Methods for reducing write error rate in voltage-induced switching having prolonged tolerance of voltage-pulse duration

Simulating the magnetization dynamics in a perpendicularly-magnetized free layer with Langevin equation, we investigated methods for reducing write error rate (WER) in voltage-induced switching with long tolerance of voltage-pulse duration (tp). The simulation results show that WER can be reduced by increasing the perpendicular anisotropy (Ku) before and after the application of voltage or by increasing both Ku and the in-plane external magnetic field.Simulating the magnetization dynamics in a perpendicularly-magnetized free layer with Langevin equation, we investigated methods for reducing write error rate (WER) in voltage-induced switching with long tolerance of voltage-pulse duration (tp). The simulation results show that WER can be reduced by increasing the perpendicular anisotropy (Ku) before and after the application of voltage or by increasing both Ku and the in-plane external magnetic field.


I. INTRODUCTION
Voltage-induced magnetization switching [1][2][3][4][5][6][7][8][9][10][11][12][13][14] has been attracting a great deal of attention because it enables low-powerconsumption writing in magnetoresistive random access memory (MRAM). In voltage-induced magnetization switching, anisotropy field is reduced through voltage control of magnetic anisotropy (VCMA) effect, [1][2][3][4][5] which induces magnetization precession around external magnetic field. 15 The switching can be completed by turning off the voltage after a half period of precession. [6][7][8][9][10][11][12] However, this dynamic switching requires the adjustment of the voltage-pulse duration (tp) to stop the precessional motion of the magnetization and to obtain low write error rate (WER). The range of tp where WER is below 1 × 10 −3 is as narrow as several hundred picoseconds. 10,14 We call this range of tp as the tolerance of tp just for convenience. Broadening the tolerance of tp is desirable because of the distribution of tp from pulsed power supply in MRAMs and the distribution of precession periods among the MRAM cells. In the dynamic switching, the tolerance of tp has been prolonged by decreasing minimum WER of perpendicular MRAMs. The WER has been improved by increasing external in-plane magnetic field (Hext), 13 or by increasing the perpendicular anisotropy constant (Ku) before and after the application of writing voltage pulse, 12,16,17 or by increasing both Ku and Hext. 10,14 In our previous study, 18 we proposed the voltage-induced magnetization switching having prolonged tolerance of tp in perpendicularly magnetized free layer at room temperature. In this switching, the switched state is kept even under the persistent application of the voltage, and so the precise adjustment of tp is not required. This prolonged tolerance of tp can be realized by the energy dissipation through damping torque during the precessional motion of magnetization. 2 However, the WER around 10 −4 in the large free-layer volume (V F ) of 140 2 π × 2 nm 3 needs further improvement. 18 In this letter, we simulated the magnetization dynamics in the free layer with smaller V F using Langevin equation and found that the WER in the switching with prolonged tolerance of tp can be reduced by increasing Ku before and after the application of writing voltage pulse and by increasing both Ku and Hext.

II. MODEL
The system we consider is schematically shown in Fig. 1(a). The lateral size of the nano-pillar is assumed to be so small that the magnetization dynamics can be described by the macrospin model. The direction of the magnetization in the free layer is represented by the unit vector m = (mx, my, mz) = (sin θ cos ϕ, sin θ sin ϕ, cos θ), where θ and ϕ are the polar and azimuthal angles of m. The x−axis is parallel to the direction of external in-plane magnetic field (Hext), hence the equilibrium azimuthal angle at 0 K (ϕ [0] ) is ϕ [0] = 0 ○ . Here and hereafter (θ [0] , ϕ [0] ) are equilibrium angles at zero-bias voltage and 0 K. The magnetization in the reference layer is fixed to align in the positive z-direction.
The energy density of the free layer is given by 19 Here, the demagnetization coefficients, Nx, Ny and Nz are assumed to satisfy Nz ≫ Nx = Ny. μ 0 is the vacuum permeability, and Ms is the saturation magnetization of the free layer. Ku and K eff are the perpendicular anisotropy constant and its effective perpendicular where the demagnetization energy is subtracted from Ku. The value of Ku can be varied by applying bias voltage, V, through the voltagecontrol of magnetic anisotropy (VCMA) effect. In this letter, K (0) eff indicates K eff at zero-bias voltage, and K (+V) eff indicates K eff during tp as illustrated in Figs. 1(b) and (c). Throughout this letter, we assume the perpendicularly magnetized free layer with the saturation magnetization of Ms = 1400 kA/m, V F = πr 2 t F = 7854 nm 3 , Nx = Ny = 0.0175, and Nz = 0.9650. 20 Here r = 50 nm is the radius of a junction area, and t F = 1 nm is the thickness of the free layer.
The thermally-agitated magnetization dynamics was simulated with the following Langevin equation, 21 h represents the thermal agitation field satisfying the following relations: where T is the temperature assumed as T = 300 K, ι, κ = x, y, z, t is time, and ⟨X⟩ denotes the statistical average of X. Unless noted otherwise, we assumed the time evolution of voltage and K eff shown in Figs. 1(b) and (c). Here, tp and the relaxation time (t relax ) before and after tp are tp = t relax = 10 ns.  Fig. 2(a). Low-WER region in K (+V) eff, c1 < K (+V) eff < K (+V) eff, c2 appears as a result of the switching with the prolonged tolerance of tp. 18 K (+V) eff, c1 and K (+V) eff, c2 are the lower and upper boundaries of K (+V) eff where the voltage-induced switching with the prolonged tolerance of tp can be induced at appropriate α, and analytical expressions of K (+V) eff, c1 and K (+V) eff, c2 were given in Ref. 18. In the low-WER region, the WER was minimized down to 1.5 × 10 −4 at the conditions of Fig. 1(d). Hereafter the minimized WER in the K (+V) eff − α dependence is indicated as WER min . Assuming constant K (0) eff = 600 kJ/m 3 , we calculated the Hext dependence of WER min , and show the results as blue solid circles on a blue curve in Fig. 2(b). WER min was reduced up to Hext(= 2.5 kOe) = 199 kA/m. This is because the magnitudes of the field torque and damping torque increase with Hext overwhelming the magnitude of the field torque by h. The other reason is shorter switching time at high Hext. On the other hand, the further increase of Hext deteriorated WER min .

