- Email: [email protected]
Field-controllable injection of virtual magnetic domain wall in discrete magnetic nanodot chains
Accepted Manuscript Field-controllable injection of virtual magnetic domain wall in discrete magnetic nanodot chains Xiao-Ping Ma, Seon-Dae Kim, Hong-Guang Piao, Dong-Hyun Kim PII:
S1567-1739(17)30294-8
DOI:
10.1016/j.cap.2017.10.013
Reference:
CAP 4609
To appear in:
Current Applied Physics
Received Date: 31 August 2017 Revised Date:
16 October 2017
Accepted Date: 19 October 2017
Please cite this article as: X.-P. Ma, S.-D. Kim, H.-G. Piao, D.-H. Kim, Field-controllable injection of virtual magnetic domain wall in discrete magnetic nanodot chains, Current Applied Physics (2017), doi: 10.1016/j.cap.2017.10.013. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
highlighted
ACCEPTED MANUSCRIPT
Field-controllable injection of virtual magnetic domain wall in discrete magnetic nanodot chains Xiao-Ping Ma1,2 , Seon-Dae Kim1 , Hong-Guang Piao1,2a , Dong-Hyun Kim1b 1
Department of Physics, Chungbuk National University,
2
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Cheongju 28644, Chungbuk, R. Korea College of Science, China Three Gorges University, Yichang 443002, P. R. China
Abstract
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Periodic injection behaviors of virtual magnetic domain wall (VDW) have been systematically investigated in asymmetrically shaped nanodot chains by means of micromagnetic simulations.
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Systematic investigation on a controllable VDW injection has been carried out. We demonstrate that precise control of VDW injection is achievable by using different nanodot shapes as well as by changing alternating magnetic field (AC field) profiles. The VDW position can be tuned by adjusting AC field frequency and amplitude. Field-controllable periodic VDW injection phenomenon is found to be sustainable over wide ranges of phase diagram spanned by AC field frequency and
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amplitude.
Key words: magnetic nanodot chain, micromagnetic simulation, domain wall injection, virtual
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domain wall
a
Corresponding author E-mail: [email protected]
b
Corresponding author E-mail: [email protected]
1
ACCEPTED MANUSCRIPT The dynamics of domain wall (DW) in ferromagnetic nanostructure is of great technological interest due to numerous possible applications in spintronics[1–3] and magnetic memory devices[4]. In order to study DW physics, one needs to initially create the DWs in a reproducible way, i.e., DW should be injected stably. There have been several approaches for DW injection, for example, by using magnetic field combined with DW nucleation pad[5–7],
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varying the nanostrip shapes[8, 9], irradiating ion beam[10, 11], or flowing currents[12, 13]. Among these approaches, a nucleation pad attached to one end of the device is widely employed as DW injector in both in-plane[5, 7] and perpendicular magnetic anisotropy systems[14, 15]. By attaching a large nucleation pad to the nano-structured device, the sta-
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tistical chance to have an imperfection where a DW could nucleate at a relatively low field increases. Thus, injection from the nucleation pad naturally depends on the quality of the fabrication process and does not always guarantee reliable and reproducible DW injections.
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Several works regarding the stochastic and complex depinning dynamics of DW have been reported[16, 17], where the origin of the stochastic and complex behavior is considered to be the significant fluctuation in injecting DWs.
On the other hand, it has been reported that small gaps in disconnected magnetic elements are able to annihilate and then re-nucleate entire DWs[18], in the same way as a
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notch-shaped defect site allows a DW to be pinned and depinned. DW-like configuration is contained in the gap of disconnected magnetic elements, where we can define a virtual domain wall (VDW) with no actual inner spin structure[18–21]. Meanwhile, it has been demonstrated that stochastic behavior of DW depinning can be dramatically suppressed
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by using VDWs with exhibiting ’single-mode depinning distributions’[21]. Even though the DW depinning can be suppressed in VDWs, it is expected that the VDW injection behavior
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also depends on the fabrication process, inevitably leading to imperfection such as roughness and defect. For instance, there have been reports on non-uniform switching field distribution in nanodots, originated from microstructural imperfection[22, 23]. Very recently, it has been reported that the thermal fluctuation and non-uniform switching behavior of nanodot system are significantly reduced in case of AC driving field[24]. Thus, it is expected that the VDW propagation with no real inner spin structure could result in much more stable injection, particularly in case of high-frequency AC field. In order to achieve timely well-defined periodic injection of VDW for future applications, VDW propagation under AC field should be investigated in details. Recently, VDW motion 2
ACCEPTED MANUSCRIPT under constant DC field is reported[19, 25], where the high average velocity with suppression of the conventional Walker breakdown behavior was explored. However, unfortunately, no study has been addressed to how to precisely and controllably inject the VDW in disconnected magnetic elements under AC magnetic field, which is essential in realization of devices operating on high frequencies. In this work, the stable field-controllable periodic
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VDW injection under AC fields is numerically demonstrated by means of micromagnetic simulations[26, 27].
