Depinning behavior of the vortex domain wall at the asymmetric triangular notch in permalloy wires

Magnetic-domain-wall-based devices, such as shift register [1], vortex domain wall trajectory [2, 3], magnetic logic devices [4–7], and the time-delay oscillator [8], have attracted considerable attention. Precise control of the wall position, stable wall structure, and adjustment of wall chiralities are critical in such applications. Creating a pinning site is a method to stabilize wall structures and control wall positions. Pinning sites are of two types, namely nongeometric and geometric. Nongeometric pinning sites are formed through a local change of magnetic properties and annealing-induced local diffusion from the nonmagnetic metal into the magnetic wire [9, 10]. Another method to generate a pinning site is through exchange bias [11, 12], which is produced from crossed ferromagnetic wires and antiferromagnetic bars [11]. In geometric pinning sites, single-notch, double-notch, antinotch, protrusion and stepped-wire methods can be used to pin the wall [6, 8, 9, 13–27]. Numerous factors, such as wall chirality, the width of magnetic wires, the thickness of magnetic wires, and the depth of notches, influence the interaction between the notch and the wall [9, 13–15, 18–20, 27] and the wall pinning potential. The competition between exchange interaction and anisotropies (e.g., crystalline anisotropy or shape-anisotropy) affect domain wall's spatial profile [8]. Domain wall length and structure are affected by the crystalline anisotropy and the magnetic exchange interaction [24]. Studies have reported the depinning field (H_D) for wires with a notch [9, 13–19, 25–29]. The walls in wide wires exhibit a lower wall energy per unit cross-sectional area. Furthermore, the H_D values in wide wires are smaller than those in narrow wires [15]. The H_D values for in-plane anisotropic wires were influenced by the potential energy of side wells (E_W) and potential energy of center barriers (E_B) in the pinning potential landscape [9, 14, 15]. The H_D provides the wall energy to overcome the E_W or E_B for artificial pinning sites, such as in the notches. A shallow E_W or small E_B results in a low H_D requirement for wire depinning from the notch [15]. The wall structures of the wires of various widths differs considerably [30]. Thick and wide wires exhibit a multi-vortex wall structure [30, 31]. However, the depinning behaviors of a wall for a wire with different symmetries of triangular notch differs considerably. As displayed in figure 1, the incoming angle (θ) is the angle between the long axis of the wire and the wall injection direction side of an asymmetric triangular notch, and the outgoing angle (ϕ) is the angle between the long axis of the wire and the wall lifting side of the asymmetric triangular notch [19]. In 2014, Brandão et al observed vortex domain wall pinning and depinning in submicron wires with an asymmetric notch with fixed θ values and various ϕ values; they revealed that H_D increased with increasing ϕ values [19]. The depinning behavior is also influenced by the shape-anisotropic energy (E_A) of the outgoing side wire [9]. Here, E_A influences E_B and the pinning potential landscape. However, few studies have discussed the influence of wall depinning behavior and physics on the various ϕ values of a notch. To understand in-plane anisotropic wall dynamics, we focused on the wall depinning behavior in a wire with an asymmetric notch.

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We designed samples of wires with various widths and an asymmetric notch, as displayed in figure 1. A 20 μm in diameter circular pad was connected with a 20 nm-thick straight magnetic wires of widths 1 and 2 μm and length of 57 μm. To eliminate the edge domain and the formation of the wall, the wire was tapered at its ends [16, 19]. The asymmetric triangular notch had a fixed notch depth ratio (notch depth divided by wire width, R_d), fixed incoming angle (θ), and various outgoing angles (ϕ). In this study, θ = 60° and ϕ was varied by 15° from 15° to 60° to determine the relationship between H_D and ϕ. To transfer the designed sample patterns onto Si wafers, we used e-beam lithography and the lift-off process. Next, we deposited a 20 nm-thick permalloy (Py) films onto the layer of pattern transferred poly(methyl methacrylate) Si wafers through DC magneto sputtering at a background pressure of 1 × 10⁻⁷ Torr. A magneto-optical Kerr effect (MOKE) microscope was then used to capture magnetic domain images and determine the depinning field (H_D) values. We used the GPU-accelerated micromagnetic simulation software Mumax³, which is based on the Landau–Lifshitz equation [32]. We eliminated magnetic charges on the left and right ends of magnetic wires to create an infinitely long wire system [30, 31] and set the wire length L ≥ 8 times the wire width. Furthermore, most of our simulations were performed at a zero temperature. We used the energy minimization function to set the wall structure and then performed the field-driven wall depinning simulation in wires of various widths [30–33]. To examine to influences of thermal fluctuation and surface defects, we run the simulation with surface defects at 300 K. The surface defects were set by GrainRoughness function of Mumax³. The GrainRoughness parameters of grain size, minimal height and maximal height are 5 nm, 0 nm, and 5 nm, respectively and the random seed of the GrainRoughness are 2997, 2998 and 3000. Py was used as the simulation material of all wires; the saturation magnetization (M_s), exchange stiffness (A_ex), magnetic crystalline anisotropic energy constant, and damping constant were 860 emu cm⁻³, 1.3 × 10⁻⁶ erg cm⁻¹, 0 erg cm⁻¹, and 0.01, respectively. To swiftly simulate our vortex domain wall depinning behavior, we ran the simulation at cell sizes of 5 nm × 5 nm × 5 nm or 5 nm × 5 nm × 20 nm.

