Impact of spin transfer torque on the write error rate of a voltage-torque-based magnetoresistive random access memory

TL;DR

This paper theoretically analyzes how spin transfer torque influences the write error rate in voltage-torque-based MRAM, revealing that STT effects are negligible below certain current densities.

Contribution

It introduces a macrospin model analysis of STT effects on write error rates in voltage-torque MRAM, identifying the current density threshold where STT becomes significant.

Findings

Write error rate is insensitive to STT below 10^{10} A/m^{2}.

Derived the characteristic current density (~5×10^{11} A/m^{2}) where STT balances external torque.

STT can either assist or suppress magnetization precession depending on initial magnetization.

Abstract

Impact of spin transfer torque (STT) on the write error rate of a voltage-torque-based magnetoresistive random access memory is theoretically analyzed by using the macrospin model. During the voltage pulse the STT assists or suppresses the precessional motion of the magnetization depending on the initial magnetization direction. The characteristic value of the current density is derived by balancing the STT and the external-field torque, which is about 5$\times$ 10$^{11}$ A/m$^{2}$. The results show that the write error rate is insensitive to the STT below the current density of $10^{10}$ A/m$^{2}$.

Equations

$$[V,K_{\mathrm{u}},J](t)=\{\begin{array}{ll}[V_{\mathrm{p}},0,J_{\mathrm{p}}]& 0\le t\le t_{\mathrm{p}}\\ [0,K_{\mathrm{u}}^{(0)},0]& \mathrm{otherwise}\end{array}.$$

$$\frac{d\mathbf{m}}{dt}=-\gamma \mathbf{m}\times {\mathbf{H}}_{\mathrm{eff}}-\gamma \chi \mathbf{m}\times \left(\mathbf{m}\times \mathbf{p}\right)+\alpha \mathbf{m}\times \frac{d\mathbf{m}}{dt},$$

$${\mathbf{H}}_{\mathrm{eff}}={\mathbf{H}}_{\mathrm{ext}}+{\mathbf{H}}_{\mathrm{anis}}+{\mathbf{H}}_{\mathrm{therm}}.$$

$${\mathbf{H}}_{\mathrm{anis}}=\frac{2K_u(t)}{{\mu }_0{M}_{\mathrm{s}}}{m}_z(t){\mathbf{e}}_z,$$

$$⟨H_{\mathrm{therm}}^i(t)⟩=0,$$

$$⟨H_{\mathrm{therm}}^i(t)H_{\mathrm{therm}}^j(t^{\prime })⟩=\mu {\delta }_{i,j}\delta (t-t^{\prime }),$$

$$\mu =\frac{2\alpha k_{\mathrm{B}}T}{\gamma {\mu }_0{M}_{\mathrm{s}}\Omega },$$

$$\chi =\frac{\hslash PJ(t)}{2e{\mu }_0{M}_{\mathrm{s}}d},$$

$${\mathbf{T}}_{H_{\mathrm{ext}}}=-\gamma \mathbf{m}\times {\mathbf{H}}_{\mathrm{ext}},$$

$${\mathbf{T}}_{\mathrm{STT}}=-\gamma \chi \mathbf{m}\times \left(\mathbf{m}\times \mathbf{p}\right).$$

$$J_{\mathrm{p}}^{(\mathrm{c})}=\frac{2e{\mu }_0{M}_{\mathrm{s}}d}{\hslash P}{H}_{\mathrm{ext}}.$$

Full text

I INTRODUCTION

Magnetoresistive random access memory (MRAM) is a kind of non-volatile memory which stores information as stable magnetic states in the magnetoresistive devicesYuasa et al. (2013); Apalkov et al. (2016); Sbiaa and Piramanayagam (2017); Cai et al. (2017). The stored information is read by measuring the resistance which strongly depends on their magnetic states. The MgO-based magnetic tunnel junction (MTJ) is widely used as a basic element of the MRAM because of its large read signalParkin et al. (2004); Yuasa et al. (2004). Several types of writing schemes have been developed. The first commercial MRAM employed the field switchingSavtchenko et al. (2001); Engel et al. (2005). The field switching requires high write energy, order of 100 pJ/bit ITR , because the field is generated by the current flowing through the wire separated from the MTJ. Discovery of the spin transfer torque (STT) switching method Slonczewski (1996); Berger (1996) substantially decreased the write energy to the order of 100 fJ/bit Cai et al. (2017). However, it is still two orders of magnitude larger than the write energy of static random-access memory, $\sim$ 1 fJ/bit. In STT switching the main contribution to the write energy is Ohmic dissipation, i.e. Joule heating. In order to decrease the write energy further much effort has been devoted to decreasing the critical current density for STT-switchingYen et al. (2008); Bosu et al. (2016); Suess et al. (2017).