Public micromagnetic simulation tools have been adopted[26, 27], where simulation results are found to be consistent for both cases. 20 asymmetric nanodots with length of
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200 nm, thickness of 5 nm, one fixed diameter of 100 nm, and the other unfixed diameter of d (changed from 0 to 75 nm) were used, as detailed dimensions of nanodot chains are illustrated in Fig. 1. The gap distance between nanodots was set to be 6 nm in order to
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ensure sufficient dipolar interaction among nanodots. With the same gap distance, a circleshaped pad, which facilitates the VDW injection, with diameter of 100 nm and thickness of 5 nm, was put on the right end of the nanodot chain, as shown in the figure. With the circle-shaped injection pad acting as a VDW injector, the VDW will be firstly generated in the gap between the pad and the first neighboring nanodot. Material parameters of
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permalloy were considered with the exchange stiffness coefficient of A = 13 × 10−12 J/m, the saturation magnetization of Ms = 8.6 × 105 A/m. No magnetocrystalline anisotropy was considered[21, 28]. In all simulations, the unit cell dimension of 2 × 2 × 5 nm3 was used, considering the exchange length of the Permalloy, which is known to be much greater than
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5 nm[29]. Gilbert damping constant α = 0.01 was used. Firstly, a series of simulations with constant magnetic field HDC were performed in or-
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der to investigate the asymmetric leftward/rightward depinning fields, thereby to confirm the unidirectional propagation behavior in the present asymmetric nanodot chain system. Without attaching the injecting pad, a head-to-head VDW has been initially positioned at the center of nanodot chain with a full relaxation of energy. Then, external magnetic fields HDC , starting from 16 kA/m with a step of 0.8 kA/m were applied rightward (+x direction). The VDW created at the center does not move even for the field strength of 20.8 kA/m. At HDC = 21.6 kA/m, a new VDW is created from the right end while the VDW at the center does not move yet. The newly created VDW is merging with the original one at the center, leading to a saturation of the whole system, as seen in Fig. 2(a). In this case, it is concluded 3
ACCEPTED MANUSCRIPT that the depinning field for VDW pinned at the center is higher than the nucleation field from the edge. When the external magnetic field is applied leftward along −x direction, the VDW is depinned at −18.4 kA/m and the magnetization reversal of the entire nanodot chain is completed by the sweeping motion of the VDW to the left end. The difference of the leftward/rightward depining field is expected from the asymmetrical shape of nanodots.
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Normalized magnetization mx (Mx /Ms ) with variation of HDC is plotted in Fig. 2(b), where it is expected that a unidirectional VDW motion is possible in the external field strength ranges of 18.4 ∼ 21.6 kA/m.
Now, the injection pad is attached to the right end of the nanodot chains to utilize
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a leftward unidirectional VDW propagation, demonstrated as in Fig. 3. AC driving field HAC with an amplitude greater than 18.4 kA/m was applied to trigger the VDW motion. Fig. 3(a) - (d) show the snapshots of VDW injection and propagation under the AC field
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of HAC = 23 kA/m with variation of frequencies (f ) from 40 to 800 M Hz. With field frequency f = 40 M Hz, as shown in Fig. 3(a), during the negative phase of AC field, a head-to-head (h2h) VDW is nucleated (t = 15.2 ns) and propagates from the right end of the chain to the left end. The whole nanodot chain system is saturated along −x direction at t = 23.0 ns. A snapshot of the h2h VDW moving to the left in the saturating process
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is shown for t = 19.2 ns. Then, during the subsequent positive phase, a tail-to-tail (t2t) VDW is created (t = 27.7 ns) and propagates to the left end of the chain until the whole chain system is saturated (t = 35.4 ns), now with the opposite magnetization. A snapshot of the t2t VDW moving to the left in the saturating process is shown for t = 31.7 ns. Thus,
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during a single cycle of AC field two different types of VDW (h2h and t2t) are injected into the nanodot chain and eventually propagate until saturating the whole system.
