Magnetization dynamics driven by a jump-noise process

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PHYSICAL REVIEW B 83, 020402(R) (2011)

Magnetization dynamics driven by a jump-noise process I. Mayergoyz Department of Electrical and Computer Engineering, UMIACS and AppEl Center, University of Maryland College Park, College Park, Maryland 20742, USA

G. Bertotti Instituto Nazionale di Ricerca Metrologica (INRiM), Torino, Italy

C. Serpico Dipartimento di Ingegneria Elettrica Universit`a di Napoli “Federico II”, Napoli, Italy (Received 30 July 2010; revised manuscript received 23 November 2010; published 10 January 2011) It is proposed to use a jump-noise process in magnetization dynamics equations to account for thermal bath effects. It is shown that in the case of a small jump-noise process, the Landau-Lifshitz and Gilbert damping terms can be analytically derived as deterministic (average) effects caused by the jump-noise process. Simple formulas for the damping constant are derived that relate it to the scattering rate of the jump-noise process and elucidate its dependence on magnetization. Generalized H -theorems for jump-noise-driven magnetization dynamics are presented. Random switching of magnetization caused by the jump-noise process is studied and it is demonstrated that the switching rate has different temperature dependence at relatively high and very low temperatures, which is traditionally attributed to the existence of phenomena of macroscopic magnetization tunneling. DOI: 10.1103/PhysRevB.83.020402

PACS number(s): 75.78.−n, 75.40.Gb, 75.60.Lr, 75.75.Jn

Stochastic magnetization dynamics has been the focus of considerable research for many years. This research is motivated by numerous scientific and technological applications that range from the study of thermally activated magnetization switching in magnetic data storage devices to the analysis of power spectral density of spin-torque nano-oscillators in the area of spintronics. Traditionally, the thermal bath effects on magnetization dynamics have been accounted for by introducing two distinct and disjoint terms in the magnetization dynamics equations1–4 : (i) a deterministic damping term and (ii) a white-noise torque term. It has been done this way because the white-noise process alone cannot fully and adequately describe the thermal bath effects since its expected value is zero. The common physical origin of these two terms is often accounted for by imposing fluctuation-dissipation relations on them, which is justifiable for close-to-equilibrium conditions but questionable when dealing with far-from-equilibrium magnetization dynamics. Furthermore, there is no internal mechanism in the traditional approach to compute the dependence of a damping parameter on magnetization. For these reasons, it is clearly desirable to describe the thermal bath effects by a single random process and to extract the damping term as its expected value. It is demonstrated in this Rapid Communication that this can be achieved by modeling the thermal bath effects by a jump-noise process. This jump process may also reflect discontinuous magnetization transitions occurring at the microscopic quantum mechanical level. It is shown in this Rapid Communication that in the case of a small jump-noise process, the Landau-Lifshitz5 and Gilbert6 damping terms can be derived as deterministic (average) effects caused by the jump-noise process. Moreover, the corresponding damping parameters can be directly related to the scattering rate of the jump-noise process. This is clearly consistent with the physical origin of damping and may serve as a bridge for connecting the damping with fundamental microscopic processes. This 1098-0121/2011/83(2)/020402(4)

approach also reveals that the damping parameter is state (magnetization) dependent, and an explicit formula for this parameter in terms of free magnetic energy is given. It is worthwhile to stress here that only in a particular case of small noise can the Landau-Lifshitz and Gilbert terms be derived from the jump-noise description of the thermal bath. However, the jump-noise description [as provided by formulas (1)–(6), (13), and (14)] is not limited to the case of small noise and can be used when the noise is not small. In other words, it can be used when the traditional small-noise approach based on damping and white-noise torque terms breaks down. Then, generalized H -theorems for magnetization dynamics driven by a jump-noise process are presented. These theorems are similar to the celebrated Boltzmann H -theorem in statistical mechanics7 and they reveal the existence of a class of Lyapunov functionals that are monotonically decreased with time during the magnetization dynamics. Finally, random magnetization switching is studied for a wide range of temperatures, and it is demonstrated that at very low temperatures the temperature dependence of the switching rate may appreciably deviate from the predictions of thermal activation theory. This is traditionally attributed to the phenomena of “macroscopic tunneling” of magnetization.8,9 To start the discussion, consider the following magnetization dynamics equation: dM (1) = −γ (M × Heff ) + Tr (t), dt where Tr (t) is a jump-noise process that accounts for thermal bath effects, while all other notations have their usual meaning. The process Tr (t) can be written in the form  mi δ(t − ti ), (2) Tr (t) = i

where mi are random jumps of magnetization occurring at random times ti . This implies that the magnetization dynamics

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©2011 American Physical Society

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I. MAYERGOYZ, G. BERTOTTI, AND C. SERPICO

PHYSICAL REVIEW B 83, 020402(R) (2011)

consists of continuous magnetization processions randomly interrupted by random jumps in magnetization. To fully describe the process Tr (t), statistics of mi and ti must be defined. In addition, process (2) must be defined in such a way that the dynamics described by Eq. (1) occurs on the sphere : |M(t)| = Ms = const.

