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Spin torque in the framework of random magnetization dynamics driven by a jump-noise process
Physica B ∎ (∎∎∎∎) ∎∎∎–∎∎∎
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Spin torque in the framework of random magnetization dynamics driven by a jump-noise process G. Bertotti a, C. Serpico b, Z. Liu c, A. Lee c, I. Mayergoyz d,n a
INRiM, Torino, Italy Dipartimento di Ingegneria Elettrica, Università di Napoli “Federico II”, Napoli, Italy c Department of Electrical and Computer Engineering, University of Maryland College Park, MD 20742 USA d Department of Electrical and Computer Engineering, UMIACS and AppEl Center, University of Maryland College Park, MD 20742 USA b
art ic l e i nf o
Keywords: Jump-noise process Spin torque effect Magnetization dynamics
a b s t r a c t It is demonstrated that the Slonczewski spin-torque term can be naturally derived from the equation for magnetization dynamics driven by a jump-noise process. The central point of the derivation is the modification of transition probability rate of the jump-noise process caused by spin-polarized current injection. This modification results in two distinct terms in the expected value of the jump-noise process: the traditional one corresponding to the Landau–Lifshitz damping and another one representing the Slonczewski spin-torque term. & 2013 Elsevier B.V. All rights reserved.
It is well known that the effects of spin-polarized current injection on magnetization dynamics can be studied by including the Slonczewski spin-torque term in the Landau–Lifshitz equation. Slonczewski derived (see Ref. [1]) this term by applying the semiclassical approach to the analysis of spin transfer phenomena. The purpose of this paper is to demonstrate that this spin-torque term can be naturally derived by using the macroscopic equation for magnetization dynamics driven by a jump-noise process. Random magnetization dynamics driven by a jump-noise process has been recently proposed (see Refs. [2–4]) as a unifying approach to the description of damping and fluctuation effects caused by interaction with a thermal bath. The jump-noise process is fully described by a transition probability rate and this rate is modified in the presence of spin-polarized current injection. This modification leads to two distinct terms in the expected value of the jump-noise process: the traditional one corresponding to the Landau–Lifshitz damping and the other representing the Slonczewski spin-torque term. It is remarkable that this approach results in a damping coefficient which is affected by the presence of spin-polarized current injection. This is quite reasonable from the physical point of view because the spin-polarized current injection affects overall thermal interactions, which are ultimately responsible for relaxation and damping. This discussion is started with a brief review of magnetization dynamics driven by a jump-noise process. This dynamics
n
is described by the following equation: dM ¼ γ ðM Heff Þ þ Tr ðtÞ; dt
ð1Þ
where Tr ðtÞ is a jump-noise process which accounts for thermal effects, γ is the gyromagnetic ratio, M is the magnetization and Heff is the effective magnetic field. The random process Tr ðtÞ can be mathematically represented in the form Tr ðtÞ ¼ ∑Mi δðtt i Þ;
ð2Þ
i
where Mi are random jumps in magnetization occurring at random times ti. It is apparent that magnetization dynamics described by formulas (1) and (2) consists of continuous precessions randomly interrupted by random jumps in magnetization. It is clear that the jump-noise process is fully defined if statistics of ti and Mi are specified. It turns out that this can be accomplished by introducing the transition probability rate SðMi ; Mi þ 1 Þ, where Mi ¼ Mðt i Þ and Mi þ 1 ¼ Mi þ Mi are magnetization vectors immediately before and after random jumps Mi , respectively. At temperature T well below the Curie temperature, magnetization dynamics occurs on the sphere Σ defined by the formula jMðtÞj ¼ M s ðTÞ ¼ const:
ð3Þ
This implies that the transition probability rate SðMi ; Mi þ 1 Þ is defined on the sphere Σ as well. By using the transition rate SðMi ; Mi þ 1 Þ, the scattering rate can be computed as follows: Corresponding author. Tel.: þ 13014053657. E-mail address: [email protected] (I. Mayergoyz).
