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Magnetic vortex dynamics induced by spin-transfer torque
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 310 (2007) 2041–2042 www.elsevier.com/locate/jmmm
Magnetic vortex dynamics induced by spin-transfer torque J. Shibataa,, Y. Nakatanib, G. Tatarac,d, H. Kohnoe, Y. Otania,f a
RIKEN-FRS, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan Department of Computer Science, University of Electro-communications, Chofu 182-8585, Tokyo, Japan c Department of Physics, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan d PRESTO, JST, 4-1-8 Honcho Kawaguchi, Saitama, Japan e Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan f Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan
b
Available online 20 November 2006
Abstract We theoretically study the dynamics of a magnetic vortex under spin-polarized electric current in ferromagnets. The equation of motion of the vortex in terms of collective coordinates is derived. We compare our theory with recent experiments for current-induced vortex displacement and resonance motion in a ferromagnetic nanodot. Our estimate for the displacement and the resonance frequency shows a good agreement with the experiment. We also study the current-induced motion of a vortex wall in a ferromagnetic thin wire. r 2006 Elsevier B.V. All rights reserved. PACS: 72.25.Ba; 75.60.Ch; 85.75.d Keywords: Magnetic vortex; Spin-transfer torque; Spin current
Current-induced domain-wall motion [1], one of the most attractive subjects in nanoscale magnetism, has opened up the possibility of new spintronic devices. Recent experiments and numerical simulation [2–4] for this wall motion in magnetic nanowire have revealed the existence of a vortex wall configuration and its importance in the current-driven dynamics was pointed out on spin-polarized current(spin current). Also, magnetic vortices in nanodots [5] have drawn much attention and have been studied extensively. In this report, we theoretically study the current-induced vortex motion in a ferromagnet. Let us denote the spin configuration of a magnetic vortex centered at the origin by a vector field nV ðxÞ with unit modulus. As a vortex profile, we take an out-of-plane vortex; nV ðx ! 0Þ ¼ pez , where p ¼ 1 is the polarization and nV ðjxjbdV Þ ¼ cosðqj þ Cp=2Þex þ sinðqj þ Cp=2Þey , where dV is the vortex core radius, j ¼ tan1 ðy=xÞ, q ¼ 1; 2; . . . ; is the vorticity and C ¼ 1 is the chirality. We introduce a collective coordinate XðtÞ ¼ X ðtÞex þ Y ðtÞey , Corresponding author. Tel.: +81 48 462 1111; fax: +81 48 467 9650.
E-mail address: [email protected] (J. Shibata). URL: http://www.riken.jp/lab-www/nanomag/index.html. 0304-8853/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2006.10.949
which represents the position of the vortex core center, and assume that a moving vortex can be written as nðx; tÞ ¼ nV ðx XðtÞÞ and ignore the deformation and spin wave excitations. Then we obtain the Lagrangian for the vortex as [6] LV ¼ 12G ðX_ XÞ UðXÞ G ðvs XÞ, (1) R where G ¼ ez _S ðd3 x=a3 Þn ðqx n qy nÞ ¼ ð_S=a3 Þ2pLpqez ; is the gyrovector with a being the lattice constant, S being the magnitude of spin and L being the thickness of the system. UðXÞ represents a potential energy of the vortex coming from the magnetostatic energy, and vs ¼ ða3 =2eSÞj s is the drift velocity of electron spins [7,8], where e40 is the elementary charge and j s is the spin-current density, which is related to the charge-current density as Pj (P is the spin polarization). The last term on the right-hand side (r.h.s) of Eq. (1) represents the contribution from the current-induced spin-transfer torque, which takes the form, ðvs rÞn in the Landau–Lifshitz–Gilbert equation. The equation of motion of the vortex in terms of collective coordinates is derived from the Euler–Lagrange _ qLV =qX ¼ qW =qX, _ where equation; ðd=dtÞðqLV =qXÞ
