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Spin-transfer torque induced domain wall ferromagnetic resonance in nanostrips
Journal of Magnetism and Magnetic Materials 332 (2013) 56–60
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Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Spin-transfer torque induced domain wall ferromagnetic resonance in nanostrips Xi-guang Wang, Guang-hua Guo n, Guang-fu Zhang, Yao-zhuang Nie, Qing-lin Xia, Zhi-xiong Li School of Physics and Electronics, Central South University, Changsha 410083, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 5 October 2012 Received in revised form 20 November 2012 Available online 12 December 2012
The frequency response of a Ne´el domain wall in a nanostrip excited by alternating spin-polarized current is studied by micromagnetic simulations. Several internal normal modes of the domain wall are excited and the corresponding spatial distributions of oscillation power are imaged. In the case of current perpendicular to the wall, the excited normal modes are mainly concentrated at the wall center and/or wall boundaries, forming edge modes, center modes, standing-wave modes and their mixed modes. The localization and spatial distribution of the modes have a close relationship with the total internal field, especially its inhomogeneity. With the decrease of saturation magnetizations Ms, the spatial inhomogeneities of the total field are gradually weakened and some domain wall normal modes disappear. In the case where the current is parallel to the wall, the wall thickness oscillation mode (or breathing mode) is excited. Furthermore, due to the geometrical confinement, high-order thickness modes such as edge- and standing wave-thickness modes are observed. The magnetization dependence of the eigen-frequency exhibits different forms for different normal modes, which can be qualitatively explained based on an approximate dispersion equation for spin-wave modes of a quasi-uniformly magnetized rectangular element. & 2012 Elsevier B.V. All rights reserved.
Keywords: Magnetic domain wall Ferromagnetic resonance Spin wave Spin-transfer torque Micromagnetic simulation Magnetic nanowire
1. Introduction The dynamic properties of a domain wall (DW) have attracted much attention recently, owing to their potential technological applications such as memory devices and logic gates [1–3]. Many significant DW dynamic characteristics, such as DW motion, have a close relation to the frequency response of a wall. It is found that the spin-wave induced DW motion in a nanowire has a strong frequency dependence resulting from the frequency dependence of the spin wave propagation through a DW [4,5]. When spin-wave frequency coincides with a DW normal mode frequency, a resonance reflection occurs. As a result, the DW moves with an extremum velocity [5]. The DW motion can be resonantly amplified also through excitation of DW internal modes by an alternating spin-polarized current [6,7]. Besides, various other phenomena, such as the spin-wave phase shift [8], spin-wave frequency doubling by DW [9], and the microwave assisted DW depining [10,11], are also correlated with the characteristics of the DW internal oscillation modes. Therefore, a comprehensive understanding about the DW intrinsic dynamic
n
Corresponding author. Tel.: þ86 731 88836443; fax: þ 86 731 88830857. E-mail address: [email protected] (G.-h. Guo).
0304-8853/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jmmm.2012.12.013
properties is important for designing the DW-based spintronic devices. Theoretically, a 1-dimensional 1801 Bloch DW has a continuous spin-wave dispersion relation [12,13]. But for a DW in a nanostrip, it was demonstrated by micromagnetic simulations that the DW normal oscillation spectrum shows discrete and localized features due to the geometrical confinement [14,15], edge modes and standing spin-wave modes are also found [15]. Standing spin waves bound inside a DW were also obtained by the analytical solution of Landau–Lifshitz–Gilbert equation [16]. Experimentally, the translational oscillation (or the Doring-type oscillation) of a pinned DW and the thickness oscillation (or breathing model) were mostly studied [10,11,17,18]. Besides, a new DW internal mode was found by Brillouin light scattering [19], and the standing spin-wave modes confined in DW in a nanostrip with zig-zag shaped magnetization configuration was detected by the ferromagnetic resonance measurement [20]. But a detailed study of the normal oscillation modes of a Ne´el wall in a nanostrip is still lacking. In this paper, by using micromagnetic simulations, we study the frequency response of a 1801 Ne´el wall in a soft nanostrip excited by the alternating spin-polarized current. Various discrete normal modes can be distinguished from the simulation results. Spatial distributions of those discrete modes are imaged. Based on the information thus obtained, the physical origin of the discrete modes is analyzed.
X.-g. Wang et al. / Journal of Magnetism and Magnetic Materials 332 (2013) 56–60
57
! ! ! ! the magnetization is described by u Ur m þ b m ½ðuUrÞm [22,23], where u¼ JPmB/eMs, J is current density, P is the spin polarization of the ferromagnet, mB is the Bohr magneton, and e is the electron charge. The dimensionless parameter b is comparable to the damping constant a. The spin-transfer torque can excite only the DW area; other parts of the nanostrip are not excited. This prohibits the coupling between the DW normal modes and the nanostrip normal modes. In order to simulate the spin torque induced ferromagnetic resonance (ST-FMR) [24,25], a sinc-function u ¼u0{sin(2pvJt)/2pvJt} is used. Here, vJ and u0 are taken to be 80 GHz and 7 m/s (the corresponding current density is J¼1.5 1011 A/m2 if Ms ¼8.6 105 A/m and P¼0.7 is assumed), ! respectively. The temporal evolution of m in each cell is recorded every 1 ps. The local FFT power is computed for each cell and then summed over the whole region. Thus, the DW resonance frequency spectrum in the range of 0–80 GHz can be calculated and the spatial distribution of the resonance oscillation can be imaged. As the excited normal modes are related to the symmetry of exciting torque, the spin-polarized currents along the x-direction (perpendicular to the DW) and y-directions (parallel to the DW) are applied to excite different types of the modes. In order to prevent the reflection of spin waves from the nanostrip ends, the 1500 nm long spin-wave absorbing areas are set at the two ends, where the damping constant is varied linearly from 0.01 to 1.
