- Email: [email protected]
Influence of temperature on current-induced domain wall motion and its Walker breakdown
Author’s Accepted Manuscript Influence of temperature on current-induced domain wall motion and its Walker breakdown Lvchao Fan, Jingguo Hu, Yuanchang Su, Jinrong Zhu www.elsevier.com/locate/jmmm
PII: DOI: Reference:
S0304-8853(15)30720-4 http://dx.doi.org/10.1016/j.jmmm.2015.10.090 MAGMA60783
To appear in: Journal of Magnetism and Magnetic Materials Received date: 10 February 2015 Revised date: 20 August 2015 Accepted date: 21 October 2015 Cite this article as: Lvchao Fan, Jingguo Hu, Yuanchang Su and Jinrong Zhu, Influence of temperature on current-induced domain wall motion and its Walker br e a k d o w n, Journal of Magnetism and Magnetic Materials, http://dx.doi.org/10.1016/j.jmmm.2015.10.090 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Influence of temperature on current-induced domain wall motion and its Walker breakdown Lvchao Fan, Jingguo Hu1 , Yuanchang Su2 and Jinrong Zhu College of Physics Science and Technology, Yangzhou University, Yangzhou 225002, People’s Republic of China
Abstract The current-driven domain wall propagation along a thin ferromagnetic strip with thermal field is studied by means of micromagnetic simulations. The results show that the velocity of domain wall is almost independent of temperature until Walker breakdown happened. However the thermal field can suppress Walker breakdown and makes domain wall move faster. Further analysis indicates that the thermal field tends to keep the out-of-plane magnetic moment of the domain wall stay in high value, which can promote domain wall motion and suppress the Walker breakdown by breaking the period of domain wall transformation. Keywords: domain wall motion, current, temperature, micromagnetic simulation 1. Introduction Current-induced domain wall (DW) dynamics along ferromagnetic nanostrip is nowadays the focus of much research, as it encompasses fundamental physics and promising novel applications [1-3]. Under the sole function of a current, the DW may be moved along the wire, it has confirmed by several experiments and Numerical Simulation [4-6]. But these applications require two primary problems be solved. Firstly, the threshold current for DW motion is too large. Experiment and micromagnetic simulation has indicated that the DW motion 1 Tel: 2 Tel:
86-0514-87970587, E-mail address: [email protected] 86-0514-87975466, E-mail address: [email protected]
Preprint submitted to Elsevier
October 23, 2015
stops eventually when the current below the threshold value, because the magnitude of the adiabatic torque is insufficient to overcome the demagnetization effect [7-10]. Secondly, it is difficult to achieve stable control to the domain wall. Some reports have pointed out that DW structural changes during the DW motion, which may lead to bidirectional displacement and the stochastic nature of DW. As for DW motion, the structural changes such as Walker breakdown, in which the DW structure transforms periodically, results in a dramatic reduction of DW average velocity [11-14] for β > α (β denotes the non-adiabatic spin torque coefficient and α is the Gilbert damping parameter). This limits DW propagation velocity and functional performance in both nanowires [15] and thin films [16]. This problem has also been shown in other references [17,18]. For practical applications, fast DW motion by suppressing DW breakdown behavior is desirable and various approaches have attempted to achieve this. For example, it has been reported that Walker breakdown can be suppressed by applying additional fields (quasi-static perpendicular fields [19,20] or oscillating axial fields [21]) or small amplitude periodic structuring [22]. In current-driven DW motion, joule heat [23] and environmental temperature can not be ignored. However ,the influence of temperature on the Walker breakdown has not been deeply reported before. In this paper, the current-driven DW motions in magnetic nanostripe system with thermal field are studied by Landau-Lifshitz-Gilbert (LLG) spin dynamics method. The result shows that the velocity of DW motion is almost independent of temperature for small current but is greatly influenced by temperature for large current where Walker breakdown happened. Namely, the Walker breakdown can be suppressed by thermal fluctuation. The detail analysis shows that thermal field can influence Mz (out-of-plane magnetic moment) which plays a key role for DW motion. The thermal fluctuation break the period of domain wall transformation, and always keep Mz stay in high value which can promote DW motion.