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With increasing f to 50 M Hz as in Fig. 3(b), the overall injection behavior is similar as in the case of f = 40 M Hz. A h2h VDW is first created at t = 12.5 ns and propagates to the left. But, interestingly, it is observed that VDW cannot propagate to the end of the nanodot chain, pinned at a certain position, as in the case of t = 18.6 to 26.0 ns, where the VDW only moved for the negative phase of driving field. While the first h2h VDW is pinned, a new t2t VDW is created as in the case of t = 22.3 ns. The t2t VDW moves to the left until it collides with the first h2h VDW, letting the whole system saturated by annihilating both VDWs as in the case of t = 27.7 ns. As f further increases to 700 M Hz, it is observed that the VDW is still injected, as seen in the case of t = 1.9 ns. After the 4
ACCEPTED MANUSCRIPT injection, the h2h VDW moves to the left by a single dot (t = 2.1 ns) and then, the second VDW is injected (t = 2.5 ns) while the first VDW is pinned. We can roughly estimate that a characteristic frequency of VDW nucleation is around 370 M Hz by calculating the nucleation time T = t2 − t1 , where t1 is the moment when the AC magnetic field changes its sign and t2 is the moment when VDW is created to the gap
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between the pad and the neighboring nanodot. Basically, this corresponds to the reversal time of the circular injection pad on the right, which might be tunable by modifying the pad geometry. In the present geometry of the pad, the ultimate frequency limit is approximated to be about two times of the VDW nucleation characteristic frequency ( 740M Hz). Below
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this value, during the half cycle of AC field, one VDW is stably injected into the nanodot chains. We have simulated, as shown in Fig. 3(d), with f up to 800 M Hz. It is observed that the VDW is still injected, but the reproducibility of the injection phenomena is substantially
configuration, as seen in the figure.
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worse. Moreover, there exist multiple VDWs with complicated and uncontrollable spin
The mx , corresponding to the position of the VDW, is plotted with respect to the time together with the driving AC field profiles in Fig. 3(e) and (f), corresponding to the case of Fig. 3(a) and (b), respectively. Stable periodic injection of VDWs is achievable with proper
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combinations of AC field amplitude and frequency. If the amplitude is too low, VDW can not be injected. If too high, the nanodot chains cannot be reversed one by one but reversed with multiple VDW nucleations. VDW injection is found to be stable for wide ranges of frequencies, once the frequency is less than the frequency limit (∼ 740M Hz). It is observed
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that a stable VDW is created periodically in these frequencies, as seen in the figure. The dotted region indicates that the h2h VDW (mx > 0) and t2t VDW (mx < 0) are created
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and subsequently injected into the nanodot chains. Note that the whole creation-injection procedures are extremely reproducible. We have examined a phase digram of VDW injection behaviors with variation of d from 25 to 75 nm, as illustrated in Fig. 4. In case that d = 0, the dot becomes sharper in the left, where the reversal of each dot requires a stronger field due to the shape anisotropy. The AC field with higher amplitudes could lead to multiple creation of DWs throughout the nanodot chain as previously discussed, resulting in a complex and unstable VDW creation and injection behavior. On the other hand, when d = 100 nm, asymmetry of the dot disappears and injection and propagation of VDWs along a preferential direction is not 5
ACCEPTED MANUSCRIPT realized. Thus, we focus on the three cases of d values as in the figure. The phase diagrams spanned by AC field frequency (f ) and amplitude (HAC ) exhibits four distinctive regions according to different VDW injection behaviors. In case of low-amplitude region, region 1, no VDW injection is observed. In case of substantially high-amplitude region, region 2, multiple VDW nucleations occur over an entire nanodot chain, where magnetization reversal
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is mostly mediated by reversal of all nanodots at the same time rather than by a sequential reversal of each dot. Very stable and reproducible periodic VDW injection is observed in region 3, where two different types of VDW (h2h and t2t) are injected alternately during one cycle of the AC field. At each cycle, the whole chain system is saturated with opposite
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magnetization alternately, corresponding to the case of Fig. 3(a) and 3(d). Compared to the continuous wire, where a propagating real DW position is inevitably fluctuating at each driving cycle[30], the discrete feature of the VDW allows devices based on the VDW to have
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a very well-defined dynamic position of the VDW. Thus, it is expected that high-precision dynamic positioning of the DW, required for applications, can be easily realized for the case of VDW. In region 4, VDW injection is still stable, but the propagation is stopped in the middle of nanodot chains, corresponding to the case of Fig. 3(b), 3(c) and 3(f). With proper combination of AC field frequencies and amplitudes, it has been found that a pinned position
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of VDW can be precisely controlled, which, again, allows us to have a high-precision control of stopping position of VDWs.