3. Results and discussion 3.1. Spin-polarized current perpendicular to the DW Fig. 1. (a) Illustration of the model nanostrip with the geometry and dimensions. A 1801 Ne´el wall is positioned at the center x¼ 0. The magnetization direction is represented by arrows. The Cartesian coordinate system is shown on the upper left. (b) The DW thickness as a function of y for Ms ¼8.6 105 A/m (black square), 4 105 A/m (red circle) and 2 105 A/m (blue triangle). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
2. Simulation details The soft magnetic nanostrip used in our simulations is 6 mm long in the x-direction, 50 nm wide in the y-direction, and 10 nm thick in the z-direction as shown in Fig. 1(a). For micromagnetic simulation, the following material parameters are used: exchange stiffness Aex ¼1.3 10-11 J/m, magnetocrystalline anisotropy K¼0, and Gilbert damping constant a ¼0.01. As the DW thickness in a nanostrip is sensitive to the saturation magnetizations Ms, in simulation we change Ms from 8.6 105 A/m to 2 105 A/m, so that the wall thickness dependence of the DW dynamic properties can be investigated. Micromagnetic simulations presented here are performed with the micromagnetic code OOMMF [21]. The simulation cell size is chosen to be 2 2 10 nm3. A 1801 Ne´el wall is at first placed at the center position (x¼0) and then relaxed to stable state. The DW thickness varies across the strip width (y-coordinate) as shown in Fig. 1(b). The thickness d changes from 35 nm at y¼0 to 83 nm at y¼50 nm for Ms ¼8.6 105 A/m. However, this variation is not monotonous, and a maximal value of 90 nm appears at y¼37 nm. The average wall thickness is about d ¼65.3 nm. The DW thickness increases with the decrease of saturation magnetization. For Ms ¼2 105 A/m, the average thickness is increased to d ¼176.9 nm. But the thickness variation along the y-direction is gradually mild. The position dependence of the wall thickness along the y-direction results from the strong inhomogeneity of the demagnetization field. The DW oscillations are excited by the injection of an alternating spin-polarized current. The spin-transfer torque acting on
The DW resonance frequency spectra excited by the spinpolarized current flowing along the nanostrip axis (perpendicular to the DW) are presented in Fig. 2(a). First consider the spectrum for Ms ¼8.6 105 A/m. It can be seen that several distinct resonance peaks are excited. This indicates that the DW in a nanostrip exhibits discrete internal normal modes. From low to high frequency these modes are labeled as C1 (6.8 GHz), CB1 (10.6 GHz), CB2 (17.2 GHz), C2 (18.1 GHz), CB3 (27.45 GHz), CB4 (43.7 GHz), C3 (55.8 GHz) and CB5 (66.4 GHz). In order to get a better understanding about the properties of these normal modes, the spatial distributions of the mode FFT power are imaged in Fig. 2(b). For all these modes, the FFT power distribution and phase (not shown here) are symmetrical with respect to the wall central line at x¼0. For the lowest frequency mode C1, oscillation power is concentrated on the top and bottom lateral edges of the DW center (edge mode). The excited area at the bottom is obviously smaller than that at the top due to the wall thickness variation across the strip width. Moreover, the oscillations at the top and bottom are out of phase with each other (not shown in Fig. 2(b)), which indicates that there is a strong dipolar coupling between the top and bottom oscillations. For the second mode CB1, besides the FFT power distributing in the top and bottom lateral edges of the wall center like mode C1, there is oscillation at the DW boundary. The oscillations at the two DW boundaries are in phase but they are out of phase with the wall center oscillation, and the oscillations at the top and bottom in the wall center are in phase. It can be seen from the frequency spectrum that the frequencies of mode CB2 and mode C2 are very close to each other, making the FFT power profile as a superposition of these 2 modes. The mode CB2 is roughly similar to the mode CB1, but the wall boundary oscillations are concentrated on the strip lateral edges. The FFT power of mode C2 is distributed mainly at the wall center. Unlike the mode C1, besides the strip lateral edges, there is strong oscillation at the central area of the strip for mode C2. With regard to the mode CB3, the wall center oscillation is concentrated at the central areas of the strip, and the lateral edge
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X.-g. Wang et al. / Journal of Magnetism and Magnetic Materials 332 (2013) 56–60
M = 8.6×10 A/m
C1
CB3
CB1
CB4
CB2
C3
C2
CB5
C1
C2
M = 4×10 A/m
M = 2×10 A/m
C2
and, in consequence, a highly inhomogeneous internal field distribution. To shed more light on the DW normal modes, we calculate ! ! the static effective field distributions Htol ¼ H ef f Um , [15,32] where ! H ef f is the total static effective field which is the sum of the exchange field and demagnetizing field. Fig. 3(a) shows the Htol variation across the strip width at x¼0. It can be seen that the Htol is inhomogeneous, especially at the strip lateral edges. The localized edge modes are formed in the area with large gradient of Htol as the strong inhomogeneity of effective field acts to confine the spin waves [15,28,32]. Furthermore, this kind of confinement exists not only at the strip lateral edges but also at the domain wall boundary as the Htol is strongly inhomogeneous also along the strip central axis as shown in Fig. 3(b). Dips at the wall boundary for Ms ¼8.6 105 A/m can be seen. These dips may act as spin-wave wells and the localized wall boundary modes are formed in these