III. RESULTS
Please note that, in the case of the switching of a micron-size magnet which holds the spatial distribution of ΔK eff = K (0) eff − K (+V) eff , the increase of Hext excites spatially-inhomogeneous magnetization oscillations. 15 The inhomogeneous oscillations tends to disturb the switching with prolonged tolerance of tp. 18 In the case of an iron garnet, 15 however, the inhomogeneous oscillations increased effective α during tp, and the increased α facilitated the switching with prolonged tolerance of tp. The combination of low α during t relax and high α during tp might be advantageous for the reduction of WER  in the switching with prolonged tolerance of tp, but the detailed analysis on micron-size magnet is the beyond the scope of this letter. Please also note that the thermal agitation of m during t relax hardly deteriorated WER min even though the thermal stability was reduced at the high Hext. In order to eliminate the thermal agitation during t relax , we conducted the simulations where t relax = 0 ns and m starts the motion from (θ, ϕ) = (θ [0] , 0 ○ ). θ [0] is given by 18 and θ [0] was varied from 3 ○ (for Hext = 0.5 kOe) to 36 ○ (for Hext = 5 kOe) in Fig. 2(b). Those results are indicated as green open squares on a green curve in Fig. 2(b). The green curve and the blue curve show similar results. This result indicates that the dominant cause of write errors for the blue curve is the thermal agitation during tp = 10 ns rather than that during t relax . The angles (θ [0] , 0 ○ ) from which m starts the motion at the beginning of tp seems to take a crucial role for the dynamics of m during tp = 10 ns and WER min .
In order to investigate the effect of θ [0] on WER min , we calculated K (0) eff dependence of WER min at constant Hext as shown in Fig. 3(a). Here, in order to decrease θ [0] from 30 ○ to 2 ○ , K (0) eff was increased from 140 to 2000 kJ/m 3 at constant Hext(= 1 kOe) = 79.6 kA/m. In Fig. 3(a), the calculated K (0) eff dependence of WER min is indicated as blue solid circles on a blue curve and the simulation results calculated at t relax = 0 ns is indicated as green open circles on a green curve. The results show that the WER min decreases with the increase of K (0) eff ( Fig. 3(a)) and the decrease of θ [0] (the inset in Fig. 3(a)). This is because, at high K (0) eff and subsequently small θ [0] , m can avoid the switching trajectories where m is sensitive to h. Especially around (θ, ϕ) = (90 ○ , 0 ○ ), the anisotropy field by K (+V) eff is small. Around this point, m is fluctuated by h rather than being driven by the fields from K (+V) eff and Hext. Varying Hext and K (0) eff and subsequently maintain θ [0] = 10 ○ , we calculated Hext dependence of WER min as shown in Fig. 3(b). Here, for example, K (0) eff = 200 kJ/m 3 was assumed for Hext = 500 Oe, and K (0) eff = 2000 kJ/m 3 was assumed for Hext = 5 kOe. The WER min monotonically decreases with increase of Hext. This is because, even at high Hext, m can avoid the switching trajectories where m is sensitive to h.

IV. CONCLUSION
We theoretically investigated methods for reducing WER in voltage-induced switching with prolonged tolerance of tp. The simulations using Langevin equation show that WER can be reduced by increasing K (0) eff or by increasing both K (0) eff and Hext.

ACKNOWLEDGMENTS
This work was partly supported by the ImPACT Program of the Council for Science, Technology and Innovation, and JSPS KAKENHI Grant No. JP19K05259.