Voltage-torque (VT) switching is another attractive method for low power writing, which is based on voltage control of magnetic anisotropy (VCMA) in a thin ferromagnetic film Weisheit et al. (2007); Maruyama et al. (2009); Nozaki et al. (2010); Shiota et al. (2011); Nozaki et al. (2014); Lin et al. (2014); Amiri et al. (2015); Kanai et al. (2016); Grezes et al. (2016); Munira et al. (2016); Nozaki et al. (2016); Shiota et al. (2016); Nozaki et al. (2017); Cai et al. (2017); Song et al. (2017); Yamamoto et al. (2018); Ikeura et al. (2018); Matsumoto et al. (2018); Miriyala et al. (2019); Yamamoto et al. (2019); Matsumoto et al. (2019). The mechanism of the VCMA in a MgO-based MTJ is considered as the combination of the selective electron/hole doping into the d-electron orbitals and the induction of a magnetic dipole moment, which affect the electron spin through the spin-orbit interactionDuan et al. (2008); Nakamura et al. (2009); Tsujikawa and Oda (2009); Niranjan et al. (2010); Miwa et al. (2017). Very recently VT switching with very small write energy of about 6 fJ/bit was demonstrated by GrezesGrezes et al. (2016) et al. and independently by Kanai et al.Kanai et al. (2016).

The basic structure of the MTJ for the VT-MRAM is the same as that for the STT-MRAM except for the value of the resistance area (RA) product. The VT-MRAM has much larger RA product than that for the STT-MRAM to suppress the current density, or energy loss by Joule heating, at the operating voltage. Although the Joule heating at the operating voltage reduces as the resistance increases, the read time increases with increase of the resistance because it is determined by the $RC$ time constant, where $R$ and $C$ are the resistance and capacitance of the MRAM cell, respectively. The resistance of the MRAM cell should be designed to balance the energy consumption and read time.

The write error rate (WER) is another key metric to characterize the performance of the MRAM cell Worledge et al. (2010); Min et al. (2010); Nowak et al. (2011); Sun et al. (2012); Apalkov et al. (2016); Shiota et al. (2016, 2017); Ikeura et al. (2018); Yamamoto et al. (2019) The WER of STT-MRAM can be lowered by increasing the applied current densityWorledge et al. (2010); Min et al. (2010); Nowak et al. (2011); Sun et al. (2012); Apalkov et al. (2016). Nowak et al. reported the WERs below 10*-11* are reported for a 4-kb STT-MRAM chipNowak et al. (2011). The WER of the VT-MRAM is still higher than that of the STT-MRAM, which ranges form 10*-3* to 10*-5*Shiota et al. (2016, 2017); Yamamoto et al. (2019). People are working to reduce the WER by improving materialsShiota et al. (2017) as well as by shaping voltage pulseIkeura et al. (2018); Yamamoto et al. (2019). Until now the WER of the VT-MRAM has been studied in the high resistance condition to eliminate the effects of STT. However, for practical application, the resistance should be lowered to decrease the read time. It is important to know the minimum current density below which the impact of STT on the WER is negligible.

In this paper, WER of a perpendicularly magnetized VT-MRAM is theoretically studied with special attention to the impact of STT. It is shown that the STT assists or suppresses the precessional motion of the magnetization depending on the direction of the initial state, i.e. up-polarized or down-polarized. There exists a characteristic value of the current density above which the precessional motion, and therefore magnetization switching, is forbidden by the STT for one direction of the switching. It is found that for typical material parameters the WER is insensitive to the STT below the current density of $10^{10}$ A/m2.

II THEORETICAL MODEL

The circular-shaped MTJ-nanopillar we consider is schematically shown in Fig. 1(a). The insulating layer is sandwiched by the two ferromagnetic layers: the free layer and the reference layer. The direction of the magnetization in the free layer is represented by the unit vector $\bm{m}=(m_{x},m_{y},m_{z})$. The magnetization unit vector in the reference layer $\bm{p}$ is fixed to align in the positive $z$ direction, i.e. $\bm{p}=(0,0,1)$. The $x$ and $y$ axes are taken to be the in-plane directions, while the $z$ axis is taken to be the out-of-plane direction. The static external-field, $\bm{H}_{\rm ext}$, is applied in the positive $x$ direction. The positive current density, $J>0$, is defined as electrons flowing from the free layer to the reference layer. The size of the nanopillar is assumed to be so small that the magnetization dynamics can be described by the macrospin model.

