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Dynamics of a magnetic moment induced by a spin-polarized current

Wonkee Kim and F. Marsiglio

Department of Physics, University of Alberta, Edmonton, Alberta, Canada, T6G 2J1 E?ects of an incoming spin-polarized current on a magnetic moment are explored. We found that the spin torque occurs only when the incoming spin changes as a function of time inside of the magnetic ?lm. This implies that some modi?cations are necessary in a phenomenological model where the coe?cient of the spin torque term is a constant, and the coe?cient is determined by dynamics instead of geometrical details. The precession of the magnetization reversal depends on the incoming energy of electrons in the spin-polarized current. If the incoming energy is smaller than the interaction energy, the magnetization does not precess while reversing its direction. We also found that the relaxation time associated with the reversal depends on the incoming energy. The coupling between an incoming spin and a magnetic moment can be estimated by measuring the relaxation time.

PACS numbers: 75.70.Cn,72.25.Ba,75.60.Jk

arXiv:cond-mat/0307633v1 [cond-mat.mtrl-sci] 25 Jul 2003

I.

INTRODUCTION

Tremendous attention has been paid to the dynamics of magnetization in recent years because this problem is of fundamental importance in understanding magnetism and because the problem is of interest to technological applications in magnetic devices.1 One of intriguing features of magnetization motion is spin transfer from a spin-polarized current to a magnetization of a ferromagnetic ?lm, theoretically proposed by Slonczewski2 and Berger3, and later experimentally veri?ed.4,5 Since this spin transfer mechanism was ?rst conceptualized, many studies6,7,8,9,10 have been performed on this phenomenon. However, the dynamics of a magnetic moment driven by a spin-polarized current has not been fully explored. In this paper we investigate the current-driven precession and reversal of a magnetic moment. This is done quantum mechanically using a simple Hamiltonian, without introducing an external magnetic ?eld. In this way one can easily distinguish contributions from the current from those induced by an externally applied ?eld. To this end, we describe the motion of a magnetic moment in the lab frame where details of the magnetization reversal are best illustrated. Since dynamics of a magnetic moment can be formally described in the local moment frame and such a description may also give some intuition

about the dynamics, we examine an interaction between a spin-polarized current and a magnetic moment in the local frame at the Hamiltonian level in section (II). Then dynamics in the lab frame is illustrated in section (III). In this section one can see details of the dynamics such as under what conditions the motion of the magnetization can be non-precessional or the relaxation time associated with the reversal is a minimum. These phenomena have not been explored in the literature so far. Section (IV) is devoted to discussions about the adiabatic approximation used to describe the motion of a magnetic moment, and we close with a summary.

II. FORMALISM IN THE LOCAL MOMENT FRAME

To describe e?ects of an incoming spin current on a magnetic moment M, as in Ref.6 one can choose a frame (X ′ Y ′ Z ′ ), where z ′ is parallel to M. Such a frame is ? called the local magnetic moment frame. Extensive work on ferromagnetism in the local moment frame has been done in Ref.11 An advantage of this frame is that it is trivial to diagonalize an interaction between an incoming spin s and a magnetic moment: ?2JH M · s, where JH is the coupling. Let us start with a simple Hamiltonian relevant to the interaction:

H=

+ dx ψα (x) ?

?2 2m

+ ψα (x) ? 2JH M(x) · s + V (x)ψα (x)ψα (x)

(1)

+ where ψα (x) creates an electron with a spin α at x, m is the electron mass and V (x) is an impurity potential. The magnitude of the magnetic moment is M0 , which remains unchanged. The electron spin can be represented + as si = (1/2)ψα σ i ψβ , where σ i is a Pauli matrix with i = x, y and z. We assume that the magnetic moment

is determined by a localized electron Ψ so that the kinetic part of the localized electron is not included in the Hamiltonian. Suppose a local magnetic moment M(x) points in the direction (θ, φ) at x as seen in Fig. 1. Then, a local rotation ( or coordinate transformation to the local moment

2 frame) is introduced: ψα (x) = Uαβ (x)χβ (x) , where

U (x) =

cos(θ/2)e?iφ/2 ? sin(θ/2)e?iφ/2 sin(θ/2)eiφ/2 cos(θ/2)eiφ/2

.

