Current-induced instability of domain walls in cylindrical nanowires
We study the current-driven domain wall (DW) motion in cylindrical nanowires using micromagnetic simulations by implementing the Landau-Lifshitz-Gilbert equation with nonlocal spin-transfer torque in a finite difference micromagnetic package. We find that in the presence of DW Gaussian wave packets (spin waves) will be generated when the charge current is applied to the system suddenly. And this effect is excluded when using the local spin-transfer torque. The existence of spin waves emission indicates that transverse domain walls can not move arbitrarily fast in cylindrical nanowires although they are free from the Walker limit. We establish an upper velocity limit for the DW motion by analyzing the stability of Gaussian wave packets using the local spin-transfer torque. Micromagnetic simulations show that the stable region obtained by using nonlocal spin-transfer torque is smaller than that by using its local counterpart. This limitation is essential for multiple domain walls since the instability of Gaussian wave packets will break the structure of multiple domain walls.
The dynamics of magnetic domain walls (DWs) in ferromagnetic nanostrips has drawn considerable attention in the past few years. An effective method to manipulate the DW is with electrical currents Berger (1986); Parkin et al. (2008); Yan et al. (2010); Thiaville et al. (2005); Hayashi et al. (2007); Thomas et al. (2010); Franchin et al. (2011). The mechanism behind this is the so-called spin transfer torque (STT); when spin-polarized electrons pass through the DW, electrons exert both adiabatic and nonadiabatic torques to the local magnetization Zhang and Li (2004); Tatara et al. (2008). Various complex phenomena make DW motion interesting from a fundamental point of view. For example, in thin nanostrips the Walker breakdown Schryer and Walker (1974) occurs when the speed of a transverse DW reaches a critical velocity due to strong driving forces such as external fields or spin-polarized currents. Hence, the Walker limit is the maximum velocity that a transverse DW can reach in thin strips (similar relativistic velocity limit can be found for antiferromagnetic DWs as well Shiino et al. (2016)). Interestingly, a transverse (head-to-head or tail-to-tail) DW does not suffer the Walker limit in cylindrical nanowires Yan et al. (2010); Goussev et al. (2010); this is because the transverse DW in cylindrical nanowires can rotate freely due to the absence of easy-plane anisotropy. Therefore, we ask if there is a similar physical limit which determines the maximum velocity of the transverse DWs in cylindrical nanowires.
The Walker breakdown is related to the instability of domain walls, for example, spin waves may be emitted when a transverse DW travels in a thin strip Hu and Wang (2013); Wang and Wang (2014). Micromagnetic simulations show that a moving DW in cylindrical nanowires has an almost vanishing mass Yan et al. (2010); Hertel and Kákay (2015). It can be seen that the mass of the transverse DW is exactly zero for an arbitrary time-dependent charge current when using the local spin transfer torque Goussev et al. (2010). The local spin transfer torque has ignored the spin diffusion effect Zhang and Li (2004); Claudio-Gonzalez et al. (2012), which is observable for sharp DWs. Moreover, when taking the spin flip into account, the spin density does not simply depends on the local magnetization Claudio-Gonzalez et al. (2012). Therefore, it is interesting to ask whether the nonlocal spin transfer torque influences the stability of domain walls in cylindrical nanowires.
The dynamics of the magnetization in the presence of a spin polarized current is governed by the extended Landau-Lifshitz-Gilbert (LLG) equation with spin transfer torque Zhang and Li (2004); Thiaville et al. (2005); Tatara et al. (2008):
where is the unit vector of magnetization, is the gyromagnetic ratio, is the total effective field and is the Gilbert damping. The local form of spin transfer torque reads
where the parameter represents the strength of spin-polarized current, and is the charge current density along axis, is the Landé factor, is the Bohr magneton, is the spin polarization rate, is the electron charge and is the saturation magnetization. The term is nonadiabatic torque, which influences the spin waves amplitude significantly, leading to either a weakened or enhanced spin wave attenuation depending on the relative direction between wave vector and charge current Seo et al. (2009); Xia et al. (2016). The current-induced spin wave instability has been reported in Refs. Bazaliy et al. (1998); Fernández-Rossier et al. (2004); Seo et al. (2009).