ARTICLE IN PRESS J. Shibata et al. / Journal of Magnetism and Magnetic Materials 310 (2007) 2041–2042
W is the R so-called dissipation 2function given by W ¼ að_S=2Þ ðd3 x=a3 Þð_n2 ðx; tÞ ¼ aDX_ =2; with a being the constant. The constant D ¼ ð_S=a3 ÞL RGilbert damping 2 2 2 D dx dyfðqi yÞ þ sin yðqi fÞ g, generally includes a factor lnðRV =dV Þ, where RV is the system size. We assume that the vortex configuration has rotational invariance around the zaxis. The equation of motion of the vortex is then given by _ ¼ G ðvs XÞ
qUðXÞ _ þ Dðbvs aXÞ. qX
(2)
Note that we have added the term bDvs , which comes from another type of current-induced spin torque, of the form, bn ðvs rÞn, noted recently [4,9–12]. The coefficient b is usually taken to be of the order 102 . If the potential UðXÞ is absent, we obtain X_ ¼ vs for a special case a ¼ b, where the vortex core moves along the spin current. However, in general, since aab [9,12], the last term on the r.h.s. of Eq. (2) causes the moving vortex to deviate in the direction of the current. Now let us apply Eq. (2) to analyze vortex displacement induced by a DC current and resonant vortex motion induced by an AC current in a single magnetic nanodot, and compare with the experimental results [13,14]. The radius and the thickness of the nanodot are denoted by R and L, respectively, where an out-of-plane vortex with vorticity q ¼ 1 is stabilized. We assume that the electric current is uniform in the nanodot, and flowing in the positive x-direction, that is, vs ¼ ða3 =2eSÞ j s ¼ vs ex for the DC current, and vs ðtÞ ¼ vs ex eiOt for the A current. We assume a full spin polarization of the current, P ¼ 1, for simplicity. The potential energy UðXÞ is assumed harmonic, i.e., UðXÞ ¼ kX 2 =2, where k is a force constant. In Ref. [15], k is evaluated in detail, revealing its dependence on the aspect ratio g ¼ L=R of the dot. In the case of DC current, the equation of motion is ~ s ; where Z ¼ X þ given by ð1 þ i~aÞZ_ ¼ ioZ þ ð1 þ ibÞv ~ iY , a~ ¼ aD=G, b ¼ bD=G and o ¼ k=G. For an initial condition Zð0Þ ¼ 0, the solution is given by ZðtÞ ¼ ið1 þ ~ s =ofexpðiot=1 þ i~aÞ 1g: Thus the vortex center ibÞv exhibits a spiral motion, whose rotational direction depends on the sign of the core polarization p ¼ 1. The final displacement position of the vortex core is given by X ð1Þ ¼ bDvs =k and Y ð1Þ ¼ Gvs =k, which also depends on p, where the transverse force G vs and the force bDvs balance the restoring force kX. Thus, the vortex displacement is proportional to the current density j. The 2 proportionality constant, jZð1Þj=j ¼ ð1 þ b~ Þ1=2 ðjGj=kÞ 19 ða3 =2eSÞP, is estimated as 0:76 10 m3 =A for g ¼ 0:03 ~ and small b51, which is in good agreement with the experimental one, 1:23 1019 m3 =A [13]. In the case of AC current, vs ðtÞ ¼ vs ex eiOt , the vortex core rotates around the dot center with frequency O. At the resonant frequency, O ¼ o=ð1 þ a~ 2 Þ, the amplitude is given qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 by jZj ¼ ð ð1 þ a~ 2 Þð1 þ b~ Þ=aÞðvs =oÞ; which is amplified by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 the factor ð1 þ a~ 2 Þð1 þ b~ Þ=a compared to the DC-case.
350 Resonant frequency (MHz) Ω/2π
2042
300 250 200 150 100 50 0 0.00
0.02
0.04 0.06 Aspect ratio g = L/R
0.08
0.1
Fig. 1. Resonance frequency O=2p as a function of the aspect ratio g ¼ L=R of the dot. The solid line and the dots represent the analytical and experimental [14] results, respectively.