(2)

In terms of χ(x), the Hamiltonian can be written as H= dx 1 + + ? χ+ Uβα · ? (Uαγ χγ ) ? JH χ+ Uβα (M · σα? )U?ν χν + V (x)χ+ χα α β β 2m (3)

Since the interaction term in the Hamiltonian is diagonalized in this χ(x) basis, we obtain H = H0 + where H0 = jαβ = Aαβ

(0)

dx Aαβ · jαβ + Aαβ ραβ

(0)

,

(4)

dx χ+ (x) ? α

?2 2m

z χα ? JH M0 χα σαβ χβ + V (x)χ+ (x)χα (x) α + Aαβ = ?iUαγ (?Uγβ ) ,

1 χ+ ?χβ ? (?χ+ )χβ , α 2im α 1 + = (?Uαγ ) · (?Uγβ ) , and 2m

ραβ = χ+ χβ . α

After diagonalizing the interaction, we have an extra (0) term H ′ = dx Aαβ · jαβ + Aαβ ραβ in Eq. (4) instead of o?-diagonal terms of the interaction in Eq. (1). Using the explicit form of U (x), we can calculate vector po(0) tentials Aαβ and Aαβ . This was the route followed in Ref.6 , which led to a monopole-like term in the energy. Those authors attributed the spin torque term to this new vector potential, which is purely geometrical. Here we follow a di?erent route, since we are interested in a simpler case, where the magnetization is not a function of position. Thus, in our case of a single-domain ferromagnet, the extra term shown above will disappear because ?U = 0. Instead, our spin torque will be present due to the dynamics of the coupled spin-moment system. In addition, we will not require an assumption regarding the magnitude of JH in order to proceed, and we will utilize an impurity potential for convergence purposes which is otherwise irrelevant to the spin transfer as in Ref.6

the introduction, we describe the motion of the magnetic moment in the lab frame. The geometry of our problem is shown in Fig. 1. We assume a single-domain ferromagnet in the Y Z plane for simplicity and consider the Hamiltonian Eq.(1). The incoming spin is along z and ? the direction of the magnetic moment is de?ned by θ(t) and φ(t), which vary as functions of time t. The equation of motion for the magnetic moment M can be obtained quantum mechanically: dM/dt = i i [H, M]. Since M i = (1/2)γ0 Ψ+ ταβ Ψβ , where Ψ and τ i α are the operator and a Pauli matrix for localized electrons, respectively, and γ0 is the gyromagnetic ratio, the equation becomes dM = 2γ0 JH (M × s) . (5) dt To analyze this equation we consider M as a classical vector and take s as its expectation value over the ferromagnet. If we decompose s into a parallel s and a perpendicular s⊥ component to M, we know that only s⊥ contributes to the equation. We can express s⊥ using any unit vector. Let us choose, for the unit vector, the initial direction of the incoming spin s0 = z . Then ? ? ? ? ? ? s s⊥ = s⊥ (M × s0 ) + s′ M × (?0 × M ) , ⊥ where s⊥ =

? s0 ·(s×M) ? 1?(?0 ·M )2 s ?

III.

DYNAMICS OF A MAGNETIC MOMENT IN THE LAB FRAME

A disadvantage of the description in the local moment frame is that the precession of the magnetic moment cannot be seen; in other words, a precessional reversal of the magnetic moment cannot be distinguished from a plain reversal. Since our goal in this paper is to investigate the dynamics of the magnetic moment as mentioned in

(6) Using

and s′ = ⊥

? s0 ·s?(?0 ·M)(s·M) ? s ? . 1?(?0 ·M)2 s ?

Eq. (6), we can rewrite Eq. (5) as follows: dM ? = ?2γ0 JH s⊥ M×(?0 ×M )+2γ0 JH s′ (M×?0 ) . (7) s s ⊥ dt

3

.

.

kY

ψ s0

Z

in

M

ψtr

k2

2m

k

ψre θ φ 0

.

2

2m

k2

2m

JHM 0

kX

X

?JHM

0

.

FIG. 1: Geometry of a quantum mechanical problem associated with the spin transfer. The incoming electron to the positive X axis are spin-polarized along z axis. The ferromag? net surface is at x = 0 and parallel to Y Z plane. The direction of the magnetic moment is de?ned by θ and φ, which are functions of time t. The ferromagnet is assumed to be su?ciently thick.

FIG. 2: An energy band and relations among k2 /2m, 2 k↑(↓) /2m, and JH M0 . In this ?gure, it is assumed that 2 k /2m > JH M0 .

