Scaling of intrinsic domain wall magnetoresistance with confinement in electromigrated nanocontacts

In this work we study the evolution of intrinsic domain wall magnetoresistance (DWMR) with domain wall confinement. Notched half-ring nanocontacts are fabricated from Permalloy using a special ultrahigh vacuum electromigration procedure to tailor the size of the wire in situ and through the resulting domain wall confinement, we tailor the domain wall width from a few tens of nm down to a few nm. Through measurements of the dependence of the resistance with respect to the applied field direction, we extract the contribution of a single domain wall to the MR of the device, as a function of the width of the domain wall in the confining potential at the notch. In this size range, an intrinsic positive MR is found which dominates over anisotropic MR, as confirmed by comparison to micromagnetic simulations. Moreover, the MR is found to scale monotonically with the size of the domain wall, ${\delta }_{\text{DW}}$, as $1/{\delta }_{\text{DW}}^b$, with $b=2.31\pm 0.39$. The experimental result is supported by quantum-mechanical transport simulations based on ab initio density functional theory calculations.

Figure 1

False color scanning electron microscope image of a typical structure. The sample consists of a notched Permalloy half-ring of radius $2.5\mu \mathrm{m}$ and 400 nm width with initial constriction size $\sim 70\mathrm{nm}$ (purple). Due to the large undercut of the resist, the sample is electrically isolated from the Permalloy on top of the resist (green), enabling direct in situ measurement. Electrical connection to the structure is made via Cr(5 nm)/Au(55 nm) contact pads (yellow). The inset shows an SEM image of a nanocontact following partial electromigration, revealing the narrowing of the contact in the vicinity of the notch (reproduced with permission from [58]). The schematic below demonstrates the mode-étoile measurement scheme, as described in the text.

Figure 2

(a) Normalized resistance of the contact at remanence as a function of angle, following saturation along the given angle. Four different base resistance levels are shown, corresponding to different resistance states for contacts without a domain wall, equivalent to 436, 496, 602, and $825\mathrm{\Omega }$ from bottom to top. The plots are offset for clarity. Three main levels are seen, depending on the state, as indicated for the $495\mathrm{\Omega }$ resistance sample. $A$ designates a high-resistance state when there is no DW in the structure. $B$ designates a lower resistance state when a DW has been nucleated in the wire, away from the notch region. $C$ designates a variable-sized local peak corresponding to the DW present in the notch. A continuous thermal drift in resistance as a result of heating from the magnet has been subtracted from the presented data in each case, from a fit to the resistance levels in the regions without a DW in the structure. (b) Extra resistance for a DW in the notch compared to a DW in the wire $[\mathrm{\Delta }R=(C-B)/A]$ as a function of the contact resistance $R$. The line is a guide to the eye. The inset shows the resistance of a narrow contact following significant electromigration as a function of the applied field along ${90}^{\circ }$. A peak at zero field is seen, again revealing a positive DWMR for a wall forming in the notch in this regime.

Figure 3

(a) A series of micromagnetic simulations showing the evolution in DW spin structure as a function of contact size. (b) SEMPA imaging of the DW spin structures in notched Py half-rings, revealing either a tilted or symmetric transverse domain wall for the different geometrically defined pinning potentials. The direction of the magnetization is indicated by the color wheel. (c) The evolution in DW width with the geometrical confinement, as extracted from the simulations. It is determined from the change in the $x$-magnetization profile due to the presence of the wall, as plotted in the inset. The dotted black line is a comparison to the previously determined experimental relation between DW width and wire width in Permalloy of $\delta \approx 1.74\times w$ [74, 75]. (d) Evolution in AMR associated with the presence of a DW in the constriction as a function of the constriction size.

Figure 4

Evolution of scaled domain wall resistivity with the calculated domain wall width. The solid line is a fit to Eq. (1). The inset shows an enlarged view of the data for small domain wall widths.

Figure 5

Band structure of ${\mathrm{Ni}}_3\mathrm{Fe}$. The width of the lines corresponds to the $s$ character of the Bloch states. The inset corresponds to the region of the band structure that was used to extract the Fermi velocity $v_{\mathrm{f}}$, the $sd$ coupling $J_{sd}$, the chemical potential $\mu$, and the Fermi wave vector $k_{\mathrm{f}}$.

Figure 6

(a) Calculated DWMR of a single channel of $k=\frac{k_{\text{f}}}{10}$ as a function of ${\delta }_{\mathrm{DW}}$. The maxima of the resistance for this single momentum scales with ${\delta }_{\mathrm{DW}}^{-1.82}$. For each value of the incident particle's momentum we obtain a different exponent, and as we sum over all contributions up to Fermi momentum to get the total resistance the exponent diverges from inverse square as a result of oscillations. (b) DWMR $\left(R_{\mathrm{DW}}\right)$ as a function of domain wall width $\left({\delta }_{\mathrm{DW}}\right)$ depicted on a log-log plot. The DWMR fits to $1/{\delta }_{\mathrm{DW}}^{2.53}$ (blue line) but for all DW widths, oscillations in the resistance occur. The red line shows a fit to the maxima of the total DWMR and scales with ${\delta }_{\mathrm{DW}}^{-2.32}$, in excellent agreement with the experimentally measured exponent. Actual data are shown by black dots.