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 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

Current-induced multiple domain wall motion modulated

by magnetic pinning in zigzag shaped nanowires

Xiaochao Zhou,1Zhaocong Huang,1Wen Zhang,1,2, aYuli Yin,1

Philipp D¨urrenfeld,3Shuai Dong,1and Ya Zhai1,3, b

1Department of Physics, Southeast University, Nanjing 211189, People’s Republic of China

2Department of Physics, National University of Singapore, 2 Science Drive 3,

117542 Singapore, Singapore

3National Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093,

People’s Republic of China

(Presented 3 November 2016; received 23 September 2016; accepted 3 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 dif-ferent 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 sug-gests 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 Experi-ments have demonstrated that the injection of an electrical current through the strip can drive DWs

in the direction of the electron flow.3 5For 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 accu-rate and stable manipulation on DW motion is also intensely concerned by industrial application

Miron et al.12have reported the multiple DW mobility in which two DWs propagate together along Pt/Co/AlOxwire 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

rea-son 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.14However, 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 works15,16,22have 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

b Electronic mail: yazhai@seu.edu.cn

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 graphics-processing-unit-based program Mumax3.17The time-dependent spin dynamics is described by the Landau–Lifshitz–Gilbert–Slonczewski equation with adiabatic and non-adiabatic terms:

∂~m

∂t =−γ~m × ~Heff + α~m × ∂~m∂t −u ∂~m

∂x +βu~m ×

∂~m

in which ~m is the unit vector of local magnetization, ~ H eff represents the effective magnetic field 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

where J is the current density, P is the spin polarization, g is the Lande factor, µ Bis the Bohr magneton,

e is the electron charge, and M Sis 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

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 struc-ture 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

Permal-loy (M S=8.60×105 Am-1, the exchange constant Aex=1.3×10-11Jm-1) and the mesh size is set to 4×4×4 nm3in the simulation A remanence state after applying a transversal magnetic field ~H init=3000

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 1stDW 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 Figure1(b) Spin polarized pulse current with polarization 0.411,14is 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 1 (a) Schematic illustration of the top view of zigzag nanowires with same dimensions: 1400 nm in length, 140 nm

in width and 8 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.

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 Figure2illustrates these two dynamic processes of DWM including the results of symmetric simulations in 150◦zigzag nanowire For current below the first threshold current density

J1

Th, 7.0×1011Am-2 for 150◦ wire, no DWMs are observed However, when the positive current is

set between J Th1 and J Th2 , i.e 7.0×1011and 8.0×1011Am-2for 150◦wire, single DWM is observed The 2ndDW 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 1stDW 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

However, for the current increased above the second threshold current density J Th2 , 8.0×1011Am-2 for 150◦wire, the double DWM occurs, in which the 1st and 2nd 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 Figure2(b) However, in the process of single DWM driven by the negative current,

it is the 1stDW that is depinned and companied with the transformation to VW with anticlockwise spin configuration, while the 2ndDW 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 Figure3) 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 Himeno20has 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

the spin polarity of VW Therefore, when the positive current between J Th1 and J Th2 is applied in a

zigzag wire, the polarity of the VW trapped in each bend is p=-1 (see in Figure2(a)), leading to an upward νyand resulting in the depining of the 2ndDW due to the 1stDW 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 TableI The fact that the threshold current densities

FIG 2 The snapshots of two dynamic processes of DWM in 150◦zigzag nanowire (a) The single DWM as the positive

current between J Th1 and J Th2 (up) and the double DWM as the positive current above J Th2 (down) (b) The single DWM as the

amplitude of negative current between J Th1 and J Th2 (up) and the double DWM as the amplitude of negative current above J2Th

(down).

056014-4 Zhou et al. AIP Advances 7, 056014 (2017)

FIG 3 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 7.0×10 11 and 7.4×10 11 Am -2 respectively, while green and blue curves are representative for double DWM driven by the current with amplitude of 8.0×1011and 9.0×1011Am-2respectively.

