## Abstract

We studied the effect of a current pulse width on current-induced magnetization switching in magnetic tunnel junctions based on a macrospin model of the free layer. We performed finite temperature Langevin simulations of the Landau-Lifshitz-Gilbert-Slonczewski equation with an additional spin-torque term. By evaluating the switching current density, we obtained the diagram in the plane of the critical current density and the pulse width at 300 K. As the pulse width increased, we observed an adiabatic regime in the shorter pulse widths, an intermediate crossover regime, and a thermally activated regime in long pulse widths. We found that the easy-plane anisotropy field shifts the crossover pulse width to the lower pulse width, suggesting that the reversed region is enhanced by controlling the device shape. Our results are consistent with those of recent experiments over the pulse widths ranging from 10-1 to 105 ns.

Original language | English |
---|---|

Article number | 07D130 |

Journal | Journal of Applied Physics |

Volume | 105 |

Issue number | 7 |

DOIs | |

Publication status | Published - 2009 |

## ASJC Scopus subject areas

- Physics and Astronomy(all)

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*Journal of Applied Physics*,

*105*(7), [07D130]. https://doi.org/10.1063/1.3075854

**Landau-Lifshitz-Gilbert study of the effect of pulse width on spin-transfer torque magnetization switching.** / Sugano, R.; Ichimura, M.; Takahashi, S.; Maekawa, S.

Research output: Contribution to journal › Article › peer-review

*Journal of Applied Physics*, vol. 105, no. 7, 07D130. https://doi.org/10.1063/1.3075854

**Landau-Lifshitz-Gilbert study of the effect of pulse width on spin-transfer torque magnetization switching**. In: Journal of Applied Physics. 2009 ; Vol. 105, No. 7.

}

TY - JOUR

T1 - Landau-Lifshitz-Gilbert study of the effect of pulse width on spin-transfer torque magnetization switching

AU - Sugano, R.

AU - Ichimura, M.

AU - Takahashi, S.

AU - Maekawa, S.