As we can see in the above equation, the ?rst term on the right hand side gives the spin torque while the second term causes a precession of the magnetic moment. We emphasize that the spin torque occurs only when s(t) changes as a function of time t. If s remains parallel to s0 , then s⊥ vanishes and no spin torque takes place. In ? this instance, the e?ect of a spin is the same as that of an external magnetic ?eld along z and the magnetic mo? ment precesses. In a phenomenological model,7 the spin ? torque is represented by M×(?0 × M ) with a proportional s constant. However, a time dependence of s⊥ is crucial as we emphasized. We also should stress that s⊥ and s′ ⊥ are determined by dynamics, not geometrical details as in Ref.10 To evaluate the expectation value of s, we need to solve the Schr¨dinger equation for the Hamiltonian Eq. (1). o Basically, the equation is one-dimensional because of translational symmetry in the Y Z plane. We choose the direction of the polarized spin to be z . Then, an incom? ing wave function |ψin with a momentum k or an energy ? = k 2 /2m is |+ eikx , where |+ is the spin-up state in the lab frame. We need to consider a normalization factor C for |ψin . Since this wave function describes an electron beam, |C|2 is the number of electrons Ne per unit length in one dimension. Intuitively, the more electrons are bombarded into the ferromagnet, the stronger is the e?ect of spin transfer. We thus expect the time scale for the reversal to scale inversely with Ne (the more the number of electrons, the faster the moment responds). Similarly, the time scale will be proportional to the magnitude of the local spin, Slocal (= M0 /γ0 ) (the larger the moment, the longer it will take to reverse it).

The re?ected (|ψre ) and transmitted (|ψtr ) wave functions are eigenstates |χ↑ and |χ↓ of the interaction 2JH M · s = JH M · σ ; namely, JH M · σ|χ↑ = JH M0 |χ↑ and JH M · σ|χ↓ = ?JH M0 |χ↓ . Therefore, |ψre = R↑ |χ↑ χ↑ |+ + R↓ |χ↓ χ↓ |+ e?ikx while |ψtr = T↑ |χ↑ χ↑ |+ eik↑ x + T↓ |χ↓ χ↓ |+ eik↓ x , (9) √ √ where k↑ = k + 2mJH M0 and k↓ = k ? 2mJH M0 as depicted in Fig. 2. If the energy of the incoming electron is less than √H M0 , k↓ = iκ↓ becomes pure imaginary J where κ↓ = 2mJH M0 ? k, and its corresponding wave function decays exponentially; e?κ↓ x . For x < 0, |ψ(x < 0) = |ψin + |ψre and for x > 0, |ψ(x > 0) = |ψtr . The coe?cients R↑(↓) and T↑(↓) are determined by matching conditions of wave functions and their derivatives at x = 0: R↑(↓) = k ? k↑(↓) k + k↑(↓) and T↑(↓) = 2k . k + k↑(↓) (10) (8)

Note that we take |ψin = |+ eikx in the above derivations. This means that the number of electrons in the incoming beam Ne is unity for simplicity; however, when we numerically solve the equation of motion for a magnetic moment, we can control this parameter. In the Hamiltonian Eq. (1), we also have an impurity potential V (x). We shall introduce mean free paths l↑ and l↓ for each channel due to the impurity, and as in Ref.3 they serve as convergence factors such as e?x/l↑ and e?x/l↓ when we average the expectation of s using |ψ(x > 0) over the ferromagnet. We assume that the thickness of the ferromagnet (L) is much larger than the

4 mean free paths: L ? l↑(↓) . One may wonder if the matching coe?cients change when the convergence factors are introduced. They do change as, for example, k↑ → k↑ + i/l↑ ; however, the conclusions we make later remain unchanged as we veri?ed. Now we can calculate the expectation value of s within the ferromagnet; si = (1/2) ψtr |σ i |ψtr with i = x , y and z. The average values of the expectation values are L ? evaluated as si = (1/2) 0 dx ψtr |σ i |ψtr . After some straightforward algebra, we obtain for incoming energy greater than JH M0