λðMðtÞÞ ¼ ∮Σ SðM; M′Þ dΣ M′ :
0921-4526/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physb.2013.08.030
Please cite this article as: G. Bertotti, et al., Physica B (2013), http://dx.doi.org/10.1016/j.physb.2013.08.030i
ð4Þ
G. Bertotti et al. / Physica B ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
It is clear that λðMðtÞÞ dt has the meaning of probability of magnetization jump occurrence from M to any M′ on Σ during the time interval dt. By using this fact, it can be concluded that the statistics of random times ti is described by the formula Z ti þ τ Probðt i þ 1 t i 4 τÞ ¼ exp λðMðtÞÞ dt ; ð5Þ ti
while the probability density χ ðMi jMi Þ of random jump Mi is given by the equation SðMi ; Mi þ mi Þ : ð6Þ χ ðmi Mi Þ ¼ λðMi Þ It can be remarked that stochastic magnetization dynamics equations described by formulas (1)–(6) are similar to semiclassical transport equations in semiconductors (see Ref. [5]). Thus, the jump-noise process is fully defined by formulas (4)–(6), provided that the transition probability rate SðMi ; Mi þ 1 Þ is specified. This rate can be always written in the form gðMÞgðM′Þ ; ð7Þ SðM; M′Þ ¼ ϕðM; M′Þexp 2kT where gðMÞ is the micromagnetic energy density and ϕðM; M′Þ is some function to be determined. It has been shown (see Ref. [2]) that in the absence of spinpolarized current injection the principle of “detailed balance” implies that ϕðM; M′Þ is a symmetric function, i.e.,
ϕðM; M′Þ ¼ ϕðM′; MÞ:
ð8Þ
Furthermore, under some rather natural assumptions the following formula (see Ref. [3]) can be derived for ϕðM; M′Þ jMM′j2 jmj2 ϕðM; M′Þ ¼ B exp ¼ B exp ; ð9Þ 2s2 2s2 where B and s are some parameters which characterize the strength of the jump-noise process. Next, we decompose the jump-noise process Tr ðtÞ ¼ E½Tr ðtÞ þ Tð0Þ r ðtÞ;
ð10Þ
where E½Tr ðtÞ is a vector equal to the expected value of Tr ðtÞ, while Tð0Þ r ðtÞ has the physical meaning of fluctuations. By substituting the last formula into Eq. (1) and neglecting the fluctuations, we end up with the following deterministic equation for magnetization dynamics: dM ¼ γ ðM Heff Þ þ E½Tr ðtÞ: dt
ð11Þ
This equation is fully defined as long as the calculation of E½Tr ðtÞ is performed. It is known (see Refs. [6,7]) that E½Tr ðtÞ ¼ λðMðtÞÞE½mðtÞ:
By substituting the last formula into Eq. (7), we find jmj2 ΦðMÞΦðM′Þ SðM; M′Þ ¼ B exp 2 þ ; 2kT 2s
ð15Þ
where
ð12Þ
ΦðMÞ ¼ gðMÞ þ Ψ ðM ep Þ:
ð13Þ
In the case of a small jump-noise process when small magnetization jumps M are most probable, the following approximation can be used:
By using formulas (6) and (12) we conclude that E½Tr ðtÞ ¼ ∮Σ mSðM; M′Þ dm:
which is proportional to the injected current density J, have spin orientation parallel or anti-parallel to ep , where ep is the magnetization orientation in the fixed layer. Therefore, the exchange energy, which is minimized through random (spin-transfer) scattering, is proportional to JM ep . The magnetization state favored as a result of this spin-transfer scattering is the one for which M ep ¼ M s when electrons flow from the fixed layer to the free layer ðJ o 0Þ; for the opposite flow ðJ 40Þ of electrons the favored magnetization state is defined by equality M ep ¼ M s . The question can be naturally asked why the spin-polarized current injection modifies random scattering in such a way that it favors the reduction of the exchange energy rather than the reduction of the total magnetic energy related to the effective field. First, the favoring of the reduction of exchange energy is consistent with the semiclassical (WKB) quantum-mechanical approach used in Ref. [1] for the derivation of the expression for spin-torque. Second, in ferromagnets the strong exchange interaction prevails over