The shape of the pulse of voltage, $V$, we assumed is shown in Fig. 1(b), where $V_{\rm p}$ and $t_{\rm p}$ represent the amplitude and the duration of the pulse, respectively. Without application of the voltage the free layer is assumed to have the out-of-plane uniaxial anisotropy which is characterized by the anisotropy constant $K_{\rm u}^{(0)}$. Here $K_{\rm u}^{(0)}$ represents the total anisotropy comprising the crystalline anisotropy, the interfacial anisotropy, and the shape anisotropy. As shown in Fig. 1(c) the anisotropy constant is assumed to decrease to zero by application of the voltage of $V_{\rm p}$ through the VCMA effect. During the voltage pulse the current with density of $J_{\rm p}$ flows through the MTJ-nanopillar as shown in Fig. 1(d). The time dependence of the voltage, anisotropy constant, and current density are summarized as

[TABLE]

The dynamics of the magnetization unit vector in the free layer is obtained by solving the Landau Lifshitz Gilbert (LLG) equation

[TABLE]

where the first, second, and third terms on the right hand side represent the torque due to the effective field $\bm{H}_{\rm eff}$, STT, and damping torque, respectively. The effective field comprises the external field, anisotropy field, $\bm{H}_{\rm anis}$, and thermal agitation field, $\bm{H}_{\rm therm}$, as

[TABLE]

The anisotropy field is defined as

[TABLE]

where $\bm{e}_{z}$ is the unit vector in the positive $z$ direction. The thermal agitation field is determined by the fluctuation-dissipation theorem Brown (1963); Callen and Welton (1951); Callen and Greene (1952); Callen et al. (1952); Greene and Callen (1952) and satisfies the following relations

[TABLE]

where 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 $\mu$ is given by

[TABLE]

where $k_{\rm B}$ is the Boltzmann constant, $T$ is temperature, and $\Omega$ is the volume of the free layer. The coefficient of the STT is defined as

[TABLE]

where $P$ is the spin polarization of the current, $e$ is the elementary charge, $d$ is the thickness of the free layerSlonczewski (1996); Stiles and Miltat (2005). Here the angle dependence of $\chi$ is neglected for simplicity.

The following parameters are assumed for numerical calculations: $\alpha$ = 0.1, $K_{\rm u}$ = 0.11 MJ/m3, $M_{\rm s}$ = 0.955 MA/m Yamamoto et al. (2019). The magnitude of the external-field is $H_{\rm ext}$ = 970 Oe. The diameter of the free layer is 40 nm. The thickness of the free layer is $d$= 1.1 nm. The spin polarization of current is $P=0.6$. The WERs are calculated from 106 trials.

III RESULTS AND DISCUSSION

Before showing the numerical results let us discuss the role of STT on the dynamics of $\bm{m}$. Since $\bm{H}_{\rm ext}$ is the static external-field, the torque due to $\bm{H}_{\rm ext}$ is exerted on $\bm{m}$ all the time. The STT exists only during the pulse, where the anisotropy constant is zero. During the pulse the following two kinds of torques give the dominant contributions to the magnetization dynamics: One is the external field torque,

[TABLE]

and the other is the STT,

[TABLE]

Neglecting the thermal agitation and damping, the trajectory of magnetization precession for switching can be well approximated by the the semi-arc on the $yz$ plane. Therefore the vector $\bm{m}\times\bm{p}$ is parallel or anti-parallel to the external-field depending on the sign of $m_{y}$.

For the switching from the up-state ($m_{z}>0$) to the down-state ($m_{z}<0$), the vector $\bm{m}\times\bm{p}$ is parallel to the external-field, and therefore $\bm{T}_{\rm STT}$ is parallel to $\bm{T}_{H_{\rm ext}}$ as shown in Fig. 1(e). The angular velocity of the precessional motion of $\bm{m}$ is increased by the STT as if the external-field is enhanced.

On the contrary, for the switching from the down-state to the up-state, $\bm{T}_{\rm STT}$ is anti-parallel to $\bm{T}_{H_{\rm ext}}$ as shown in Fig. 1(f). The angular velocity of $\bm{m}$ is decreased by the STT as if the external field is reduced. There exists a characteristic current density above which the STT overcomes the external field torque, which is obtained by solving $T_{H_{\rm ext}}=T_{\rm STT}$ as

[TABLE]

For the parameters stated before the value of the characteristic current density is $J_{\rm p}^{(\rm c)}=5.4\times 10^{11}$ A/m2, which is as large as the typical value of the critical current density for the STT switching.