? sx ? sy ? sz

l↓ β? γ + δ?α l↑ Re [α? γ] + Re [β ? δ] + Re 2 2 (1/l↑ + 1/l↓ ) ? i(k↑ ? k↓ ) l↑ l↓ δ? α ? β? γ = ? Im [γ ? α] ? Im [δ ? β] ? Im 2 2 (1/l↑ + 1/l↓ ) ? i(k↑ ? k↓ ) l↑ l↓ β? α ? δ?γ = |α|2 ? |γ|2 + |β|2 ? |δ|2 + Re 4 4 (1/l↑ + 1/l↓ ) ? i(k↑ ? k↓ ) =

(11) (12) , (13)

where α = (1/2)T↑ (1 + mz ), β = (1/2)T↓ (1 ? mz ), γ = (1/2)T↑ (mx + imy ), and δ = ?(1/2)T↓ (mx + imy ). Here m (= M/γ0 Slocal ) is the unit vector of the magnetic moment; namely, mz = cos(θ) and mx +imy = sin(θ)eiφ . In our treatment, the incoming energy ? = k 2 /2m is a control parameter and JH M0 is a scaling parameter. Experimentally, ? can be controlled by adjusting the applied voltage while JH M0 is uncontrollable because JH is a microscopic parameter. If ? = ηJH M0 , 2 2 then k↑ /2m = (η + 1)JH M0 and k↓ /2m = (η ? 1)JH M0 . 2 De?ning k0 /2m = JH M0 , √↑ and k↓ can be written as k √ k↑ = η + 1 k0 and k↓ = η ? 1 k0 . Since the current density is in energy units in 1D (? ≡ 1), using j0 = k0 /m h with one electron per unit length we can de?ne a dimensionless time τ = j0 t, which will be used √ the nuin merical calculations. When η < 1, k↓ = i 1 ? η k0

as mentioned earlier. In this case ? changes to re?ect s √ k↓ = i 1 ? η k0 . We do not present equations for η < 1 here because the derivation is parallel to the above case and expressions are similar with those for η > 1. Since we attribute the impurity potential to the mean free paths, it is natural to assume l↑ = l↓ ≡ l. We also introduce a parameter a = lk0 . In the numerical calculations, we vary a from 0.5 to 2. Qualitative behaviors of m are not sensitive to the value of a. A dimensionless equation of motion for the magnetic moment is dm Ne /2 (m × h) , = dτ Slocal where (14)

√ √ a a 4Ai /a ? 2( η + 1 ? η ? 1)Bi 2 2 hi = |T↑ | (1 + mz )mi ? |T↓ | (1 ? mz )mi + √ √ 2 4 4 4/a2 + η+1? η?1 1 ? Im T↑ T↓ 2

(15)

with (i = x, y, and z)

Bz =

m2 + m2 . x y

Ax = Bx = Ay = By = Az =

1 ? ? ? Re T↑ T↓ mx mz + Im T↑ T↓ my 2 1 ? ? Re T↑ T↓ my ? Im T↑ T↓ mx mz 2 1 ? ? ? Re T↑ T↓ my mz ? Im T↑ T↓ mx 2 1 ? ? Re T↑ T↓ mx + Im T↑ T↓ my mz 2 1 ? Re T↑ T↓ m2 + m2 x y 2

Clearly the factor Ne /2Slocal could be absorbed into the time (already dimensionless). Since its e?ect is obvious, we set Ne /2Slocal = 4 for all our results. We choose various values of η between 0.25 and 4, and show mi (τ ) vs. τ and a locus of m in the (mx , my , mx ) coordinate. For an initial condition of m we choose θ0 = π/1.01 and φ0 = π/4 to see the magnetic moment reversal. Because of a rotational symmetry, the initial value of φ is not important. It is obvious that if θ0 = 0 or π, the spin polarized current has no e?ect on m. In Fig. 3(a), we show the locus (dotted curve) of m for η = 2

5

3(a)

1 0.5 0 ?0.5 ?1

η=2

4(a)

1 0.5 0 ?0.5 ?1

1

η = 0.5

1 0.5 ?1 ?0.5 0 0.5 1 ?1 ?0.5 0

0.5 ?1 ?0.5 0 0.5 1 ?1 ?0.5 0

1

1

3(b)

mx ( dotted ) my ( dashed )

4(b)

mx ( dotted ) my ( dashed ) 0.5 mz ( solid )

0.5

mz ( solid )

mx, my, mz

mx, my, mz

0

0

-0.5

-0.5

-1

0

10

τ

20

30

-1

0

5

τ

10

15

FIG. 3: Precessional reversal of the magnetic moment for η = 2 and a = 1. Fig. 3(a) shows the locus (dotted curve) of m and Fig. 3(b) is for mi (τ ) vs. τ . The initial direction of m is given by θ0 = π/1.01 and φ0 = π/4. Thin circles de?ne a uni-sphere.