2
2. Model The sample used in this paper has a width of 40nm, a thickness of 5nm and a length of 2048nm. In order to minimize the effects that appear when DW approaches the end of the stripe, a scheme that keeps the DW centered in the stripe has been used. The stripe is placed in x-y plane. The horizontal axis, the vertical axis and the perpendicular axis are defined as x axis, y axis and z axis, respectively. In initial state, a transverse wall (TW) is placed in the center of the stripe by the helping of the mathematical equation given in Ref. 24, and evaluated by numerically solving the LLG equation augmented by the adiabatic and non-adiabatic spin-polarized torques [18,22,25]. We show the modified LLG equation, again obtained in the continuous limit. − → − → dM → − − → − → dM → − = γ0 H ef f + H th × M + α M × dt dt − → − → − → → → −(− u · ∇)M + β M × (− u · ∇)M
(1)
− → → − where M is the unit magnetization vector. H ef f is the effective magnetic field including the exchange coupling field and the magnetostatic field. γ is the gyromagnetic ratio. α is the Gilbert damping constant. The parameter β is the dimensionless constant describing the degree of non-adiabaticity between the spin → of conduction electrons and the local magnetization [25]. − u is a vector directed along the direction of electron motion, with an amplitude u = JP gμB /(2eMs ) [18]. J is the current density and P its polarization rate (u is positive for P > 0, i.e. for carriers polarized along the majority spin direction). Hth is the thermal field [23], which is a Gaussian random process with the following statistical properties [26]: → Hth (− r , t) = 0
(2)
αKB T − → → → r , t) Hth (− r , t) = 2 δ(→ r −− r )δ (t − t) Hth (− γ 0 μ0 M s
(3)
Where KB is the Boltzmann constant and T represents the temperature. The average thermal field taken over different stochastic realizations vanishes in each direction i : x, y, z in space (Eq. (2)), and it is assumed to be uncorrelated in 3
− − time (δ (t − t)) and uncorrelated at different points (δ → r −→ r ) of the finite difference mesh (Eq. (3)). The following values have been considered as the intrinsic parameters of the model, all of which are used in many references [18,20,24]. In simulations, the unit cell size a × a × b is 4nm × 4nm × 5nm. The saturation magnetization Ms = 8 × 105 A/m, and the exchange stiffness A = 1.0 × 10−11 J/m. In this paper, we choose α = 0.02 and β = 0.1, which are mentioned in many references [2,18,20]. The code used in this paper is made by ourself, which is similar to our previous works [27,28]. 3. Results and discussions Fig. 1(a) shows the temporal evolution of the TW displacement driven by different current with T = 0K (thin line) and T = 100K (bold line). When u = 88m/s, which is below the critical current uw (Walker breakdown happened), TW exhibit a stationary behavior and moves rigidly with stationary velocity. In this case, the velocity of TW is almost independent of temperature. However when current (u = 132m/s) is larger than the critical current at T = 0K, the TW moves with a oscillation behavior. In this case the internal structure of TW is no longer stable, namely the TW precesses around the x-axis as TW moves, we called Walker breakdown. At T = 100K, oscillation disappeared and TW moves faster. In order to have a better sense of thermal influence to TW motion, the time-dependent velocity of TW and the precession angle ϕ of the local magnetic moment for TW as a function of time for different current and temperature are shown in Fig. 1(b) and Fig. 1(c), respectively. After all, for small current with different temperature, TW motion changes little. But for larger current TW motion has a great difference between 0K and 100K. Specially, in the region of the Walker breakdown, Fig. 1(c) shows that the precession of the local magnetic moment for TW can be suppressed by temperature. Fig. 2 shows the average velocity of TW as a function of current with different temperature. When u < uw and T = 0K, the average velocity can be described as v = βu/α [17]. If there is temperature, TW velocity has a slight
4
Figure 1: (a) Displacement, (b) time-dependent velocity and (c) Precession angle ϕ of TW as a function of time under different current and temperature. The precession angle is defined by ϕ = arccos(
My
2 +M 2 My z
), where My and Mz are y-axis and z-axis components of the magnetic
moment of TW, respectively. Both My and Mz denote the local magnetic moments for the center of TW in x axis and have been added up in y direction.
5
Figure 2: Average velocity of TW as a function of current with different temperature. The inset are the amplified scale of the small current region.
Figure 3: Displacement of TW as a function of time with different temperature. The current is u = 132m/s, which is in the region of Walker breakdown.