areas. Therefore, the characteristics of the normal modes of the Ne´el wall in a nanostrip are determined by the distribution of the total field Htol, especially its inhomogeneity. With decreasing Ms and hence increasing DW thickness, the spatial inhomogeneities of the total field Htol in DW are gradually weakened. This can be seen from the field variation curves for Ms ¼4 105 A/m and 2 105 A/m (Fig. 3(a) and (b)). In these cases, the boundary modes disappear. This is because the field gradient along the wall thickness direction is not large enough to localize a spin wave at the wall boundary. Besides, the increasing of the DW width and the reduction of the saturation magnetization may change the spin-wave dispersion relation in the DW as discussed below so that the frequency of the same DW mode can increase a lot with the decrease of Ms. All these factors influence the DW normal mode distribution as well as its frequency. Moreover, the reduction of the magnetization leads to
Fig. 2. (a) Resonance spectra of the domain wall excited by a spin-polarized current flowing perpendicularly to the wall for Ms ¼ 8.6 105 A/m (black solid line), Ms ¼ 4 105 A/m (red dash–dotted line) and Ms ¼ 2 105 A/m (blue short dashed line). (b) Spatial distributions of the FFT power corresponding to the resonance peaks. Dashed white lines in these figures represent the domain wall width shown in Fig. 1(b). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
oscillations disappear. The wall boundary oscillations come into being in standing-wave mode with two nodes along y-axis. The high frequency normal modes display more complicated FFT power profile. As for mode C3, a standing-wave mode with more than 2 nodes along y-axis is formed at the wall center. For modes CB4 and CB5, standing-wave modes are formed both at the wall center and boundaries. In addition, the spin waves emitted from the domain wall are clearly observed for the modes with frequency higher than 12 GHz, which is the cut-off frequency of spin wave propagation in the nanostrip. In Fig. 2(a) the resonance frequency spectra of the domain wall for Ms ¼4 105 A/m and 2 105 A/m are also presented. In these cases only the wall center oscillation modes C1 and C2 are excited and this can be seen from the corresponding FFT power distributions shown in Fig. 2(b). To sum up, the resonance oscillation modes of a 1801 Ne´el wall in a nanostrip excited by the spin-polarized current perpendicular to the wall are mainly concentrated at the wall center and/or wall boundaries, forming edge modes, center modes, standing-wave modes and their mixed modes. The edge mode and the standingwave mode are observed and studied comprehensively in uniformly or quasi-uniformly magnetized elements with different shapes [26–30]. It is believed that the edge mode results from the strong inhomogeneous internal field at the edges of elements [28,31]. The lateral confinement causes the quantization of spin wave and forms the standing-wave modes [26,31]. But compared with the quasi-uniformly magnetized element, the Ne´el wall in a nanostrip exhibits much more complicated oscillation modes due to the more inhomogeneous equilibrium magnetization configuration
Fig. 3. (a) Variation of the total internal field Htol across the nanostrip’s width at x¼ 0 nm for Ms ¼ 2 105 A/m (black square), Ms ¼4 105 A/m (red circle) and Ms ¼ 8.6 105 A/m (blue triangle). (b) Variation of the total internal field Htol along the nanostrip’s central axis (y¼ 25 nm) for Ms ¼2 105 A/m (black solid line), Ms ¼ 4 105 A/m (red dash–dotted line) and Ms ¼ 8.6 105 A/m (blue short dashed line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
X.-g. Wang et al. / Journal of Magnetism and Magnetic Materials 332 (2013) 56–60
M = 8.6×10 A/m
TM
TS-3
TE
TS-4
TS-2 M = 4×10 A/m
TM
TS-2
TE
TS-3
TM
TS-2
M = 2×10 A/m
TE Fig. 4. (a) Resonance spectra of the domain wall excited by the spin-polarized current flowing parallel to the wall for Ms ¼8.6 105 A/m (black solid line), Ms ¼ 4 105 A/m (red dash–dotted line) and Ms ¼2 105 A/m (blue short dashed line). (b) Spatial distributions of the FFT power corresponding to the resonance peaks, and dashed white lines in these figures represent the domain wall width shown in Fig. 1(b). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
the disappearance of the lateral edge mode C1 for the DW with Ms ¼2 105 A/m. 3.2. Spin-polarized current parallel to DW In contrary to the excitation of DW by the spin-polarized current perpendicular to the DW (x-direction), where the STT acts mainly on the central part of the wall, the spin-polarized current parallel to the DW (y-direction) mainly stimulates the two sides of the wall, and different kinds of resonance oscillation modes are expected. Fig. 4(a) shows the resonance frequency spectra of domain walls with Ms ¼ 8.6 105 A/m, 4 105 A/m and 2 105 A/m. Several normal mode resonance peaks are observed. The spatial distributions of FFT power corresponding to these normal modes are shown in Fig. 4(b). It can be seen that oscillation powers of these modes are located mainly in both sides of the domain wall and symmetrical with respect to the wall central line at x¼0. But different from the boundary oscillation excited by the x-direction current, where the oscillations at the 2 wall boundaries are in phase, here the oscillation at the 2 sides are out of phase (not shown here). Actually, these modes are the DW thickness oscillation modes (or breathing modes) [33], where the DW thickness oscillates. Furthermore, due to the geometrical confinement in the y-direction, except for the typical fundamental thickness mode (labeled as TM), highorder thickness modes are observed. These modes can be described as the edge-thickness mode (labeled as TE) and standing wavethickness modes labeled as TS-n (n denotes the node number) as
