Current-driven inertial domain wall dynamics in ferromagnet

We investigate the inertial domain wall (DW) dynamics driven by spin-polarized current in ferromagnets. The exact solutions reveal an upper limit for DW velocity, given by $V\le 1/\sqrt{\alpha \tau }$. This indicates that damping and inertia become the key factors in achieving higher DW speeds. For the case of uniaxial anisotropy, we analyze the effects of inertia and current on DW dynamics. Due to inertia, the DW velocity, width, rotation frequency, and wave number are mutually coupled. When the DW width varies slightly, the velocity decreases rapidly while the magnetization precession frequency increases sharply with the inertia term. However, once the rotation frequency exceeds its maximum value, both the DW velocity and rotation frequency gradually decline. Regarding current-driven dynamics, we identify a critical current j_1c that directly triggers the Walker breakdown. For currents below this threshold j₁ < j_1c, the absolute DW velocity increases with current, whereas it decreases for j₁ > j_1c. During this process, the DW velocity rapidly peaks under current drive, accompanied by the magnetization rotation frequency nearing its maximum and minimal variation in DW width. These results suggest that the DW behaves like a classical rigid body, reaching its maximum velocity as it approaches peak rotational speed. For biaxial anisotropy, we derive analytical solutions. The competition between hard-axis anisotropy and inertia causes the DW magnetization to lose its spiral structure and rotational symmetry. The inertia effect leads to a slow initial decrease followed by a rapid increase in DW width, whereas current modulation gradually widens the DW. The analytical solution also reveals another critical current, $j_{1\max }=\sqrt{\alpha /\tau }/\beta$, which scales with the square root of the inertia-to-damping ratio and is inversely proportional to the nonadiabatic spin-transfer torque parameter β.