6
decline with the increasing of temperature (shown in the inset), which agrees with Ref. 20. When u > uw and T = 0K, Walker breakdown happened and TW velocity declined precipitously. In this case the TW moves turbulently by precessing clockwise around the x-axis. But with the increasing of temperature, TW velocity increases abnormally in the Walker breakdown region. Especially, when T = 200K, TW velocity is increased without any decline. Fig. 3 shows the displacement of TW as a function of time with different temperature in the region of Walker breakdown. It shows that, with the increasing of temperature, the oscillation period increases and correspondingly the displacement increases. When T > 200K the oscillation disappeared and TW moves with stationary velocity. It means that the Walker breakdown is suppressed completely. In order to analysis the mechanism of the suppression, the change of magnetic moment with time is calculated, as shown in Fig. 4. Fig. 4(a) shows the time-dependent velocity and the magnetic moment Mz of TW as a function of time with T = 0K and T = 50K in the region of u < uw . Here, Mz denote z-axis component of the local magnetic moment for the center of TW in x axis and have been added up in y direction. In this small-current case, which Walker breakdown does not happen, the velocity of TW increases with the increasing of Mz and is almost independent of the temperature. Fig. 4(b) shows the velocity and the magnetic moment Mz as a function of time with T = 0K and u = 132m/s. In this case, TW velocity and Mz oscillate synchronously with a constant period. The oscillation of Mz displays well the precession of the TW. Noted that the velocity of TW increases and decreases with the increasing and decreasing of the amplitude of Mz , respectively. The oscillation of the velocity lead to dramatic reduction in the average velocity of TW. Fig. 4(c) and Fig. 4(d) show the velocity and Mz as a function of time with T = 50K and T = 100K in the region of u > uw , respectively. In this cases, the periodic oscillation is broken, namely Mz stays in high value longer and the average velocity of TW is faster than the case of T = 0K. It also shows that the velocity of TW increases and decreases with the increasing and decreasing of the amplitude of Mz , respectively. So, the reason for suppressing Walker breakdown comes form 7
Figure 4: time-dependent velocity and the z-axis component of the magnetic moment Mz of TW as a function of time with (a) T=0K, T=50K and u=88m/s; (b) T=0K and u=132m/s; (c) T=50K and u=132m/s; (d) T=100K and u=132m/s. The dash lines are auxiliary lines.
8
Figure 5: Typical micromagnetic configurations at different times. (a) and (b) correspond to the Fig. 4(b) and Fig. 4(c), respectively.
the thermal field control the magnetic moment and break the transformation period of domain wall magnetization. In fact, When temperature is lower than 50K, the period of domain wall transformation changes little. However, when temperature get higher, the thermal fluctuation breaks the period of domain wall transformation, and always keep Mz stay in high value. For example, when T = 100K, Mz keep the high value for about 10ns. The oscillation period get longer with the increasing of temperature, which means that the Walker breakdown is suppressed gradually. Fig. 5 shows typical micromagnetic configurations (corresponding to the Fig. 4(b) and Fig. 4(c)) at different times. It shows that the transformations of the local magnetic moments in TW for different y are not synchronous. Noted that the width of TW varies with y. As the width of TW increases, the transformation of the local magnetic moment becomes more difficult. Comparing Fig. 5(a) with Fig. 5(b), one will find that temperature can hinder the transformation of the local magnetic moments in TW, which lead to longer the transformation period.