The nonlocal (full version) spin transfer torque reads Claudio-Gonzalez et al. (2012)
where is the nonequilibrium spin density and is - exchange time. The spin density is obtained by solving the equation:
where is the diffusion constant, is the spin-flip relaxation time. Two characteristic lengths related to and can be defined: – the diffusion length during the exchange time and –the diffusion length during the spin-flip time Claudio-Gonzalez et al. (2012). In the limit case and no time derivative, the torque reduces to the local torque and .
The nonlocal spin transfer torque influences the vortex DW dynamics significantly Claudio-Gonzalez et al. (2012). However, it does not have a large influence for the transverse DWs. In this paper, we will show that the nonlocal spin transfer torque induces spin wave (Gaussian wave packet) emission in the presence of transverse domain wall when we apply the charge current to the system suddenly, as sketched in Fig. 1.
We consider a quasi-1D nanowire with the exchange interaction and an uniaxial anisotropy along the axis. The total micromagnetic energy density of the system reads
where is the exchange constant, and is the anisotropy constant. The demagnetization field is included in as an effective anisotropy for the cylindrical nanowire.
Ii Generation of Gaussian wave packets
In this study, we perform micromagnetic simulation by solving the coupled LLG equation (1) and spin-density equation (4) simultaneously Wang et al. (). A fourth order accurate finite difference discretization in space is used to compute the effective fields and the diffusion equation (4). We make use of the parameters of NiFe alloy Claudio-Gonzalez et al. (2012): the exchange constant , anisotropy , the saturation magnetization and the damping coefficient . For given parameters, the domain wall width nm. In the simulation, the discretization size is chosen to be nm.
A head-to-head DW is placed in the middle of wire, as shown in Fig. 1. We apply a charge current in the direction with amplitude m/s, and we can estimate that m/s corresponds to a current density A/m if . Interestingly, Gaussian wave packets are generated when we apply the charge current to the system suddenly. Fig. 2(a) shows the snapshots of the -component magnetization at ns and ns, from which we can see the emergence of spin wave packets. The parameters used in the simulation are nm, nm and m/s. As a comparison, we performed the micromagnetic simulation using the local spin transfer torque . Fig. 2(b) shows the snapshots of at ns and ns, clearly, the DW moves smoothly without spin waves emission. The detailed animation [I.mp4] can be found in the Supplemental Material Sup ().
The fact that no spin waves are emitted for the local spin transfer torque can be understood using the Walker DW profile
where we have written the magnetization unit vector as . Eq. (6) describes a head-to-head domain wall and is the typical DW width. The Walker solution for the LLG equation with is
Note that equation (8) is an exact spatiotemporal solution for the extended LLG equation including the local STT, and thus the solution allows an arbitrary time-dependence function . The procedure to show equation (8) is an exact spatiotemporal solution can be found in the literature Goussev et al. (2010).
The generation of spin waves only happens if we use the full version of spin transfer torques. Moreover, the Walker solution is not an exact solution for the LLG equation with nonlocal spin transfer torques. Fig. 3(a) shows the stable DW velocities as a function of for the both local and nonlocal cases. It is found that the DW velocity is very close to the Walker solution. Figure 3(b) plots the velocity difference between the two cases, and the amplitude ratio is .
Iii Beyond the stable motion.
From the animation [I.mp4], we can see that the amplitude of the wave packet is decreasing slowly, where m/s is used. There are two reasons responsible for the amplitude decrease: the existence of Gilbert damping and the intrinsic delocalization due to the quadratic dispersion relation of spin waves.
