Common universal behavior of magnetic domain walls driven by spin-polarized electrical current and magnetic field

We explore the universal behaviors of a magnetic domain wall driven by the spin-transfer torque of an electrical current, in a ferromagnetic (Ga,Mn)(As,P) thin film with perpendicular magnetic anisotropy. For a current transverse to the domain wall, the dynamics of the thermally activated creep regime and the depinning transition are found to be compatible with a self-consistent universal description of magnetic-field-induced domain-wall dynamics. This common universal behavior, characteristic of the so-called quenched Edwards-Wilkinson universality class, is confirmed by an independent analysis of domain-wall roughness. Complementary investigations reveal the directional properties of interaction between current and domain walls which result in the instability of their transverse orientation.

Figure 1

Time evolution of DW shape. Successive positions of a DW, at $T=28$ K, driven by (a) magnetic field $(H=0.16$ mT, delay between images $\mathrm{\Delta }t=0.5$ s, total duration 60 s) and (b) current $(j\approx 0.5\mathrm{GA}/{\mathrm{m}}^2,\mathrm{\Delta }t=0.5$ s, total duration 16 s) observed at the same sample location. The DW move in the direction opposite to the current density, which is indicated by the arrow. The initial DW position is underlined by a thick dashed line. The triangles indicate the strongest DW pinning positions.

Figure 2

Analysis of domain-wall roughness. Roughness exponents ${\zeta }_j$ and ${\zeta }_H$ as a function of temperature, measured for $j<j_d$ and $H<H_d$, respectively. Inset (a): Definition of the domain-wall displacements $u\left(x\right)$ and $u(x+L)$ used to calculate the displacement-displacement correlation function [see Eq. (1)]. Inset (b): Typical correlation functions $w$ vs DW segment length $L$ in log-log scale for current-driven (diamonds) and field-driven (triangles) DWs $\left(T=14.2\text{K}\right)$. The dashed lines are a fit of $w\sim L^{2\zeta }$ used to determine the roughness exponents ${\zeta }_j$ and ${\zeta }_H$.

Figure 3

STT-induced domain-wall dynamics. DW velocity vs current density $j$ for different temperatures. The solid, dashed, and dotted lines are predictions for the creep, depinning, and flow regimes, respectively, obtained from simultaneous fits of Eqs. (2). The lower and upper stars correspond to the velocities at the depinning threshold $v\left(j_d\right)$ and $v_T\left(j_d\right)$, respectively. A sliding average over five points is used to smooth the curves. The unexpected large velocities measured for $T=54$ K and $j>7.5{\mathrm{GA}/\mathrm{m}}^2$ were not used for the fit. Inset: Semilog plot of the same velocity curves vs $j^{-1/4}$ highlighting the thermally activated creep regime and its fit.

Figure 4

Magnetic-field-induced domain-wall dynamics. DW velocity vs magnetic field $H$ for different temperatures and their fit with Eqs. (2). The solid, dashed, and dotted lines are predictions for the creep $[H<H_d\left(T\right)]$, depinning $[H\gtrsim H_d\left(T\right)]$, and flow $[H\gg H_d\left(T\right)]$ regime, respectively. The lower and upper stars correspond to the velocities $v\left(H_d\right)$ and $v_T\left(H_d\right)$ at the depinning threshold $H_d$, respectively. The slightly nonlinear variations observed for $T=59$ K and $H>33$ mT were not used for the fit of flow regime.

Figure 5

Current-driven domain-wall motion. (a–f) Evolution of an almost rectangular magnetic domain submitted to pulses of current (duration $\mathrm{\Delta }t=6\mu \text{s}$ and amplitude $j=11\mathrm{GA}/{\mathrm{m}}^2)$ for $T=45$ K. The gray levels correspond to opposite directions of magnetization $\vec{M}$ perpendicular to the sample. The back DW conserves its shape and velocity while the front DW becomes pointed, which reduces its velocity and leads to the collapse of the domain (see images e and f). (g, h) Two successive images of domains $(\mathrm{\Delta }t=1\mu \text{s}$ and $j=11.1\mathrm{GA}/{\mathrm{m}}^2)$ showing, for $T=55$ K, the displacement $\mathrm{\Delta }{x}_n$ of a tilted DW (see the dashed and solid segments) along its normal direction $\vec{n}$. $\theta$ is the tilting angle between $\vec{n}$ and $-\vec{j}$. (i) Curve of the velocity $v_n$ along $\vec{n}$ vs $\left|j\right|cos\theta$ obtained for a constant value of $j\left(=11.1\mathrm{GA}/{\mathrm{m}}^2\right)$ for $T=55$ K, and $\mathrm{\Delta }t=1\mu \text{s}$, and comparison with the velocity curves vs current density $\left(\theta =0\right)$ taken from Fig. 3. The error bars reflect the uncertainties of tilting angle and displacement.

Figure 6

Samples and velocity measurements. (a) Optical image of a set of devices showing the rectangular (Ga,Mn)As/(Ga,Mn)(As,P) bilayer of three different sizes, the connecting Ti/Au electrodes (in yellow) used to generate current pulses, and an example of the location where domain-wall motion is observed. (b) Schematic of the stack. The direction of red arrow is the same as in (a). The DW motion is observed through the ${\mathrm{SiO}}_2$ and the thinnest part of the Au/Ti electrode deposited above. The latter is not connected for the presented experiments. (c, d) Differential polar Kerr images showing two successive positions of a DW, which separates regions with opposite magnetization directions perpendicular to the bilayer. The displacement of the DW was produced by $0.45{\mathrm{GA}/\mathrm{m}}^2$ current-density pulse of 10-s duration. (e) Subtraction of images (d) and (c) highlighting the DW displacement $u\left(x\right)$.

Figure 7

Temperature rise produced by Joule heating. (a) Temperature rise vs time for different current densities. (b) Normalized temperature rise for a $1{\mathrm{GA}/\mathrm{m}}^2$ pulse for the device $(228\times 302$ $\mu {\text{m}}^2)$ connected to the current source (solid diamonds) and for a device with the same dimensions placed $\sim 1\text{mm}$ away (empty diamonds). The solid line corresponds to the best fit of the function $\mathrm{\Delta }T(t,1{\mathrm{GA}/\mathrm{m}}^2)={\mathrm{aT}}^{\mathrm{b}}$ for $t<50\text{s}$.

Figure 8

Roughness exponent vs time for field-induced DW motion. (a)–(l) Temporal evolution of the roughness exponent at different temperatures and constant magnetic field amplitude within the creep regime. In every case $\frac{H}{H_d}<1$. The average ${\zeta }_H$ is indicated for each temperature.

Figure 9

Roughness exponent vs time for current-induced DW motion. (a)–(l) Temporal evolution of the roughness exponent at different temperatures and constant current density within the creep regime. In every case $\frac{j}{j_d}<1$. The average ${\zeta }_J$ is indicated for each temperature.