9
4. Conclusions In conclusion, there are two cases for temperature influence on currentinduced TW motion. For small current density without the Walker breakdown, the temperature fluctuation has a little influence on TW motion. But for large current density case where the Walker breakdown occurs, the temperature fluctuation can suppress the Walker breakdown, and the effect is more obvious with the increasing of temperature. The reason is that, in the small current density case, the out-of-plane magnetization (Mz ) of the TW induced by the current can keep its saturation states, so the TW motion can almost independent on the temperature fluctuation. But for large current density case, the TW cannot keep its Mz which can closely control the TW motion. Interestingly, the Mz can flip periodically with the current increase. So the TW motion has a regular changing where Walker breakdown occurs. Interestingly, in this case, the Mz periodic inversion can be suppressed by temperature fluctuation, and the suppression become obvious with the increasing of the temperature. Finally, the Mz inversion disappears, the TW motion approach stabilization, the Walker breakdown has been suppressed completely. Acknowledgments This work is supported by the National Natural Science Foundation of China under the Grant Nos. 11374253 and 11247026. [1] O. Boulle, G. Malinowski, and M. Klaui, Mater. Sci. Eng. R. 72(2011)159. [2] E. Martinez, J. Appl. Phys. 111(2012)07D302. [3] A. Yamaguchi, T. Ono, and S. Nasu, Phys. Rev. Lett. 92(2004)077205. [4] J. Ryu, K.-J. Lee, and H.-W. Lee, Appl. Phys. Lett. 102(2013)172404. [5] M. Kl¨aui, C. A. F. Vaz, J. A. C. Bland, W. Wernsdorfer, G. Faini, E. Cambril, and L. J. Heyderman, Appl. Phys. Lett. 83(2003)105. 10
[6] J. Ryu and H.-W. Lee, J. Appl. Phys. 105(2009)093929. [7] G. Tatara and H. Kohno, Phys. Rev. Lett. 92(2004)086601. [8] N. Vernier, J. P. Adam, A. Thiaville, V. Jeudy, A. Lemaˆıtre, J. Ferr´e, and G. Faini, Phys. Rev. B 88(2013)224415. [9] M. Tsoi, R. E. Fontana, and S. S. P. Parkin, Appl. Phys. Lett. 83(2003)2617. [10] D. Ravelosona, S. Mangin, J. A. Katine, E. E. Fullerton, and B. D. Terris, Appl. Phys. Lett. 90(2007)072508. [11] S. Glathe,
R. Mattheis,
and D. V. Berkov, Appl. Phys. Lett.
93(2008)072508. [12] N. L. Schryer and L. R. Walker, J. Appl. Phys. 45(1974)5406. [13] M. Yan, A. K´akay, S. Gliga, and R. Hertel, Phys. Rev. Lett. 104(2010)057201. [14] J. -Y. Lee, K. -S. Lee, and S. -K. Kim, Appl. Phys. Lett. 91(2007)122513. [15] H. Tanigawa, T. Koyama, M. Bartkowiak, S. Kasai, K. Kobayashi, T. Ono, and Y. Nakatani, Phys. Rev. Lett. 101(2008)207203. [16] D. M. Burn and D. Atkinson, J. Appl. Phys. 108(2010)073926; [17] C. Schieback, M. Kl¨aui, U. Nowak, U. R¨ udiger, and P. Nielaba, Eur. Phys. J. B 59(2007)429. [18] A. Thiaville, Y. Nakatani, J. Miltat, and Y. Suzuki, Europhys. Lett. 69(2005)990. [19] K. Richter, R. Varga, G. A. Badini-Confalonieri, and M. V´ azquez, Appl. Phys. Lett. 96(2010)182507. [20] M. T. Bryan, T. Schrefl, D. Atkinson, and D. A. Allwood, J. Appl. Phys. 103(2008)073906. 11
[21] K. Weerts, W. Van Roy, G. Borghs, and L. Lagae, Appl. Phys. Lett. 96(2010)062502. [22] D. M. Burn and D. Atkinson, Appl. Phys. Lett. 102(2013)242414. [23] J. Curiale, A. Lemaˆıtre, T. Niazi, G. Faini, and V. Jeudy, J. Appl. Phys. 112(2012)103922. [24] J. He, Z. Li, and S. Zhang, Phys. Rev. B 73(2006)184408. [25] S. Zhang and Z. Li, Phys. Rev. Lett. 93(2004)127204. [26] J. L. Garcia-Palacios and F. J. Lazaro, Phys. Rev. B 58(1998)14937. [27] Y. Su, J. Sun, J. Hu, and H. Lei, Europhys. Lett. 103(2013)67004. [28] J. Zhu, Z. Han, Y. Su, and J. Hu, J. Magn. Magn. Mater. 369(2014)96.
