Non-equilibrium thermodynamic study of magnetization dynamics in the presence of spin-transfer torque

# Non-equilibrium thermodynamic study of magnetization dynamics in the presence of spin-transfer torque

## Abstract

The dynamics of magnetization in the presence of spin-transfer torque was studied. We derived the equation for the motion of magnetization in the presence of a spin current by using the local equilibrium assumption in non-equilibrium thermodynamics. We show that, in the resultant equation, the ratio of the Gilbert damping constant, , and the coefficient, , of the current-induced torque, called non-adiabatic torque, depends on the relaxation time of the fluctuating field . The equality holds when is very short compared to the time scale of magnetization dynamics. We apply our theory to current-induced magnetization reversal in magnetic multilayers and show that the switching time is a decreasing function of .

###### pacs:

Spin-transfer torque-induced magnetization dynamics such as current-induced magnetization reversal Slonczewski (1996); Berger (1996); Katine et al. (2000), domain wall motion Kläui et al. (2003), and microwave generation Kiselev et al. (2003) have attracted a great deal of attention because of their potential applications to future nano-spinelectronic devices. In the absence of spin-transfer torque, magnetization dynamics is described by either the Landau-Lifshitz (LL) equation Landau and Lifshitz (1935) or the Landau-Lifshitz-Gilbert (LLG) equation Gilbert (). It is known that the LL and LLG equations become equivalent through rescaling of the gyromagnetic ratio.

However, this is not the case in the presence of spin-transfer torque. For domain wall dynamics, the following LLG-type equation has been studied by several groups Thiaville et al. (2005); Barnes and Maekawa (2005); Kohno et al. (2006):

 ∂t⟨M⟩+v⋅∇⟨M⟩=γH×⟨M⟩ +αM⟨M⟩×∂t⟨M⟩+βM⟨M⟩×[(v⋅∇)⟨M⟩], (1)

where represents the magnetization, is the velocity, is the gyromagnetic ratio and is the Gilbert damping constant. The second term on the left-hand side represents the adiabatic contribution of spin-transfer torque. The first and the second terms on the right-hand side are the torque due to the effective magnetic field and the Gilbert damping. The last term on the right-hand side of Eq. (1) represents the current-induced torque, called “non-adiabatic torque” or simply the term. The directions of the adiabatic contribution of spin-transfer torque and non-adiabatic torque are shown in Fig. 1 (a).

As shown by Thiaville et al., the value of the coefficient strongly influences the motion of the domain wall Thiaville et al. (2005). However, the value of the coefficient is still controversial, and different conclusions have been drawn from different approaches Barnes and Maekawa (2005); Tserkovnyak et al. (2006, 2008); Kohno et al. (2006); Xiao et al. (2006); Stiles et al. (2007); Duine et al. (2007). For example, Barnes and Maekawa showed that the value of should be equal to that of the Gilbert damping constant to satisfy the requirement that the relaxation should cease at the minimum of electrostatic energy, even under particle flow. Kohno et al. performed microscopic calculations of spin torques in disordered ferromagnets and showed that the and terms arise from the spin relaxation processes and that in general Kohno et al. (2006). Tserkovnyak et al. Tserkovnyak et al. (2006) derived the term using a quasiparticle approximation and showed that within a self-consistent picture based on the local density approximation.

In the current-induced magnetization dynamics in the magnetic multilayers shown in Fig. 1 (b) Zhang et al. (2004); Tulapurkar et al. (2005); Kubota et al. (2008), the non-adiabatic torque exerts a strong effect, and therefore affects the direct-current voltage of the spin torque diode, as shown in Refs. Tulapurkar et al. (2005); Kubota et al. (2008). The magnetization dynamics of the free layer, , has been studied by using the following LLG-type equation,

 ∂tS2−Iegℏ(S2×S1)×S2=γH×S2+αS2S2×∂tS2 +ηIS2×S1, (2)

where is the charge current density, is the amplitude of the spin torque introduced by Slonczewski Slonczewski (1996), is the Dirac constant and represents the magnitude of the “non-adiabatic torque” which is sometimes called the field-like torque Tulapurkar et al. (2005); Kubota et al. (2008).

In this paper, we study the magnetization dynamics induced by spin-transfer torque in the framework of non-equilibrium thermodynamics. We derive the equation of motion of the magnetization in the presence of a spin current by using the local equilibrium assumption. In the resultant equation, the Gilbert damping term and the term are expressed as memory terms with the relaxation time of the fluctuating field . We show that the value of the coefficient is not equal to that of the Gilbert damping constant in general. However, we also show that the equality holds if . We apply our theory to the current-induced magnetization reversal in magnetic multilayers and show that the switching time is a decreasing function of .