We show the resonance frequency as a function of aspect ratio in Fig. 1, which shows good agreement with experimental result [14]. Finally, we study the current-induced motion of a vortex wall in a magnetic thin wire. In this case, the potential energy is assumed to be UðXÞ ¼ k0 Y 2 yðw jY jÞ=2, where k0 is a force constant, yðY Þ is the step function, and w is the wire width. This potential leads to a restoring force along the transverse direction of the wire. If we apply a DC current ðvs ¼ vs ex Þ along the wire, a steady-state motion along the wire is eventually attained with X_ ¼ bvs =a and Y ð1Þ ¼ ð1 þ b=aÞGvs =k0 for 2ð1 þ b=aÞGvs =k0 ow. For 2ð1 þ b=aÞGvs =k0 4w, namely, for weak restoring force or large current, vortex center can reach the wire edge and moves out of the wire, resulting in the transformation into a transverse wall [3,4]. Recently, another recent study [16] of current driven vortex walls reaches similar conclusions. References [1] C.H. Marrows, Adv. Phys. 53 (2005) 585. [2] A. Yamaguchi, T. Ono, S. Nasu, K. Miyake, K. Mibu, T. Shinjo, Phys. Rev. Lett. 92 (2004) 077205. [3] M. Kla¨ui, P.O. Jubert, R. Allenspach, A. Bischof, J.A.C. Bland, G. Faini, U. Ru¨diger, C.A.F. Vaz, L. Vila, C. Vouille, Phys. Rev. Lett. 95 (2005) 026601. [4] A. Thiaville, Y. Nakatani, J. Miltat, Y. Suzuki, Europhys. Lett. 69 (2005) 990. [5] T. Shinjo, T. Okuno, R. Hassdorf, K. Shigeto, T. Ono, Science 289 (2000) 930. [6] J. Shibata, Y. Nakatani, G. Tatara, H. Kohno, Y. Otani, Phys. Rev. B 73 (2006) R020403. [7] G. Tatara, H. Kohno, Phys. Rev. Lett. 92 (2004) 086601. [8] J. Shibata, G. Tatara, H. Kohno, Phys. Rev. Lett. 94 (2005) 076601. [9] S. Zhang, Z. Li, Phys. Rev. Lett. 93 (2004) 127204. [10] S.E. Barnes, S. Maekawa, Phys. Rev. Lett. 95 (2005) 107204. [11] Y. Tserkovnyak, A. Brataas, G.E.W. Bauer, cond-mat/0512715. [12] H. Kohno, G. Tatara, J. Shibata, cond-mat/0605186. [13] T. Ishida, T. Kimura, Y. Otani, condmat/0511040. [14] S. Kasai, Y. Nakatani, K Kobayashi, H. Kohno, T. Ono, condmat/ 0604123. [15] K.Yu. Guslienko, B.A. Ivanov, V. Novosad, Y. Otani, H. Shima, K. Fukamichi, J. Appl. Phys. 91 (2002) 8037. [16] J. He, Z. Li, S. Zhang, Phys. Rev. B. 73 (2006) 184405.
Journal of Magnetism and Magnetic Materials 310 (2007) 2041–2042 www.elsevier.com/locate/jmmm
Magnetic vortex dynamics induced by spin-transfer torque J. Shibataa,, Y. Nakatanib, G. Tatarac,d, H. Kohnoe, Y. Otania,f a
RIKEN-FRS, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan Department of Computer Science, University of Electro-communications, Chofu 182-8585, Tokyo, Japan c Department of Physics, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan d PRESTO, JST, 4-1-8 Honcho Kawaguchi, Saitama, Japan e Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan f Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan
b
Available online 20 November 2006
Abstract We theoretically study the dynamics of a magnetic vortex under spin-polarized electric current in ferromagnets. The equation of motion of the vortex in terms of collective coordinates is derived. We compare our theory with recent experiments for current-induced vortex displacement and resonance motion in a ferromagnetic nanodot. Our estimate for the displacement and the resonance frequency shows a good agreement with the experiment. We also study the current-induced motion of a vortex wall in a ferromagnetic thin wire. r 2006 Elsevier B.V. All rights reserved. PACS: 72.25.Ba; 75.60.Ch; 85.75.d Keywords: Magnetic vortex; Spin-transfer torque; Spin current
Current-induced domain-wall motion [1], one of the most attractive subjects in nanoscale magnetism, has opened up the possibility of new spintronic devices. Recent experiments and numerical simulation [2–4] for this wall motion in magnetic nanowire have revealed the existence of a vortex wall configuration and its importance in the current-driven dynamics was pointed out on spin-polarized current(spin current). Also, magnetic vortices in nanodots [5] have drawn much attention and have been studied extensively. In this report, we theoretically study the current-induced vortex motion in a ferromagnet. Let us denote the spin configuration of a magnetic vortex centered at the origin by a vector field nV ðxÞ with unit modulus. As a vortex profile, we take an out-of-plane vortex; nV ðx ! 0Þ ¼ pez , where p ¼ 1 is the polarization and nV ðjxjbdV Þ ¼ cosðqj þ Cp=2Þex þ sinðqj þ Cp=2Þey , where dV is the vortex core radius, j ¼ tan1 ðy=xÞ, q ¼ 1; 2; . . . ; is the vorticity and C ¼ 1 is the chirality. We introduce a collective coordinate XðtÞ ¼ X ðtÞex þ Y ðtÞey , Corresponding author. Tel.: +81 48 462 1111; fax: +81 48 467 9650.