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indicated in Fig. 4(a) and (b). Besides, because of the spatial inhomogeneity of the effective field in the wall, the mode profiles are distorted along the y-axis. With decreasing saturation magnetization Ms, the field inhomogeneity is gradually weakened, and the distribution of the oscillation power becomes more uniform across the strip width. It is worth noting that the dependence of the resonance frequency on the magnetization Ms exhibits different variation patterns for different modes. For the fundamental thickness mode TM, the frequency decreases with Ms. While for the edgethickness mode TE, its frequency first decreases and then increases with the decrease of Ms. For all the standing wavethickness modes, the frequency increases monotonously with the decrease of Ms. For the mode TS-4, the corresponding frequencies for Ms ¼4 105 A/m and 2 105 A/m are even beyond the range of 0–80 GHz in our simulations. The dependence of resonance frequency on the saturation magnetization for the fundamental breathing mode can be explained by the harmonic thickness oscillation model proposed by Liu and Grutter [34]. According to this model, the oscillation frequency is inversely proportional to the wall width. In our case, the decrease of magnetization leads to the decrease of the shape magnetic anisotropy and, hence the increase of the wall thickness. To understand the f–Ms relation for the edge- and standing wave-thickness modes, the influence of the edge mode and standing-wave mode must be taken into account. The edge mode and standing-wave mode are studied extensively in magnetic elements with different shapes. The eigen-frequency for a rectangular element can be approximately described by the following formula [29]: ffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 f sw ¼ oH þ aoM k2mn oH þaoM k2mn þ oM F ðkmn ,tÞÞ ð1Þ 2p and 2 2 km kn M ð t Þ P k F ðkmn ,t Þ ¼ 1 þ Pðkmn t Þð1P ðkmn t ÞÞ oH þoao mn 2 M kmn kmn k2mn pðkmn t Þ ¼ 1
1expðkmn t Þ kmn t
Here, oH ¼ gHint describes the internal static field contribution, in which Hint is the sum of the static exchange and static demagnetizing fields. The term aoMk2mn is the dynamic exchange contribution, where oM ¼ gMs, and a¼2Aex/(m0Ms). F(kmn,t) describes the dynamic dipole–dipole interaction, t is the thickness in the z-direction, the 2-dimensional quantization wave vector k is k¼ kmex þkney and k2mn ¼k2m þk2n. Eq. (1) describes well the spin-wave modes in a uniformly or quasi-uniformly magnetized rectangular element. But for the Ne´el wall in a nanostrip, the magnetization configuration is inhomogeneous. Even though, this formula can still qualitatively and even quasi-quantitatively account for the DW oscillation modes. For numerical calculation of the frequency from Eq. (1), a key point is to determine the quantization wave numbers km and kn; km and kn can be approximately taken to be km ¼mp/wm and kn ¼np/wn. Taking the dipolar pinning boundary condition into account, wm is suppossed to be dd(p)/(d(p) 2) [35], where d(p)¼2p/p[1þ2ln(1/p)] is the effective pinning parameter and p¼t/d; t ¼10 nm and d are thickness of the nanostrip and the DW width, respectively. As for wn, it is approximately taken to be the effective localization length of the spin-wave mode along the y-direction. Taking the values of wm, wn and the internal static field Hint from the micromagnetic simulation results, the DW resonance frequencies can be calculated from Eq. (1). As Hint is inhomogeneous in the DW, it is taken to be the average of the internal field in the oscillatory area. Fig. 5 shows the magnetization dependence of the resonance frequency for the
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X.-g. Wang et al. / Journal of Magnetism and Magnetic Materials 332 (2013) 56–60
References
Fig. 5. Dependence of the resonance frequency on magnetization for the DW thickness modes. The solid dots and open dots denote the simulated and calculated data, respectively.
thickness oscillation modes. It can be seen that the frequencies calculated from Eq. (1) are in reasonable agreement with the micro-magnetically simulated data. Furthermore, the magnetization dependence of the frequency for the normal oscillation modes excited by the spin-polarized current perpendicular to DW, such as mode C1 and mode C2, can be qualitatively explained in the framework of above-mentioned theory.
4. Conclusion We have studied the spin-transfer torque induced ferromagnetic resonance of the 1801 Ne´el domain wall in a nanostrip using the micromagnetic simulations. A series of resonance peaks are observed. If the spin-polarized current flows along the strip’s long axis, the excited normal oscillation modes are mainly concentrated at the wall center and/or wall boundaries, forming edge mode, center mode, standing-wave mode and their mixed modes. The spin-polarized current flowing along the y-direction excites the wall’s thickness oscillation modes (or breathing modes). In addition to the fundamental thickness mode, high-order thickness modes such as edge- and standing wave-thickness modes are observed.
Acknowledgments Critical comments and discussions from D. Wang are gratefully acknowledged. This work was supported by the National Natural Science Foundation of China (No. 60571043), the Doctoral Fund of Ministry of Education of China (No. 20120162110020) and the Scientific Plane Project of Hunan Province of China (No. 2011FJ3193).
