Domain Walls in Bent Nanowires

The influence of geometric parameters on the magnetic fine structure of domain walls in bent nanowires is investigated. The domain pattern in the soft-magnetic ${\mathrm{Co}}_{39}{\mathrm{Fe}}_{54}{\mathrm{Si}}_7$ alloy is studied via scanning electron microscopy with polarization analysis and modeled via micromagnetic simulations. It is demonstrated that the bending angle affects details of the microstructure as well as the preponderant domain-wall type. A phenomenological model is developed that provides the global energy minimum of individual types of domain walls as a function of the geometric parameters of the wire. The results can be directly transferred to permalloy wires, as permalloy and ${\mathrm{Co}}_{39}{\mathrm{Fe}}_{54}{\mathrm{Si}}_7$ alloy have a comparable magnetostatic exchange length.

Figure 1

(a) Sketch of a nanowire and definition of the geometric parameters. In (b)–(g), SEM micrographs of wire arrays with varied geometries are shown. Bright areas indicate magnetic material. Each array consists of six wires, which are carved out of the magnetic film by FIB milling. The area where the film material is removed appears dark. Outside the dark region, the pristine magnetic film can be recognized. In (b)–(d) the width while in (e)–(g) the angle is varied. Each ($\alpha$,$w$) set is realized in three different film thicknesses (10, 20, and 30 nm).

Figure 2

(a) Magnetization behavior of the unstructured film with a thickness of 20 nm for a magnetic field applied along two perpendicular in-plane directions with the highest and lowest remanence. The curves are obtained utilizing the longitudinal magneto-optical Kerr effect. (b) SEMPA image of a 20-nm-thick nanowire and the adjacent magnetic film. The magnetization orientation is color coded and indicated by arrows. In the bright area, which is the area where the magnetic film is removed, no magnetic signal is obtained.

Figure 3

SEMPA micrographs obtained with one polarization-sensitive axis of the polarization analyzer. The polarization-sensitive axis (indicated by $P$) is along the vertical of the SEM micrographs. (a) displays the magnetic structure in the “as-fabricated” state, while the micrograph in (b) gives the domain pattern after applying an external field of $B_{\mathrm{ext}}\approx 60\mathrm{mT}$ along the direction indicated by the white arrow.

Figure 4

(a)–(c) SEMPA images and (d)–(f) micromagnetic simulations of the three predominant wall types found for $100\mathrm{nm}\le w\le 400\mathrm{nm}$, $10\mathrm{nm}\le t\le 30\mathrm{nm}$, and $20°\le \alpha \le 160°$. The topological points are labeled by red circles (edge defects) and blue circles (volume defects), and the associated topological charge is given next to the circles. (a),(d) Transverse wall in a nanowire with $\alpha =100°$, $w=400\mathrm{nm}$, $t=20\mathrm{nm}$; (b),(e) vortex wall in a nanowire with $\alpha =120°$, $w=400\mathrm{nm}$, $t=20\mathrm{nm}$; and (c),(f) asymmetric transverse wall in a nanowire with $\alpha =150°$, $w=600\mathrm{nm}$, $t=10\mathrm{nm}$. The orientation of magnetization is color coded according to the color wheel.

Figure 5

SEMPA images of domain walls observed in nanowires with widths of $w\ge 600\mathrm{nm}$. In this transition region to film (a) cross-tie-like domain walls and (b) zigzaglike domain walls are found, referred to hereinafter as filmlike domain walls. The orientation of magnetization is color coded according to the color wheel.

Figure 6

SEMPA images of domain walls in bent nanowires ($\alpha =150°$) for different wire widths and thicknesses [(a) 10, (b) 20, and (c) 30 nm]. The wire width is given in the images in nanometers next to the dashed line. The orientation of magnetization is color coded according to the color wheel.

Figure 7

Phase diagram displaying the wall structure as a function of the thickness and wire width. The bending angle is fixed to $\alpha =150°$. The calculated points of equal energy of the transverse and vortex wall (red stars) are shown together with the experimental results, whereby triangles represent transverse walls, circles vortex walls, diamonds asymmetric transverse walls, and squares filmlike walls (see Fig. 6). The fit of the numerical data (red line) is based on Eq. (6) [$t(w)\propto 1/w$] and represents the line of equal energy. To guide the eye, we plot similar lines [$t(w)\propto 1/w$] for the experimentally found transition between the transverse and vortex wall, and vortex and filmlike wall, respectively.

Figure 8

SEMPA images of domain walls in bent nanowires ($w=400\mathrm{nm}$) for different bending angles $\alpha$ and thicknesses [(a) 10, (b) 20, and (c) 30 nm]. The bending angles are indicated in the top right corner of the micrographs. The orientation of magnetization is color coded according to the color wheel. The different positions of the vortex core are due to a change of the seeding field orientation.

Figure 9

(a) Histogram of observed wall types as a function of the bending angle for wires with $t=20\mathrm{nm}$ and $w=400\mathrm{nm}$. “Others” contain zigzag, cross-tie, and not assignable wall types. (b) Phase diagram displaying the domain-wall type as a function of the thickness and bending angle for nanowires with a wire width of $w=400\mathrm{nm}$. The experimental results are indicated by triangles for transverse, circles for vortex, and diamonds for asymmetric transverse walls. The simulated points of equal energy of the transverse and vortex wall (red stars) are shown. The red line represents the line of equal energy, which is a fit to the numerical data based on the phenomenological model [Eq. (6)]. The dashed line [also based on Eq. (6)] is a guide to the eye indicating the experimentally found transition between vortex and transverse walls.

Figure 10

Micromagnetic calculation of the curl of the magnetization ${(\nabla \times \mathbf{M})}_z$ for different wall types in straight (a)–(c) and bent (d)–(f) nanowires. (a),(d) Transverse wall; (b),(e) vortex wall; (c),(f) asymmetric transverse wall. The arrows indicate the magnetization orientation, while the dashed lines highlight the additional Néel wall in the bent wires for the vortex and asymmetric transverse walls.

Figure 11

Sketch and corresponding energy contributions of the important differences between the transverse wall compared to the vortex wall (upper panel) and vice versa (three lower panels). For explanations, see the text.

Figure 12

Numerically calculated (oommf) total energy differences between the transverse wall and vortex wall in dependence of the bending angle for a nanowire of width $w=250\mathrm{nm}$ and thickness $t=20\mathrm{nm}$. The solid line represents the plot utilizing Eq. (5) and the obtained fit parameters.

Figure 13

Phase diagrams. (a) Phase diagram displaying the global energy minimum of wall type as a function of the geometric parameters $t$, $w$, and $\alpha$. The diagram is created utilizing Eq. (6) and the obtained fit parameters (see the text). (b) For selected angles $\alpha$, the line of equal energy is shown in the $t(w)$ diagram. For the sake of comparison with calculations for permalloy, the results for the straight permalloy wire taken from Ref. [12] are added.