Fig. 4 shows the simulation results for the propagation of the Gaussian wave packets using nonlocal spin transfer torque with m/s. As can be seen by comparing Fig. 4(a) and Fig. 4(b), in this scenario the Gaussian wave packet is growing. The corresponding animation [II.mp4] can be found in Supplemental Material Sup (). There are two panels in the animation show the same data but with different scale. The amplitude of the packet increases exponentially, leading to a magnetization reversal and ending with a chaotic dynamics Seo et al. (2009),
A chaotic dynamics of the domains gives an upper velocity limit for the DWs. We now investigate the critical current density that determines whether the Gaussian wave packet grows or decreases using the local spin transfer torque . For the given energy density [Eq. (5)], the corresponding effective field is where and . Substituting the effective field into Eq. (1), one obtains
where , and . From Fig. 4(b) we can find that the wave packet is far away from DW, so we will consider the wave packet moves in uniform domains. For example, the head-to-head DW separates two domains with magnetization and . By introducing the complex transformation Li et al. (2007); Zhao et al. (2012)
and linearizing the LLG equation around , we arrive at a Schrödinger-type equation
The complex conjugate of this corresponds to the linearized equation around with transformation . A typical solution of Eq. (11) is the travelling wave in the form where and are dimensionless wave vector and frequency. However, in the presence of damping (i.e., ), a complex frequency or wave vector must be introduced, which corresponds to a finite linewidth or amplitude decay Seo et al. (2009); Sekiguchi et al. (2012) of spin waves, respectively. For spin waves with localized shape, we chose the former Covington et al. (2002), and thus we look for a solution in the form
where gives the dispersion relation of spin waves and indicates the energy dissipation during the spin wave propagation. Fig. 5(a) shows the dispersion relations for different values of . We can see that the wave vector is shifted in the presence of spin current, and the spin current is proportional to Bazaliy et al. (1998); Fernández-Rossier et al. (2004). This wave vector shift is similar to case that induced by the DMI Moon et al. (2013), resulting in an asymmetric spin-wave dispersion. Note that the spin wave frequency is negative if . It is estimated that corresponds to m/s for the parameters we used above. The domain wall velocity can be obtained using Eq. (8) and is of similar magnitude.
Fig. 5(b) shows a contour plot of the dissipation rate as a function of and for and , it is found that in the top right corner is negative, which indicates that term may leads to a negative dissipation rate if . For the case that and , the solution of reduces to
To analyze the stability of the wave packets, we define the total energy . For the case , it is convenient to compute in space, i.e., where since Eq. (12) can be considered as a Fourier transformation of . Hence, we obtain
where , and . It is clear that the sign of determines the total energy of the Gaussian wave packet over longer time scales; a positive suppresses the Gaussian wave packet while a negative leads to a growing wave packet. Therefore, the critical current density is determined by
which gives and thus the critical current density reads
It is perhaps surprising that this critical current density is independent from the group velocity and the Gaussian wave width while for the spin wave case the amplitude decaying length is influenced by the wave vector of spin waves for a given current density. Using the typical parameters of NiFe, one finds that is in the range of 10–10 A/m. As a comparison, the critical for Walker breakdown in the presence of easy-plane anisotropy is approximately established as Thiaville et al. (2005) where . For a typical one obtains .
Fig. 6 plots the critical current density for . The dashed blue line is plotted using Eq. (16), which is obtained by checking the stability of the Gaussian wave packets using local spin transfer torque. The green line is extracted from the micromagnetic simulation with the nonlocal spin transfer torque. In detail, we monitor the maximum amplitude of the wave packet for given charge current density , if the maximum amplitude always increase we think the corresponding charge current is in the unstable region. In the simulation, we fixed fs and vary according to the value of . As we can see, the stable region obtained using nonlocal spin transfer torque is much smaller than that using its local counterpart. Moreover, for the local case, when the total energy of the Gaussian wave packet can be simplified to . We find that is independent from the charge current and decreases with for arbitrary , which means that the wave packet is always suppressed if . However, this conclusion is not true when we using the nonlocal spin transfer torque.
In summary, we have studied the current-driven domain wall motion in a cylindrical nanowire using micromagnetic simulation with nonlocal spin transfer torque. We show that in the presence of domain wall a Gaussian wave packet will be generated when the charge current is applied to the system suddenly. The generation of wave packet only happens when we use the nonlocal spin transfer torque. By analyzing the stability of the Gaussian wave packet, we give an upper velocity limit for the domain wall motion driven with spin currents. The limitation is especially important for multiple domain walls motion since the instability of Gaussian wave packets will break their structure. Moreover, the limitation also indicates that transverse domain walls can not move arbitrarily fast in cylindrical nanowires though they are not subject to the Walker limit.
We acknowledge the financial support from National Natural Science Foundation of China (Grants No. 11404280 and No. 11604169) and EPSRC under Centre for Doctoral Training Grant EP/L015382/1. This work is sponsored by K.C.Wong Magna Fund in Ningbo University.
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