N1 - Funding Information: Sugano R. 1,2 a) Ichimura M. 1,2 Takahashi S. 2,3 Maekawa S. 2,3 1 Advanced Research Laboratory , Hitachi, 1-280, Higashi-Koigakubo, Kokubunji-shi, Tokyo 185-8601, Japan 2 JST, CREST , 5, Sanbancho, Chiyoda-ku, Tokyo 102-0075, Japan 3 Institute for Materials Research, Tohoku University , Sendai 980-8577, Japan a) Electronic mail: ryoko.sugano.qq@hitachi.com . 01 04 2009 105 7 07D130 11 11 2008 08 10 2008 15 12 2008 24 03 2009 2009-03-24T15:43:55 2009 American Institute of Physics 0021-8979/2009/105(7)/07D130/3/ $25.00 We studied the effect of a current pulse width on current-induced magnetization switching in magnetic tunnel junctions based on a macrospin model of the free layer. We performed finite temperature Langevin simulations of the Landau–Lifshitz–Gilbert–Slonczewski equation with an additional spin-torque term. By evaluating the switching current density, we obtained the diagram in the plane of the critical current density and the pulse width at 300 K. As the pulse width increased, we observed an adiabatic regime in the shorter pulse widths, an intermediate crossover regime, and a thermally activated regime in long pulse widths. We found that the easy-plane anisotropy field shifts the crossover pulse width to the lower pulse width, suggesting that the reversed region is enhanced by controlling the device shape. Our results are consistent with those of recent experiments over the pulse widths ranging from 10 − 1 to 10 5 ns . PROCEEDINGS OF THE 53RD ANNUAL CONFERENCE ON MAGNETISM AND MAGNETIC MATERIALS Austin, Texas (USA) 10-14 November 2008 Spin transfer torque phenomena in magnetic nanostructures have received much attention because of their potentiality for spintronic device applications as well as academic interest. In particular, current induced magnetization switching (CIMS) by a spin-polarized current in the free layer of magnetic tunnel junctions (MTJs) is expected to be used as a universal random access memory. CIMS behaviors in MTJs are controlled by two factors: input current density J and current pulse width τ p . The device operation should have both higher switching speed and a lower J . However, these two factors are mutually competitive because the higher switching speed requires a higher J . For device applications, J and τ p need to be optimized. Recent experiments on CIMS in CoFeB/MgO/CoFeB MTJs have revealed two typical behaviors of τ p , which shows a short τ p regime in the nanosecond order and a relatively long τ p regime in the microsecond order. 1,2 These behaviors have been theoretically discussed in relation with thermal activation and adiabatic precession, respectively. 3–5 However, the crossover region between the thermally activated regime and the adiabatic regime is not clear. In this work, we numerically studied the modified Landau–Lifshitz–Gilbert–Slonczewski (LLGS) equation with an additional spin-torque term to study the effect of τ p on CIMS in the MTJs. 6 We performed finite temperature Langevin simulations based on a macrospin model 3 of the free layer with the in plane anisotropy. We started with a Stoner–Wohlfarth single domain magnetic body with magnetization M as a model of a uniformly magnetized free layer of MTJs. A free layer with a volume V 0 and a saturation magnetization M s were assumed to have a uniaxial anisotropy field H k with an easy axis along the e z direction and an easy-plane anisotropy field H p with a hard axis along the e y direction. As shown in Fig. 1(a) , the magnetic field H = H h a = H k ( h x e x + h z e z ) , where | h a | = 1 , was applied in the easy plane of e x − e z , making an angle of Ψ with the easy-axis e z . Using a normalized magnetization m = M / | M | and a uniaxial anisotropy energy scale E 0 = μ 0 M s H k V 0 / 2 , the potential energy E for M is given by E = − E 0 ( m ⋅ e z ) 2 + μ 0 M s 2 V 0 2 ( m ⋅ e y ) 2 − μ 0 M s V 0 H ( m ⋅ h a ) . (1) When a spin-polarized current enters the free layer in the e y direction through the pinned layer, the time evolution of m at finite temperature is described as the following equation with the Gilbert damping constant α : d m d τ = m × ( − ( 1 + α 2 ) h eff + α d m d τ ) . (2) Here τ = γ 0 H k t / ( 1 + α 2 ) is a unit of time with the gyromagnetic constant γ 0 . The effective magnetic field, h eff , is given by h eff = − 1 2 E 0 ∇ m E − J g T ( η T , m ⋅ m p ) J T ( m × m p ) + h f l ( τ ) . (3) The second term in Eq. (3) is the spin-angular-momentum transfer with a spin polarization factor η T ⋅ g T ( η T , m m p ) = η T / 2 [ 1 + η T 2 ( m m p ) ] and J T = 2 E 0 γ 0 e / S μ B with an easy plane area. m p is a unit vector pointing to the spin polarization direction along the e z direction S . To include the heat-bath effect of temperature T , we introduce random field h f l ( τ ) = D T ζ f l ( t ) , which obey the distribution laws h f l ( τ ) = D T ζ f l ( t ) , ⟨ ζ f l ( τ ) ⟩ = 0 , ⟨ ζ f l ( τ ) ζ f l ( τ ′ ) ⟩ = 2 δ ( τ − τ ′ ) of D T = α k B T ( 1 + α 2 ) E 0 . In this work, we examined this free layer in the CoFeB/MgO/CoFeB MTJs using the experimental values for M s , H k , η T , V 0 , H , h x , h z , T , and α . These were set using μ 0 M s = 0.65 T , μ 0 H k = 0.03 T , η T = 0.65 , V 0 = 100 × 150 × 3 nm 3 , μ 0 H = 0.01 T , ψ = 3 π / 4 , h x = ( 1 / 3 ) sin ψ , h z = ( 1 / 3 ) cos ψ , T = 300 K , and α = 0.007 . In this situation, E 0 ∼ 25 000 K and h p = M s / H k ∼ 21.7 We solved this set of equations using the Heun scheme. The time step Δ t when integrating Eq. (2) was set to 1–2 ps. We performed 512 independent samples with a fixed J , in which we used different sets of random numbers generating their initial configurations, as well as fluctuation fields. For each sample, Eq. (2) was integrated from t = − 10 ns to t = 0 without spin torque to reach a condition of thermal equilibrium at t = 0 , and subsequently we integrated over 0 < t < τ p in the presence of the spin torque term. Systematically varying J , we numerically calculated the time evolution of m . The observables Q at each current were averaged over 512 different samples, and they are denoted by ⟨ Q ⟩ . We compared three typical cases: (A) uniaxial anisotropy only ( h x = h z = h p = 0 ) , (B) easy-plane anisotropy included ( h x = h z = 0 , h p = M s / H k ) , and (C) both easy-plane anisotropy and an applied magnetic field included [ h z = ( 1 / 3 ) cos χ , h x = ( 1 / 3 ) sin χ , h p = M s / H k ] . We first discuss the dependence on J of the switching probability P for various τ p . Stochastic variable P is given by P = ( 1 ± ⟨ m ⋅ m p ⟩ ) / 2 , where plus and minus signs correspond to the “antiparallel to parallel (AP to P)” and “parallel to antiparallel (P to AP)” CIMSs, respectively. Figures 1(b) and 1(c) display P ( J , τ p ) for AP to P CIMS as a function of J at various τ p for the three cases. As shown in Figs. 1(b) and 1(c) , the easy-plane anisotropy elevates a critical current density J c for τ p > 8.5 ns . Comparing Figs. 1(c) and 1(d) , we can see that the applied field also