Dynamics of single-domain magnetic particles at elevated temperatures

A stochastic differential equation that describes the dynamics of single-domain magnetic particles at any temperature is derived using a classical formalism. The deterministic terms recover existing theory and the stochastic process takes the form of a mean-reverting random walk. In the ferromagnetic state diffusion is predominantly angular and the relevant diffusion coefficient increases linearly with temperature before saturating at the Curie point ($T_c$). Diffusion in the macrospin magnitude, while vanishingly small at room temperature, increases sharply as the system approaches $T_c$. Beyond $T_c$, in the paramagnetic state, diffusion becomes isotropic and independent of temperature. The stochastic macrospin model agrees well with atomistic simulations.

The dynamics of single-domain magnetic particles at elevated temperatures are a fundamental physics problem that encompasses a phase transition at the Curie point (T c ). While the behavior near T c can be described using atomistic [1][2][3][4] and renormalization 5 methods, more efficient approaches are needed for modeling macroscopic systems and for developing technologies based on magnetic phase transitions. The Landau-Lifshitz-Bloch (LLB) equation, derived by ensemble averaging a thermal distribution of interacting atomic spins using either a quantum-mechanical 6 or a classical 7 framework, bridges the gap between room temperature, where a ferromagnet is described by the Landau-Lifshitz equation 8 , and temperatures above T c , where the material becomes paramagnetic and its behavior is modeled by the Bloch equation 9 . Nevertheless, the deterministic LLB equation is insufficient to capture the dynamics of single domain magnetic particles, because a degree of stochasticityowing to the thermal fluctuations of individual atomic spins-survives and plays a critical role in determining temporal evolution.
Thermal fluctuations in single-domain magnetic particles were first studied by augmenting the external magnetic field in the Landau-Lifshitz-Gilbert equation with a stochastic term 10 , thereby introducing randomness in the angle of the macrospin but not in its magnitude. The validity of this approach is limited to temperatures much lower than T c , where the individual atomic spins are locked together by exchange interactions and the magnitude of the resulting macrospin is constant. Recently, stochastic fields were incorporated into the LLB equation in a number of different ways [11][12][13][14][15] in order to include the effect of thermal fluctuations at elevated temperatures. These formulations are distinct, the physics they describe are different, and they exhibit markedly diverse stochastic behaviors when implemented in simulation codes.
In this letter we present an improved stochastic LLB a) Electronic mail: mtzoufras@physics.ucla.edu equation derived by considering the required behavior of macrospin thermal fluctuations. First we find the macrospin probability distribution for a system of atomic spins in thermal equilibrium by invoking the mean field approximation and the central limit theorem. We then formulate an advection-diffusion equation such that this probability distribution is the stationary solution and deduce the corresponding stochastic differential equation. We benchmark the new stochastic equation against mean field atomistic LLG calculations and find excellent agreement. In a classical framework 7 an atomic spin is described by a unit vector S with magnetic moment µ = µ 0 S. Under the influence of thermal fluctuations the spin direction at any point in time can vary, and we may use a distribution function f (s) on the unit sphere |s| = 1 to represent it. When an isolated classical spin is in thermal equilibrium, the function f (s) takes the form of a Boltzmann distribution f 0 (s) ∝ exp[−H(s)/(k B T )], where H(S) = −µ 0 H · S is the Hamiltonian and H the total magnetic field. Using the normalized magnetic field ξ 0 = µ 0 H/(k B T ) and assuming, without loss of generality, that the magnetic field is along the z-axis, i.e. H ≡ê z , we rewrite the equilibrium distribution as: To calculate the moments of f 0 on the unit sphere we write s in spherical coordinates (s x , s y , s z ) = (sin ϑ cos ϕ, sin ϑ sin ϕ, cos ϑ) with 0 ≤ ϕ < 2π and 0 ≤ ϑ ≤ π. The averages S x and S y , as well as the covariances S i S j i =j , vanish identically, and we obtain the average S along the z-axis and a diagonal covariance matrix Σ = diag(σ 2 ⊥ , σ 2 ⊥ , σ 2 ): The normalized magnetization of a single-domain particle comprised of n unit cells is M (n) = 1 n n i=1 S (i) . To render the random vectors S (i) independent from one another we drop the spin-spin exchange interactions in favor of a mean field (mean field approximation). Moreover, we assume that both the mean field and temperature are constant throughout the magnetic particle such that S (i) = m 0 and Σ (i) = Σ for all unit cells. If the number of unit cells is large enough their average follows the multivariate normal distribution with √ n(M (n) − m 0 ) D − → N {x,y,z} (0, Σ), according to the central limit theorem, regardless of the individual S (i) -distributions. We may thus write the macrospin probability distribution: We seek the stochastic process for which F 0 from Eq.
we need the advective (vF ) and diffusive (D · ∂ m F ) fluxes to cancel out when the system is stationary and identify the first term as the advective flux and the second as the diffusive flux. The conservation law can be written as: with τ s a coefficient in units of time.