The value of $J_{\rm p}^{(\rm c)}$ gives a rough estimation of the current density above which the switching from the down-state to the up-state is forbidden. Even if the current density is smaller than $J_{\rm p}^{(\rm c)}$ the STT can affect the precessional motion and increase or decrease the WER. For quantitative understanding of the impact of STT on the WER we perform numerical simulations based on Eq. (2). There are two approaches to obtain the WER starting from Eq. (2). One is the Fokker-Planck-equation approachBrown (1963); Apalkov and Visscher (2005, 2005); Tzoufras (2018) and the other is the Langevin-equation approachShiota et al. (2016); Ikeura et al. (2018); Yamamoto et al. (2018); Matsumoto et al. (2018); Yamamoto et al. (2019); Matsumoto et al. (2019). In principle these two approaches give the same results because they are based on the same LLG equation. Here we employ the Langevin-equation approach because we have many experiences on this approach and have reproduced the experimentally observed WER very well as reported in Refs. 29 and 33.

Figure 2(a) shows the $t_{\rm p}$ dependence of the WER for the switching from the up-state to the down-state. The initial states are prepared by relaxing the magnetization from the equilibrium direction at $T=0$ with $m_{z}>0$ for 5 ns before the beginning of the pulse. The success or failure of switching is determined by the sign of $m_{z}$ at 5 ns after the end of the pulse. During this 5 ns the voltage is not applied, and therefore the magnetization relaxes to the equilibrium directions. The distributions of $m_{z}$ at the beginning of the pulse and at 5 ns after the end of the pulse are well localized around the equilibrium directions (see APPENDIX A). The results for $J_{\rm p}$ =0 are represented by the circles. Since the WER satisfies the binomial distribution, the standard deviation of the WER is given by $\sqrt{q(1-q)/N}$, where $q$ is the switching probability, $1-p$ is the WER, and $N$ is the number of trials. For $J=0$ the WER takes the minimum value of 5.46$\times$10*-4* at $t_{\rm p}=0.18$ ns. The corresponding standard deviation for $N=10^{6}$ trials is 2.34$\times$10*-5*, which is smaller than the radius of circles plotted in Figs. 2(a) and 2(b). The results for the current density of $J_{\rm p}=10^{9}$, $10^{10}$, $10^{11}$, $10^{12}$ are represented by the solid, dotted, dashed, and dot-dashed curves. Below $10^{10}$ A/m2 the $t_{\rm p}$ dependence of WER is almost the same as that for $J_{\rm p}$ =0 because $T_{\rm STT}$ is much smaller than $T_{H_{\rm ext}}$. Above $10^{11}$ A/m2 the pulse width at which the WER is minimized decreases with increase of the current density, because the STT assists the precessional motion around the external-field.

Figure 2(b) shows the $t_{\rm p}$ dependence of the WER for the switching from the down-state to the up-state. Similar to Fig. 2(a) the $t_{\rm p}$ dependence of the WER is almost the same as that for $J_{\rm p}$ =0 for $J_{\rm p}\leq 10^{10}$ A/m2. At $J_{\rm p}=10^{11}$ A/m2 the pulse width at which the WER is minimized increases with increase of the current density, because the STT suppresses the precessional motion around the external-field. At $J_{\rm p}=10^{12}$ A/m2 the dip in the $t_{\rm p}$ dependence of WER disappears as shown by the dot-dashed curve because $T_{\rm STT}$ exceeds $T_{H_{\rm ext}}$ much earlier than one half of the precession period. The magnitude of the STT is proportional to the sine of the relative angle, $\theta_{\rm r}$, between $\bm{m}$ and $\bm{p}$. The relative angle is $\theta_{\rm r}=\pi$ at the initial down-state and decreases as the magnetization precesses toward the up-sate ($\theta_{\rm r}=0$). The precession stops once $\theta_{\rm r}$ reaches a certain critical angle where the external-field torque is canceled with the STT. For the switching from the down-state to the up-state the critical angle increases with increase of the current density.

In Fig. 3 the minimum values of WER, WERmin, are plotted as a function of the current density. The results for the switching from the up-state to the down-state are shown by the blue triangles. The value of WERmin at $J_{\rm p}=0$ is indicated by the dotted line as a guide. Below the current density of $10^{11}$ A/m2 the WERmin takes almost the same value as that at $J_{\rm p}=0$. It shows a shallow decrease above $10^{11}$ A/m2. The red circles show WERmin for the switching from the down-state to the up-state. One can easily confirm that below the current density of $10^{10}$ A/m2 the WERmin takes almost the same value as that at $J_{\rm p}=0$. It shows a shallow increase until $J_{\rm p}$ reaches $10^{11}$ A/m2. Above the current density of $10^{11}$ A/m2 it shows a rapid increase and reaches almost unity at $J_{\rm p}$ = $2\times 10^{12}$ A/m2. At around $J_{\rm p}=J_{\rm p}^{(\rm c)}$ there appears a plateau where the WER is insensitive to the variation of $J_{\rm p}$.