One of the domain and depinning images of the MOKE microscope for a 2 μm-wide wire with an asymmetric triangular notch are displayed in figure 2. The θ and ϕ values of the notch of the wire were 60° and 30°, respectively. Figure 2(a) displays the images of the injection field and H_D of the tail-to-tail-type clockwise (TT-CW) wall, which had values of −5.18 and −19.94 Oe, respectively. The injection field and the H_D of the tail-to-tail-type counterclockwise (TT-CCW) wall were −5.38 and −11.94 Oe, respectively, as displayed in figure 2(b). The head-to-head-type counterclockwise (HH-CCW) wall and the TT-CW wall exhibited the same symmetry of wall configuration and the same interaction between the domain wall and the notch. Therefore, their H_D values were almost the same. Similarly, the H_D values of the TT-CCW and HH-CW walls were identical [19]. This study focused on the TT-type domain walls. Figure 3 depicts H_D as a function of ϕ for the TT-walls for 1- and 2 μm-wide wires. H_D value of TT-walls was negative because the applied field was toward to left. Here, we defined H_D value as positive value. The maximum H_D values of the TT-CW wall in these wires were approximately 17.48 and 19.64 Oe, respectively, at ϕ = 30°. The H_D of the TT-CCW wall was lower than that of the TT-CW wall, as displayed in figures 3(a) and (b). To understand the wall depinning behavior, we simulated the vortex wall propagation for wires with an asymmetric notch. The micromagnetic simulation revealed that the depinning behaviors of the HH-CCW and TT-CW walls in the 1 μm-wide wire were the same, as displayed in figure 3(c). This behavior was consistent with the experimental results. The maximum H_D value of the TT-CW and HH-CCW walls in the simulated H_D–ϕ curves were approximately 74 Oe in the 1 μm-wide wire at ϕ = 35°, as displayed in figure 3(c). The simulated results were consistent with the experimental depinning behaviors and exhibited the same trend for the H_D–ϕ curves. The ϕ value of the maximum H_D is defined as the transition angle (ϕ_T). The quantitative differences in H_D between the experimental and simulation results were caused by the thermal fluctuation and sample defects. To confirm the room-temperature and surface defects influences, we ran simulations of the 1000 nm-wide wire with surface defects at 300 K, as shown in figure 3(d). We chose the ϕ equal to 30°, 35°, and 40° to run the simulations. Blue line of figure 3(d) displayed averaged H_D values that were calculated by the wires with different random seeds of GrainRoughness function. The averaged H_D values with surface defects at 300 K are around 47, 56.2, and 46.7 Oe at ϕ equal to 30°, 35°, and 40°, respectively. Figure 3(d) revealed that room-temperature and surface defects reduced H_D values, but unobviously affected ϕ_T value. We utilized EdgeSmooth function of Mumax³ to observe straircase effect for the notch. The simulated results with EdgeSmooth function are similar to those without EdgeSmooth function. It indicates that the straircase effect of asymmetric notches has a weak influence on our study.