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Spin-transfer torque induced domain wall ferromagnetic resonance in nanostrips Xi-guang Wang, Guang-hua Guo n, Guang-fu Zhang, Yao-zhuang Nie, Qing-lin Xia, Zhi-xiong Li School of Physics and Electronics, Central South University, Changsha 410083, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 5 October 2012 Received in revised form 20 November 2012 Available online 12 December 2012
The frequency response of a Ne´el domain wall in a nanostrip excited by alternating spin-polarized current is studied by micromagnetic simulations. Several internal normal modes of the domain wall are excited and the corresponding spatial distributions of oscillation power are imaged. In the case of current perpendicular to the wall, the excited normal modes are mainly concentrated at the wall center and/or wall boundaries, forming edge modes, center modes, standing-wave modes and their mixed modes. The localization and spatial distribution of the modes have a close relationship with the total internal field, especially its inhomogeneity. With the decrease of saturation magnetizations Ms, the spatial inhomogeneities of the total field are gradually weakened and some domain wall normal modes disappear. In the case where the current is parallel to the wall, the wall thickness oscillation mode (or breathing mode) is excited. Furthermore, due to the geometrical confinement, high-order thickness modes such as edge- and standing wave-thickness modes are observed. The magnetization dependence of the eigen-frequency exhibits different forms for different normal modes, which can be qualitatively explained based on an approximate dispersion equation for spin-wave modes of a quasi-uniformly magnetized rectangular element. & 2012 Elsevier B.V. All rights reserved.
Keywords: Magnetic domain wall Ferromagnetic resonance Spin wave Spin-transfer torque Micromagnetic simulation Magnetic nanowire
1. Introduction The dynamic properties of a domain wall (DW) have attracted much attention recently, owing to their potential technological applications such as memory devices and logic gates [1–3]. Many significant DW dynamic characteristics, such as DW motion, have a close relation to the frequency response of a wall. It is found that the spin-wave induced DW motion in a nanowire has a strong frequency dependence resulting from the frequency dependence of the spin wave propagation through a DW [4,5]. When spin-wave frequency coincides with a DW normal mode frequency, a resonance reflection occurs. As a result, the DW moves with an extremum velocity [5]. The DW motion can be resonantly amplified also through excitation of DW internal modes by an alternating spin-polarized current [6,7]. Besides, various other phenomena, such as the spin-wave phase shift [8], spin-wave frequency doubling by DW [9], and the microwave assisted DW depining [10,11], are also correlated with the characteristics of the DW internal oscillation modes. Therefore, a comprehensive understanding about the DW intrinsic dynamic
n
Corresponding author. Tel.: þ86 731 88836443; fax: þ 86 731 88830857. E-mail address: [email protected] (G.-h. Guo).
0304-8853/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jmmm.2012.12.013
properties is important for designing the DW-based spintronic devices. Theoretically, a 1-dimensional 1801 Bloch DW has a continuous spin-wave dispersion relation [12,13]. But for a DW in a nanostrip, it was demonstrated by micromagnetic simulations that the DW normal oscillation spectrum shows discrete and localized features due to the geometrical confinement [14,15], edge modes and standing spin-wave modes are also found [15]. Standing spin waves bound inside a DW were also obtained by the analytical solution of Landau–Lifshitz–Gilbert equation [16]. Experimentally, the translational oscillation (or the Doring-type oscillation) of a pinned DW and the thickness oscillation (or breathing model) were mostly studied [10,11,17,18]. Besides, a new DW internal mode was found by Brillouin light scattering [19], and the standing spin-wave modes confined in DW in a nanostrip with zig-zag shaped magnetization configuration was detected by the ferromagnetic resonance measurement [20]. But a detailed study of the normal oscillation modes of a Ne´el wall in a nanostrip is still lacking. In this paper, by using micromagnetic simulations, we study the frequency response of a 1801 Ne´el wall in a soft nanostrip excited by the alternating spin-polarized current. Various discrete normal modes can be distinguished from the simulation results. Spatial distributions of those discrete modes are imaged. Based on the information thus obtained, the physical origin of the discrete modes is analyzed.
X.-g. Wang et al. / Journal of Magnetism and Magnetic Materials 332 (2013) 56–60
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! ! ! ! the magnetization is described by u Ur m þ b m ½ðuUrÞm [22,23], where u¼ JPmB/eMs, J is current density, P is the spin polarization of the ferromagnet, mB is the Bohr magneton, and e is the electron charge. The dimensionless parameter b is comparable to the damping constant a. The spin-transfer torque can excite only the DW area; other parts of the nanostrip are not excited. This prohibits the coupling between the DW normal modes and the nanostrip normal modes. In order to simulate the spin torque induced ferromagnetic resonance (ST-FMR) [24,25], a sinc-function u ¼u0{sin(2pvJt)/2pvJt} is used. Here, vJ and u0 are taken to be 80 GHz and 7 m/s (the corresponding current density is J¼1.5 1011 A/m2 if Ms ¼8.6 105 A/m and P¼0.7 is assumed), ! respectively. The temporal evolution of m in each cell is recorded every 1 ps. The local FFT power is computed for each cell and then summed over the whole region. Thus, the DW resonance frequency spectrum in the range of 0–80 GHz can be calculated and the spatial distribution of the resonance oscillation can be imaged. As the excited normal modes are related to the symmetry of exciting torque, the spin-polarized currents along the x-direction (perpendicular to the DW) and y-directions (parallel to the DW) are applied to excite different types of the modes. In order to prevent the reflection of spin waves from the nanostrip ends, the 1500 nm long spin-wave absorbing areas are set at the two ends, where the damping constant is varied linearly from 0.01 to 1.