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PII: DOI: Reference:
S0304-8853(15)30720-4 http://dx.doi.org/10.1016/j.jmmm.2015.10.090 MAGMA60783
To appear in: Journal of Magnetism and Magnetic Materials Received date: 10 February 2015 Revised date: 20 August 2015 Accepted date: 21 October 2015 Cite this article as: Lvchao Fan, Jingguo Hu, Yuanchang Su and Jinrong Zhu, Influence of temperature on current-induced domain wall motion and its Walker br e a k d o w n, Journal of Magnetism and Magnetic Materials, http://dx.doi.org/10.1016/j.jmmm.2015.10.090 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Influence of temperature on current-induced domain wall motion and its Walker breakdown Lvchao Fan, Jingguo Hu1 , Yuanchang Su2 and Jinrong Zhu College of Physics Science and Technology, Yangzhou University, Yangzhou 225002, People’s Republic of China
Abstract The current-driven domain wall propagation along a thin ferromagnetic strip with thermal field is studied by means of micromagnetic simulations. The results show that the velocity of domain wall is almost independent of temperature until Walker breakdown happened. However the thermal field can suppress Walker breakdown and makes domain wall move faster. Further analysis indicates that the thermal field tends to keep the out-of-plane magnetic moment of the domain wall stay in high value, which can promote domain wall motion and suppress the Walker breakdown by breaking the period of domain wall transformation. Keywords: domain wall motion, current, temperature, micromagnetic simulation 1. Introduction Current-induced domain wall (DW) dynamics along ferromagnetic nanostrip is nowadays the focus of much research, as it encompasses fundamental physics and promising novel applications [1-3]. Under the sole function of a current, the DW may be moved along the wire, it has confirmed by several experiments and Numerical Simulation [4-6]. But these applications require two primary problems be solved. Firstly, the threshold current for DW motion is too large. Experiment and micromagnetic simulation has indicated that the DW motion 1 Tel: 2 Tel:
86-0514-87970587, E-mail address: [email protected] 86-0514-87975466, E-mail address: [email protected]
Preprint submitted to Elsevier
October 23, 2015
stops eventually when the current below the threshold value, because the magnitude of the adiabatic torque is insufficient to overcome the demagnetization effect [7-10]. Secondly, it is difficult to achieve stable control to the domain wall. Some reports have pointed out that DW structural changes during the DW motion, which may lead to bidirectional displacement and the stochastic nature of DW. As for DW motion, the structural changes such as Walker breakdown, in which the DW structure transforms periodically, results in a dramatic reduction of DW average velocity [11-14] for β > α (β denotes the non-adiabatic spin torque coefficient and α is the Gilbert damping parameter). This limits DW propagation velocity and functional performance in both nanowires [15] and thin films [16]. This problem has also been shown in other references [17,18]. For practical applications, fast DW motion by suppressing DW breakdown behavior is desirable and various approaches have attempted to achieve this. For example, it has been reported that Walker breakdown can be suppressed by applying additional fields (quasi-static perpendicular fields [19,20] or oscillating axial fields [21]) or small amplitude periodic structuring [22]. In current-driven DW motion, joule heat [23] and environmental temperature can not be ignored. However ,the influence of temperature on the Walker breakdown has not been deeply reported before. In this paper, the current-driven DW motions in magnetic nanostripe system with thermal field are studied by Landau-Lifshitz-Gilbert (LLG) spin dynamics method. The result shows that the velocity of DW motion is almost independent of temperature for small current but is greatly influenced by temperature for large current where Walker breakdown happened. Namely, the Walker breakdown can be suppressed by thermal fluctuation. The detail analysis shows that thermal field can influence Mz (out-of-plane magnetic moment) which plays a key role for DW motion. The thermal fluctuation break the period of domain wall transformation, and always keep Mz stay in high value which can promote DW motion.