E-mail address: [email protected] (J. Shibata). URL: http://www.riken.jp/lab-www/nanomag/index.html. 0304-8853/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2006.10.949
which represents the position of the vortex core center, and assume that a moving vortex can be written as nðx; tÞ ¼ nV ðx XðtÞÞ and ignore the deformation and spin wave excitations. Then we obtain the Lagrangian for the vortex as [6] LV ¼ 12G ðX_ XÞ UðXÞ G ðvs XÞ, (1) R where G ¼ ez _S ðd3 x=a3 Þn ðqx n qy nÞ ¼ ð_S=a3 Þ2pLpqez ; is the gyrovector with a being the lattice constant, S being the magnitude of spin and L being the thickness of the system. UðXÞ represents a potential energy of the vortex coming from the magnetostatic energy, and vs ¼ ða3 =2eSÞj s is the drift velocity of electron spins [7,8], where e40 is the elementary charge and j s is the spin-current density, which is related to the charge-current density as Pj (P is the spin polarization). The last term on the right-hand side (r.h.s) of Eq. (1) represents the contribution from the current-induced spin-transfer torque, which takes the form, ðvs rÞn in the Landau–Lifshitz–Gilbert equation. The equation of motion of the vortex in terms of collective coordinates is derived from the Euler–Lagrange _ qLV =qX ¼ qW =qX, _ where equation; ðd=dtÞðqLV =qXÞ
ARTICLE IN PRESS J. Shibata et al. / Journal of Magnetism and Magnetic Materials 310 (2007) 2041–2042
W is the R so-called dissipation 2function given by W ¼ að_S=2Þ ðd3 x=a3 Þð_n2 ðx; tÞ ¼ aDX_ =2; with a being the constant. The constant D ¼ ð_S=a3 ÞL RGilbert damping 2 2 2 D dx dyfðqi yÞ þ sin yðqi fÞ g, generally includes a factor lnðRV =dV Þ, where RV is the system size. We assume that the vortex configuration has rotational invariance around the zaxis. The equation of motion of the vortex is then given by _ ¼ G ðvs XÞ
qUðXÞ _ þ Dðbvs aXÞ. qX
(2)
Note that we have added the term bDvs , which comes from another type of current-induced spin torque, of the form, bn ðvs rÞn, noted recently [4,9–12]. The coefficient b is usually taken to be of the order 102 . If the potential UðXÞ is absent, we obtain X_ ¼ vs for a special case a ¼ b, where the vortex core moves along the spin current. However, in general, since aab [9,12], the last term on the r.h.s. of Eq. (2) causes the moving vortex to deviate in the direction of the current. Now let us apply Eq. (2) to analyze vortex displacement induced by a DC current and resonant vortex motion induced by an AC current in a single magnetic nanodot, and compare with the experimental results [13,14]. The radius and the thickness of the nanodot are denoted by R and L, respectively, where an out-of-plane vortex with vorticity q ¼ 1 is stabilized. We assume that the electric current is uniform in the nanodot, and flowing in the positive x-direction, that is, vs ¼ ða3 =2eSÞ j s ¼ vs ex for the DC current, and vs ðtÞ ¼ vs ex eiOt for the A current. We assume a full spin polarization of the current, P ¼ 1, for simplicity. The potential energy UðXÞ is assumed harmonic, i.e., UðXÞ ¼ kX 2 =2, where k is a force constant. In Ref. [15], k is evaluated in detail, revealing its dependence on the aspect ratio g ¼ L=R of the dot. In the case of DC current, the equation of motion is ~ s ; where Z ¼ X þ given by ð1 þ i~aÞZ_ ¼ ioZ þ ð1 þ ibÞv ~ iY , a~ ¼ aD=G, b ¼ bD=G and o ¼ k=G. For an initial condition Zð0Þ ¼ 0, the solution is given by ZðtÞ ¼ ið1 þ ~ s =ofexpðiot=1 þ i~aÞ 1g: Thus the vortex center ibÞv exhibits a spiral motion, whose rotational direction depends on the sign of the core polarization p ¼ 1. The final displacement position of the vortex core is given by X ð1Þ ¼ bDvs =k and Y ð1Þ ¼ Gvs =k, which also depends on p, where the transverse force G vs and the force bDvs balance the restoring force kX. Thus, the vortex displacement is proportional to the current density j. The 2 proportionality constant, jZð1Þj=j ¼ ð1 þ b~ Þ1=2 ðjGj=kÞ 19 ða3 =2eSÞP, is estimated as 0:76 10 m3 =A for g ¼ 0:03 ~ and small b51, which is in good agreement with the experimental one, 1:23 1019 m3 =A [13]. In the case of AC current, vs ðtÞ ¼ vs ex eiOt , the vortex core rotates around the dot center with frequency O. At the resonant frequency, O ¼ o=ð1 þ a~ 2 Þ, the amplitude is given qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 by jZj ¼ ð ð1 þ a~ 2 Þð1 þ b~ Þ=aÞðvs =oÞ; which is amplified by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 the factor ð1 þ a~ 2 Þð1 þ b~ Þ=a compared to the DC-case.