3. Results and discussion 3.1. Spin-polarized current perpendicular to the DW Fig. 1. (a) Illustration of the model nanostrip with the geometry and dimensions. A 1801 Ne´el wall is positioned at the center x¼ 0. The magnetization direction is represented by arrows. The Cartesian coordinate system is shown on the upper left. (b) The DW thickness as a function of y for Ms ¼8.6 105 A/m (black square), 4 105 A/m (red circle) and 2 105 A/m (blue triangle). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
2. Simulation details The soft magnetic nanostrip used in our simulations is 6 mm long in the x-direction, 50 nm wide in the y-direction, and 10 nm thick in the z-direction as shown in Fig. 1(a). For micromagnetic simulation, the following material parameters are used: exchange stiffness Aex ¼1.3 10-11 J/m, magnetocrystalline anisotropy K¼0, and Gilbert damping constant a ¼0.01. As the DW thickness in a nanostrip is sensitive to the saturation magnetizations Ms, in simulation we change Ms from 8.6 105 A/m to 2 105 A/m, so that the wall thickness dependence of the DW dynamic properties can be investigated. Micromagnetic simulations presented here are performed with the micromagnetic code OOMMF [21]. The simulation cell size is chosen to be 2 2 10 nm3. A 1801 Ne´el wall is at first placed at the center position (x¼0) and then relaxed to stable state. The DW thickness varies across the strip width (y-coordinate) as shown in Fig. 1(b). The thickness d changes from 35 nm at y¼0 to 83 nm at y¼50 nm for Ms ¼8.6 105 A/m. However, this variation is not monotonous, and a maximal value of 90 nm appears at y¼37 nm. The average wall thickness is about d ¼65.3 nm. The DW thickness increases with the decrease of saturation magnetization. For Ms ¼2 105 A/m, the average thickness is increased to d ¼176.9 nm. But the thickness variation along the y-direction is gradually mild. The position dependence of the wall thickness along the y-direction results from the strong inhomogeneity of the demagnetization field. The DW oscillations are excited by the injection of an alternating spin-polarized current. The spin-transfer torque acting on
The DW resonance frequency spectra excited by the spinpolarized current flowing along the nanostrip axis (perpendicular to the DW) are presented in Fig. 2(a). First consider the spectrum for Ms ¼8.6 105 A/m. It can be seen that several distinct resonance peaks are excited. This indicates that the DW in a nanostrip exhibits discrete internal normal modes. From low to high frequency these modes are labeled as C1 (6.8 GHz), CB1 (10.6 GHz), CB2 (17.2 GHz), C2 (18.1 GHz), CB3 (27.45 GHz), CB4 (43.7 GHz), C3 (55.8 GHz) and CB5 (66.4 GHz). In order to get a better understanding about the properties of these normal modes, the spatial distributions of the mode FFT power are imaged in Fig. 2(b). For all these modes, the FFT power distribution and phase (not shown here) are symmetrical with respect to the wall central line at x¼0. For the lowest frequency mode C1, oscillation power is concentrated on the top and bottom lateral edges of the DW center (edge mode). The excited area at the bottom is obviously smaller than that at the top due to the wall thickness variation across the strip width. Moreover, the oscillations at the top and bottom are out of phase with each other (not shown in Fig. 2(b)), which indicates that there is a strong dipolar coupling between the top and bottom oscillations. For the second mode CB1, besides the FFT power distributing in the top and bottom lateral edges of the wall center like mode C1, there is oscillation at the DW boundary. The oscillations at the two DW boundaries are in phase but they are out of phase with the wall center oscillation, and the oscillations at the top and bottom in the wall center are in phase. It can be seen from the frequency spectrum that the frequencies of mode CB2 and mode C2 are very close to each other, making the FFT power profile as a superposition of these 2 modes. The mode CB2 is roughly similar to the mode CB1, but the wall boundary oscillations are concentrated on the strip lateral edges. The FFT power of mode C2 is distributed mainly at the wall center. Unlike the mode C1, besides the strip lateral edges, there is strong oscillation at the central area of the strip for mode C2. With regard to the mode CB3, the wall center oscillation is concentrated at the central areas of the strip, and the lateral edge
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X.-g. Wang et al. / Journal of Magnetism and Magnetic Materials 332 (2013) 56–60
M = 8.6×10 A/m
C1
CB3
CB1
CB4
CB2
C3
C2
CB5
C1
C2
M = 4×10 A/m
M = 2×10 A/m
C2
and, in consequence, a highly inhomogeneous internal field distribution. To shed more light on the DW normal modes, we calculate ! ! the static effective field distributions Htol ¼ H ef f Um , [15,32] where ! H ef f is the total static effective field which is the sum of the exchange field and demagnetizing field. Fig. 3(a) shows the Htol variation across the strip width at x¼0. It can be seen that the Htol is inhomogeneous, especially at the strip lateral edges. The localized edge modes are formed in the area with large gradient of Htol as the strong inhomogeneity of effective field acts to confine the spin waves [15,28,32]. Furthermore, this kind of confinement exists not only at the strip lateral edges but also at the domain wall boundary as the Htol is strongly inhomogeneous also along the strip central axis as shown in Fig. 3(b). Dips at the wall boundary for Ms ¼8.6 105 A/m can be seen. These dips may act as spin-wave wells and the localized wall boundary modes are formed