(3)

The latter constraint is the consequence of strong local exchange interaction. To specify process (2), the transition probability rate S(Mi ,Mi+1 ) is introduced, where Mi = M(ti− ) and Mi+1 = M(ti+ ) = Mi + mi . To satisfy the constraint |M(t)| = Ms , the function S(Mi ,Mi+1 ) is defined on the sphere . By using this function, the random timing of magnetization jumps is characterized by the formula   ti +τ  λ(M(t)) dt , (4) Prob(ti+1 − ti > τ ) = exp − ti

where λ(M(t)) is the scattering rate defined as  λ(M(t)) = S(M(t),M )d  .

(5)



It is apparent that λ(M(t)) dt is the probability of occurrence of a magnetization jump during the time interval dt. Assuming that a jump event occurs at some time ti , the probability density function of magnetization jump mi is given by the formula S(Mi ,Mi + mi ) . χ (mi |Mi ) = λ(Mi )

(7)

has the where E() denotes the expected value, while T(0) r meaning of fluctuations. It is known11 that E(Tr (t)) = λ(M(t))E(m(t)).

(8)

If the process Tr (t) is small in the sense that only small jumps m(t) have a non-negligible probability of occurrence, then it can be shown that M(t) · E(m(t))  0.

(9)

Indeed, from formula (3) and relation |M(t)| = |M(t − )| = |M(t − ) + m|, we find that [2M(t) + m] · m = 0. Since for jumps with non-negligible probability the inequality 2Ms  |m| holds, we conclude that M(t) · m  0,

M(t) · E(Tr (t))  0.

(10)

The latter implies that the expected value E(Tr (t)) is in the plane normal to M(t). By using the basis vectors M × Heff and M × (M × Heff ) in that plane, we find E(Tr (t)) ≈ −γ  (M × Heff ) − αM × (M × Heff ).

(11)

This means that Eq. (1) can be transformed into the randomly perturbed Landau-Lifshitz equation dM = −γ˜ (M × Heff ) − αM × (M × Heff ) + T(0) r (t), (12) dt where γ˜ = γ + γ  . It is easy to see that if basis vectors M × Heff and M × dM dt are chosen in the plane normal to M(t), then the randomly perturbed Landau-Lifshitz-Gilbert dynamics equation emerges. Thus, the choice of different basis vectors in the plane normal to M(t) leads to different (but mathematically equivalent) forms of the magnetization dynamics equation. The stochastic magnetization dynamics defined by Eq. (1) can be also studied on the level of transition probability density w(M,t; M0 ,t0 ). For the sake of notational simplicity, the “backward variables” M0 and t0 will be suppressed (omitted) in the sequel. It can be shown12 that w(M,t) is the solution of the following equation: ∂w ˆ = −γ div [(M × ∇ g)w] + C(w), ∂t

(6)

Formulas (3)–(6) completely define the jump process (2) provided that S(Mi ,Mi+1 ) is known. It can be also remarked that stochastic magnetization dynamics equations described by formulas (1)–(6) are (in many ways) similar to the semiclassical transport equations used in semiconductor physics.10 Process (2) can be written in the form Tr (t) = E(Tr ) + T(0) r (t),

which implies formula (9). From formulas (8) and (9), we obtain

(13)

ˆ where C(w) is the so-called “collision integral”:  ˆ C(w) = [S(M ,M)w(M ,t) − S(M,M )w(M,t)]d  . 