2
2. Model The sample used in this paper has a width of 40nm, a thickness of 5nm and a length of 2048nm. In order to minimize the effects that appear when DW approaches the end of the stripe, a scheme that keeps the DW centered in the stripe has been used. The stripe is placed in x-y plane. The horizontal axis, the vertical axis and the perpendicular axis are defined as x axis, y axis and z axis, respectively. In initial state, a transverse wall (TW) is placed in the center of the stripe by the helping of the mathematical equation given in Ref. 24, and evaluated by numerically solving the LLG equation augmented by the adiabatic and non-adiabatic spin-polarized torques [18,22,25]. We show the modified LLG equation, again obtained in the continuous limit. − → − → dM → − − → − → dM → − = γ0 H ef f + H th × M + α M × dt dt − → − → − → → → −(− u · ∇)M + β M × (− u · ∇)M
(1)
− → → − where M is the unit magnetization vector. H ef f is the effective magnetic field including the exchange coupling field and the magnetostatic field. γ is the gyromagnetic ratio. α is the Gilbert damping constant. The parameter β is the dimensionless constant describing the degree of non-adiabaticity between the spin → of conduction electrons and the local magnetization [25]. − u is a vector directed along the direction of electron motion, with an amplitude u = JP gμB /(2eMs ) [18]. J is the current density and P its polarization rate (u is positive for P > 0, i.e. for carriers polarized along the majority spin direction). Hth is the thermal field [23], which is a Gaussian random process with the following statistical properties [26]: → Hth (− r , t) = 0
(2)
αKB T − → → → r , t) Hth (− r , t) = 2 δ(→ r −− r )δ (t − t) Hth (− γ 0 μ0 M s
(3)
Where KB is the Boltzmann constant and T represents the temperature. The average thermal field taken over different stochastic realizations vanishes in each direction i : x, y, z in space (Eq. (2)), and it is assumed to be uncorrelated in 3
− − time (δ (t − t)) and uncorrelated at different points (δ → r −→ r ) of the finite difference mesh (Eq. (3)). The following values have been considered as the intrinsic parameters of the model, all of which are used in many references [18,20,24]. In simulations, the unit cell size a × a × b is 4nm × 4nm × 5nm. The saturation magnetization Ms = 8 × 105 A/m, and the exchange stiffness A = 1.0 × 10−11 J/m. In this paper, we choose α = 0.02 and β = 0.1, which are mentioned in many references [2,18,20]. The code used in this paper is made by ourself, which is similar to our previous works [27,28]. 3. Results and discussions Fig. 1(a) shows the temporal evolution of the TW displacement driven by different current with T = 0K (thin line) and T = 100K (bold line). When u = 88m/s, which is below the critical current uw (Walker breakdown happened), TW exhibit a stationary behavior and moves rigidly with stationary velocity. In this case, the velocity of TW is almost independent of temperature. However when current (u = 132m/s) is larger than the critical current at T = 0K, the TW moves with a oscillation behavior. In this case the internal structure of TW is no longer stable, namely the TW precesses around the x-axis as TW moves, we called Walker breakdown. At T = 100K, oscillation disappeared and TW moves faster. In order to have a better sense of thermal influence to TW motion, the time-dependent velocity of TW and the precession angle ϕ of the local magnetic moment for TW as a function of time for different current and temperature are shown in Fig. 1(b) and Fig. 1(c), respectively. After all, for small current with different temperature, TW motion changes little. But for larger current TW motion has a great difference between 0K and 100K. Specially, in the region of the Walker breakdown, Fig. 1(c) shows that the precession of the local magnetic moment for TW can be suppressed by temperature. Fig. 2 shows the average velocity of TW as a function of current with different temperature. When u < uw and T = 0K, the average velocity can be described as v = βu/α [17]. If there is temperature, TW velocity has a slight
4
Figure 1: (a) Displacement, (b) time-dependent velocity and (c) Precession angle ϕ of TW as a function of time under different current and temperature. The precession angle is defined by ϕ = arccos(
My
2 +M 2 My z
), where My and Mz are y-axis and z-axis components of the magnetic
moment of TW, respectively. Both My and Mz denote the local magnetic moments for the center of TW in x axis and have been added up in y direction.
5
Figure 2: Average velocity of TW as a function of current with different temperature. The inset are the amplified scale of the small current region.
Figure 3: Displacement of TW as a function of time with different temperature. The current is u = 132m/s, which is in the region of Walker breakdown.