in these areas. Therefore, the characteristics of the normal modes of the Ne´el wall in a nanostrip are determined by the distribution of the total field Htol, especially its inhomogeneity. With decreasing Ms and hence increasing DW thickness, the spatial inhomogeneities of the total field Htol in DW are gradually weakened. This can be seen from the field variation curves for Ms ¼4 105 A/m and 2 105 A/m (Fig. 3(a) and (b)). In these cases, the boundary modes disappear. This is because the field gradient along the wall thickness direction is not large enough to localize a spin wave at the wall boundary. Besides, the increasing of the DW width and the reduction of the saturation magnetization may change the spin-wave dispersion relation in the DW as discussed below so that the frequency of the same DW mode can increase a lot with the decrease of Ms. All these factors influence the DW normal mode distribution as well as its frequency. Moreover, the reduction of the magnetization leads to
Fig. 2. (a) Resonance spectra of the domain wall excited by a spin-polarized current flowing perpendicularly to the wall for Ms ¼ 8.6 105 A/m (black solid line), Ms ¼ 4 105 A/m (red dash–dotted line) and Ms ¼ 2 105 A/m (blue short dashed line). (b) Spatial distributions of the FFT power corresponding to the resonance peaks. Dashed white lines in these figures represent the domain wall width shown in Fig. 1(b). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
oscillations disappear. The wall boundary oscillations come into being in standing-wave mode with two nodes along y-axis. The high frequency normal modes display more complicated FFT power profile. As for mode C3, a standing-wave mode with more than 2 nodes along y-axis is formed at the wall center. For modes CB4 and CB5, standing-wave modes are formed both at the wall center and boundaries. In addition, the spin waves emitted from the domain wall are clearly observed for the modes with frequency higher than 12 GHz, which is the cut-off frequency of spin wave propagation in the nanostrip. In Fig. 2(a) the resonance frequency spectra of the domain wall for Ms ¼4 105 A/m and 2 105 A/m are also presented. In these cases only the wall center oscillation modes C1 and C2 are excited and this can be seen from the corresponding FFT power distributions shown in Fig. 2(b). To sum up, the resonance oscillation modes of a 1801 Ne´el wall in a nanostrip excited by the spin-polarized current perpendicular to the wall are mainly concentrated at the wall center and/or wall boundaries, forming edge modes, center modes, standing-wave modes and their mixed modes. The edge mode and the standingwave mode are observed and studied comprehensively in uniformly or quasi-uniformly magnetized elements with different shapes [26–30]. It is believed that the edge mode results from the strong inhomogeneous internal field at the edges of elements [28,31]. The lateral confinement causes the quantization of spin wave and forms the standing-wave modes [26,31]. But compared with the quasi-uniformly magnetized element, the Ne´el wall in a nanostrip exhibits much more complicated oscillation modes due to the more inhomogeneous equilibrium magnetization configuration
Fig. 3. (a) Variation of the total internal field Htol across the nanostrip’s width at x¼ 0 nm for Ms ¼ 2 105 A/m (black square), Ms ¼4 105 A/m (red circle) and Ms ¼ 8.6 105 A/m (blue triangle). (b) Variation of the total internal field Htol along the nanostrip’s central axis (y¼ 25 nm) for Ms ¼2 105 A/m (black solid line), Ms ¼ 4 105 A/m (red dash–dotted line) and Ms ¼ 8.6 105 A/m (blue short dashed line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
X.-g. Wang et al. / Journal of Magnetism and Magnetic Materials 332 (2013) 56–60
M = 8.6×10 A/m
TM
TS-3
TE
TS-4
TS-2 M = 4×10 A/m
TM
TS-2
TE
TS-3
TM
TS-2
M = 2×10 A/m
TE Fig. 4. (a) Resonance spectra of the domain wall excited by the spin-polarized current flowing parallel to the wall for Ms ¼8.6 105 A/m (black solid line), Ms ¼ 4 105 A/m (red dash–dotted line) and Ms ¼2 105 A/m (blue short dashed line). (b) Spatial distributions of the FFT power corresponding to the resonance peaks, and dashed white lines in these figures represent the domain wall width shown in Fig. 1(b). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
the disappearance of the lateral edge mode C1 for the DW with Ms ¼2 105 A/m. 3.2. Spin-polarized current parallel to DW In contrary to the excitation of DW by the spin-polarized current perpendicular to the DW (x-direction), where the STT acts mainly on the central part of the wall, the spin-polarized current parallel to the DW (y-direction) mainly stimulates the two sides of the wall, and different kinds of resonance oscillation modes are expected. Fig. 4(a) shows the resonance frequency spectra of domain walls with Ms ¼ 8.6 105 A/m, 4 105 A/m and 2 105 A/m. Several normal mode resonance peaks are observed. The spatial distributions of FFT power corresponding to these normal modes are shown in Fig. 4(b). It can be seen that oscillation powers of these modes are located mainly in both sides of the domain wall and symmetrical with respect to the wall central line at x¼0. But different from the boundary oscillation excited by the x-direction current, where the oscillations at the 2 wall boundaries are in phase, here the oscillation at the 2 sides are out of phase (not shown here). Actually, these modes are the DW thickness oscillation modes (or breathing modes) [33], where the DW thickness oscillates. Furthermore, due to the geometrical confinement in the y-direction, except for the typical fundamental thickness mode (labeled as TM), highorder thickness modes are observed. These modes can be described as the edge-thickness mode (labeled as TE) and standing wavethickness modes labeled as TS-n (n denotes the node number) as