FIG. 4: Plain reversal of the magnetic moment for η = 0.5 and a = 1. Fig. 4(a) shows the locus (dotted curve) of m and Fig. 4(b) is for mi (τ ) vs. τ . The initial direction of m is the same as in Fig. 3. Note that there are no oscillations in mx and my . Thin circles de?ne a uni-sphere.

and a = 1, and plot mi (τ ) vs. τ in Fig. 3(b). Thin circles de?ne a uni-sphere. Oscillations in mx and my imply precession of m. For η = 2, m shows a precessional reversal. On the other hand, for η = 0.5 it has a plain reversal without precession as we can see in Fig. 4(a) and (b). In this instance, mx and my do not show oscillations. The precessional reversal takes place only when η ≥ 1. This remains true for a = 0.5 or 2. We plot these results in Fig. 5(a) and 5(b) for η = 0.25 and 4. One can de?ne the relaxation time τ0 of the reversal as an elapsed time during the reversal between θ ? π and θ ? 0. When mz ? 1, we can parametrize ln [1 ? mz (τ )] = c1 ? c2 τ /τ0 , where c1 ≈ 8.9 and c2 ≈ 13.6. We found these values are independent of η and

a. For given η and a, we can determine τ0 by comparing numerical results with c1 ? c2 τ /τ0 . For example, τ0 ? 7.9 for η = 0.9 and a = 1. In general, the smaller a (or l) is, the longer τ0 is for a given η. This can be understood because the wave function |ψtr decays faster if l is shorter so that the spin transfer is relatively less e?ective and, thus, it takes a longer time to reverse m. In Fig. 6, we plot mz vs. τ for η = 4 (main frame) and for η = 0.25 (inset) with a = 0.5 (solid) 1 (dashed), and 2 (dotted curve). In this ?gure, we can see the relation between a and τ0 mentioned above. For η ≥ 1, as a increases, weak precession occurs because τ0 decreases as seen in the main frame of Fig. 6; in other words, m does not have enough time to precess strongly. We can also see

6

5(a)

1 0.5 0

a = 0.5

1

η=4

0.5 1 0.5

mz

?0.5 ?1 1 0.5 ?1 ?0.5 0 0.5 1 ?1 ?0.5 0

0 -0.5 -1 -0.5 0 10

η = 0.25

20 30

a = 0.5 ( solid ) a = 1 ( dashed ) a = 2 ( dotted )

-1 0 50

τ

100

150

5(b)

1 0.5 0 ?0.5 ?1

a=2

FIG. 6: mz as a function of τ . In the main frame, η = 4 while in the inset η = 0.25 with a = 0.5 (solid) 1 (dashed), and 2 (dotted curve).

20

a=1

1 0.5 ?1 ?0.5 0 0.5 1 ?1 ?0.5 0

15

τ0

10

6 5.5 5

FIG. 5: Locus of m for η = 0.5 (thick dotted curve)and η = 4 (dotted curve). In Fig. 5(a), a = 0.5 while a = 2 in Fig. 5(b). Regardless of a, no precession occurs when η = 0.25. Thin circles de?ne a uni-sphere.

a=2

5 4.5 4 0.8 0 0 0.5 1 0.9 1.5 1 1.1 2 1.2 2.5

such a behavior in Fig. 5 comparing a = 0.5 and a = 2 for η = 4. We plot τ0 vs. η in Fig. 7 for a given a. The relaxation time is evaluated using the parameterization: ln [1 ? mz (τ )] = c1 ? c2 τ /τ0 . In the main frame, a = 1 while in the inset a = 2. At τ = τ0 , mz (τ0 ) ? 0.99 for all plots. Interestingly, τ0 is minimum at η ? 1. Therefore it is possible to estimate the microscopic coupling parameter JH between an incoming spin and a magnetic moment by measuring τ0 (η), because τ0 has a minimum for a given mean free path.

η

FIG. 7: The relaxation time τ0 vs. η. In the main frame, a = 1 while in the inset a = 2. τ0 has a minimum value at η ? 1.

IV.