It is interesting to note that there exists an optimal condition for d values in achieving the stable phase of VDW injection and propagation, represented by region 3 and 4. As seen
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in the figure, the region 3 and 4 are maximized in case of d = 50 nm. In terms of stable unidirectional motion of VDW, higher degree of nanodot asymmetry is helpful. In case of
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d = 25 nm, degree of asymmetry is larger compared to the cases of d = 50 and 75 nm. On the other hand, the shape anisotropy also matters since higher depinning field is required for the case of larger asymmetry as in the case of d = 25 nm, resulting in complex multiple reversal behaviors over nanodot chains rather than a stable sequential reversal of each dot. Balanced by the two factors, wider regions of stable VDW motions (region 3 and 4) in the phase diagram are observed in the case of d = 50 nm. Rather monotonic increase of the boundary between the region 3 and 4 is due to the increase of VDW velocity with increase of HAC , which is observed for all cases. With increase of d, the left part of nanodots become larger, allowing to have more complex spin configurations during the VDW propagation, 6
ACCEPTED MANUSCRIPT which leads to rather rough boundary between the region 3 and 4 in cases of d = 50 and 75 nm. In conclusion, the VDW injection/propagation under AC magnetic field has been systematically investigated, where the controllability of the periodic injection and propagation has been demonstrated by means of micromagnetic simulations. Stable periodic VDW in-
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jections under various AC field are observed to be achievable in the asymmetrical nanodot chains combined with a circle-shaped pad. The pad is found to function as a reliable VDW injector, providing much improved stability compared to the cases of continuous wires with real domain wall motion. Wide operation region of the VDW injection and propagation is
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explored with systematic variation of frequencies and amplitudes of AC magnetic field. ACKNOWLEDGMENT
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This work was supported by the National Research Foundation of Korea (NRF) (Grant No. 2015R1D1A3A01020686) and the KBSI Grant D37614. This work was also supported by the National Natural Science Foundation of China (Grant No. 11474183) and the National Key R&D Program of China (Grant No.2017YFB0903700).
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ACCEPTED MANUSCRIPT http://math.nist.gov/oommf 2002. [27] A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia-Sanchez, and B. V. Waeyenberge, AIP Adv.4, 107133 (2014). [28] R. M. Bozorth, Rev. Mod. Phys. 25, 42(1953). [29] N. Kikuchi, S. Okamoto, O. Kitakami, Y. Shimada, S. G. Kim, Y. Otani, and K. Fukamichi,
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192512 (2011).
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FIG.1. The geometry and dimension of the asymmetrically shaped nanodot chain with variation of d. A circle-shaped pad is located at the right end. Detailed dimension of the nanodot and pad is on the bottom. Horizontal AC field direction is denoted by the arrow
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on the top.
FIG.2. (a) A sequence of snapshots of VDW motion in case of d = 50 nm under DC
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magnetic field of HDC = 21.6 and −18.4 kA/m. Color code represents the magnetization direction on the xy-plane. The arrows on the top indicate the magnetic field direction. (b)
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Normalized magnetization mx = Mx /Ms profile with respect to HDC .
FIG.3. Snapshots of VDW motion under AC field with HAC = 23 kA/m and (a) f = 40, (b) f = 50, (c) f = 700 M Hz and (d) f = 800 M Hz. Color code represents the magne-
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tization direction on the xy-plane. The magnetic field strength at each moment is stated in the right. The positions of VDWs are denoted by dotted lines with configuration of h2h or t2t. Normalized magnetization mx and driving AC field profile with respect to time for the case of HAC = 23 kA/m and (e) f = 40 M Hz and (f) f = 50 M Hz. The region-
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s where VDW injection occurs are denoted as dotted ellipses with configuration of h2h or t2t.