acts to reduce J c for pulse widths ranging from 10 − 1 to 10 5 ns . As τ p decreases, the slope of P ( J = J c , τ p ) tends to decrease. CIMS is completely achieved in the long τ p limit due to thermal fluctuations, thus the slope approaches infinity: d P ( J c , τ p → ∞ ) / d J → ∞ . On the other hand, d P ( J c , τ p → 0 ) / d J → 0 in the short τ p limit. We see substantial decreases in d P ( J c , τ p ) / d J around τ p = τ cross . As will be discussed later, this change in the slope means that the magnetization dynamics change from the thermal activation regime to the adiabatic precession amplification regime via the crossover region around τ cross as τ p decreases. This tendency remains valid for the in-plane applied field, as shown in Fig. 1(d) . In the presence of an easy-plane anisotropy field in Figs. 1(c) and 1(d) , d P ( J c , τ p ) / d J becomes less than 1 × 10 − 6 cm 2 / A for τ p > 8.7 ns and d P ( J c , τ p ) / d J < 1 × 10 − 6 cm 2 / A for τ p < 8.7 ns . Here, we set τ cross ∼ 10 ns in the presence of the easy-plane anisotropy. Now, using the criterion P ( J c , τ p ) = 1 / 2 , we have numerically obtained the J c - τ p diagram of CIMS in the CoFeB/MgO/CoFeB MTJs at 300 K throughout the pulse widths ranging from 10 − 1 to 10 5 ns . Of note is that our J c - τ p diagram takes into account only the heat-bath effect by solving the stochastic LLGS equations based on the macrospin model and does not include any other additional heating effect, such as Joule heating. 7 The inset of Fig. 2 shows the semilog plot of the J c - τ p diagram. In each case, we can see a downward bending for P to AP and an upward bending for AP to P. The bending appears at τ p = τ bend ∼ 300 ns for the uniaxial case and τ bend ∼ 10 ns for the case with the easy-plane anisotropy field. τ bend coincides with the crossover pulse width τ cross in Figs. 1(b)–1(d) : τ bend ∼ τ cross . The asymmetric behavior of J c between AP to P and P to AP, i.e., J c ( P to AP ) ≠ − J c ( AP to P ) , originates from the spin polarization factor η T and g T ( η T , m ⋅ m p ) in MTJs. By introducing the scaling J c ∗ = g T ( η T , m ⋅ m p ) J c , we can symmetrize J c ( P to AP ) ∗ = − J c ( AP to P ) ∗ . This rescaling enables us to classify the dependence on τ p of J c independent of AP to P or P to AP” In the longer regime for τ p > τ cross , we have a straight line of several orders of magnitude, indicating a thermal activation regime with a logarithmic dependence of τ p , 3,4 J c J c 0 = ( 1 + h z ) − 1 [ 1 − k B T E 0 ( 1 + h z ) − 2 ln ( τ p τ 0 L ) ] , (4) where J c 0 is the zero-temperature switching current density: J c 0 = 2 E 0 J T α ( 1 + h p / 2 + h z ) / g T and τ 0 L is an attempt frequency associated with the frequency of a quasiperiodic stable state. Figure 2 shows this rescaled J c ∗ - τ p diagram with the absolute values of J c ∗ adopted. This J c ∗ - τ p curve makes a clear separation between thermal activation and adiabatic precession regimes by τ cross as indicated by black arrows. For τ p < τ cross , the slope of the J c - τ p curves is nearly equal to −1, suggesting J c ∝ 1 / τ p . This relationship is consistent with theoretical predictions of the adiabatic precession that take into account thermally distributed initial conditions, 5 J c J c 0 ≈ τ 0 H τ p + 1. (5) Here, τ 0 H is a characteristic time scale in the adiabatic regime: 1 / τ 0 H ∝ α ( 1 + h z + h p / 2 ) ln ( h p ) × 1 + h z / [ h p ln ( E / k B T ) ] . The global behavior of this diagram is consistent with experimental results. 1,2 The obtained τ cross ∼ O ( 10 ns ) in the presence of the easy-plane anisotropy field also coincides with the measurements of the CoFeB/MgO/CoFeB MTJs. 1 We note that τ cross is shifted by the easy-plane anisotropy field, suggesting the possibility of a reversed region enhancement by controlling the easy-plane anisotropy due to the device shape. We next observed the crossover regime, focusing on the simple uniaxial case A where J c 0 ∼ 0.17 × 10 6 A / cm 2 . Around τ cross , J c begins to deviate from the logarithmic dependence as τ p decreases. Figure 3 shows the time evolution of the potential energy ⟨ E ⟩ and switching probability P ( J , τ p ) for various current densities J . Solid curves indicate the uniaxial energy and dotted curves show P ( J , τ p ) . In the high current region J ⪢ J c 0 , the peaks in the curves of ⟨ E ⟩ steepened, and their values approximate E max , which is the maximum value of Eq. (1) . This suggests magnetization dynamics along the potential energy surface. In this regime, CIMS occurs at the moment when E becomes a maximum at t = τ peak . As J decreases, only the maximum value of E decreases. Around J c ∼ J c 0 , the peaks begin to broaden and deviate from the symmetrical structure as J decreases. Moreover, in such a broader peak region, CIMS occurs behind τ peak and ⟨ E ⟩ ⪡ E max . This suggests that the delay time between the time of P = 1 / 2 and τ peak corresponds to the relaxation time over the barrier E max − ⟨ E ⟩ , which can be considered as thermal activation. In conclusion, we studied the effect of τ p on CIMS based on a macrospin model of the free layer with the in plane anisotropy. We performed finite temperature Langevin simulations of the LLGS equation. Systematically varying J , we numerically evaluated switching current density J c . The calculated J c - τ p diagram at 300 K, in which only the heat-bath effect is taken into account, makes a clear separation between thermal activation and adiabatic precession regimes by the crossover pulse width τ cross . We found that τ cross is shifted to a lower τ p by the easy-plane anisotropy field, suggesting that the reversed region is enhanced by controlling the device shape. These results are consistent with recent experiments on CIMS over pulse widths ranging from 10 − 1 to 10 5 ns . In the crossover regime, we observed the delay between the CIMS time and the time when the potential energy becomes a maximum in the evolution of ⟨ E ⟩ . This work was supported by the Next Generation Supercomputing Project, Nanoscience Program, MEXT, Japan. FIG. 1. (a) Schematic views of macrospin model of free layer of MTJs. [(b)–(d)] Switching probability P ( J , τ p ) of AP to P CIMS as a function of J for various τ p . (b) Uniaxial case A: h x = h z = h p = 0 . (c) Easy-plane anisotropy case B: h x = h z = 0 , h p = M s / H k . (d) In plane-field case C: h z = ( 1 / 3 ) cos ψ , h x = ( 1 / 3 ) sin ψ , h p = M s / H k . FIG. 2. Double logarithmic plot of J c - τ p diagram for three different cases corresponding to cases in Figs. 1(b)–1(d) . The absolute values of J c ∗ = g T J c were plotted by using J c in the inset. Solid and open symbols indicate AP to P and P to AP, respectively. Diamonds, circles, and triangles correspond to the cases A, B, and C. Inset: semilog plot of the same J c - τ p diagram without correction by g T . FIG. 3. Time evolution of potential energy ⟨ E ⟩ and switching probability P ( J , τ p ) for various current densities J in uniaxial case A.