For a system rotating around a magnetic field H we can write the rotation flux −(m × H)F , which is diver-genceless when F = F 0 . The resulting precession equation is: and we recognize γ as the gyromagnetic ratio with γµ 0 S the mechanical moment of an atomic spin. The final Fokker-Planck equation for the distribution F combines both the stochastic [Eq. (6)] and precession [Eq. (7)] processes: The second term on the right-hand-side (RHS) of Eq. (8) corresponds to an Ornstein-Uhlenbeck (OU) stochastic process with stationary solution F 0 in Eq. (5). This process describes a mean-reverting random walk, a typical example of which is the velocity of a massive Brownian particle under the influence of friction. Here the stochasticity is due to the finite number (n) of unit cells contained in a magnetic grain, and τ −1 s incorporates an effective friction coefficient 16 . For an infinitely large particle (n → ∞) the diffusive flux vanishes and F 0 becomes a delta function. The diffusional relaxation time τ s is related to the phenomenological damping parameter (λ) in the LLG equation (see below) and is a function of both temperature and material properties.
In order to simplify Eq. (8) we express the tensor product in terms of vectors perpendicular and parallel to the axis imposed by the total magnetic fieldĤ ≡ e z (which includes exchange, magnetic anisotropy and externally applied fields) to obtain: where n has been substituted by M 0 s V /µ 0 , with M 0 s the saturation magnetization per unit volume at T = 0K. The first two terms on the RHS of Eq. (9) yield the deterministic dynamics and the last two terms the diffusive behavior. The third and forth terms describe the stochasticity in terms of transverse and longitudinal diffusion with respect to theĤ-axis, with diffusion coef- s V , rather than them-axis as in Refs. [11][12][13][14] . Below T c , where the exchange field dominates, the macrospin points roughly along the equilibrium directionm ≃m 0 ≡Ĥ, and the distinction between the decomposition used here and the one in the literature disap- , the decomposition along and acrossĤ is redundant. Consequently, the differentiation betweenm andĤ is only meaningful in the immediate vicinity of the Curie temperature.
A numerical scheme that solves Eq. (9) would have to keep track of the three-dimensional probability distribution F (m). To formulate a simpler and less computationally intensive approach we write a stochastic differential equation in accordance with the Fokker-Planck equation [Eq. (9)]: where m 0 = m 0Ĥ , and the coefficients σ , σ ⊥ are given in Eqs.
To further elucidate the diffusive behavior we substitute the normalized magnetic field ξ 0 with µ 0 H/(k B T ) and rewrite the diffusion coefficient This is identical to the angular diffusion coefficient in Ref. 13 , such that angular diffusion in Eqs. (9)-(10) and in Ref. 13 are the same everywhere except near T c , wherem(T c ) ≈ H(T c ). To obtain an expression for the temperature dependence of the diffusion coefficient D H ⊥ we write the total magnetic field as H ≃ (3k B T c /µ 0 )m + H eff , where H eff includes the anisotropy and external fields. Below T c the exchange field dominates, µ 0 |H eff | ≪ 3k B T c |m|, and the angular diffusion is a linear function of temper- Physically, above T c the exchange field vanishes and (assuming H eff ≃ 0) atomic spins have no preferred direction, such that the spin distribution f 0 (s) fills the entire spherical shell |s| ≡ 1 homogeneously. The resulting variance is then identical in all directions V ar(S {x,y,z} )(T > T c ) = 4π cos 2 ϑdΩ = 1/3. An approximate expression for D H ⊥ with temperature may be written as: The diffusion in the macrospin magnitude D H saturates at the same value as D H ⊥ above T c . However, below the Curie point, D H is not a linear function of temperature. Instead, its properties can be found by using the expression σ 2 = ∂m 0 /∂ξ 0 ≡ L ′ (ξ 0 ), where L(ξ 0 ) = coth(ξ 0 ) − 1/ξ 0 is the Langevin function and the prime denotes differentiation with respect to ξ 0 . For temperatures much lower than T c the slope of m 0 (T ) is very small (especially if one considers the Brillouin 17 instead of the Langevin function), and as a result D H (T ≪ T c ) can be neglected when compared to D H ⊥ (T ≪ T c ). As the temperature approaches T c from below, the diffusion in the magnetization magnitude D H increases sharply following the magnitude of the slope of m 0 (T ). At the Curie point, the slope of m 0 (T ) vanishes abruptly and D H saturates. Beyond T c the system is isotropic and the overall diffusive flux in the Fokker-Planck equation [Eq. (9)] reduces to (3M 0 s V /µ 0 ) −1 ∂ m F . This is consistent with the physical picture of a paramagnetic grain, for which all directions are equivalent and the expected stochastic behavior is isotropic. The deterministic terms of Eq. (10) are identical to those in Ref. 13 and may be expressed in the same form as Eq. (2.17) in Ref. 7 : with λ the phenomenological damping parameter from the LLG-Langevin equation. This discrepancy arises because the present derivation, unlike the one in Ref. 7 , does not assume that individual atoms obey LLG-Langevin dynamics. The classical treatment of dissipation in atomic spins, in which all dissipative processes (e.g. spin interactions with photons and spin waves) are approximated by a single damping term, has been employed in many theoretical and numerical investigations. With that in mind, for T T c , the expressions from Ref.