FIG.4. Phase diagram of VDW injection categorized as region 1, 2, 3 and 4 for the cases of (a) d = 25 nm, (b) d = 50 nm, and (c) d = 75 nm.
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FIG. 1
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FIG. 2
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FIG. 3
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S1567-1739(17)30294-8
DOI:
10.1016/j.cap.2017.10.013
Reference:
CAP 4609
To appear in:
Current Applied Physics
Received Date: 31 August 2017 Revised Date:
16 October 2017
Accepted Date: 19 October 2017
Please cite this article as: X.-P. Ma, S.-D. Kim, H.-G. Piao, D.-H. Kim, Field-controllable injection of virtual magnetic domain wall in discrete magnetic nanodot chains, Current Applied Physics (2017), doi: 10.1016/j.cap.2017.10.013. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
highlighted
ACCEPTED MANUSCRIPT
Field-controllable injection of virtual magnetic domain wall in discrete magnetic nanodot chains Xiao-Ping Ma1,2 , Seon-Dae Kim1 , Hong-Guang Piao1,2a , Dong-Hyun Kim1b 1
Department of Physics, Chungbuk National University,
2
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Cheongju 28644, Chungbuk, R. Korea College of Science, China Three Gorges University, Yichang 443002, P. R. China
Abstract
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Periodic injection behaviors of virtual magnetic domain wall (VDW) have been systematically investigated in asymmetrically shaped nanodot chains by means of micromagnetic simulations.
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Systematic investigation on a controllable VDW injection has been carried out. We demonstrate that precise control of VDW injection is achievable by using different nanodot shapes as well as by changing alternating magnetic field (AC field) profiles. The VDW position can be tuned by adjusting AC field frequency and amplitude. Field-controllable periodic VDW injection phenomenon is found to be sustainable over wide ranges of phase diagram spanned by AC field frequency and
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amplitude.
Key words: magnetic nanodot chain, micromagnetic simulation, domain wall injection, virtual
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domain wall
a
Corresponding author E-mail: [email protected]
b
Corresponding author E-mail: [email protected]
1
ACCEPTED MANUSCRIPT The dynamics of domain wall (DW) in ferromagnetic nanostructure is of great technological interest due to numerous possible applications in spintronics[1–3] and magnetic memory devices[4]. In order to study DW physics, one needs to initially create the DWs in a reproducible way, i.e., DW should be injected stably. There have been several approaches for DW injection, for example, by using magnetic field combined with DW nucleation pad[5–7],
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varying the nanostrip shapes[8, 9], irradiating ion beam[10, 11], or flowing currents[12, 13]. Among these approaches, a nucleation pad attached to one end of the device is widely employed as DW injector in both in-plane[5, 7] and perpendicular magnetic anisotropy systems[14, 15]. By attaching a large nucleation pad to the nano-structured device, the sta-
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tistical chance to have an imperfection where a DW could nucleate at a relatively low field increases. Thus, injection from the nucleation pad naturally depends on the quality of the fabrication process and does not always guarantee reliable and reproducible DW injections.
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Several works regarding the stochastic and complex depinning dynamics of DW have been reported[16, 17], where the origin of the stochastic and complex behavior is considered to be the significant fluctuation in injecting DWs.
On the other hand, it has been reported that small gaps in disconnected magnetic elements are able to annihilate and then re-nucleate entire DWs[18], in the same way as a
TE D
notch-shaped defect site allows a DW to be pinned and depinned. DW-like configuration is contained in the gap of disconnected magnetic elements, where we can define a virtual domain wall (VDW) with no actual inner spin structure[18–21]. Meanwhile, it has been demonstrated that stochastic behavior of DW depinning can be dramatically suppressed
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by using VDWs with exhibiting ’single-mode depinning distributions’[21]. Even though the DW depinning can be suppressed in VDWs, it is expected that the VDW injection behavior
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also depends on the fabrication process, inevitably leading to imperfection such as roughness and defect. For instance, there have been reports on non-uniform switching field distribution in nanodots, originated from microstructural imperfection[22, 23]. Very recently, it has been reported that the thermal fluctuation and non-uniform switching behavior of nanodot system are significantly reduced in case of AC driving field[24]. Thus, it is expected that the VDW propagation with no real inner spin structure could result in much more stable injection, particularly in case of high-frequency AC field. In order to achieve timely well-defined periodic injection of VDW for future applications, VDW propagation under AC field should be investigated in details. Recently, VDW motion 2
ACCEPTED MANUSCRIPT under constant DC field is reported[19, 25], where the high average velocity with suppression of the conventional Walker breakdown behavior was explored. However, unfortunately, no study has been addressed to how to precisely and controllably inject the VDW in disconnected magnetic elements under AC magnetic field, which is essential in realization of devices operating on high frequencies. In this work, the stable field-controllable periodic
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VDW injection under AC fields is numerically demonstrated by means of micromagnetic simulations[26, 27].
