^{1,a)}, A. D. Kent

^{1}and D. L. Stein

^{1,2}

### Abstract

We consider the general Landau-Lifshitz-Gilbert (LLG) dynamical theory underlying the magnetization switching rates of a thin film uniaxial magnet subject to spin-torque effects and thermal fluctuations. After discussing the various dynamical regimes governing the switching phenomena, we present analytical results for the mean switching time behavior. Our approach, based on explicitly solving the first passage time problem, allows for a straightforward analysis of the thermally assisted, low spin-torque, switching asymptotics of thin film magnets. To verify our theory, we have developed an efficient Graphics Processing Unit (GPU)-based micromagnetic code to simulate the stochastic LLG dynamics out to millisecond timescales. We explore the effects of geometrical tilts between the spin-current and uniaxial anisotropy axes on the thermally assisted dynamics. We find that even in the absence of axial symmetry, the switching times can be functionally described in a form virtually identical to the collinear case.

The authors would like to acknowledge A. MacFadyen, Aditi Mitra, and J. Z. Sun for many useful discussions and comments leading to this paper. This research was supported by NSF-DMR-100657 and PHY0965015.

I. INTRODUCTION

II. GENERAL FORMALISM

III. THERMAL EFFECTS

IV. SWITCHING DYNAMICS

V. COLLINEAR SPIN-TORQUE MODEL

A. Collinear high current regime

VI. UNIAXIAL TILT

VII. THERMALLY ACTIVATED REGIME

VIII. SWITCHING TIME PROBABILITY CURVES

IX. CONCLUSION

### Key Topics

- Boltzmann equations
- 13.0
- Magnetic anisotropy
- 13.0
- Ballistics
- 12.0
- Critical currents
- 10.0
- Anisotropy
- 9.0

## Figures

Histogram distribution of mz after letting the magnetic system relax to thermal equilibrium (103 natural time units). The overlayed red dashed line is the theoretical equilibrium Boltzmann distribution. In the inset, we show a semilog-plot of the probability vs. dependency. As expected, the data scale linearly.

Histogram distribution of mz after letting the magnetic system relax to thermal equilibrium (103 natural time units). The overlayed red dashed line is the theoretical equilibrium Boltzmann distribution. In the inset, we show a semilog-plot of the probability vs. dependency. As expected, the data scale linearly.

Blue line shows the fit of the ballistic limit to the numerical data (in blue crosses). Red line shows the improvement obtained by including diffusion gradient terms. Times are shown in units of ( ) where T stands for Tesla: real time is obtained upon division by HK .

Blue line shows the fit of the ballistic limit to the numerical data (in blue crosses). Red line shows the improvement obtained by including diffusion gradient terms. Times are shown in units of ( ) where T stands for Tesla: real time is obtained upon division by HK .

: green , red for applied current I = 5. The plane dissecting the sphere is perpendicular to the uniaxial anistropy axis. Its intersection with the sphere selects the regions with highest uniaxial anisotropy energy.

: green , red for applied current I = 5. The plane dissecting the sphere is perpendicular to the uniaxial anistropy axis. Its intersection with the sphere selects the regions with highest uniaxial anisotropy energy.

Mean switching time behavior for various angular tilts above critical current obtained by numerically solving (7) . Each set of data is rescaled by its critical current such that all data plotted has . Angular tilts are shown in the legend in units of such that the smallest angular tilt is 0 and the largest is . Times are shown in units of ( ) where T stands for Tesla: real time is obtained upon division by HK .

Mean switching time behavior for various angular tilts above critical current obtained by numerically solving (7) . Each set of data is rescaled by its critical current such that all data plotted has . Angular tilts are shown in the legend in units of such that the smallest angular tilt is 0 and the largest is . Times are shown in units of ( ) where T stands for Tesla: real time is obtained upon division by HK .

Mean switching time behavior in the sub-critical low current regime obtained by numerically solving (7) . Times are shown in units of ( ) where T stands for Tesla: real time is obtained upon division by HK . The red andgreen lines are born by fitting to the data the functional form , where μ is the debated exponent (either 1 or 2) and C is deduced numerically. The red curve fits the numerical data asymptotically better the green curve. The difference between the red line and (21) is that our theoretical prediction includes a current dependent prefactor, which was not fitted numerically. The differences between numerical data and (20) are due to numerical inaccuracies out to such long time regimes. The differences between (20) and (21) quantify the reach of the crossover regime.

Mean switching time behavior in the sub-critical low current regime obtained by numerically solving (7) . Times are shown in units of ( ) where T stands for Tesla: real time is obtained upon division by HK . The red andgreen lines are born by fitting to the data the functional form , where μ is the debated exponent (either 1 or 2) and C is deduced numerically. The red curve fits the numerical data asymptotically better the green curve. The difference between the red line and (21) is that our theoretical prediction includes a current dependent prefactor, which was not fitted numerically. The differences between numerical data and (20) are due to numerical inaccuracies out to such long time regimes. The differences between (20) and (21) quantify the reach of the crossover regime.

Mean switching time behavior in the sub-critical low current regime obtained by numerically solving (7) . Various uniaxial tilts are compared by rescaling all data by the appropriate critical current value. Times are shown in units of ( ) where T stands for Tesla: real time is obtained upon division by HK .

Mean switching time behavior in the sub-critical low current regime obtained by numerically solving (7) . Various uniaxial tilts are compared by rescaling all data by the appropriate critical current value. Times are shown in units of ( ) where T stands for Tesla: real time is obtained upon division by HK .

Influence of precessional orbits on transient switching as seen from the switching time probability curve in the supercritical current regime. The case shown is that of an angular tilt of subject to a current intensity of 2.0 times the critical current. Data were gathered by numerically solving (7) . The non-monotonicity in the probability curve shows the existence of transient switching. Times are shown in units of ( ) where T stands for Tesla: real time is obtained upon division by HK .

Influence of precessional orbits on transient switching as seen from the switching time probability curve in the supercritical current regime. The case shown is that of an angular tilt of subject to a current intensity of 2.0 times the critical current. Data were gathered by numerically solving (7) . The non-monotonicity in the probability curve shows the existence of transient switching. Times are shown in units of ( ) where T stands for Tesla: real time is obtained upon division by HK .

Spin-torque induced switching time probability curves for various angular configurations of uniaxial tilt (a sample normalized current of 10 was used) obtained by numerically solving (7) . A log-log y-axis is used following (28) to make the tails of the probability distributions visible.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content