Comparison with Heisenberg Simulation

In order to simulate the domain walls in constrictions an extended clas-sical Heisenberg model was employed which can reproduce the changes

in the spin structure at very short length scales as found in geometri-cally confined domain walls [Bru99]. In the conventional micromagnetic approach used in [NTM05], the exchange energy is approximated by (∇ ·~ m)~ ²(m~ local magnetization), which is the first order Taylor expansion of the dot product and only valid for small angles between neighboring cells [KF03, Aha96].

In order to address this problem, C. Schieback from the Universität Konstanz performed simulations of domain walls in constrictions us-ing an atomistic / semi-classical spin model approach. In this model the magnetic moments are located on a cubic lattice with nearest neighbors having ferromagnetic exchange coupling, a dipole-dipole interaction and coupling to an external magnetic field [SKN⁺07]. The exchange energy is calculated as the dot product of magnetic moments. The radius of cur-vature of the magnetic element was kept constant to 1µm as in the ex-periments and the element widthw_ewas varied between 120 and 400 nm with a thickness of 4 nm. The domain wall configuration for constriction widthsw_cin the range 20 - 200 nm were simulated. The notch had a trian-gular shape with an angle of 70^◦ in line with the experimental samples.

The parameters of the Heisenberg simulations were deduced according to [Aha96] from the material parameters of permalloy; damping constant α= 0.02, exchange constantA= 13×10⁻¹²J/mand saturation magnetiza-tionM_s = 800×10³A/m. A cell size of 2 nm and 4 nm was used with no significant difference in the results. In the experiment, domain walls are formed after reducing an external magnetic field from saturation in the y-direction (see Fig. 4.3) to zero. Since there is never a perfect alignment of the field to the constriction in the experiments, the field was tilted by 5.7^◦to the y-axis in the simulation.

In the simulations, two types of transverse domain walls are found within the constriction as shown in Fig. 4.11: symmetric (Fig. 4.11(a)) and asymmetric transverse domain walls (Fig. 4.11(b)). The symmet-ric transverse domain wall is obtained in small constsymmet-rictions, w_c= 20 -80 nm, and exhibits an elliptical shape also observed in [MD97]. For wider constrictions, w_c ≥160 nm, an asymmetric spin structure is

fa-Figure 4.11: Magnetization directions taken from computer simulations of (a) symmet-rical and (b) asymmetsymmet-rical transverse domain walls with thickness 4 nm; constriction widthwc/element widthwe(a) 80 nm/400 nm and (b) 200 nm/400 nm. The color code in the inset of (a) and the arrows indicate the magnetization direction.

vored with the direction of the asymmetry (to the right in Fig 4.11(b)) governed by the initial field angle. For intermediate w_c= 120 nm both types are found depending on w_e. The key energy contributions to the domain walls are the exchange energy, which favors large wall widths, and the stray field energy (shape anisotropy), which favors alignment of the spins parallel to the element edges. The increasing influence of the stray field energy results in smaller w_DW for smaller constrictions. For symmetric walls, w_DW is comparable with the experimental values for 0 <w_c< 150 nm (Fig. 4.10(c+d)). However forw_c> 150 nm,w_DW extracted from the simulation increases to much larger values, since the exact tilt in experiment depends on irregularities such as edge roughness, which are inherently not well known and thus not taken into account into the simulation. In addition, the difficulty in determining the opening angle leads to the observed discrepancy ofw_DW for the highly asymmetric sim-ulated walls in that range of w_c (see Fig. 4.11(b)). The increase in the opening angle means that w_DW according to Eq. 4.3 increases more than linearly with increasing wc. This is obvious in Fig. 4.10(d), but less so in Fig. 4.10(c), since in the experiment the increase in the angle is smaller than in the simulated data as discussed above. But even for the large con-striction widths, the same qualitative trend as in the experimental data is observed.

4.2.6 Conclusion

In this section, to our knowledge for the first time symmetric and asym-metric transverse walls are directly imaged in constrictions down to 30 nm using electron holography. The observed transverse wall types are in agreement with micromagnetic predictions and Heisenberg sim-ulations. In the measurements it was found that, depending on the con-striction widthw_c, the asymmetric walls could be subdivided into tilted and buckled walls, the latter being an intermediate state just before the appearance of a vortex. It was also confirmed that the domain wall width w_DW depends strongly on the constriction widthw_c and decreases with decreasing wc. In agreement with simulations, the wall opening angle decreases with decreasing constriction width. This results in a faster than linear decrease of the wall width withw_cwhich will facilitate the fabrica-tion of very narrow domain walls.

On such small length scales exciting physical effects are expected. In field-induced domain wall motion, the velocity of the domain walls de-pends critically on the spin structure [KJA⁺05]. Furthermore, for very narrow domain wall widths non-adiabatic contributions to the electron transport become significant [ZL04, TNMS05, XZS06]. This can lead to an increase in current-induced domain wall velocity and a decrease of critical current density which is important for applications. Finally, the domain wall magnetoresistance depends critically on the spin structure and therefore the width of the domain walls [LZ97, KRP01]. Direct imag-ing and characterization of domain walls in narrow constrictions carried out in this work will provide essential understanding of experimental re-sults obtained by indirect measurement methods, e.g. magnetoresistance measurements.