Public micromagnetic simulation tools have been adopted[26, 27], where simulation results are found to be consistent for both cases. 20 asymmetric nanodots with length of
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200 nm, thickness of 5 nm, one fixed diameter of 100 nm, and the other unfixed diameter of d (changed from 0 to 75 nm) were used, as detailed dimensions of nanodot chains are illustrated in Fig. 1. The gap distance between nanodots was set to be 6 nm in order to
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ensure sufficient dipolar interaction among nanodots. With the same gap distance, a circleshaped pad, which facilitates the VDW injection, with diameter of 100 nm and thickness of 5 nm, was put on the right end of the nanodot chain, as shown in the figure. With the circle-shaped injection pad acting as a VDW injector, the VDW will be firstly generated in the gap between the pad and the first neighboring nanodot. Material parameters of
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permalloy were considered with the exchange stiffness coefficient of A = 13 × 10−12 J/m, the saturation magnetization of Ms = 8.6 × 105 A/m. No magnetocrystalline anisotropy was considered[21, 28]. In all simulations, the unit cell dimension of 2 × 2 × 5 nm3 was used, considering the exchange length of the Permalloy, which is known to be much greater than
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5 nm[29]. Gilbert damping constant α = 0.01 was used. Firstly, a series of simulations with constant magnetic field HDC were performed in or-
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der to investigate the asymmetric leftward/rightward depinning fields, thereby to confirm the unidirectional propagation behavior in the present asymmetric nanodot chain system. Without attaching the injecting pad, a head-to-head VDW has been initially positioned at the center of nanodot chain with a full relaxation of energy. Then, external magnetic fields HDC , starting from 16 kA/m with a step of 0.8 kA/m were applied rightward (+x direction). The VDW created at the center does not move even for the field strength of 20.8 kA/m. At HDC = 21.6 kA/m, a new VDW is created from the right end while the VDW at the center does not move yet. The newly created VDW is merging with the original one at the center, leading to a saturation of the whole system, as seen in Fig. 2(a). In this case, it is concluded 3
ACCEPTED MANUSCRIPT that the depinning field for VDW pinned at the center is higher than the nucleation field from the edge. When the external magnetic field is applied leftward along −x direction, the VDW is depinned at −18.4 kA/m and the magnetization reversal of the entire nanodot chain is completed by the sweeping motion of the VDW to the left end. The difference of the leftward/rightward depining field is expected from the asymmetrical shape of nanodots.
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Normalized magnetization mx (Mx /Ms ) with variation of HDC is plotted in Fig. 2(b), where it is expected that a unidirectional VDW motion is possible in the external field strength ranges of 18.4 ∼ 21.6 kA/m.
Now, the injection pad is attached to the right end of the nanodot chains to utilize
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a leftward unidirectional VDW propagation, demonstrated as in Fig. 3. AC driving field HAC with an amplitude greater than 18.4 kA/m was applied to trigger the VDW motion. Fig. 3(a) - (d) show the snapshots of VDW injection and propagation under the AC field
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of HAC = 23 kA/m with variation of frequencies (f ) from 40 to 800 M Hz. With field frequency f = 40 M Hz, as shown in Fig. 3(a), during the negative phase of AC field, a head-to-head (h2h) VDW is nucleated (t = 15.2 ns) and propagates from the right end of the chain to the left end. The whole nanodot chain system is saturated along −x direction at t = 23.0 ns. A snapshot of the h2h VDW moving to the left in the saturating process
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is shown for t = 19.2 ns. Then, during the subsequent positive phase, a tail-to-tail (t2t) VDW is created (t = 27.7 ns) and propagates to the left end of the chain until the whole chain system is saturated (t = 35.4 ns), now with the opposite magnetization. A snapshot of the t2t VDW moving to the left in the saturating process is shown for t = 31.7 ns. Thus,
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during a single cycle of AC field two different types of VDW (h2h and t2t) are injected into the nanodot chain and eventually propagate until saturating the whole system.