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In order to examine the depinning process, we saved the Mumax³ output of the TT-CW wall parameter in the 1 μm-wide wire before the depinning state at the near H_D and performed the time evolution simulations at the H_D. Figure 4 displays the simulated TT-CW wall depinning process for the wires with notches of ϕ = 30°, 35°, and 40°, respectively, as displayed in figure 3(c). The top of figure 4 displays the state of the wall near H_D values before depinning and the state of the wall with applied fields of −68, −72, and −64 Oe for ϕ values of 30°, 35°, and 40°, respectively. When the wall approached the notch, the applied field compressed the wall at the incoming side of notch, as displayed at the top of figure 4. During the depinning process, when the wall arrived at the notch, a boundary line that was leaning to the right appeared at the right of the wall, as denoted by the black arrow in figure 4(a). Because the wall was compressed, the boundary line of the wall became an arc at the notch sharp to easily pass through the notch and go to the outgoing side of the notch with ϕ = 30° or 35°. A new domain (green color) gradually formed from 0 to 4.6 ns in the outgoing side, as displayed in figures 4(a) and (b), and then the wall was depinned from the notch. The simulated depinning images are displayed in figure 4 at 19.8 ns. However, the time evolutions for the wire with ϕ = 40° in figure 4(c) differed from those for the wires with ϕ = 30° and 35°, as displayed in figures 4(a) and (b). The wall depinning process shown in figure 4(c) revealed that the new domain was generated on the outgoing side of the notch at an applied field of −64 Oe.

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The wall depinning behavior is related to the wall pinning potential energy landscape [9, 15, 28]. As indicated in [9], the wall pinning potential well or barrier is generated by the asymmetric notch. Both the shape of the notch and the wire width influence the wall depinning behavior. Therefore, the depinning behaviors in figure 4 are influenced by the potential energy of the side well (E_W) at the incoming side of the notch and the pinning potential energy barrier (E_B) at the outgoing side of notch. For notches with the same depth and θ, the E_W values are almost the same [9]. Therefore, the influence of E_W on the depinning behavior can be ignored, and the E_B dominates the H_D. The E_B value is composed of the local exchange energy (E_Ex) and E_A. E_Ex denotes the exchange energy between the right side of the wall and the magnetic moments located at the outgoing side region of the notch, and E_A indicates the shape-anisotropic energy of the outgoing side wire. E_Ex is affected by the dot product between right side the wall and magnetic moments located at the outgoing side region. Direction of moments located at outgoing side region is aligned along outgoing side. Therefore, E_Ex increased with ϕ increased. The various ϕ changed both E_A and E_Ex. Thus, E_B can affect the depinning behavior. During the depinning process of the wall, the applied field caused the wall to overcome the notch pinning. The wires with higher ϕ values exhibited larger E_Ex values because of the greater interaction between the right side of the wall and the magnetic moments located at the outgoing side region of the notch. However, E_A became smaller as the ϕ value increased because the local demagnetic energy density increased and a new wall was created from the outgoing side region of the notch, which facilitated depinning. E_A and E_Ex competed for the influence on depinning. When the ϕ was smaller than ϕ_T, E_Ex increased with increases in ϕ, but E_A slowly decreased at the same time. These results increased E_B, which resulted in a higher H_D. E_A was strongly dependent on ϕ. E_A decreased slowly for small angles but decreased rapidly for large angles, with ϕ_T as the crossover point. As the ϕ was further increased, the influence of E_A exceeded that of E_Ex. E_B and H_D were considerably reduced by the small E_A when ϕ was larger than the ϕ_T. To reduce the local demagnetic energy density, a vortex wall core and a new wall were created at the outgoing side of the notch, and the H_D was then reduced, as displayed in figures 3 and 4(c). We ran the simulation of 400- and 500 nm-wide wires with θ = 45° and 60° to examine the influences of θ. Figure 5 revealed that the incoming side of notch affected E_W and H_D values, but both ϕ_T values appeared at ϕ = 35°. It indicates that the incoming side of the notch has less influence on E_A and E_Ex.

