Strong current actions on ferrimagnetic domain walls in the creep regime

We observe domain-wall (DW) motion in ferrimagnetic TbFe wires with perpendicular anisotropy under combined field and current in the creep regime. The current action on the DW is double: Joule heating and spin-transfer torque. We propose a genuinely robust analysis of velocity, separating thermal effects and spin-transfer torque, quantifying the latter as an equivalent field in the so-called one-dimensional (1D) model. Its efficiency is much larger than in transition-metal ferromagnets above room temperature. The equivalent field reveals the large polarization-to-magnetization ratio in ferrimagnets despite a vanishing $M_s$. The usual 1D DW model is extended to mimic creep and predicts that, in low net magnetization systems, the internal DW structure precesses with currents above a field-independent threshold, leading to two propagation regimes with different mobilities. This is another example of how the internal DW magnetization is relevant in creep. We could not detect experimentally these two regimes, possibly because of the dispersion of the data.

Figure 1

Magnetic properties of TbFe film in temperature: net magnetization $M_s$ obtained from VSM hysteresis loop amplitude (green circles), MOKE hysteresis loops (red diamonds), and Hall resistivity (red circles) amplitudes in arbitrary units chosen so they superimpose. The continuous lines are the result of mean-field calculations.

Figure 2

(a) DW velocities on a logarithmic scale as a function of magnetic field: linear representation (top) and $H^{-1/4}$ scale (bottom) at different $T_{sp}$ indicated in the figure with $J=0$. Continuous lines are fits using the Eq. (3) signature of creep. (b) Creep parameters $C_H\left(T\right)$ and $C_V\left(T\right)$ ($J=0$). The solid lines are polynomial fits used in the following fitting process.

Figure 3

(a) Illustrative superposition of Kerr images (taken every 1 s) at $T_{sp}=328\mathrm{K}$ for field and current indicated in each numbered picture. Dark (bright) regions correspond to up (down) magnetization. A sequence of magnetic fields was used to nucleate and drive two DWs to the initial configuration. The following images appear with an orange contrast. The difference between $v_{\mathrm{fast}}$ and $v_{\mathrm{slow}}$ is illustrated by the center of the remaining central domain (marked by a triangle in the images). (b) Time line of the experiments and of the corresponding parameters (c) Velocity ($v_{\mathrm{fast}}$ up triangles, $v_{\mathrm{slow}}$ down triangles) versus $H$ at different $J$ at $T_{sp}=328\mathrm{K}$. Continuous lines are fits using the creep law given by Eq. (4).

Figure 4

(a) $T_{\mathrm{pulse}}$ and $T_{\mathrm{cool}}$ as a function of the current density $J$. The light red (blue) line corresponds to the Joule heating amplitude determined experimentally (the set point temperature). (b) $H_{eq}$ versus $J$ (averaged for each value of $\left|J\right|$). The solid line corresponds to a single behavior over the entire current range considering $u\left(T\right)/J\propto P/M_s$ as determined in Fig. 1. The dotted line corresponds to the situation where $P/M_s$ is constant. The current (bottom axis) and temperature (top axis) are related by using the Joule heating amplitude.

Figure 5

(a) Evolution of the velocity of fast and slow DWs ($v_{\mathrm{fast}}$ up triangles, $v_{\mathrm{slow}}$ down triangles) as a function of $t_{pw}$. (b) Instantaneous velocity difference used to establish the heating time; note the change in the timescale (linear vs log).

Figure 6

(a) Evolution of MOKE hysteresis loops without current at increasing temperatures and at fixed set-point temperature with increasing current density. (b) Coercive fields as a function of temperature (bottom scale) and current density (top scale) allowing us to determine the relation between $T$, $T_{sp}$ and $J$.

Figure 7

Creep parameters $C_H$ and $C_V$ as a function of current density (bottom axis) and temperature (top axis) calculated from the $T_{\mathrm{pulse}}$ temperatures extracted from the simultaneous fit of fast and slow DWs velocities. The lines represent $C_H(T_{sp},J=0)$ and $C_V(T_{sp},J=0)$ determined without current for different $T_{sp}$ shown in Fig. 2.