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indicated in Fig. 4(a) and (b). Besides, because of the spatial inhomogeneity of the effective field in the wall, the mode profiles are distorted along the y-axis. With decreasing saturation magnetization Ms, the field inhomogeneity is gradually weakened, and the distribution of the oscillation power becomes more uniform across the strip width. It is worth noting that the dependence of the resonance frequency on the magnetization Ms exhibits different variation patterns for different modes. For the fundamental thickness mode TM, the frequency decreases with Ms. While for the edgethickness mode TE, its frequency first decreases and then increases with the decrease of Ms. For all the standing wavethickness modes, the frequency increases monotonously with the decrease of Ms. For the mode TS-4, the corresponding frequencies for Ms ¼4 105 A/m and 2 105 A/m are even beyond the range of 0–80 GHz in our simulations. The dependence of resonance frequency on the saturation magnetization for the fundamental breathing mode can be explained by the harmonic thickness oscillation model proposed by Liu and Grutter [34]. According to this model, the oscillation frequency is inversely proportional to the wall width. In our case, the decrease of magnetization leads to the decrease of the shape magnetic anisotropy and, hence the increase of the wall thickness. To understand the f–Ms relation for the edge- and standing wave-thickness modes, the influence of the edge mode and standing-wave mode must be taken into account. The edge mode and standing-wave mode are studied extensively in magnetic elements with different shapes. The eigen-frequency for a rectangular element can be approximately described by the following formula [29]: ffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 f sw ¼ oH þ aoM k2mn oH þaoM k2mn þ oM F ðkmn ,tÞÞ ð1Þ 2p and 2 2 km kn M ð t Þ P k F ðkmn ,t Þ ¼ 1 þ Pðkmn t Þð1P ðkmn t ÞÞ oH þoao mn 2 M kmn kmn k2mn pðkmn t Þ ¼ 1
1expðkmn t Þ kmn t
Here, oH ¼ gHint describes the internal static field contribution, in which Hint is the sum of the static exchange and static demagnetizing fields. The term aoMk2mn is the dynamic exchange contribution, where oM ¼ gMs, and a¼2Aex/(m0Ms). F(kmn,t) describes the dynamic dipole–dipole interaction, t is the thickness in the z-direction, the 2-dimensional quantization wave vector k is k¼ kmex þkney and k2mn ¼k2m þk2n. Eq. (1) describes well the spin-wave modes in a uniformly or quasi-uniformly magnetized rectangular element. But for the Ne´el wall in a nanostrip, the magnetization configuration is inhomogeneous. Even though, this formula can still qualitatively and even quasi-quantitatively account for the DW oscillation modes. For numerical calculation of the frequency from Eq. (1), a key point is to determine the quantization wave numbers km and kn; km and kn can be approximately taken to be km ¼mp/wm and kn ¼np/wn. Taking the dipolar pinning boundary condition into account, wm is suppossed to be dd(p)/(d(p) 2) [35], where d(p)¼2p/p[1þ2ln(1/p)] is the effective pinning parameter and p¼t/d; t ¼10 nm and d are thickness of the nanostrip and the DW width, respectively. As for wn, it is approximately taken to be the effective localization length of the spin-wave mode along the y-direction. Taking the values of wm, wn and the internal static field Hint from the micromagnetic simulation results, the DW resonance frequencies can be calculated from Eq. (1). As Hint is inhomogeneous in the DW, it is taken to be the average of the internal field in the oscillatory area. Fig. 5 shows the magnetization dependence of the resonance frequency for the
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References
Fig. 5. Dependence of the resonance frequency on magnetization for the DW thickness modes. The solid dots and open dots denote the simulated and calculated data, respectively.
thickness oscillation modes. It can be seen that the frequencies calculated from Eq. (1) are in reasonable agreement with the micro-magnetically simulated data. Furthermore, the magnetization dependence of the frequency for the normal oscillation modes excited by the spin-polarized current perpendicular to DW, such as mode C1 and mode C2, can be qualitatively explained in the framework of above-mentioned theory.
4. Conclusion We have studied the spin-transfer torque induced ferromagnetic resonance of the 1801 Ne´el domain wall in a nanostrip using the micromagnetic simulations. A series of resonance peaks are observed. If the spin-polarized current flows along the strip’s long axis, the excited normal oscillation modes are mainly concentrated at the wall center and/or wall boundaries, forming edge mode, center mode, standing-wave mode and their mixed modes. The spin-polarized current flowing along the y-direction excites the wall’s thickness oscillation modes (or breathing modes). In addition to the fundamental thickness mode, high-order thickness modes such as edge- and standing wave-thickness modes are observed.
Acknowledgments Critical comments and discussions from D. Wang are gratefully acknowledged. This work was supported by the National Natural Science Foundation of China (No. 60571043), the Doctoral Fund of Ministry of Education of China (No. 20120162110020) and the Scientific Plane Project of Hunan Province of China (No. 2011FJ3193).
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