Current induced multiple domain wall motion modulated by magnetic pinning in zigzag shaped nanowires Current induced multiple domain wall motion modulated by magnetic pinning in zigzag shaped nanowire[.]
Current-induced multiple domain wall motion modulated by magnetic pinning in zigzag shaped nanowires Xiaochao Zhou, Zhaocong Huang, Wen Zhang, Yuli Yin, Philipp Dürrenfeld, Shuai Dong, and Ya Zhai Citation: AIP Advances 7, 056014 (2017); doi: 10.1063/1.4975129 View online: http://dx.doi.org/10.1063/1.4975129 View Table of Contents: http://aip.scitation.org/toc/adv/7/5 Published by the American Institute of Physics AIP ADVANCES 7, 056014 (2017) Current-induced multiple domain wall motion modulated by magnetic pinning in zigzag shaped nanowires Xiaochao Zhou,1 Zhaocong Huang,1 Wen Zhang,1,2,a Yuli Yin,1 Shuai Dong,1 and Ya Zhai1,3,b ă Philipp Durrenfeld, Department of Physics, Southeast University, Nanjing 211189, People’s Republic of China of Physics, National University of Singapore, Science Drive 3, 117542 Singapore, Singapore National Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, People’s Republic of China Department (Presented November 2016; received 23 September 2016; accepted November 2016; published online 30 January 2017) Using micromagnetic simulation, we investigate the current-induced multiple domain wall motion (CIDWM) in zigzag nanowires with different bar angles (θ=90◦ , 120◦ and 150◦ ) Two dynamic processes of single DWM and double DWM are found in different regimes of current density identified by two thresholds in all zigzag nanowires The decreasing threshold current is found in the zigzag nanowires with increased bar angles, indicating the angular-dependence of the magnetic pinning This work suggests a possibility of manipulating the single/multiple DWM in future DW devices by introducing the shape anisotropy © 2017 Author(s) All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/) [http://dx.doi.org/10.1063/1.4975129] I INTRODUCTION Due to the potential for the development of logic and memory devices,1,2 the current-induced domain wall motion (CIDWM) in different metallic and semi-conductive nanostructures has become an attractive subject on fundamental research and technological interest The CIDWM along Permalloy (Py) ferromagnetic strips has been extensively studied during the last decade Experiments have demonstrated that the injection of an electrical current through the strip can drive DWs in the direction of the electron flow.3–5 For technological purposes, the most urgent issue is to reduce the threshold current and achieve high DW velocity.2,4–7 The threshold current in experiments so far on metallic wires under dc or slow pulse current is mostly of 1011 –1012 Am-2 8–10 Smaller value of 1010 Am-2 is reported when a pulse of order of ns is applied.11 Furthermore, the accurate and stable manipulation on DW motion is also intensely concerned by industrial application Miron et al.12 have reported the multiple DW mobility in which two DWs propagate together along Pt/Co/AlOx wire with structural inversion asymmetry (SIA) However, the pinning sites of two DWs are random and the manipulation on one of the DWs pinned in the wire is impossible for the reason of weak pinning Torrejon et al.13 have reported the single/multiple CIDWM under the effect of thermal gradient in curved Py nanowire and the same structure is also used to investigate the effects on two DWs propagation in Py strips with uniform and non-uniform cross section reported by Raposo.14 However, the stable manipulation of DWM, especially single/multiple DW motions cannot be accomplished simply by thermal gradient along the wire or with sole strip structure Numerous works15,16,22 have indicated that the shape of nanowires plays an unneglectable role in the DW pinning/depinning process due to shape anisotropy Therefore, shape anisotropy is suggested for a new way to improve the operation on DW dynamics In this paper, we present zigzag-shaped Py nanowires with two DWs trapped in two bends respectively and investigate the dependence a Electronic mail: xiaotur@gmail.com Electronic mail: yazhai@seu.edu.cn b 2158-3226/2017/7(5)/056014/6 7, 056014-1 © Author(s) 2017 056014-2 Zhou et al AIP Advances 7, 056014 (2017) of