Let us first briefly introduce the non-equilibrium statistical theory of magnetization dynamics in the absence of spin current Miyazaki and Seki (1998). The LLG equation describing the motion of magnetization under an effective magnetic field is given by

 ∂tM=γH×M+αMM×∂tM. (3)

The equivalent LL equation is expressed as

 ∂tM=γ1+α2H×M−αγM(1+α2)M×(M×H). (4)

The Langevin equations leading to Eqs. (3) and (4) by taking the ensemble average of magnetization , are

 ∂tm=γHtot×m (5) ∂tδH=−1τc(δH−χsm)+R(t), (6)

where the total magnetic field is the sum of the effective magnetic field and the fluctuating magnetic field and is the susceptibility of the local magnetic field induced at the position of the spin. According to Eq. (6) the fluctuating magnetic field relaxes toward the reaction field with the relaxation time . The random field satisfies and the fluctuation-dissipation relation, , where is the Boltzmann constant, is the temperature, denotes the ensemble average, and represents the Cartesian components. It was shown that Eqs. (5) and (6) lead to Kawabata’s extended Landau-Lifshitz equation Kawabata (1972) derived by the projection operator method Miyazaki and Seki (1998). In the Markovian limit, i.e., , we can obtain the LLG equation (3) and the corresponding LL equation (4) with Miyazaki and Seki (1998).

In order to consider the flow of spins, i.e., spin current, we introduce the positional dependence. Since we are interested in the average motion, it is convenient to introduce the mean velocity of the carrier, . The average magnetization, , is obtained by introducing the positional dependence and taking the ensemble average of Eq. (5). In terms of the mean velocity, the ensemble average of the left-hand side of Eq. (5) leads to

 ∂t⟨m⟩+(v⋅∇)⟨m⟩. (7)

Assuming , which is applicable when the thermal fluctuation is small compared to the mean value, we obtain

 ∂t⟨m⟩+(v⋅∇)⟨m⟩=γ⟨Htot(x,t)⟩×⟨m(x,t)⟩. (8)

The mean magnetization density is expressed as , i.e., by the product of the scalar and vectorial components both of which depend on the position of the spin carrier at time . The spin carrier density satisfies the continuity equation,

 ∂tρ(x,t)+∇⋅(vρ(x,t))=0. (9)

By multiplying the left-hand side of Eq. (8) by and using the continuity equation (9), the closed expression for the mean magnetization is obtained as de Groot and Mazur (1962)

 ρ(∂t⟨m⟩+v⋅∇⟨m⟩) =∂tρ⟨m⟩+⟨m⟩∇⋅vρ+ρv⋅∇⟨m⟩ =∂t⟨M⟩+Divv⟨M⟩, (10)

where is defined by

 Divv⟨M⟩=3∑i=1∂vi⟨M⟩∂xi=⟨M⟩(∇⋅v)+(v⋅∇)⟨M⟩. (11)

By multiplying the right-hand side of Eq. (8) by and using Eq. (10), we obtain

 ∂t⟨M⟩+Divv⟨M⟩=γ(H+⟨δH⟩)×⟨M⟩. (12)

Equation (12) takes the standard form of a time-evolution equation for extensive thermodynamical variables under flow de Groot and Mazur (1962). The average of Eq. (6) with the positional dependence is given by

 ∂t⟨δH(x,t)⟩=−1τc[⟨δH(x,t)⟩−χ⟨M(x(t),t)⟩], (13)

where is the mean position at time of the spin carrier, which flows with velocity and is assumed to be a constant independent of the position. Equations (12) and (13) constitute the basis for the subsequent study of magnetization dynamics in the presence of spin-transfer torque.

The formal solution of Eq. (13) is expressed as

 ⟨δH(x,t)⟩=χτc∫t−∞ψ(t−t′)⟨M(x(t′),t′)⟩dt′, (14)

where the memory kernel is given by . Using partial integration, we obtain

 ⟨δH(x,t)⟩=χ⟨M⟩−∫t−∞ψ(t−t′)χ⟨˙M(t′)⟩dt′, (15)

where the explicit expression for is given by the convective derivative,

 ˙M(t) =∂tM(x(t),t)+(v⋅∇)M(x(t),t). (16)

Substituting Eq. (15) into Eq. (12), we obtain the equation of motion for the mean magnetization density,

 ∂t⟨M⟩+Divv⟨M⟩=γH×⟨M⟩ +γ∫t−∞dt′ψ(t−t′)χ⟨M(t)⟩×⟨˙M(t′)⟩. (17)

Equation (17) supplemented by Eq. (16) is the principal result of this paper.

When the relaxation time of the fluctuating field, , is very short compared to the time scale of the magnetization dynamics, the memory kernel is decoupled and Eq. (17) can be written in the form of an LLG-type equation as

 ∂t⟨M⟩+Divv⟨M⟩=γH×⟨M⟩+αM⟨M⟩×˙⟨M⟩, (18)

where is the Gilbert damping constant. Substituting the explicit form of the convective derivative, Eq. (16), into Eq. (18) and using Eq.(11) we obtain the following LLG-type equation:

 ∂t⟨M⟩+⟨M⟩(∇⋅v)+(v⋅∇)⟨M⟩=γH×⟨M⟩ +αM⟨M⟩×∂t⟨M⟩+αM⟨M⟩×[(v⋅∇)⟨M⟩]. (19)