all other interactions at the small spatial scales compatible with the continuum hypothesis. For this reason, it is only natural to conclude that the spin transfer is mostly driven by the exchange interaction. It is worthwhile to point out that a somewhat similar situation occurs in micromagnetics where the exchange interaction constraint jMðtÞj ¼ M s ðTÞ ¼ const is taken into account before other interactions favoring the reduction of the total magnetic energy are accounted for. In addition to the exchange interaction, there also exist interface phenomena which affect polarization and scattering of electrons and eventually determine the portion of spin-polarized electrons in the injected current. For this reason, the energy of interaction JM ep has to be scaled by a function which takes into account the interface processes. This function depends on the mutual orientation of M and ep , and thus it can be taken as a function ηðM ep Þ. As a result, the effective energy of interaction between injected electrons and free layer magnetization can be written as J ηðM ep ÞM ep or simply as Ψ ðM ep Þ where Ψ ðuÞ ¼ JuηðuÞ. In analogy with the exponential term in formula (7), which describes thermal scattering, one is naturally led to write the function ϕðM; M′Þ in the following form, reflecting the nature of spin-transfer and interface scattering: Ψ ðM ep ÞΨ ðM′ ep Þ jmj2 : ð14Þ ϕðM; M′Þ ¼ B exp 2 exp 2kT 2s
Next, we consider how the transition probability rate SðM; M′Þ is modified in the presence of spin-polarized current injection. This injection is a non-equilibrium process whose steady state may be quite different from thermodynamic equilibrium. This must lead to the breaking of the symmetry of the function ϕðM; M′Þ. Indeed, as mentioned previously, the transition rate given by formula (7) leads to thermodynamic equilibrium for any symmetric function ϕðM; M′Þ. In determining the form of the symmetry breaking of ϕðM; M′Þ, it must be kept in mind that the spin-polarized current injection modifies random scattering in such a way that it favors the reduction of the exchange energy between the “free-layer” magnetization M and the magnetic moment carried by the injected polarized electrons. These polarized electrons, the number of
ΦðMÞΦðM′Þ C m ∇Σ Φ;
ð16Þ
ð17Þ
which leads to the following expression for the transition probability rate: jmj2 m ∇Σ Φ : ð18Þ SðM; M′Þ C B exp 2 2kT 2s By substituting the last formula into Eq. (13), we find Z jmj2 m ∇Σ Φ E½Tr ðtÞ C B m exp 2 dm: 2kT 2s
ð19Þ
By completing the square in the power of the exponent in formula (19), the integral in this formula can be reduced to the Gaussian
Please cite this article as: G. Bertotti, et al., Physica B (2013), http://dx.doi.org/10.1016/j.physb.2013.08.030i
G. Bertotti et al. / Physica B ∎ (∎∎∎∎) ∎∎∎–∎∎∎
integral. This eventually leads to the following result: ( 2 ) π Bs4 1 sj∇Σ Φj exp E½Tr ðtÞ ¼ ∇Σ Φ; 2 2kT kT
By using this formula in Eq. (24), we derive ð20Þ
where ∇Σ Φ ¼ ∇Σ g þ ∇Σ Ψ ðM ep Þ:
ð21Þ
It is known that the first term in the right-hand side of formula (21) can be written in the form ∇Σ g ¼
μ0 V M 2s
M ðM Heff Þ;
ð22Þ
where V is the volume of the free layer. It can be easily shown that the second term in formula (21) can be transformed as follows: ∇Σ Ψ ðM ep Þ ¼
Ψ ′ðM ep Þ M 2s
M ðM ep Þ;
where Ψ ′ is the derivative of Ψ . Next, we introduce the damping coefficient ( 2 ) πμ VBs4 1 sj∇Σ Φj exp α¼ 02 : 2 2kT γ Ms kT
ð23Þ
ð24Þ
E½Tr ðtÞ ¼ αγ M ðM Heff Þ
ð25Þ
By substituting the last formula into Eq. (11), we end up with dM ¼ γ ðM Heff Þαγ M ðM Heff Þ dt þ
αγ Ψ ′ðM ep ÞM ðM ep Þ: μ0 V
ð26Þ