CIDWM along zigzag nanowires on the shape anisotropy by the method of micromagnetic simulation II THEORETICAL METHOD Micromagnetic simulations are performed for strip and zigzag nanowires with the graphicsprocessing-unit-based program Mumax3.17 The time-dependent spin dynamics is described by the Landau–Lifshitz–Gilbert–Slonczewski equation with adiabatic and non-adiabatic terms: ∂~ m ∂~ m ∂~ m ∂~ m ~ eff + α~ = −γ~ m×H m× −u + βu~ m× , ∂t ∂t ∂x ∂x (1) ~ eff represents the effective magnetic field ~ is the unit vector of local magnetization, H in which m including the external field, the anisotropy field, the demagnetization field, and the exchange field γ is the gyromagnetic ratio, α is the Gilbert damping, which is set to 0.02,11,18,19 β is the non-adiabatic factor which is selected as 0.1, u is the velocity of spin-polarized electrons which is defined as:18 u = JPgµB /2eMS (2) where J is the current density, P is the spin polarization, g is the Lande factor, µB is the Bohr magneton, e is the electron charge, and MS is the saturation magnetization For β , 0, numerical calculation on one dimensional model shows the relation between DW velocity ν and the velocity of spin current u should have the form below:11,18 ν = βu/α (3) Combined Equations (2) and (3), the DW velocity is proportional to the current density J By using the program of Mumax3 based on the Equation (1), the center-symmetric zigzag-shaped nanowires with different angles between adjacent bars are investigated Figure 1(a) shows the detailed structure of the zigzag nanowire in which bar angle (θ) is designed as 90◦ , 120◦ and 150◦ to construct different transversal shape anisotropy In addition, the material parameters are those typical of Permalloy (M S =8.60×105 Am-1 , the exchange constant Aex =1.3×10-11 Jm-1 ) and the mesh size is set to ~ init =3000 4×4×4 nm3 in the simulation A remanence state after applying a transversal magnetic field H Oe along +y direction is considered as the initial configuration of magnetic moments where three magnetic domains are divided by two DWs at the bends, i.e 1st DW with head-to-head wall type (HtH) and 2nd DW with tail-to-tail wall type (TtT), trapped in the left and right bend respectively as shown in Figure 1(b) Spin polarized pulse current with polarization 0.411,14 is set to flow along the wire where the current distribution in wire is obtained by solving the electric field distribution in program COMSOL Multiphysics The positive current is defined when the current flows along the wire from the left to the right extreme FIG (a) Schematic illustration of the top view of zigzag nanowires with same dimensions: 1400 nm in length, 140 nm in width and nm in thickness for each bar, and the bar angle θ between adjacent bars is designed as 90◦ , 120◦ and 150◦ respectively (b) The initial configuration of magnetic moments for 90◦ , 120◦ and 150◦ zigzag nanowires in the simulations 056014-3 Zhou et al AIP Advances 7, 056014 (2017) III RESULTS AND DISCUSSION Micromagnetic simulations are performed for the DWM as a function of time via various positive current densities It has been found that two dynamic processes of DWM occur in different regimes of current density Figure illustrates these two dynamic processes of DWM including the results of symmetric simulations in 150◦ zigzag nanowire For current below the first threshold current density , 7.0×1011 Am-2 for 150◦ wire, no DWMs are observed However, when the positive current is JTh and J , i.e 7.0×1011 and 8.0×1011 Am-2 for 150◦ wire, single DWM is observed set between JTh Th The 2nd DW begins to be depinned from the right bend, transforming from transversal wall (TW) to vortex wall (VW) with clockwise spin configuration, and propagates along the bar (electron flowing direction) And at the same time, the 1st DW shifts a little away from the bend center also companied by the transformation to VW with clockwise spin configuration, and yet is still pinned in the left bend Finally, these two DWs meet and annihilate together near the left bend, leading to a ground state in which all local magnetic moments are aligned along the bar and deflect to +x direction , 8.0×1011 Am-2 However, for the current increased above the second threshold current density JTh ◦ st nd for 