PY - 2009

Y1 - 2009

N2 - We studied the effect of a current pulse width on current-induced magnetization switching in magnetic tunnel junctions based on a macrospin model of the free layer. We performed finite temperature Langevin simulations of the Landau-Lifshitz-Gilbert-Slonczewski equation with an additional spin-torque term. By evaluating the switching current density, we obtained the diagram in the plane of the critical current density and the pulse width at 300 K. As the pulse width increased, we observed an adiabatic regime in the shorter pulse widths, an intermediate crossover regime, and a thermally activated regime in long pulse widths. We found that the easy-plane anisotropy field shifts the crossover pulse width to the lower pulse width, suggesting that the reversed region is enhanced by controlling the device shape. Our results are consistent with those of recent experiments over the pulse widths ranging from 10-1 to 105 ns.

AB - We studied the effect of a current pulse width on current-induced magnetization switching in magnetic tunnel junctions based on a macrospin model of the free layer. We performed finite temperature Langevin simulations of the Landau-Lifshitz-Gilbert-Slonczewski equation with an additional spin-torque term. By evaluating the switching current density, we obtained the diagram in the plane of the critical current density and the pulse width at 300 K. As the pulse width increased, we observed an adiabatic regime in the shorter pulse widths, an intermediate crossover regime, and a thermally activated regime in long pulse widths. We found that the easy-plane anisotropy field shifts the crossover pulse width to the lower pulse width, suggesting that the reversed region is enhanced by controlling the device shape. Our results are consistent with those of recent experiments over the pulse widths ranging from 10-1 to 105 ns.

UR - http://www.scopus.com/inward/record.url?scp=65249133610&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=65249133610&partnerID=8YFLogxK

U2 - 10.1063/1.3075854

DO - 10.1063/1.3075854

M3 - Article

AN - SCOPUS:65249133610

VL - 105

JO - Journal of Applied Physics

JF - Journal of Applied Physics

SN - 0021-8979

IS - 7

M1 - 07D130

ER -