The last magnetization dynamics equation attains the form which is mathematically similar to the Slonczewski equation in the case of the following particular choice of function Ψ : bμ V Ψ ðM ep Þ ¼ ~ 0 lnð1 þ c~ p M ep Þ; cp
ð27Þ
where b is a parameter proportional to the spin-polarized current density, c~ p ¼ cp =M s and cp depends only on polarizing factor P (see Refs. [1,8]). Indeed, in this case
Ψ ′ðM ep Þ ¼
bμ0 V 1 þ c~ p M ep
α ¼ λðMÞ
μ0 Vs2 : 2 2γ kTM s
ð28Þ
ð32Þ
The last two formulas clearly reveal the dependence of damping coefficient α on properties of the jump-noise process as well as on M. Formulas (16) and (31)–(32) also suggest that the damping coefficient α is affected by the presence of spin-polarized current injection. The latter can be expected on physical grounds because the spin-polarized current injection affects overall random thermal scattering which is ultimately responsible for damping. Finally, it is instructive to represent the magnetization dynamics equation (26) in the following equivalent form: dM s2 ¼ γ M Heff λðMÞ ∇ Φ: dt 2kT Σ
ð33Þ
This equation describes the magnetization dynamics on the sphere Σ . According to the Helmholtz decomposition theorem, any dynamics on a sphere can be fully described in terms of two potentials (see Ref. [7]) dM ¼ M ∇Σ Γ ∇Σ Ω: dt
Now, by using formulas (20)–(24), we obtain
αγ Ψ ′ðM ep ÞM ðM ep Þ: þ μ0 V
3
ð34Þ
It is clear from formulas (32)–(33) that in the case of constant (magnetization-independent) scattering rate λ, potentials Γ and Ω are scaled versions of g and Φ, respectively. In the case of magnetization-dependent λ, the identification of potentials Γ and Ω is much more involved. It is interesting to point out that in the case of uniaxial symmetry magnetization-dependent scattering rate λðMÞ does not affect the analysis of magnetization oscillations caused by spin-polarized current injection. According to Eq. (33), these microwave magnetization oscillations will occur along precessional trajectories with the property that at each point of these trajectories ∇Σ Φ ¼ 0:
ð35Þ
By using the last formula, the analysis of magnetization oscillations in uniaxial systems can be performed in the same way as in Ref. [8]. For non-uniaxial systems, the analysis of magnetization oscillations is based on the mathematical machinery of Melnikov functions (see Ref. [8]) and in this case magnetization dependence of the scattering rate may actually affect oscillation frequencies.
and the dynamics equation (26) is reduced to dM ¼ γ ðM Heff Þαγ M ðM Heff Þ dt M ðM ep Þ : þ αγ b 1 þ c~ p M ep
Acknowledgments ð29Þ
In general, the factor Ψ ′ðM ep Þ in Eq. (26) accounts for interface effects and should be identified from experiments. Next, we give the expression for the damping coefficient α in terms of the scattering rate λðMðtÞÞ. To this end, we substitute formula (18) into Eq. (4) and find Z jmj2 m ∇Σ Φ dm: ð30Þ λðMðtÞÞ C B exp 2 2kT 2s By evaluating the integral in the last formula through completing of the square, we obtain ( 2 ) 1 sj∇Σ Φj λðMÞ ¼ 2π Bs2 exp : ð31Þ 2 2kT
This research has been supported by NSF and ONR, as well as by the Grant PRIN 2010-2011 N.2010ECA8P3.
References [1] [2] [3] [4] [5] [6]
J.C. Slonczewski, J. Magn. Magn. Mater. 159 (1996) L1. I. Mayergoyz, G. Bertotti, C. Serpico, Phys. Rev. B 83 (2011) 020402(R). I. Mayergoyz, G. Bertotti, C. Serpico, J. Appl. Phys. 109 (2011) 07D312. I. Mayergoyz, G. Bertotti, C. Serpico, J. Appl. Phys. 109 (2011) 07D327. C.E. Korman, I.D. Mayergoyz, Phys. Rev. B 54 (1996) 17620. I.I. Gihman, A.V. Skorohod, Stochastic Differential Equations, Springer-Verlag, New York, 1972. [7] D. Kannan, An Introduction to Stochastic Processes, North-Holland, New York, 1979. [8] G. Bertotti, I. Mayergoyz, C. Serpico, Nonlinear Magnetization Dynamics in Nanosystems, Elsevier, 2009.