150 wire, the double DWM occurs, in which the and DWs are depinned together, also companied with the transformation to VW with clockwise spin configuration, and propagate along the wire all the way, leading to the similar ground state mentioned in former process Furthermore, if we change the current direction, same thresholds and symmetric dynamics are still confirmed illustrated in Figure 2(b) However, in the process of single DWM driven by the negative current, it is the 1st DW that is depinned and companied with the transformation to VW with anticlockwise spin configuration, while the 2nd DW is pinned in the right bend Since the symmetric simulations indicate that the dynamic processes of HtH and TtT DWs are the same from the view of system energy (see in Figure 3) We assume that the intrinsic mechanism for different depinning thresholds may originates from two possibilities below rather than the difference of system energy between HtH and TtT configurations One is the asymmetry of structure in the two sides of each bend for zigzag wires Himeno20 has reported that the asymmetric structure works as an asymmetric pinning potential against the DW propagation Another is the propagation direction of the vortex wall (VW) with different polarity p=+1 (anticlockwise) or -1 (clockwise) driven by the current According to He,23 the velocity of VW has a y component νy of which the direction is only dependent on and J is applied in a the spin polarity of VW Therefore, when the positive current between JTh Th zigzag wire, the polarity of the VW trapped in each bend is p=-1 (see in Figure 2(a)), leading to an upward νy and resulting in the depining of the 2nd DW due to the 1st DW being blocked by the wire boundary Two dynamic processes are also found similarly in both 90◦ and 120◦ zigzag nanowires and the detailed parameters are obtained as shown in Table I The fact that the threshold current densities FIG The snapshots of two dynamic processes of DWM in 150◦ zigzag nanowire (a) The single DWM as the positive and J (up) and the double DWM as the positive current above J (down) (b) The single DWM as the current between JTh Th Th and J (up) and the double DWM as the amplitude of negative current above J amplitude of negative current between JTh Th Th (down) 056014-4 Zhou et al AIP Advances 7, 056014 (2017) FIG The energy density of system as the function of pulse time for different amplitudes of positive and negative current densities in 150◦ zigzag nanowires Black and red curves are representative for single DWM driven by the current with amplitude of 7.0×1011 and 7.4×1011 Am-2 respectively, while green and blue curves are representative for double DWM driven by the current with amplitude of 8.0×1011 and 9.0×1011 Am-2 respectively and J decrease with increasing bar angle indicates the angular-dependence of the depinning JTh Th process, i.e the pinning force in the bends is weakened when the bar angle increases We propose that the decreasing shape anisotropy is supposed to take responsibility for the weakening of pinning and J is found decreases with increasing bar angle, force In addition, the difference between JTh Th which is also consequent on the reduced pinning force For better understanding of the decrease of threshold current, the pinning potential is taken into consideration In general, each bend is a potential well for DW and the well depth is determined by the pinning force Higher pinning force results in deeper potential well Therefore, when the bar angle increases, the spin configuration of DW pinned in each bend gets less stable and as a consequence, smaller driven current is required to trigger the DWM There is one thing should be stressed here The dynamic process and the threshold current for double DWM are in good agreement with that demonstrated in experiments.2023 For example, Klăaui20 has reported a threshold current of 2.0×1012 Am-2 in 120◦ zigzag Permalloy wires, while it is 1.9×1012 Am-2 for 120◦ zigzag wires in our simulation However, the single DWM in zigzag wires is less studied in experiments and our work may have an enlightened guidance on the study of single/multiple DWM in shaped nanowires Depinning process is found strongly dependent on the structure of the bends in zigzag nanowires