Simplified Fokker-Plank Equation Treatment of Finite-temperature Spin-torque Problems

A Legendre function expansion method is proposed to solve the simplified Fokker-Plank equation to study the dynamics of a macrospin under spin-torque-driven magnetic reversal at finite temperature. The first and second eigenvalues (λτ0)1 and (λτ0)2 as functions of I/Ic and Hk are obtained, in agreement with the previous results using the Taylor series expansion method. The Legendre function expansion method compared with the Taylor series expansion method has faster convergence properties and clear physical means. Besides, it is found that in some case, especially the second eigenvalue (λτ0)2 can become complex, that means that the probability density P not only decays with time, but also oscillates with time.


Introduction
In the spin-torque-induced switching the write and read time is essential. [1][2][3] There are two aspects which will affect the reversal time τ s . First, the initial position of the magnetic moment is thermally distributed at the time the reversal field or current is applied, causing a variation in switching time. Secondly, during reversal, thermal fluctuation would modify the orbit, [4][5][6][7] causing additional fluctuation for τ s even for identical initial conditions. The dynamics of a macrospinunder spin-torque-driven magnetic reversal has been extensively studied. [8][9] In the limit of uniaxial anisotropy only and with finite temperature at large drive amplitude I >> I c0 , with I being the current passing through the junction, I c0 the zero-temperature spin-torque current instability threshold, the "long-time" super-threshold asymptotic form for the probability of not switching at time t can be expressed as: π ξ π ξ τ (1) (for I >> I c0 , E r << 1 and ξ >> 1) where ξ = mH k /2k B T is the normalized thermal activation energy barrier height, m is the total magnetic moment of the free layer, and H k the uniaxial anisotropy field. τ I = τ 0 / (I/I c0 -1) is the characteristic time scale for spin-torque-induced reversal, τ 0 = 1/γH k α is the natural unit of time with γ ≈ 2µ B /ℏ as the gyromagnetic constant, and α the LLG damping coefficient. The comparison of Eq. (1) with experimental results suggests the presence of sub volume magnetic excitations which often dominate the switching process and which degrade the spin-torque switching efficiency. [9] He et al. presented a Fokker-Plank formulation for the full problem including both thermalized initial condition and reversal orbit with estimates for the reversal time and its distribution. [10] In the case of uniaxial anisotropy they reduced the Fokker-Plank equation (FPE) to an ordinary differential equation in which the lowest eigenvalue λ 1 determines the slowest switching events. They calculated λ 1 using both analytical and numerical methods. It is found that the previous model [Eq. (1)] based on thermally distributed initial magnetization states can be accurately justified in some useful limiting conditions.
In this paper we use the simplified FPE to study the switch time at the finite temperature by solving the eigenvalue equation derived from FPE.

Simplified Fokker-Plank Equation
Fokker-Plank equation describes the time-dependent evolution of the ensemble-average probability distribution of a system in a given environment and initial condition. [10] For a macrospin, one defines a probability density function P (n m , t) = P (θ, ϕ, t) that is the time-dependent probability of finding the macrospin in the solid angle of sinθdθdϕ with a spherical coordinate set (θ,ϕ) describing the magnet's direction, n m . The Fokker-Plank equation describes the dynamic flow of this probability as a function of space (on the surface of a unit sphere) and time in the form of: is the ballistic (zero-temperature) part of the probability current and D∇ 2 P = ∇⋅J D is the diffusive part of the probability current, with J D = D∇P. The constant D is the diffusion rate in the probability phase space, D = γαk B T/m [10]. dn m /dt is determined by the LLG equation including the spin-torque term, n H n n n (4) where n s is the unit vector of the magnetization direction in the pinned layer. Eqs. (1), (2) and (3) give a set of partial equations that describes the magnetic moment dynamics at the finite temperature in the spin-torque condition. The problem can be solved numerically in principle, here we refer a simplified solution of the set of Fokker-Plank equations [11]. Consider an ensemble time-dependent magnetization probability density P (n m , t). The unit vector of magnetic moment n m is characterized by the polar angle (θ,φ). Before the magnetic field and the current are turned on, the probability density takes the equilibrium value. For a uniaxial anisotropy situation, (5) where P 0 is the normalization factor, determined by.
The probability current is derived by the LLG Eq. (4), where H s is the spin torque term, I is the current density, p is the spin polarization coefficient, q is the electron charge. In the simplified treatment of the Fokker-Plank equation [10], it is assumed that the magnetic field H and n s are parallel to the anisotropy axis, i.e., H e = (H+H k cosθ) e z and n s = e z where H k = 2K/M s .
In the polar coordinate, ( ) We obtain the second term in Eq. (1), where x = cosθ. In the polar coordinate, Because P is only function of θ, we obtain the third term in Then Eq. (1) reduces to.
( ) Eq. (14) can be solved by the method of separation of Eq. (15) can be written as.
The original Fokker-Plank equation (2) is now reduced to the standard eigenvalue problem. If determining eigenfunction By use of the paper [12].
Therefore, the matrix elements of the secular equation include two parts: the diagonal parts, Eq. (26); the diagonal and non-diagonal parts, Eq. (31).
It is noticed that the right side part of the secular equation Eq. (21) is dependent on n, so it is not the standard eigen-value equation AX = cX. Take the normalized Legendre functions