Please cite this article as: G. Bertotti, et al., Physica B (2013), http://dx.doi.org/10.1016/j.physb.2013.08.030i
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Spin torque in the framework of random magnetization dynamics driven by a jump-noise process G. Bertotti a, C. Serpico b, Z. Liu c, A. Lee c, I. Mayergoyz d,n a
INRiM, Torino, Italy Dipartimento di Ingegneria Elettrica, Università di Napoli “Federico II”, Napoli, Italy c Department of Electrical and Computer Engineering, University of Maryland College Park, MD 20742 USA d Department of Electrical and Computer Engineering, UMIACS and AppEl Center, University of Maryland College Park, MD 20742 USA b
art ic l e i nf o
Keywords: Jump-noise process Spin torque effect Magnetization dynamics
a b s t r a c t It is demonstrated that the Slonczewski spin-torque term can be naturally derived from the equation for magnetization dynamics driven by a jump-noise process. The central point of the derivation is the modification of transition probability rate of the jump-noise process caused by spin-polarized current injection. This modification results in two distinct terms in the expected value of the jump-noise process: the traditional one corresponding to the Landau–Lifshitz damping and another one representing the Slonczewski spin-torque term. & 2013 Elsevier B.V. All rights reserved.
It is well known that the effects of spin-polarized current injection on magnetization dynamics can be studied by including the Slonczewski spin-torque term in the Landau–Lifshitz equation. Slonczewski derived (see Ref. [1]) this term by applying the semiclassical approach to the analysis of spin transfer phenomena. The purpose of this paper is to demonstrate that this spin-torque term can be naturally derived by using the macroscopic equation for magnetization dynamics driven by a jump-noise process. Random magnetization dynamics driven by a jump-noise process has been recently proposed (see Refs. [2–4]) as a unifying approach to the description of damping and fluctuation effects caused by interaction with a thermal bath. The jump-noise process is fully described by a transition probability rate and this rate is modified in the presence of spin-polarized current injection. This modification leads to two distinct terms in the expected value of the jump-noise process: the traditional one corresponding to the Landau–Lifshitz damping and the other representing the Slonczewski spin-torque term. It is remarkable that this approach results in a damping coefficient which is affected by the presence of spin-polarized current injection. This is quite reasonable from the physical point of view because the spin-polarized current injection affects overall thermal interactions, which are ultimately responsible for relaxation and damping. This discussion is started with a brief review of magnetization dynamics driven by a jump-noise process. This dynamics
n
is described by the following equation: dM ¼ γ ðM Heff Þ þ Tr ðtÞ; dt
ð1Þ
where Tr ðtÞ is a jump-noise process which accounts for thermal effects, γ is the gyromagnetic ratio, M is the magnetization and Heff is the effective magnetic field. The random process Tr ðtÞ can be mathematically represented in the form Tr ðtÞ ¼ ∑Mi δðtt i Þ;
ð2Þ
i
where Mi are random jumps in magnetization occurring at random times ti. It is apparent that magnetization dynamics described by formulas (1) and (2) consists of continuous precessions randomly interrupted by random jumps in magnetization. It is clear that the jump-noise process is fully defined if statistics of ti and Mi are specified. It turns out that this can be accomplished by introducing the transition probability rate SðMi ; Mi þ 1 Þ, where Mi ¼ Mðt i Þ and Mi þ 1 ¼ Mi þ Mi are magnetization vectors immediately before and after random jumps Mi , respectively. At temperature T well below the Curie temperature, magnetization dynamics occurs on the sphere Σ defined by the formula jMðtÞj ¼ M s ðTÞ ¼ const:
ð3Þ
This implies that the transition probability rate SðMi ; Mi þ 1 Þ is defined on the sphere Σ as well. By using the transition rate SðMi ; Mi þ 1 Þ, the scattering rate can be computed as follows: Corresponding author. Tel.: þ 13014053657. E-mail address: [email protected] (I. Mayergoyz).
