Thermal effects in spin-torque assisted domain wall depinning

We have investigated the magnetization reversal in V-shaped permalloy (Py) nanowires under high dc currents via anisotropic magnetoresistance (AMR) measurements. Utilizing a diamond substrate as heat sink, current densities up to $2\times {10}^{12}{\text{A}/\text{m}}^2$ can be applied. These high current densities enabled us to observe spin-torque assisted switching in thermal equilibrium, without the influence of transient effects. At high currents, the field driven magnetization reversal process is influenced by Joule heating, Oersted fields, and spin-torque effects. These contributions can be identified in our experiment due to their different symmetry properties under magnetization and current reversal. We obtain two important results when evaluating the spin-torque efficiency $\epsilon$, which is proportional to the nonadiabaticity parameter $\beta$. First, we find different values for $\epsilon$ within the same sample, obtained with just a slight variation of the experimental parameters. Second, within the temperature range of 77 to 327 K, $\epsilon$ is found to be constant.

Figure 1

Experimental geometry. (a) SEM micrograph of the V-shaped wire used for the measurements. (b) Sketch of the experimental geometry. The blue arrow indicates the direction of the external magnetic field. The Oersted field (green arrows) is caused by a vertically nonuniform current distribution (green arrows). The corresponding current direction is given in yellow. This direction of the electron flow is taken as negative in the paper. (c) Zoom into the bend region. (d) Domain structure at the kink of the wire measured with SEMPA. Color wheel according to Fig. 3c.

Figure 2

Resistance vs field curve obtained for the wire shown in Fig. 1 ($T_{\mathrm{sample}}=77\text{K})$. The black and gray arrows indicate the directions of the field sweep. Blue and red circles mark jumps in resistance, which correspond to irreversible magnetic switching of the arm on the left-hand side. The difference in resistance at 0 mT between up and down sweep ($R_{\text{DW}}=0.2\Omega$) is caused by the presence of a domain wall at the kink during the up sweep [see Fig. 1d].

Figure 3

Micromagnetic structure (simulated via OOMMF) as function of field and the resulting field dependent AMR. (a) A zoom into the MR curve of Fig. 2 for increasing field. (b) The calculated AMR vs field curve that is estimated based on the simulated micromagnetic structure [panel (c)].

Figure 4

Switching fields as a function of temperature at lowest current density. The irreversible jumps that appear in the AMR loops (Fig. 2) are plotted as a function of temperature. The color coding is as indicated in Fig. 2.

Figure 5

Switching fields as a function of current density. In Fig. 5a the result for the magnetic configuration shown in Figs. 1 and 2 is plotted, which is referred to as configuration “A” throughout the text. (b) Displays the result of the configuration “B” where the magnetization of the arm on the right-hand side is reversed while the same measurement routine is applied. The inset in Fig. 5b displays the temperature increase at the kink due to Joule heating. The gray line is a parabolic fit to the data (black dots).

Figure 6

Switching field as a function of current density for a different field orientation. The field is tilted 7${}^{\circ }$ to the bisecting line of the two arms (i.e., 2° more than in Fig. 5). Configurations “A”/“B” are given by triangles and circles, respectively. The spin-torque signature can be observed for almost all positive current densities. It reduces the critical field. Data points not lying on the dashed linear slope are marked with open symbols.

Figure 7

Height of the resistance jumps (blue/red symbols, left ordinate) and overall AMR signal [$\Delta R_{\text{AMR}}=R\left(0\text{mT}\right)-R\left(54\text{mT}\right)$, green symbols, right ordinate] vs temperature. The data from Fig. 4 (open symbols) are measured with the whole system in thermal equilibrium. The relative change versus temperature is almost the same for all three signals (AMR: green diamonds, annihilation: blue triangles, nucleation: red triangles). For comparison, the same signals (AMR, annihilation, nucleation) are plotted in the case when a current is driven through the wire (taken from Fig. 5, full symbols) which causes Joule heating. The temperature in the latter case is a local temperature at the kink where the domain wall is located before release. The almost constant signal height indicates that the MR signal is generated at a location where the temperature is not strongly changing, i.e., in the arm of the wire on the left-hand side. Lines are meant as guide to the eye.

Figure 8

Height of resistance jumps (annihilation) as function of critical field. The projection of the switching field onto an imginated straight wire perpendicular to the bisecting line of the two arms (in plane) is used as abscissa. For low fields the height of the resistance jumps is constant and comparable to the domain wall resistance (see also Fig. 2). As with increasing field the domain wedge leaves the contact pad the resistance jump increases. These switching fields can be attributed to spin-torque-assisted depinning (data points of configuration A/B in light/dark blue, open/closed symbols as in Fig. 6). Green crosses represent resistance jumps due to the pure field-driven annihilation (blue parabolas in Fig. 6).

Figure 9

The projected critical field versus current density. The results for configuration “A”/“B” are given as triangles/circles, respectively. The filled symbols belong to the data points that fit to the lines given in the plots of critical field vs current density. Open symbols correspond to the data points that cluster at high current. The blue symbols are the results obtained in the field that is stronger tilted. The lines are least square fits to the data points with different geometries and configurations. The dashed lines give the fit to all data points of the same angle. The dark/light blue lines yield spin-torque efficiencies of $\epsilon =(1.07\pm 0.02)\times {10}^{-15}{\text{Tm}}^2{\text{A}}^{-1}\text{and}(0.98\pm 0.02)\times {10}^{-15}{\text{Tm}}^2{\text{A}}^{-1}$, respectively. The light/dark red symbols correspond to the experiment with smaller angle of field orientation and yield $\epsilon =(0.34\pm 0.04)\times {10}^{-15}{\text{Tm}}^2{\text{A}}^{-1}\mathrm{and}(0.45\pm 0.03)\times {10}^{-15}{\text{Tm}}^2{\text{A}}^{-1}$. The averaged values that are obtained for configuration “A”/“B” are $\epsilon =(0.4\pm 0.04)\times {10}^{-15}{\text{Tm}}^2{\text{A}}^{-1}$ for 5° and $\epsilon =(1.03\pm 0.02)\times {10}^{-15}{\text{Tm}}^2{\text{A}}^{-1}$ for 7°, respectively.