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With increasing f to 50 M Hz as in Fig. 3(b), the overall injection behavior is similar as in the case of f = 40 M Hz. A h2h VDW is first created at t = 12.5 ns and propagates to the left. But, interestingly, it is observed that VDW cannot propagate to the end of the nanodot chain, pinned at a certain position, as in the case of t = 18.6 to 26.0 ns, where the VDW only moved for the negative phase of driving field. While the first h2h VDW is pinned, a new t2t VDW is created as in the case of t = 22.3 ns. The t2t VDW moves to the left until it collides with the first h2h VDW, letting the whole system saturated by annihilating both VDWs as in the case of t = 27.7 ns. As f further increases to 700 M Hz, it is observed that the VDW is still injected, as seen in the case of t = 1.9 ns. After the 4
ACCEPTED MANUSCRIPT injection, the h2h VDW moves to the left by a single dot (t = 2.1 ns) and then, the second VDW is injected (t = 2.5 ns) while the first VDW is pinned. We can roughly estimate that a characteristic frequency of VDW nucleation is around 370 M Hz by calculating the nucleation time T = t2 − t1 , where t1 is the moment when the AC magnetic field changes its sign and t2 is the moment when VDW is created to the gap
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between the pad and the neighboring nanodot. Basically, this corresponds to the reversal time of the circular injection pad on the right, which might be tunable by modifying the pad geometry. In the present geometry of the pad, the ultimate frequency limit is approximated to be about two times of the VDW nucleation characteristic frequency ( 740M Hz). Below
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this value, during the half cycle of AC field, one VDW is stably injected into the nanodot chains. We have simulated, as shown in Fig. 3(d), with f up to 800 M Hz. It is observed that the VDW is still injected, but the reproducibility of the injection phenomena is substantially
configuration, as seen in the figure.
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worse. Moreover, there exist multiple VDWs with complicated and uncontrollable spin
The mx , corresponding to the position of the VDW, is plotted with respect to the time together with the driving AC field profiles in Fig. 3(e) and (f), corresponding to the case of Fig. 3(a) and (b), respectively. Stable periodic injection of VDWs is achievable with proper
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combinations of AC field amplitude and frequency. If the amplitude is too low, VDW can not be injected. If too high, the nanodot chains cannot be reversed one by one but reversed with multiple VDW nucleations. VDW injection is found to be stable for wide ranges of frequencies, once the frequency is less than the frequency limit (∼ 740M Hz). It is observed
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that a stable VDW is created periodically in these frequencies, as seen in the figure. The dotted region indicates that the h2h VDW (mx > 0) and t2t VDW (mx < 0) are created
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and subsequently injected into the nanodot chains. Note that the whole creation-injection procedures are extremely reproducible. We have examined a phase digram of VDW injection behaviors with variation of d from 25 to 75 nm, as illustrated in Fig. 4. In case that d = 0, the dot becomes sharper in the left, where the reversal of each dot requires a stronger field due to the shape anisotropy. The AC field with higher amplitudes could lead to multiple creation of DWs throughout the nanodot chain as previously discussed, resulting in a complex and unstable VDW creation and injection behavior. On the other hand, when d = 100 nm, asymmetry of the dot disappears and injection and propagation of VDWs along a preferential direction is not 5
ACCEPTED MANUSCRIPT realized. Thus, we focus on the three cases of d values as in the figure. The phase diagrams spanned by AC field frequency (f ) and amplitude (HAC ) exhibits four distinctive regions according to different VDW injection behaviors. In case of low-amplitude region, region 1, no VDW injection is observed. In case of substantially high-amplitude region, region 2, multiple VDW nucleations occur over an entire nanodot chain, where magnetization reversal
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is mostly mediated by reversal of all nanodots at the same time rather than by a sequential reversal of each dot. Very stable and reproducible periodic VDW injection is observed in region 3, where two different types of VDW (h2h and t2t) are injected alternately during one cycle of the AC field. At each cycle, the whole chain system is saturated with opposite
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magnetization alternately, corresponding to the case of Fig. 3(a) and 3(d). Compared to the continuous wire, where a propagating real DW position is inevitably fluctuating at each driving cycle[30], the discrete feature of the VDW allows devices based on the VDW to have
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a very well-defined dynamic position of the VDW. Thus, it is expected that high-precision dynamic positioning of the DW, required for applications, can be easily realized for the case of VDW. In region 4, VDW injection is still stable, but the propagation is stopped in the middle of nanodot chains, corresponding to the case of Fig. 3(b), 3(c) and 3(f). With proper combination of AC field frequencies and amplitudes, it has been found that a pinned position
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of VDW can be precisely controlled, which, again, allows us to have a high-precision control of stopping position of VDWs.