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To confirm the effect of E_A, we simulated half wires with a single arm of outgoing side wire. The single-arm wires exhibited a left half wire with a half notch with depth R_d fixed at 50%. Figure 6(a) displays simulations of a single-arm wire. We simulated the single-arm wires without a wall initially, as displayed in figure 6. The simulations revealed a single-arm wire with a reversal field (H_R), which caused the single arm to shift from a negative saturation state to a positive saturation state. The H_R–ϕ curves were plotted for single-arm wires to compare the simulated H_D for whole wires with a notch. Figure 6(b) displays the results of 300-, 400-, and 500 nm-wide single-arm and whole wires. For single-arm wires, the H_R value decreased with ϕ. Furthermore, the H_R values of single-arm wires differed considerably from those of the whole wires. However, the decreasing trends of H_D and H_R were quite similar between ϕ values of 35° and 90° for all simulated wires. The shape of the single-arm wire was the same as the right side of the whole wires but differed on the left side. The reversal of the magnetic saturation state depended on E_A for the single-arm wires. For the single-arm wire, magnetic switching behavior was only affected by outgoing side of notch. The H_R for the single-arm wires only depended on E_A and decreased as ϕ increased. These results displayed a same trend when ϕ larger than the ϕ_T for both single-arm wire and whole wire. It displayed the E_A that was dominant in the depinning behavior for ϕ larger than the ϕ_T. For the whole wire simulation, the wall H_D values increased with ϕ increased when the ϕ smaller than ϕ_T. It indicated that the E_Ex increased with ϕ increased because E_B value was composed of E_Ex and E_A. The H_D values were affected by E_Ex for the range of ϕ to ϕ_T. As displayed in figure 6(b), the competition between E_A and E_Ex is independent on the wire width.

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Moreover, the R_d of the notch was simulated to confirm the influence of E_A and the ϕ. We varied the R_d to adjust the competition between E_A and E_Ex for the 400- and 500 nm-wide wires with asymmetric notches and various R_d values, as displayed in figure 7. In these simulations, the θ of the notch was the same and the ϕ of notch was varied. The ϕ_T value in the H_D–ϕ curves was dependent on the R_d for the wires, as illustrated in figures 7(a) and (b). With the same wire asymmetry, the notch with the larger R_d resulted in a smaller ϕ_T value. For both 400- and 500 nm-wide wires simulations, the ϕ_T values for the wires with R_d between 30% to 85% were approximately 50° to 10°, respectively, which revealed that the ϕ_T value was almost independent on the width of the wires, as illustrated in figure 7(c). Because of anisotropy, the moments on both sides of the notch tend to align the edge of the notch. Deeper notches exhibit higher magnetic moments aligning the sides of the notch. Large R_d values enhanced the energy depth of E_W and increased the H_D value. The Large R_d changed ϕ_T value because depth of notch affected the competition between E_A and E_Ex. The notch with a high R_d extended the length of the outgoing side of the notch, which produced a strong influence on E_A for the HH-CCW wall depinning. A small E_A due to the large R_d leads to the reduced H_D value. The anisotropy of notch with high R_d lead the E_A was easily to dominate the depinning behavior. Moreover, a high R_d resulted in a large local demagnetic energy density at the outgoing side of the notch. Therefore, a new domain was easily generated at the outgoing side of the notch for reducing the total energy.

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We performed MOKE microscopy and micromagnetic simulations on Py wires with asymmetric triangular notches to investigate the factor influencing the wall depinning behavior. The maximum of the H_D–ϕ curves, which is known as the ϕ_T, is well observed for all wires. The experiments revealed that the ϕ_T was independent on wire widths. Micromagnetic simulations revealed that the ϕ_T values in the H_D–ϕ curves of the TT-CW and HH-CCW walls for wires with an asymmetric triangular notch were influenced by E_A and E_Ex. Simulations of wires with asymmetric triangular notches revealed the competition between E_A and E_Ex for influencing the curves. The single-arm and various R_d values simulations also revealed the affection of E_A for depinning behavior. E_Ex dominated H_D in the HH-CCW and TT-CW walls for ϕ < ϕ_T, and E_A dominated H_D for ϕ > ϕ_T. The simulation of the wall depinning for the notch with various R_d values revealed that the ϕ_T value decreased as R_d increased and was almost independent of the wire width.

This work was supported by Ministry of Science and Technology, Taiwan under Grant Nos. MOST-106-2112-M-018-007 and MOST-107-2112-M-018-004.

The data generated and/or analysed during the current study are not publicly available for legal/ethical reasons but are available from the corresponding author on reasonable request.