6
decline with the increasing of temperature (shown in the inset), which agrees with Ref. 20. When u > uw and T = 0K, Walker breakdown happened and TW velocity declined precipitously. In this case the TW moves turbulently by precessing clockwise around the x-axis. But with the increasing of temperature, TW velocity increases abnormally in the Walker breakdown region. Especially, when T = 200K, TW velocity is increased without any decline. Fig. 3 shows the displacement of TW as a function of time with different temperature in the region of Walker breakdown. It shows that, with the increasing of temperature, the oscillation period increases and correspondingly the displacement increases. When T > 200K the oscillation disappeared and TW moves with stationary velocity. It means that the Walker breakdown is suppressed completely. In order to analysis the mechanism of the suppression, the change of magnetic moment with time is calculated, as shown in Fig. 4. Fig. 4(a) shows the time-dependent velocity and the magnetic moment Mz of TW as a function of time with T = 0K and T = 50K in the region of u < uw . Here, Mz denote z-axis component of the local magnetic moment for the center of TW in x axis and have been added up in y direction. In this small-current case, which Walker breakdown does not happen, the velocity of TW increases with the increasing of Mz and is almost independent of the temperature. Fig. 4(b) shows the velocity and the magnetic moment Mz as a function of time with T = 0K and u = 132m/s. In this case, TW velocity and Mz oscillate synchronously with a constant period. The oscillation of Mz displays well the precession of the TW. Noted that the velocity of TW increases and decreases with the increasing and decreasing of the amplitude of Mz , respectively. The oscillation of the velocity lead to dramatic reduction in the average velocity of TW. Fig. 4(c) and Fig. 4(d) show the velocity and Mz as a function of time with T = 50K and T = 100K in the region of u > uw , respectively. In this cases, the periodic oscillation is broken, namely Mz stays in high value longer and the average velocity of TW is faster than the case of T = 0K. It also shows that the velocity of TW increases and decreases with the increasing and decreasing of the amplitude of Mz , respectively. So, the reason for suppressing Walker breakdown comes form 7
Figure 4: time-dependent velocity and the z-axis component of the magnetic moment Mz of TW as a function of time with (a) T=0K, T=50K and u=88m/s; (b) T=0K and u=132m/s; (c) T=50K and u=132m/s; (d) T=100K and u=132m/s. The dash lines are auxiliary lines.
8
Figure 5: Typical micromagnetic configurations at different times. (a) and (b) correspond to the Fig. 4(b) and Fig. 4(c), respectively.
the thermal field control the magnetic moment and break the transformation period of domain wall magnetization. In fact, When temperature is lower than 50K, the period of domain wall transformation changes little. However, when temperature get higher, the thermal fluctuation breaks the period of domain wall transformation, and always keep Mz stay in high value. For example, when T = 100K, Mz keep the high value for about 10ns. The oscillation period get longer with the increasing of temperature, which means that the Walker breakdown is suppressed gradually. Fig. 5 shows typical micromagnetic configurations (corresponding to the Fig. 4(b) and Fig. 4(c)) at different times. It shows that the transformations of the local magnetic moments in TW for different y are not synchronous. Noted that the width of TW varies with y. As the width of TW increases, the transformation of the local magnetic moment becomes more difficult. Comparing Fig. 5(a) with Fig. 5(b), one will find that temperature can hinder the transformation of the local magnetic moments in TW, which lead to longer the transformation period.
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4. Conclusions In conclusion, there are two cases for temperature influence on currentinduced TW motion. For small current density without the Walker breakdown, the temperature fluctuation has a little influence on TW motion. But for large current density case where the Walker breakdown occurs, the temperature fluctuation can suppress the Walker breakdown, and the effect is more obvious with the increasing of temperature. The reason is that, in the small current density case, the out-of-plane magnetization (Mz ) of the TW induced by the current can keep its saturation states, so the TW motion can almost independent on the temperature fluctuation. But for large current density case, the TW cannot keep its Mz which can closely control the TW motion. Interestingly, the Mz can flip periodically with the current increase. So the TW motion has a regular changing where Walker breakdown occurs. Interestingly, in this case, the Mz periodic inversion can be suppressed by temperature fluctuation, and the suppression become obvious with the increasing of the temperature. Finally, the Mz inversion disappears, the TW motion approach stabilization, the Walker breakdown has been suppressed completely. Acknowledgments This work is supported by the National Natural Science Foundation of China under the Grant Nos. 11374253 and 11247026. [1] O. Boulle, G. Malinowski, and M. Klaui, Mater. Sci. Eng. R. 72(2011)159. [2] E. Martinez, J. Appl. Phys. 111(2012)07D302. [3] A. Yamaguchi, T. Ono, and S. Nasu, Phys. Rev. Lett. 92(2004)077205. [4] J. Ryu, K.-J. Lee, and H.-W. Lee, Appl. Phys. Lett. 102(2013)172404. [5] M. Kl¨aui, C. A. F. Vaz, J. A. C. Bland, W. Wernsdorfer, G. Faini, E. Cambril, and L. J. Heyderman, Appl. Phys. Lett. 83(2003)105. 10
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