λðMðtÞÞ ¼ ∮Σ SðM; M′Þ dΣ M′ :
0921-4526/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physb.2013.08.030
Please cite this article as: G. Bertotti, et al., Physica B (2013), http://dx.doi.org/10.1016/j.physb.2013.08.030i
ð4Þ
G. Bertotti et al. / Physica B ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
It is clear that λðMðtÞÞ dt has the meaning of probability of magnetization jump occurrence from M to any M′ on Σ during the time interval dt. By using this fact, it can be concluded that the statistics of random times ti is described by the formula Z ti þ τ Probðt i þ 1 t i 4 τÞ ¼ exp λðMðtÞÞ dt ; ð5Þ ti
while the probability density χ ðMi jMi Þ of random jump Mi is given by the equation SðMi ; Mi þ mi Þ : ð6Þ χ ðmi Mi Þ ¼ λðMi Þ It can be remarked that stochastic magnetization dynamics equations described by formulas (1)–(6) are similar to semiclassical transport equations in semiconductors (see Ref. [5]). Thus, the jump-noise process is fully defined by formulas (4)–(6), provided that the transition probability rate SðMi ; Mi þ 1 Þ is specified. This rate can be always written in the form gðMÞgðM′Þ ; ð7Þ SðM; M′Þ ¼ ϕðM; M′Þexp 2kT where gðMÞ is the micromagnetic energy density and ϕðM; M′Þ is some function to be determined. It has been shown (see Ref. [2]) that in the absence of spinpolarized current injection the principle of “detailed balance” implies that ϕðM; M′Þ is a symmetric function, i.e.,
ϕðM; M′Þ ¼ ϕðM′; MÞ:
ð8Þ
Furthermore, under some rather natural assumptions the following formula (see Ref. [3]) can be derived for ϕðM; M′Þ jMM′j2 jmj2 ϕðM; M′Þ ¼ B exp ¼ B exp ; ð9Þ 2s2 2s2 where B and s are some parameters which characterize the strength of the jump-noise process. Next, we decompose the jump-noise process Tr ðtÞ ¼ E½Tr ðtÞ þ Tð0Þ r ðtÞ;
ð10Þ
where E½Tr ðtÞ is a vector equal to the expected value of Tr ðtÞ, while Tð0Þ r ðtÞ has the physical meaning of fluctuations. By substituting the last formula into Eq. (1) and neglecting the fluctuations, we end up with the following deterministic equation for magnetization dynamics: dM ¼ γ ðM Heff Þ þ E½Tr ðtÞ: dt
ð11Þ
This equation is fully defined as long as the calculation of E½Tr ðtÞ is performed. It is known (see Refs. [6,7]) that E½Tr ðtÞ ¼ λðMðtÞÞE½mðtÞ:
By substituting the last formula into Eq. (7), we find jmj2 ΦðMÞΦðM′Þ SðM; M′Þ ¼ B exp 2 þ ; 2kT 2s
ð15Þ
where
ð12Þ
ΦðMÞ ¼ gðMÞ þ Ψ ðM ep Þ:
ð13Þ
In the case of a small jump-noise process when small magnetization jumps M are most probable, the following approximation can be used:
By using formulas (6) and (12) we conclude that E½Tr ðtÞ ¼ ∮Σ mSðM; M′Þ dm:
which is proportional to the injected current density J, have spin orientation parallel or anti-parallel to ep , where ep is the magnetization orientation in the fixed layer. Therefore, the exchange energy, which is minimized through random (spin-transfer) scattering, is proportional to JM ep . The magnetization state favored as a result of this spin-transfer scattering is the one for which M ep ¼ M s when electrons flow from the fixed layer to the free layer ðJ o 0Þ; for the opposite flow ðJ 40Þ of electrons the favored magnetization state is defined by equality M ep ¼ M s . The question can be naturally asked why the spin-polarized current injection modifies random scattering in such a way that it favors the reduction of the exchange energy rather than the reduction of the total magnetic energy related to the effective field. First, the favoring of the reduction of exchange energy is consistent with the semiclassical (WKB) quantum-mechanical approach used in Ref. [1] for the derivation of the expression for spin-torque. Second, in ferromagnets the strong exchange interaction prevails over all other interactions at the small spatial scales compatible with the continuum hypothesis. For this reason, it is only natural to conclude that the spin transfer is mostly driven by the exchange interaction. It is worthwhile to point out that a somewhat similar situation occurs in micromagnetics where the exchange interaction constraint jMðtÞj ¼ M s ðTÞ ¼ const is taken into account before other interactions favoring the reduction of the total magnetic energy are accounted for. In addition to the exchange interaction, there also exist interface phenomena which affect polarization and scattering of electrons and eventually determine the portion of spin-polarized electrons in the injected current. For this reason, the energy of interaction JM ep has to be scaled by a function which takes into account the interface processes. This function depends on the mutual orientation of M and ep , and thus it can be taken as a function ηðM ep Þ. As a result, the effective energy of interaction between injected electrons and free layer magnetization can be written as J ηðM ep ÞM ep or simply as Ψ ðM ep Þ where Ψ ðuÞ ¼ JuηðuÞ. In analogy with the exponential term in formula (7), which describes thermal scattering, one is naturally led to write the function ϕðM; M′Þ in the following form, reflecting the nature of spin-transfer and interface scattering: Ψ ðM ep ÞΨ ðM′ ep Þ jmj2 : ð14Þ ϕðM; M′Þ ¼ B exp 2 exp 2kT 2s
Next, we consider how the transition probability rate SðM; M′Þ is modified in the presence of spin-polarized current injection. This injection is a non-equilibrium process whose steady state may be quite different from thermodynamic equilibrium. This must lead to the breaking of the symmetry of the function ϕðM; M′Þ. Indeed, as mentioned previously, the transition rate given by formula (7) leads to thermodynamic equilibrium for any symmetric function ϕðM; M′Þ. In determining the form of the symmetry breaking of ϕðM; M′Þ, it must be kept in mind that the spin-polarized current injection modifies random scattering in such a way that it favors the reduction of the exchange energy between the “free-layer” magnetization M and the magnetic moment carried by the injected polarized electrons. These polarized electrons, the number of
ΦðMÞΦðM′Þ C m ∇Σ Φ;
ð16Þ
ð17Þ
which leads to the following expression for the transition probability rate: jmj2 m ∇Σ Φ : ð18Þ SðM; M′Þ C B exp 2 2kT 2s By substituting the last formula into Eq. (13), we find Z jmj2 m ∇Σ Φ E½Tr ðtÞ C B m exp 2 dm: 2kT 2s
ð19Þ
By completing the square in the power of the exponent in formula (19), the integral in this formula can be reduced to the Gaussian
Please cite this article as: G. Bertotti, et al., Physica B (2013), http://dx.doi.org/10.1016/j.physb.2013.08.030i
G. Bertotti et al. / Physica B ∎ (∎∎∎∎) ∎∎∎–∎∎∎
integral. This eventually leads to the following result: ( 2 ) π Bs4 1 sj∇Σ Φj exp E½Tr ðtÞ ¼ ∇Σ Φ; 2 2kT kT
By using this formula in Eq. (24), we derive ð20Þ
where ∇Σ Φ ¼ ∇Σ g þ ∇Σ Ψ ðM ep Þ:
ð21Þ
It is known that the first term in the right-hand side of formula (21) can be written in the form ∇Σ g ¼
μ0 V M 2s
M ðM Heff Þ;
ð22Þ
where V is the volume of the free layer. It can be easily shown that the second term in formula (21) can be transformed as follows: ∇Σ Ψ ðM ep Þ ¼
Ψ ′ðM ep Þ M 2s
M ðM ep Þ;
where Ψ ′ is the derivative of Ψ . Next, we introduce the damping coefficient ( 2 ) πμ VBs4 1 sj∇Σ Φj exp α¼ 02 : 2 2kT γ Ms kT
