Fokker-Planck calculation of spintorque switching rates: comparison with telegraph-noise data

Transcription

1 Fokker-Planck calculation of spintorque switching rates: comparison with telegraph-noise data P. B.Visscher and D. M. Apalkov Department of Physics and Astronomy The University of Alabama This project was supported by NSF grant # ECS , DOE Computational Materials Sciences Network, and DMR grant #

2 Motivation Current-driven spin torque is of interest for future memory devices, because it has been shown to cause switching. It is also an important source of read-head noise. The most interesting phenomena are thermal in nature, occurring on a time scale too long for convenient Landau- Lifshitz simulation. Arrhenius theory cannot be used to estimate the switching rate in presence of spin torque (in the Slonczewski model) because the torque is not conservative. We have derived a generalization of the Arrhenius rate formula, which fits experimental switching rates.

3 3-5 nm Geometry of the System Although the theory is general, we will apply it to a specific system consisting of two FM layers: Thin layer, which is switched by the current; Thick layer, whose magnetization is pinned along the easy axis e. The two layers are separated by a nonmagnetic exchange-break layer. We assume the magnetization of each layer is uniform.

4 Micromagnetic Calculation Landau-Lifshitz (LL) equation for M (magnetization of free layer) is modified to include spin torque M & = M& + M& + M& + M& M & = γ M cons H cons M & = αγm mˆ LL S cons ( mˆ H ) LL Slon rand (conservative torque -- γ is the gyromagnetic ratio; H cons is the gradient of the Stoner-Wohlfarth energy) cons ( mˆ mˆ ) M = γjm mˆ (LL torque -- α is the LL damping constant, M s is the saturation magnetization) & (Slonczewski form of the spin torque -- Slon S p J ~ current density) mˆ = M/M s = unit vector along magnetization of the free (switching) layer; ˆm p = unit vector along magnetization of the pinned layer. M& rand is the torque due to thermal (random) forces

5 Landau-Lifshitz Trajectories on the M-sphere The magnetization M of the free layer is represented by a point on a sphere of radius M s ; the Stoner- Wohlfarth orbit (no damping or spin torque) is a closed curve on the sphere. It is convenient to draw the orbits (constant-energy contours) on a planar projection of the sphere: Unswitched energy well Switched energy well M M& Switching process: spin torque increases energy in left (red) well until M is on black orbit, which carries it to right well where LL damping traps it. The LL damping torque moves M toward lower energies The conservative torque moves M along the orbit The Slonczewski spin torque moves M toward higher energies

6 Fokker-Planck Equation in Energy The 2D Fokker-Planck (FP) equation for ρ(m,t) is very difficult to solve; we have for the first time derived a FP equation in energy. This also takes the form of a continuity equation for a 1D probability distribution ρ (E): γm ( ) ρ '( E, t ) S P E = j E t where j E (E,t) is the number of systems per unit time crossing a constant-energy contour. Computing the energy-current j E from the 2D probability current j, and setting j E (E,t)=0 (steady state) gives ln ρ' ( E) γm S = [ α + η( E) J ] Vβ ( E) E D Here η ( E) = mˆ p [ dm mˆ ] H dm cons µ 0 t E ( ) E, is a spin-torque damping coefficient, ratio of work done by spin torque to work done by LL damping. Note that η(e) is nearly independent of energy: If η(e) is independent of E, the FP equation above has the trivial (Maxwell-Boltzmann) solution ρ = exp(-βev) where β is an effective inverse temperature.

7 Check of FP Theory with LL Simulation Even if the effective temperature is not exactly constant, it is easy to compute the probability distribution in energy, which is Boltzmann-like with an effective temperature Solid lines are FP equation prediction Symbols are simulation data. T eff ( E) T 1 ( ) η E = J α 1 Note that if the current J=0, T eff (E)=T, and the Boltzmann distribution is recovered. For positive J the effective temperature is greater than the (LL noise) temperature, leading to an increase in switching rate.

8 Switching rates and telegraph noise We can calculate the dwell time in each well (reciprocal of the switching rate out of the well) in the TST (transition state theory) approximation: τ = population switching current = orbit period e [1 ηj / α ] E where the barrier E b depends on the applied field H, and J is proportional to the current density through the device. In some ranges of H and J the system jumps back and forth thermally (telegraph noise) and the dwell time has been measured as a function of H and J by the Michigan State group (Urazhdin, Birge, Pratt, and Bass). We have compared the predictions of our Fokker-Planck theory with their results. b / k B T

9 Fit to telegraph-noise experiment Dwell time in well #1 Dwell times (the time a system stays in one of the energy wells, along an easy axis) were measured by S. Urazhdin et al τ 1, τ 2, ns The system is a multilayer Py(20 nm)/cu(10)/py(6) (Py=Ni 84 Fe 16 ) with lateral dimensions 130x60 nm 2. Dwell time in well #2 [Phys. Rev. Lett. 91, (2003)] (squares and circles). Three poorly known parameters were adjusted (H K, volume, current scale I/J) to exactly fit the crossing point and the slope difference. Average slope is a genuine prediction of the theory, as is the upward curvature. Current I, ma The fit gave H K =217 Oe, volume~(geometric volume)/3.

10 Conclusions Fokker-Planck theory has been extended to include non-conservative current-induced spin torque The effect of the torque is better described by an increase of the effective spin temperature T eff (E) than by a decrease of the energy barrier Calculated dwell times in energy wells agree with experiments on telegraph noise.