Dynamics of a Domain Wall and Spin-Wave Excitations Driven by a Mesoscopic Current

The dynamics of a domain wall driven by a spin-polarized current in a mesoscopic system is studied numerically. Spin mixing in the states of the conduction electrons is fully taken into account. When the Fermi energy of the electrons is larger than the exchange energy ($E_{\mathrm{F}}>J_{\mathrm{sd}}$), the spin precession induces spin-wave excitations in the local spins which contribute towards the displacement of the domain wall. The resulting average velocity is found to be much smaller than the one obtained in the adiabatic limit. For $E_{\mathrm{F}}<J_{\mathrm{sd}}$, the results are consistent with the adiabatic approximation except for the region below the critical current where a residual domain wall velocity is found.

Figure 1

Top: the initial $s\mathrm{\text{−}}d$ exchange field of the conduction electrons, $⟨\mathbf{s}⟩$, calculated by the transfer matrix method. Bottom: the local spin configuration for $t=0$ (dashed dotted lines) and after time evolution ($t=100$). Parameters: $H_{\mathrm{ex}}=3.0$, $s\mathrm{\text{−}}d$ exchange $H_{\mathrm{sd}}=0.1$, anisotropy field $H_{\mathrm{K}}=0.1$; $E_{\mathrm{F}}=0.11$, $J_{\mathrm{sd}}=0.2$ (dotted lines); $E_{\mathrm{F}}=0.21$, $J_{\mathrm{sd}}=0.2$, $\alpha =0.03$ (full lines).

Figure 2

The position of the center of the domain wall ($X_{\mathrm{DW}}$) as a function of time for the $E_{\mathrm{F}}<J_{\mathrm{sd}}$ (dotted line) and the $E_{\mathrm{F}}>J_{\mathrm{sd}}$ (full line). Parameters: $s\mathrm{\text{−}}d$ exchange $H_{\mathrm{sd}}=0.04$, anisotropy field $H_{\mathrm{K}}=0.1$, $\alpha =0.03$.

Figure 3

The domain wall velocity ($V$) as a function of the $s\mathrm{\text{−}}d$ exchange constant ($H_{\mathrm{sd}}$) that is related to the amplitude of the current. The initial velocity is plotted for $E_{\mathrm{F}}<J_{\mathrm{sd}}$ ($E_{\mathrm{F}}=0.11$, $J_{\mathrm{sd}}=0.2$, bullets and dotted line) while the average velocity is shown for $E_{\mathrm{F}}>J_{\mathrm{sd}}$ ($E_{\mathrm{F}}=0.21$, $J_{\mathrm{sd}}=0.2$, circles). Error bars: square root of the variance of the velocity. Solid line: the analytical result $\propto \sqrt{H_{\mathrm{sd}}^2-H_{\mathrm{cr}}^2}$, where $H_{\mathrm{cr}}$ is the critical $s\mathrm{\text{−}}d$ exchange constant that is estimated as $H_{\mathrm{cr}}\sim 0.014$.