ð23Þ
ð24Þ
E½Tr ðtÞ ¼ αγ M ðM Heff Þ
ð25Þ
By substituting the last formula into Eq. (11), we end up with dM ¼ γ ðM Heff Þαγ M ðM Heff Þ dt þ
αγ Ψ ′ðM ep ÞM ðM ep Þ: μ0 V
ð26Þ
The last magnetization dynamics equation attains the form which is mathematically similar to the Slonczewski equation in the case of the following particular choice of function Ψ : bμ V Ψ ðM ep Þ ¼ ~ 0 lnð1 þ c~ p M ep Þ; cp
ð27Þ
where b is a parameter proportional to the spin-polarized current density, c~ p ¼ cp =M s and cp depends only on polarizing factor P (see Refs. [1,8]). Indeed, in this case
Ψ ′ðM ep Þ ¼
bμ0 V 1 þ c~ p M ep
α ¼ λðMÞ
μ0 Vs2 : 2 2γ kTM s
ð28Þ
ð32Þ
The last two formulas clearly reveal the dependence of damping coefficient α on properties of the jump-noise process as well as on M. Formulas (16) and (31)–(32) also suggest that the damping coefficient α is affected by the presence of spin-polarized current injection. The latter can be expected on physical grounds because the spin-polarized current injection affects overall random thermal scattering which is ultimately responsible for damping. Finally, it is instructive to represent the magnetization dynamics equation (26) in the following equivalent form: dM s2 ¼ γ M Heff λðMÞ ∇ Φ: dt 2kT Σ
ð33Þ
This equation describes the magnetization dynamics on the sphere Σ . According to the Helmholtz decomposition theorem, any dynamics on a sphere can be fully described in terms of two potentials (see Ref. [7]) dM ¼ M ∇Σ Γ ∇Σ Ω: dt
Now, by using formulas (20)–(24), we obtain
αγ Ψ ′ðM ep ÞM ðM ep Þ: þ μ0 V
3
ð34Þ
It is clear from formulas (32)–(33) that in the case of constant (magnetization-independent) scattering rate λ, potentials Γ and Ω are scaled versions of g and Φ, respectively. In the case of magnetization-dependent λ, the identification of potentials Γ and Ω is much more involved. It is interesting to point out that in the case of uniaxial symmetry magnetization-dependent scattering rate λðMÞ does not affect the analysis of magnetization oscillations caused by spin-polarized current injection. According to Eq. (33), these microwave magnetization oscillations will occur along precessional trajectories with the property that at each point of these trajectories ∇Σ Φ ¼ 0:
ð35Þ
By using the last formula, the analysis of magnetization oscillations in uniaxial systems can be performed in the same way as in Ref. [8]. For non-uniaxial systems, the analysis of magnetization oscillations is based on the mathematical machinery of Melnikov functions (see Ref. [8]) and in this case magnetization dependence of the scattering rate may actually affect oscillation frequencies.
and the dynamics equation (26) is reduced to dM ¼ γ ðM Heff Þαγ M ðM Heff Þ dt M ðM ep Þ : þ αγ b 1 þ c~ p M ep
Acknowledgments ð29Þ
In general, the factor Ψ ′ðM ep Þ in Eq. (26) accounts for interface effects and should be identified from experiments. Next, we give the expression for the damping coefficient α in terms of the scattering rate λðMðtÞÞ. To this end, we substitute formula (18) into Eq. (4) and find Z jmj2 m ∇Σ Φ dm: ð30Þ λðMðtÞÞ C B exp 2 2kT 2s By evaluating the integral in the last formula through completing of the square, we obtain ( 2 ) 1 sj∇Σ Φj λðMÞ ¼ 2π Bs2 exp : ð31Þ 2 2kT
This research has been supported by NSF and ONR, as well as by the Grant PRIN 2010-2011 N.2010ECA8P3.
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Please cite this article as: G. Bertotti, et al., Physica B (2013), http://dx.doi.org/10.1016/j.physb.2013.08.030i