Figure 4(a) shows typical curves of DW displacement as a function of pulse time for single DWM of different zigzag wires in corresponding current density Each curve has three parts marked with solid circles, open circles and solid triangles, representing depinning, propagating and annihilating process respectively Note that the depinning process takes a large proportion in the whole dynamic TABLE I The detailed numerical results for zigzag wires with different bar angles Threshold (×1011 Am-2 ) Bar angle θ(◦ ) JTh JTh 90 22.5 26.0 J v 22.5a 284.0 24.0 314.2 25.2 330.0 26.0b 340.8 27.0 350.1 28.0 357.3 120 17.0 19.0 J v 17.0a 215.9 18.1 237.5 18.4 246.8 19.0b 255.3 19.6 256.8 20.0 260.0 150 7.0 8.0 J v 7.0a 89.5 7.2 89.9 7.6 90.6 8.0b 104.3 9.0 118.0 10.0 130.2 a threshold current for single DWM JTh current for double DWM JTh b threshold DW Velocity ν (ms-1 ) under Pulse Current J (×1011 Am-2 ) 056014-5 Zhou et al AIP Advances 7, 056014 (2017) FIG (a) DW displacement versus pulse time of zigzag nanowires with different bar angles: blue curve and symbols represent the displacement in 150◦ wire with J=7.4×1011 Am-2 , red for the displacement in 120◦ wire with J=1.8×1012 Am-2 and green 90◦ for the displacement in 90◦ wire with J=2.5×1012 Am-2 (b) The snapshots for spin configuration of the right bend in 120◦ wire and 150◦ wire just before the DW being depinned from the bend marked with black cross in (a) (c) The wire, ratio of tdep /ttotal as a function of current density for zigzag wires with bar angle of 90◦ , 120◦ and 150◦ plotted with blue, red and green solid rectangles respectively (d) The DW velocity (ν) versus current density (J) in nanostrip and zigzag wires with different bar angles process of DWM for all zigzag wires and another interesting finding is that the minimum displacement for depinning increases with the decreasing bar angle which is diagrammed in Figure 4(b) It seems that the pinning force in the bend with smaller bar angle strongly “drags” the DW, blocking it from depinning Further research on depinning shows that the ratio between depinning time t dep and total time t total of the whole dynamic process depends on not only the wire structure but also the current density injected into the wires which is shown in Figure 4(c) In addition, the DW velocity as a function of current density for 90◦ , 120◦ and 150◦ zigzag wires is also obtained as shown in Figure 4(d) For comparison, the DWM driven by polarized current pulse (P=0.4) in nanostrip with the dimension same as a single bar in zigzag wire is simulated and the results are plotted with black rectangles in Figure 4(d) The solid triangles are the simulation results in zigzag wires and the solid line is the fitting for the linear increasing part of DW velocity in nanostrip We find that after depinning, the DW velocity in bars for different zigzag wires is almost the same with that in nanostrip Furthermore, the linear relation between ν and J should meet the formula of v = JP βgµB /2αeMS derived from Equations (2) and (3) and the factor has P gàB /2eMS the value of 1.40ì10-10 m3 C-1 for the parameters chosen in this work And for the fitting line in Figure 4(d), the obtained slope is 1.31×10-10 m3 C-1 which matches the theoretical factor mentioned above very well, suggesting that the bend with high pinning force has a direct impact on the depinning process, but not the propagation of DW on the bars IV CONCLUSIONS In this work, the CIDWM in zigzag nanowires with different bar angles is investigated It is suggested that the shape anisotropy plays an important role in the DW mobility for it enhances the pinning force in the bends Two dynamic processes of single DWM and double DWM identified by two threshold current densities are found in each wires, which offers a possibility of manipulation 056014-6 Zhou et al AIP Advances 7, 056014 (2017) on single/multiple DWM in future DW devices There are two interesting findings related to the weakened pinning force in the bends The first one is that it results in the decrease of threshold current The second one is that it decreases the proportion of depinning process in both space and time scale in the dynamic process of DWM 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bars for different zigzag