Secular Equation of Dimensionless form
Multiplying the numerator and denominator of the eigenvalue c, Eq. (16) by (H+H k ), and using. From the above discussion it found that the temperature relation is only included in the diagonal matrix elements, divided by ξ, Eq.(37). If we take m = 4π×10 -24 Vsm, H = 10 6 A/m, H k = 10 4 A/m, then it can obtain ξ = 9.2×10 5 /T (K).
Hence, the diagonal matrix elements are of 10 -6 ×T order of magnitude, the effect of temperature is very small.

Some Physical Quantities
Magnetic moment m = µ 0 MV, where µ 0 is the vacuum magnetic permeability, M is the magnetization, and V is the volume of the free layer. We take M = 10 6 A/m, V = 10 -23 m 3 , obtain m = 4π×10 -24 Vsm.
Spin torque field.

Convergence of the Expansion Method
There are input parameters: η' and ζ' (Eq. (38)) and 1/ξ (Eq. (37)). As a test, we take η' = -0.01 and ζ' = 2.0, 1/ξ = 0.01, calculate the eigenvalues λτ 0 for different numbers of the expansion Legendre functions N, the results are listed in Table  1. 1. The lowest eigenvalue is 1.86045, which is in agreement with the value in Ref. [12]. When N = 40, the lowest two eigenvalues are complex, the second rows are their imagine parts. 2. When N ≥ 60, the two lowest eigenvalues are real, and become convergent to five decimal place. Afterward, we will take N = 100 in our calculation. In Ref. [12] they expand F (x) via Taylor series, F (x) = Σ n a n x n , the convergent results are obtained for larger N = 300. 3. The corresponding eigenvectors include about 50 lowest basic functions (Legendre functions) to five decimal places.
It means that the probability density P not only decays with time, but also oscillates with time if the eigenvalue is complex. In the region 0.3 ≤ I/I c -1 ≤ 0.7 the eigenvalues (λτ 0 ) 2 are listed in Table 2.  From Figure 3 we see that (λτ 0 ) 1 decrease with -η' (H k , if H+H k = const.), except for I/I c = 1.0. The whole trend is in agreement with Fig. 1 in Ref. [12]. From Figure 4 we see that (λτ 0 ) 2 increase with -η' (H k , if H+H k = const.). Some curves are also not smooth, because the eigenvalues are complex, the curves only show the real parts of (λτ 0 ) 2 . For I/I c = 1.5, the complex (λτ 0 ) 2 exists in the region -0.07≤η'≤-0.19. For I/I c = 2.0, the complex (λτ 0 ) 2 exists in the region -0.13≤η'. For I/I c = 2.5, the complex (λτ 0 ) 2 exists in the region -0.19≤η'.

Summary
We proposed a Legendre function expansion method to solve the simplified Fokker-Plank equation to study the dynamics of a macrospin under spin-torque-driven magnetic reversal at finite temperature. We obtained the first and second eigenvalues (λτ 0 ) 1 and (λτ 0 ) 2 as functions of I/I c and H k . The Legendre function expansion method compared with the previous Taylor series expansion method has faster convergence properties and clear physical means. Besides, it is found that in some case, especially the second eigenvalue (λτ 0 ) 2 can become complex, that means that the probability density P not only decays with time, but also oscillates with time.