(14) Here g is the magnetic free energy related to Heff by the formula Heff = −∇ g. It is apparent that Eq. (13) contains the collision integral ˆ term C(w) instead of a “diffusion” term as in the case of the white-noise process. Equation (13) is convenient for the derivation of constraints on S(M,M ) that follow from the consistency of this equation with thermodynamics. At 0 = thermal equilibrium w(M,t) = w0 (M) = Ae−g(M)/kT , ∂w ∂t 0 and div [(M × ∇ g)w0 ] = 0. Thus, w0 (M) satisfies Eq. (13) if the “detailed balance” condition S(M,M )w0 (M) = S(M ,M)w0 (M )

(15)

is fulfilled. This condition is quite natural from the physical point of view. From the mathematical point of view, the detailed balance condition (15) can be used for the symmetrization of the kernel of the collision integral when the ratio w(M,t)/w0 (M) is treated as an unknown function. Indeed, by using symmetric function K(M,M ) = S(M,M )w0 (M) > 0, ˆ the collision integral C(w) can be represented as follows:    w(M,t) w(M ,t)  ˆ C(w) = − K(M,M ) − d  . (16) w0 (M) w0 (M ) 

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MAGNETIZATION DYNAMICS DRIVEN BY A JUMP-NOISE . . .

By using (16) and the fact that a value of an integral does not depend on the notation of the variable of integration, it is straightforward to derive the identity

 w(M,t) ˆ d C(w(M,t))G w0 (M)      1 w(M,t) w(M ,t)  =− − K(M,M ) 2   w0 (M) w0 (M )



   w(M ,t) w(M,t) −G d  d, (17) × G w0 (M) w0 (M ) where G is an arbitrary function. From this identity, the uniqueness of the thermal equilibrium solution w0 (M) to Eq. (13) can be easily established by using G(w/w0 ) = w/w0 . More importantly, this identity can be used for the derivation of the following generalized H -theorem for the random magnetization dynamics governed by Eq. (1):

 d w(M,t) d < 0, (18) w0 (M)F dt  w0 (M) where F(x) is any function with the property that its derivative F  (x) = G(x) is a monotonically increasing function. The main details of the derivation of (18) can be outlined as follows. Both sides of Eq. (13) are multiplied by an arbitrary monotonically increasing function G( ww0 ) and integrated over . Then, it can be shown that the first term on the right-hand side is equal to zero, while the second term on the same side is negative according to the identity (17). Finally, by using the “product rule” for differentiation, we arrive at the inequality (18). In a particular case, when the function F(w/w0 ) = (w/w0 )ln(w/w0 ), formula (18) is reduced to

 w(M,t) d d < 0, (19) w(M,t)ln dt  w0 (M) which is quite similar in form to the celebrated Boltzmann H -theorem in  statistical mechanics. By introducing average energy U =  g(M)w(M,t) d, entropy, and free energy F , it is easy to show by using formula (19) that dF < 0, and the dt rate of decrease of F can be estimated by using the right-hand side of (17). It can be remarked that results similar to (18) and (19) can be obtained for the case of random magnetization dynamics driven by spin-polarized current injection. In this case w0 (M) has the meaning of stationary distribution. It is easy to derive from the detailed balance condition that 



S(M,M ) = φ(M,M )e

g(M)−g(M ) 2kT

,

|M−M |2 2σ 2

e

g(M)−g(M ) 2kT

,

where σ 2 = φ(0)/|φ  (0)|. It must be remarked that the identical expression for S was postulated in the study of nucleation rates (see Ref. 13). By using the last formula and formulas (6), (8), and (11) as well as the smallness of σ 2 , explicit expressions can be derived for α and λ(M). The derivation proceeds as follows: According to formulas (6), (8), and (21), we have  |M−M |2 g(M)−g(M ) E(Tr ) = A m e− 2σ 2 e 2kT d. (22) By taking into account that M − M = m and g(M) − g(M ) ≈ −m · ∇ g, we end up with the following Gaussiantype integral:

   |m|2 m · ∇ g E(Tr )  A m exp − d. (23) + 2σ 2 2kT By evaluating this integral, we find 

 1 σ |∇ g| 2 πσ4 E(Tr ) = − ∇ g. A exp kT 2 2kT

(21)

(24)

At the same time, using formula (21) in Eq. (5), in a similar way we derive

  1 σ |∇ g| 2 . (25) λ(M) = 2π σ 2 A exp 2 2kT By substituting formula (25) into (24) and taking into account that ∇ g = M12 M × (M × Heff ), we finally arrive at the s formula α = λ(M)

σ2 . 2kT Ms2

(26)