It is interesting to note that there exists an optimal condition for d values in achieving the stable phase of VDW injection and propagation, represented by region 3 and 4. As seen
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in the figure, the region 3 and 4 are maximized in case of d = 50 nm. In terms of stable unidirectional motion of VDW, higher degree of nanodot asymmetry is helpful. In case of
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d = 25 nm, degree of asymmetry is larger compared to the cases of d = 50 and 75 nm. On the other hand, the shape anisotropy also matters since higher depinning field is required for the case of larger asymmetry as in the case of d = 25 nm, resulting in complex multiple reversal behaviors over nanodot chains rather than a stable sequential reversal of each dot. Balanced by the two factors, wider regions of stable VDW motions (region 3 and 4) in the phase diagram are observed in the case of d = 50 nm. Rather monotonic increase of the boundary between the region 3 and 4 is due to the increase of VDW velocity with increase of HAC , which is observed for all cases. With increase of d, the left part of nanodots become larger, allowing to have more complex spin configurations during the VDW propagation, 6
ACCEPTED MANUSCRIPT which leads to rather rough boundary between the region 3 and 4 in cases of d = 50 and 75 nm. In conclusion, the VDW injection/propagation under AC magnetic field has been systematically investigated, where the controllability of the periodic injection and propagation has been demonstrated by means of micromagnetic simulations. Stable periodic VDW in-
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jections under various AC field are observed to be achievable in the asymmetrical nanodot chains combined with a circle-shaped pad. The pad is found to function as a reliable VDW injector, providing much improved stability compared to the cases of continuous wires with real domain wall motion. Wide operation region of the VDW injection and propagation is
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explored with systematic variation of frequencies and amplitudes of AC magnetic field. ACKNOWLEDGMENT
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This work was supported by the National Research Foundation of Korea (NRF) (Grant No. 2015R1D1A3A01020686) and the KBSI Grant D37614. This work was also supported by the National Natural Science Foundation of China (Grant No. 11474183) and the National Key R&D Program of China (Grant No.2017YFB0903700).
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FIG.1. The geometry and dimension of the asymmetrically shaped nanodot chain with variation of d. A circle-shaped pad is located at the right end. Detailed dimension of the nanodot and pad is on the bottom. Horizontal AC field direction is denoted by the arrow
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on the top.
FIG.2. (a) A sequence of snapshots of VDW motion in case of d = 50 nm under DC
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magnetic field of HDC = 21.6 and −18.4 kA/m. Color code represents the magnetization direction on the xy-plane. The arrows on the top indicate the magnetic field direction. (b)
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Normalized magnetization mx = Mx /Ms profile with respect to HDC .
FIG.3. Snapshots of VDW motion under AC field with HAC = 23 kA/m and (a) f = 40, (b) f = 50, (c) f = 700 M Hz and (d) f = 800 M Hz. Color code represents the magne-
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tization direction on the xy-plane. The magnetic field strength at each moment is stated in the right. The positions of VDWs are denoted by dotted lines with configuration of h2h or t2t. Normalized magnetization mx and driving AC field profile with respect to time for the case of HAC = 23 kA/m and (e) f = 40 M Hz and (f) f = 50 M Hz. The region-
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s where VDW injection occurs are denoted as dotted ellipses with configuration of h2h or t2t.
FIG.4. Phase diagram of VDW injection categorized as region 1, 2, 3 and 4 for the cases of (a) d = 25 nm, (b) d = 50 nm, and (c) d = 75 nm.
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