the pulse width is $t_{\rm p}=0.18$ ns. The magnitude of the external field is $H_{\rm ext}$ = 1 kOe. The diameter and thickness of the free layer are 40 nm and 1.1 nm, respectively. The Gilbert damping constant $\alpha$ = 0.1, and temperature is $T=$ 300 K.

Figure 2 shows $K_{0}$ dependence of the WER, where the dot represents the mean of the WER, and the error bar stands for the standard deviation of the WER. Each data points are obtained by averaging the results of $10^{7}$ trials. The WER is about 0.5 for small $K_{0}$ $(<0.8)$ because the anisotropy energy is too small to retain the direction of the initial magnetization for 5 ns. The magnetization is almost equally distributed around the equilibrium directions with $m_{z}>0$ and $m_{z}<0$ at the beginning of the pulse. In other words, the retention time is less than 5 ns. For $K_{0}>0.8$, the WER exponentially decreases with increase of $K_{0}$ and reaches below $10^{-6}$ at $K_{0}=1.6\times 10^{5}$ J/m³.

Figure 3 shows an example of the typical $t_{\rm r}$-dependence of the error rate. The effective anisotropy constant is assumed to be $K_{0}=1.0\times 10^{5}$ J/m³. The dots connected by the solid curve represents the results obtained by numerically solving the LLG equation. The gray horizontal solid line indicates the value of the WER: $7.3\times 10^{-3}$. The Dashed curve represents the RER of Eq. (8). For short $t_{\rm r}$, the error rate is much larger than the RER and is dominated by the WER. For large $t_{\rm r}$, the error rate converges to the RER, i.e., the RER dominates the error rate. We introduce the the characteristic time, $t_{\rm c}$, for transition from the write-error-dominant region to the retention-error-dominant region as the intersection point of the dashed curve and the gray horizontal solid line, as shown in Fig. 3.

Since the RER equals to WER at $t_{\rm r}=t_{\rm c}$, we have

where $w$ denotes the WER. Assuming that $w\ll 1$, Eq. (14) is approximated as

The $K_{0}$-dependence of $t_{\rm c}$ is determined by the $K_{0}$-dependence of $w$ and $\tau$. The $K_{0}$-dependence of $w$ is already shown in Fig. 2. $w$ decreases exponentially with increase of $K_{0}$.

The $K_{0}$-dependence of $\tau$ is shown in Fig. 4(a). On contrary to the $K_{0}$-dependence of $w$, $\tau$ increases with increase of $K_{0}$. The $K_{0}$-dependence $t_{\rm c}$ is determined by competition between the decreasing contribution from $w$ and the increasing contribution from $\tau$. Figure 4(b) shows the $K_{0}$-dependence of $t_{\rm c}$. The WER at each point shown in Fig. 2 is indicated by color. Although $t_{\rm c}$ is almost independent of $K_{0}$ for small $K_{0}\,(<1.0)$, it exponentially increases with increase of $K_{0}$ for large $K_{0}\,(>1.0)$ due to the exponential dependence of $\tau$ on $K_{0}$ shown in Fig. 4(a).

We focused on the influence of $K_{0}$, which is the most important parameter of VC MRAM because of its operating principle. We briefly comment on the influence over $t_{\rm c}$ from another key parameter, the damping constant, $\alpha$. With increase of $\alpha$, the WER becomes larger. The present values for the pulse width and the pulse amplitude are the optimal values that minimize the WER. The increase of the WER results in the increase of $t_{\rm c}$ (see Fig. 3).

We also briefly comment on the RER of the other MRAM. The spin-transfer torque (STT) MRAM and spin-orbit torque (SOT) MRAM are principally does not required the application of an external magnetic field, unlike the VC MRAM. The analytical formula of the RER for STT/SOT-MRAM has been derived by Brown [42].

In summary, we study the characteristic time, $t_{\rm c}$, of the transition from write-error-dominant region to retention-error-dominant region in VC MRAM by paying special attention to the dependence of $t_{\rm c}$ on the effective anisotropy constant, $K_{0}$. We show that the characteristic time is approximately expressed as $t_{\rm c}=2\,w\,\tau$, where $w$ is the write error rate, and $\tau$ is the relaxation time derived by Kalmkov [37]. The $K_{0}$-dependence of $t_{\rm c}$ is determined by competition between the $K_{0}$-dependence of $w$ and $\tau$. We show that for large $K_{0}$, $t_{\rm c}$ increases with increase of $K_{0}$ similar to $\tau$. The characteristic time is a key parameter for designing the VC MRAM for the variety of applications such as machine learning and artificial intelligence because the working frequency should be higher than $1/t_{\rm c}$ to ensure the practical recognition accuracy required.