If , Eq. (19) reduces to Eq. (14) of Ref. Barnes and Maekawa (2005), which is derived by replacing the time derivative of magnetization on both sides of the LLG equation (3) by the convective derivative . The term appears not on the right-hand side of Eq. (19) but on the left-hand side, which means we cannot obtain Eq. (19) using the same procedure used in Ref. Barnes and Maekawa (2005). As shown in Refs. Barnes and Maekawa (2005); Tserkovnyak et al. (1935), Eq. (19) with leads to a steady-state solution in the comoving frame, , where denotes the stationary solution in the absence of domain wall motion. However, if , the steady-state solution may break the Galilean invariance. The situation can be realized, for example, in magnetic semiconductors Ohno (1998); Dietl and Ohno (2006), where the spin carrier density is spatially inhomogeneous, i.e., .

The last term of Eq. (19) represents the non-adiabatic component of the current-induced torque, which is also known as the “ term”. By comparing Eq. (19) with Eq. (1), one can see that the coefficient of the last term is equal to the Gilbert damping constant . However, Eq. (19) is valid when the relaxation time of the fluctuating field, , is very short compared to the time scale of the magnetization dynamics. It should be noted that the general form of the equation describing the magnetization dynamics is given by Eq. (17) where the last term on the right-hand side is the origin of the and terms. It is possible to project the torque represented by the memory function onto the direction of the and terms. This projection leads to in general.

In order to observe the effect of on the magnetization dynamics we applied our theory to the current-induced magnetization switching in the magnetic multilayer shown in Fig.1 (b). We assumed that the fixed and free layers are single-domain magnetic layers acting as a large spin characterized by the total magnetization vector defined as , where for the fixed (free) layer and denotes the volume integration over the fixed (free) layer. Both the magnetization vector of the fixed layer and the effective magnetic field, , acting on the free layer lie in the plane.

Integrating Eqs. (12) and (13) over the volume of the free layer, we obtain the equations,

 Missing or unrecognized delimiter for \left (20) ∂t⟨δH⟩=−1τc(⟨δH⟩−χVS2), (21)

where is the spin current tensor represents the surface integration over the free layer, is the unit normal vector of the surface, and is defined by the volume of the free layer .

The same procedure used to derive Eq. (17) yields

 ∂tS2+∫dS^n⋅J=γH×S2 +γ∫t−∞dt′ψ(t−t′)χVS2(t)×∂t′S2(t′), (22)

where .

When the relaxation time of the fluctuating field is short compared to the time scale of magnetization dynamics, the LLG-type equation in the presence of the spin-transfer torque is obtained as

 ∂tS2+∫dS^n⋅J=γH×S2+αS2S2×∂tS2, (23)

where . By introducing the conventional form of the spin-transfer torque Slonczewski (1996), we obtain the following LLG-type equation:

 ∂tS2−Iegℏ(S2×S1)×S2=γH×S2+αS2S2×∂tS2. (24)

However, Eq. (24) is valid only when . As mentioned before, the torque represented by using the memory function generally has a component parallel to the non-adiabatic torque. In order to observe the effect of the non-adiabatic torque induced by the memory function on the magnetization dynamics, we performed numerical simulation using Eqs. (20) and (21).

For the simulation, we used the following conditions. At the initial time of , we assumed that the magnetization of the free layer is aligned parallel to the effective magnetic field and the angle between the magnetizations of the fixed and the free layers is 45. This arrangement corresponds to the recent experiment on a magnetic tunnel junction system Kubota et al. (2008). We also assumed that the fluctuation field has zero mean value at , i.e., .

In Fig. 2, we plot the time dependence of the component of the magnetization of the free layer, , under the large-enough spin current to flip the magnetization of the free layer, . The value of is varied while the value of is maintained. The solid, dotted, and dot-dashed lines correspond to , and 10.0, respectively. As shown in Fig. 2, the time required for the magnetization of the free layer to flip decreases with increasing , which can be understood by considering the non-adiabatic torque induced by the spin current. The non-adiabatic torque induced by the spin current is obtained by projecting the torque given by the last term of Eq. (22) onto the direction of , which results in the positive contribution to the spin-flip motion of . Since the last term of Eq. (22) includes a memory function, the non-adiabatic torque induced by the spin current increases with increasing . Therefore, the time required for to flip decreases with increasing . For we observe no further decrease of the time required for to flip because the memory function is an integral of the vector and the contributions from the memory at is eliminated.

In conclusion, we derived the equation for the motion of magnetization in the presence of a spin current by using the local equilibrium assumption in non-equilibrium thermodynamics. We demonstrated that the value of the coefficient is not equal to that of the Gilbert damping constant in general. However, we also show that the equality holds if . We then applied our theory to current-induced magnetization reversal in magnetic multilayers and showed that the switching time is a decreasing function of .

The authors would like to acknowledge the valuable discussions they had with S.E. Barnes, S. Maekawa, P. M. Levy, K. Kitahara, K. Matsushita, J. Sato and T. Taniguchi. This work was supported by NEDO.

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