350 Resonant frequency (MHz) Ω/2π
2042
300 250 200 150 100 50 0 0.00
0.02
0.04 0.06 Aspect ratio g = L/R
0.08
0.1
Fig. 1. Resonance frequency O=2p as a function of the aspect ratio g ¼ L=R of the dot. The solid line and the dots represent the analytical and experimental [14] results, respectively.
We show the resonance frequency as a function of aspect ratio in Fig. 1, which shows good agreement with experimental result [14]. Finally, we study the current-induced motion of a vortex wall in a magnetic thin wire. In this case, the potential energy is assumed to be UðXÞ ¼ k0 Y 2 yðw jY jÞ=2, where k0 is a force constant, yðY Þ is the step function, and w is the wire width. This potential leads to a restoring force along the transverse direction of the wire. If we apply a DC current ðvs ¼ vs ex Þ along the wire, a steady-state motion along the wire is eventually attained with X_ ¼ bvs =a and Y ð1Þ ¼ ð1 þ b=aÞGvs =k0 for 2ð1 þ b=aÞGvs =k0 ow. For 2ð1 þ b=aÞGvs =k0 4w, namely, for weak restoring force or large current, vortex center can reach the wire edge and moves out of the wire, resulting in the transformation into a transverse wall [3,4]. Recently, another recent study [16] of current driven vortex walls reaches similar conclusions. References [1] C.H. Marrows, Adv. Phys. 53 (2005) 585. [2] A. Yamaguchi, T. Ono, S. Nasu, K. Miyake, K. Mibu, T. Shinjo, Phys. Rev. Lett. 92 (2004) 077205. [3] M. Kla¨ui, P.O. Jubert, R. Allenspach, A. Bischof, J.A.C. Bland, G. Faini, U. Ru¨diger, C.A.F. Vaz, L. Vila, C. Vouille, Phys. Rev. Lett. 95 (2005) 026601. [4] A. Thiaville, Y. Nakatani, J. Miltat, Y. Suzuki, Europhys. Lett. 69 (2005) 990. [5] T. Shinjo, T. Okuno, R. Hassdorf, K. Shigeto, T. Ono, Science 289 (2000) 930. [6] J. Shibata, Y. Nakatani, G. Tatara, H. Kohno, Y. Otani, Phys. Rev. B 73 (2006) R020403. [7] G. Tatara, H. Kohno, Phys. Rev. Lett. 92 (2004) 086601. [8] J. Shibata, G. Tatara, H. Kohno, Phys. Rev. Lett. 94 (2005) 076601. [9] S. Zhang, Z. Li, Phys. Rev. Lett. 93 (2004) 127204. [10] S.E. Barnes, S. Maekawa, Phys. Rev. Lett. 95 (2005) 107204. [11] Y. Tserkovnyak, A. Brataas, G.E.W. Bauer, cond-mat/0512715. [12] H. Kohno, G. Tatara, J. Shibata, cond-mat/0605186. [13] T. Ishida, T. Kimura, Y. Otani, condmat/0511040. [14] S. Kasai, Y. Nakatani, K Kobayashi, H. Kohno, T. Ono, condmat/ 0604123. [15] K.Yu. Guslienko, B.A. Ivanov, V. Novosad, Y. Otani, H. Shima, K. Fukamichi, J. Appl. Phys. 91 (2002) 8037. [16] J. He, Z. Li, S. Zhang, Phys. Rev. B. 73 (2006) 184405.