DISCUSSION AND SUMMARY

In this section we would like to discuss the adiabatic approximation, which we tacitly used to study the dynamics of a magnetic moment. First we summarize

the procedure we followed. We calculated s using |ψ(x) ; namely, s = (1/2) ψ(x)|σ|ψ(x) for x > 0 to solve dM/dt = 2γ0 JH (M × s ). Here we mention that |ψ(x > 0) is obtained by considering the Hamiltonian at a given time t following Ref.3 Since the incoming wave function |ψin ? |+ is not an eigenstate of the Hamiltonian for x > 0, we have a linear combination of |+ and |? for |ψ(x > 0) and |ψ(x < 0) . The matching conditions of wave functions at x = 0 allow us to express the coe?cients of the combination for |ψ(x > 0) in terms of M(t) (see Eqs. (9) and (10)). Now s is a function of

7 M(t), and the time dependence of s is given exclusively by M(t). This means that the time evolution of the wave function for x > 0 is not fully taken into account. In addition to the equation for dM/dt, one can derive the time derivative of the spin operator using ds/dt = i [H, s]: ds + ? · J = 2JH (s × M) , dt (16) parallel to its initial direction, no spin torque takes place. This implies that some modi?cations are necessary in a phenomenological model where the coe?cient of the spin torque term is a constant. Moreover, the coe?cient is determined by dynamics instead of geometrical details. The magnetization reversal can be precessional as well as non-precessional depending on the incoming energy of electrons in the spin-polarized current. If the incoming energy is greater than the interaction energy (JH M0 ), the magnetization precesses while reversing its direction. For the incoming energy smaller than JH M0 , the magnetization reversal is non-precessional. We also found that the relaxation time associated with the reversal depends on the incoming energy for a given mean free path. Our numerical calculations imply the coupling between an incoming spin and a magnetic moment JH can be estimated by measuring the relaxation time.

where J is the spin-current tensor. It is obvious that when we calculate an expectation value of s in Eq.(16) we need to use |ψ(x, t) ; s t = (1/2) ψ(x, t)|σ|ψ(x, t) , d where |ψ(x, t) is obtained from i dt |ψ(x, t) = H|ψ(x, t) . Rigorously speaking, one has to solve the two coupled equations for M and s using |ψ(x, t) to calculate the expectation value of s and ? · J . However, if we compare Eq. (5) or (14) with Eq. (16), we see that Eq. (14) has a factor 1/Slocal while Eq. (16) does not. This means that if we treat the magnetic moment semiclassically, i.e. Slocal ? 1, then the time scale of Eq. (14) is much longer than that of Eq. (16). Therefore, the adiabatic approximation is applicable to our analysis. In summary, we have studied the e?ect of an incoming spin-polarized current on a local magnetic moment in a magnetic thin ?lm. We found that the spin torque occurs only when the incoming spin changes as a function of time inside of the magnetic ?lm. If the incoming spin remains

Acknowledgments

We thank Mark Freeman for interest and helpful discussions. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC), by ICORE (Alberta), and by the Canadian Institute for Advanced Research (CIAR).

1

2 3 4

5 6

See, for example, Spin Dynamics in Con?ned magnetic Structure I edited by B. Hillebrands and K. Ounadjela (Springer-Verlag, 2002). J.C. Slonczewski, J. Magn. Magn. 159, L1 (1996); 195, L261 (1999) L. Berger, Phys. Rev. B 54, 9353 (1996). E.B. Myers, D.C. Ralph, J.A. Katine, R.N. Louie, and R.A. Buhrman, Science 285, 867 (1999). J.A. Katine, F.J. Albert, R.A. Buhrman, E.B. Myers, and D.C. Ralph, Phys. Rev. Lett. 84, 3149 (2000). Y. Bazaliy, B. A. Jones, and S. -C. Zhang, Phys. Rev. B

7 8 9 10

11

57, R3213 (1998). J.Z. Sun, Phys. Rev. B 62, 570 (2000). X. Waintal, E.B. Myers, P.W. Brouwer, and D.C. Ralph, Phys. Rev. B 62, 12317 (2000). M. D. Stile and A. Zangwill, Phys. Rev. B 66, 014407 (2002). S. Zhang, P.M. Levy, and A. Fert, Phys. Rev. Lett. 88, 236601 (2002). V. Koreman, J. L. Murray, and R. E. Prange, Phys. Rev. B 16, 4032 (1977).

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