T /µ 0 in agreement with the deterministic terms in Eq. (10). Hence, for magnetic particles near the Curie point, we may set the relaxation rate to τ −1 To verify the macrospin model in Eq. (10) we benchmarked it against atomistic simulations that solve the LLG-Langevin equation for every unit cell. We consider systems of FePt nanoparticles, which are of great technological interest especially for mangetic recording 18,19 , with f ct crystal structure that comprises 2 Fe atoms per unit cell of volume v = 3.7Å × (3.88Å) 2 , anisotropy energy density K 1 = 7.64 × 10 7 erg/cm 3 , and magnetic moment µ 0 = 3.23µ B (µ B being the Bohr magneton). The total particle volume is set to V = 7744v ≃ 431(nm) 3 , a mean field H = 3k B T c /µ 0 S + H eff is used, where H eff includes the anisotropy (H k ) and external (H w ) fields with the Curie temperature set to T c = 646K, and we chose the phenomenological damping parameter for the LLG-Langevin equation λ = 0.1. The same parameters are used for benchmarking the stochastic macrospin models in Refs. 12,13 and Eq. (10). We chose τ −1 s = Γ 1 both for Eq. (10) and for the angular diffusion model from Ref. 13 .
We examined the cooling of single-domain magnetic particles through T c for a wide range of parameters. For each parameter choice 4096 particles are initialized at 750K and their temperature decays exponentially T [K] = 300 + 450 × exp(−t/τ ). The benchmarks include two time constants, τ = 100ps and τ = 1ns, and a parameter scan over the external field magnitude (H w ) and its angle (θ) with the anisotropy axis. In each simulation we measure the percentage of particles (P ↑ ) for which M · H w > 0. At the end of the cooling process, when the macrospin is frozen, the final P ↑ (T ≪ T c ) represents the probability that a particle has been written correctly, i.e. the "write probability". Simulations use the Heun numerical scheme 20 with 0.5f s time-step.
The LLB equation is used extensively to model systems for magnetic data storage, such as Heat-Assisted Magnetic Recording (HAMR) 21-23 and magneto-optical recording 24,25 . Current HAMR media employ FePt grains that contain more than 10 4 unit cells. Even for fu-ture media with reduced grain-sizes, the number of unit cells can be expected to exceed 10 3 , sufficient to justify application of the central limit theorem in deriving the macrospin distribution [Eq. (5)]. However, the underlying assumption that the mean field and temperature are constant across the magnetic grain, is weaker. For example, for unit cells on the surface of the grain the overall magnetic field can be considerably smaller because there are fewer neighboring atoms to interact with. For systems where the number of surface unit cells is comparatively large, that is for small or irregularly-shaped grains, this effect becomes significant. Moreover, temperature variations within the grain influence the strength of the exchange and anisotropy fields, thereby undermining the accuracy of the macrospin model. The mean field approximation also precludes the excitation of internal modes of the grain that reflect the properties of the exchange interaction. Atomistic models can be used to lift the restrictions due to the mean field approximation, and they offer the flexibility of modeling heterogenous magnetic materials, albeit at considerable computational cost. Benchmarking against such models is beyond the scope of the present work.
We have developed a stochastic Landau-Lifshitz-Bloch equation [Eq. (10)] for single-domain magnetic particles, in which the stationary state is determined directly from the underlying atomic spin equilibrium and the stochastic process is a mean-reverting random walk. At room temperature the stochastic behavior reduces to angular diffusion in agreement with the stochastic LLG equation. The angular diffusion coefficient increases linearly with temperature before saturating at T c . Diffusion in the magnetization magnitude increases rapidly with temperature when T T c and saturates at T c . The stochastic LLB model is in excellent agreement with mean-field atomistic LLG-Langevin simulations and is immediately applicable for modeling a variety of magnetic systems.