The last two formulas clearly reveal the dependence of α on properties of the jump-noise process as well as on M and can be used, for instance, to estimate the possible range of variation of α during the magnetization switching dynamics. The presented calculations reveal that E(Tr ) has only a component along the vector M × (M × Heff ) and, consequently, γ  in formula (11) is equal to zero. This occurs because function φ in Eq. (20) has been chosen to be isotropic, that is, it depends on m = |M − M |. It is expected that when φ is anisotropic, γ  may be different from zero. This means that the thermal bath effects may result in modification of γ . In our calculations, the first-order approximation for g(M) − g(M ) has been used. More accurate results can be obtained in the axially symmetric case by using the secondorder approximation for g(M) − g(M ). The relevant formulas are presented below: σ2 , 2kT Ms2 Q2    1 σ ∂g 2 2π σ 2 A , exp λ(θ ) = Q 2 2kT Q ∂θ α = λ(θ )

(20)

where φ(M,M ) = φ(M ,M). In general, function φ(M,M ) is expected to be found through some identification procedure based on experimental data. However, it is natural to assume on the physical grounds that φ(M,M ) = φ(|M − M |) and it is narrow peaked at M = M . Then, using the formula φ(x) = eln(φ(x)) and three terms in the Taylor expansion for ln(φ(x)), formula (20) can be transformed as follows: S(M,M ) = Ae−

PHYSICAL REVIEW B 83, 020402(R) (2011)

(27) (28)

σ 2 ∂ 2g . (29) 2kT ∂θ 2 Next, we consider the problem of random switching of magnetization in uniaxial nanoparticles with only two possible equilibrium (minimum energy) states, located in two energy wells D1 and D2 such that D1 + D2 = . More complex

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Q=1−

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PHYSICAL REVIEW B 83, 020402(R) (2011)

energy landscapes can be treated in a similar way. We shall use the Kramers-Brown quasilocal equilibrium approximation for transition probability density function:2,14 w(M,t) 

2  Pi (t) i=1

Zi

e−

g(M)−gi kT

=



Pi (t)w0i (M),

(30)

i

where Pi (t) is the probability of M ∈ Di , Zi are normalization constants, and gi energy minima. By integrating both sides of Eq. (13) over Dk (k = 1,2), taking into account that the integral of the first term on the right-hand side of (13) is equal to zero and using formula (30), the following master equation can be rigorously derived: 2 2   dPk = λki Pi − Pk λik , dt i=1 i=1



 λki = Dk

S(M ,M) d  w0i (M ) d.

(31)

(32)

Di

By using formulas (21) and (30), we find

  g(M)+g(M )−2gi |M−M |2 A 2kT λki = e− 2σ 2 e− d  d. Zi Dk Di

(33)

master equation (32) we find dP2 (34)  λ21 . dt It is clear from (33) and (34) that there are two distinct regimes that are controlled by σ 2 and T . For sufficiently large T and small σ 2 , the integrand in (33) is strongly peaked in the narrow region near the boundary between D1 and D2 . In this narrow region, the second factor of the integrand in (33) is close to exp{−(gmax − gi )/kT }. This leads to the classical Arrhenius law for the switching rate, and its temperature dependence is typical for thermally activated switching phenomena. Another distinct case is when at very low temperatures the second factor in the integrand in (33) dominates and it is strongly peaked for M and M being around respective energy minima. This phenomenon is especially pronounced if g2 < g1 , which corresponds to the presence of bias magnetic fields. It is apparent that this will result in a different temperature dependence of the switching rate at very low temperatures. This is consistent with experimental observations of low-temperature magnetization switching, and it is usually attributed to the phenomena of macroscopic tunneling of magnetization. It is also apparent that for intermediate values of T the crossover between the above distinct regimes may occur,9,15 which may be used for the identification of σ 2 [or function φ(M,M ) in (20)].

At the initial stage of random switching from well D1 into well D2 , we have P1  1 and P2  0. Consequently, from the

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This research has been supported by NSF and by ONR.

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L. Thomas, F. Lionti, R. Ballou, D. Gatteschi, R. Sessoli, and B. Barbara, Nature (London) 383, 145 (1996). 10 C. E. Korman and I. D. Mayergoyz, Phys. Rev. B 54, 17620 (1996). 11 D. Kannan, An Introduction to Stochastic Processes (NorthHolland, New York, 1979) 12 I. I. Gihman and A. V. Skorohod, Stochastic Differential Equations (Springer-Verlag, New York, 1972) 13 J. S. Langer, Phys. Rev. Lett. 21, 973 (1968). 14 H. A. Kramers, Physica 7, 284 (1940). 15 W. Wernsdorfer, E. Bonet Orozco, K. Hasselbach, A. Benoit, D. Maily, O. Kubo, H. Nakano, and B. Barbara, Phys. Rev. Lett. 79, 4014 (1997).

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