In order to understand the mechanism for appearance of the plateau, let us look at the $t_{\rm p}$-dependence of the WER at the current density around $J_{\rm p}^{(\rm c)}$. Figure 4(a) shows the $t_{\rm p}$-dependence of WER at $J_{\rm p}$=0.3 and 0.4 TA/m2, where the WERmin shows a rapid increase. The WER takes the minimum value at the second dip which corresponds to one half of the precession period. The appearance of the first dip, or the appearance of the bump between the first and the second dips, is originated from the thermally induced precession-orbit transition of magnetization as discussed in Ref. 29. The position of the bump corresponds to one quarter of the precession period, at which the magnetization is on the equator plane on the Bloch sphere, i.e. $m_{z}=0$. Since the magnetization around this direction has high anisotropy energy in the relaxation process it takes long time for the magnetization to relax to the equilibrium direction, the up-state or the down-state. Therefore the probability of switch failure or the WER is enhanced around the pulse width of one quarter of the precession period. As the current density increases from $J_{\rm p}$=0.3 to 0.4 TA/m2 the position of the second dip moves to the longer $t_{\rm p}$ and the minimum value increases.

Further increase of current density eliminates the second dip and moves the position of the WERmin to the longer $t_{\rm p}$ as shown in Fig. 4(b). From $J_{\rm p}$=0.5 to 0.7 TA/m2 the increase of the current density does not change the value of the WERmin very much but decreases WER at $t_{\rm p}$ longer than the first dip because in this range of the current density the STT exceeds external-field torque around one quarter of the precession period. At $J_{\rm p}$= 0.8 TA/m2 the first dip, or the bump, disappears. Above the current density of 0.9 TA/m2 the WERmin increases with increase of $J_{\rm p}$ as shown in Fig. 4(c).

IV SUMMARY

In summary the impact of STT on the WER of a VT-MRAM is theoretically investigated. The characteristic value of the current density above which the precessional motion is forbidden by the STT is derived by balancing the STT and the external-field torque. The WER is insensitive to the STT at the current density below $10^{10}$ A/m2.

Acknowledgements.

This work was partly supported by JSPS KAKENHI Grant Number 19H01108, and the ImPACT Program of the Council for Science, Technology and Innovation.

Appendix A Distribution of $m_{z}$

In this section we discuss the distributions of $m_{z}$ at the beginning of pulse and at the 5 ns after the end of pulse. The states at the beginning of the pulse are prepared by relaxing the magnetization from the equilibrium direction at $T=0$ for 5 ns. The success or failure of switching is determined by the sign of $m_{z}$ at 5 ns after the end of the pulse. The relaxation time of 5 ns is set to be long enough for magnetization to be relaxed around the equilibrium directions. To confirm the validity of this procedure we show the distributions of $m_{z}$ for the switching from the up-state to the down-state as histograms in Figs. 5(a) – 5(d).

The distribution of $m_{z}$ at the beginning of pulse ($t=0$) is shown in Fig. 5(a) where the values of $m_{z}$ of the equilibrium directions at $T=0$ are indicated by the vertical dotted lines. The initial distribution is independent of the value of $J_{\rm p}$ because it is prepared by relaxing the magnetization without applying current. The distribution is well localized in the vicinity of the positive equilibrium value.

Application of the voltage pulse induces the precessional motion of magnetization and switches the magnetization direction with a certain probability. Then the magnetization relaxes to the equilibrium directions because $J=0$ and $K_{\rm u}=K_{\rm u}^{(0)}$ for $t\geq t_{\rm p}$. The success or failure of switching is determined by the sign of $m_{z}$ at $t=t_{\rm p}$ + 5 ns. In Figs. 5(b), (c), and (d) the distributions at $t=t_{\rm p}$ + 5 ns are plotted for $J_{\rm p}$ = 0, $10^{11}$, and $10^{12}$ A/m2 , respectively. In these figures the pulse width is assumed to be $t_{\rm p}=0.18$ ns at which the WER for $J_{\rm p}$ = 0 is minimized. As shown in Figs. 5 (b) – (d) the distributions are well localized in the vicinity of the equilibrium values, which enables to clearly determine the success or failure of switching.