J Th1 and J Th2 decrease with increasing bar angle indicates the angular-dependence of the depinning 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

force In addition, the difference between J Th1 and J Th2 is found decreases with increasing bar angle, 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.20 – 23For example,

Kl¨aui20has reported a threshold current of 2.0×1012 Am-2 in 120◦ zigzag Permalloy wires, while

it is 1.9×1012Am-2for 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 Figure4(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 (×10 11 Am -2 )

Bar angle θ( ◦ ) J1

Th DW Velocity ν (ms -1) under Pulse Current J (×1011 Am -2 )

athreshold current for single DWM J1

Th.

bthreshold current for double DWM J2

FIG 4 (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 for the displacement in 90 ◦wire with J=2.5×1012Am-2 (b) The snapshots for spin configuration of the right bend in 1 ◦

wire, 2 ◦wire and 3 ◦wire just before the DW being depinned from the bend marked with black cross in (a) (c) The

ratio of t dep /t totalas 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 Figure4(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 depand total

time t totalof the whole dynamic process depends on not only the wire structure but also the current density injected into the wires which is shown in Figure4(c) In addition, the DW velocity as a func-tion of current density for 90◦, 120◦and 150◦zigzag wires is also obtained as shown in Figure4(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 Figure4(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 veloc-ity 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 βgµ B /2αeM Sderived from Equations (2) and (3) and the factor has P βgµ B /2αeM S the value of 1.40×10-10m3C-1for the parameters cho-sen in this work And for the fitting line in Figure 4(d), the obtained slope is 1.31×10-10m3C-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 However, the pinning force has no effect on the DW velocity in bars of different zigzag wires

ACKNOWLEDGMENTS

This work was supported by NSFC (Nos 61427812, 51571062, 11504047, 61306121), NSF of Jiangsu Province of China (No BK20141328)

1 D A Allwood, G Xiong, C Faulkner, D Atkinson, D Petit, and R P Cowburn, Science309, 1688 (2005).

2 S S P Parkin, M Hayashi, and L Thomas, Science320, 190 (2008).

3 A Yamaguchi, T Ono, S Nasu, K Miyake, K Mibu, and T Shinjo, Phys Rev Lett.96, 179904 (2006).

4 G S D Beach, C Knutson, C Nistor, M Tsoi, and J L Erskine, Phys Rev Lett.97, 057203 (2006).

5 M Hayashi, L Thomas, C Rettner, R Moriya, Y B Bazaliy, and S S P Parkin, Phys Rev Lett.98, 037204 (2007).

6V Uhl´ıˇr, S Pizzini, N Rougemaille, J Novotn´y, V Cros, E Jim´enez et al.,Phys Rev B.81, 224418 (2010).

7 T A Moore, I M Miron, G Gaudin, G Serret, S Auffret, B Rodmacq, A Schuhl, S Pizzini, J Vogel, and M Bonfim,

Appl Phys Lett.93, 262504 (2008).

8 M Kl¨aui, P.-O Jubert, R Allenspach, A Bischof, J A C Bland, G Faini, U R¨udiger, C A F Vaz, L Vila, and C Vouille,

Phys Rev Lett.95, 026601 (2005).

9 A Yamaguchi, T Ono, S Nasu, K Miyake, K Mibu, and T Shinjo, Phys Rev Lett.92, 077205 (2004).

10 M Hayashi, L Thomas, C Rettner, R Moriya, Y B Bazaliy, and S S P Parkin, Phys Rev Lett.98, 037204 (2007).

11 J H Ai, B F Miao, L Sun, B You, A Hu, and H F Ding, J Appl Phys.110, 093913 (2011).

12I M Miron, T Moore, H Szambolics, L D Buda-Prejbeanu et al.,Nat Mater.10, 419 (2011).

13J Torrejon, G Malinowski, M Pelloux, R Weil, A Thiaville et al.,Phys Rev Lett.109, 106601 (2012).

14 V Raposo, S Moretti, M A Hernandez, and E Martinez, Appl Phys Lett.108, 042405 (2016).

15 Y Akinobu, Y Kuniaki, T Hironobu, K Shinya, and O Teruo, Jpn J Appl Phys.45, 5A (2006).

16 G Tatara and H Kohno, Phys Rev Lett.92, 086601 (2004).

17 A Vansteenkiste, J Leliaert, M Dvornik, M Helsen, F Garcia-Sanchez, and B V Waeyenberge, AIP Adv.4, 107133 (2014).

18 A Thiaville, Y Nakatani, J Miltat, and Y Suzuki, Europhys Lett.69, 990 (2005).

19 G S D Beach, M Tsoi, and J L Erskine, J Magn Magn Mater.320, 1272 (2008).

20 A Himeno, T Okuno, S Kasai, T Ono, S Nasu, K Mibu, and T Shinjo, J Appl Phys.97, 066101 (2005).

21 M Kl¨aui, M Laufenberg, L Heyne, D Backes, and U R¨udiger, Appl Phys Lett.88, 232507(2006).

22 Y Togawa, T Kimura, K Harada, T Matsuda, A Tonomura, Y Otani, and T Akashi, Appl Phys Lett.92, 012505 (2008).

23 J He, Z Li, and S Zhang, Phys Rev B73, 184408 (2006).