Current-driven periodic domain wall creation in ferromagnetic nanowires

We predict the electrical generation and injection of domain walls into a ferromagnetic nanowire without the need of an assisting magnetic field. Our analytical and numerical results show that above a critical current $j_c$ domain walls are injected into the nanowire with a period $T\sim {(j-j_c)}^{-1/2}$. Importantly, domain walls can be produced periodically even in a simple exchange ferromagnet with uniaxial anisotropy, without requiring any standard “twisting” interaction such as Dzyaloshinskii-Moriya or dipole-dipole interactions. We show analytically that this process and the period exponents are universal and do not depend on the peculiarities of the microscopic Hamiltonian. Finally we give a specific proposal for an experimental realization.

Figure 1

Magnetization texture for increasing currents. Without current ($j=0$, top) the configuration is half of a planar DW centered at $x=0$; for $j<j_c$ the domain wall is tilted out of the $xz$ plane; for currents larger than $j_c$ (bottom panel) domains move along the nanowire. The black arrow at one end of the wire is fixed. The color code is chosen such that gray arrows lie in the $xz$ plane, while red (blue) arrows have a finite component out of the $xz$ plane in $\pm y$ direction, respectively.

Figure 2

Upper graph (a): The solid black line is a sketch of a potential $P\left(\tilde{x}\right)$ for $j<j_c$. The allowed region for the particle in the potential with total energy zero is shown by the blue interval. The dotted line shows the same function for $j>j_c$. Lower graph (b): Sketch of the function $\tilde{x}\left(t\right)$ or, translated into the language of the magnetic model, of $M_x\left(x\right)$.

Figure 3

Numeric results based on MicroMagnum [18] for the square of the frequency of texture formation as a function of current for different $\alpha$. For these simulations we used [16] ${\lambda }_{\mathrm{bulk}}={10}^4\mathrm{J}/{\mathrm{m}}^3,J_{\mathrm{bulk}}=1.6\times {10}^6\mathrm{J}/\mathrm{m}$, and we have fixed the magnetization at the start of the wire by a local magnetic field. For these values, numerically we obtain for the critical current density $j_c^{\mathrm{num}}/\sigma \approx -3.78\times {10}^{12}\mathrm{A}/{\mathrm{m}}^2$ which agrees within four percent with the analytical result of $j_c\approx -3.92\times {10}^{12}\mathrm{A}/{\mathrm{m}}^2$. The small discrepancy is probably due to the method of how we fixed the spin in the numerics. Inset: Slope of the main figure vs damping constant $\alpha$. At $\alpha =0$ we can compare to the analytical result of $-2.5\times {10}^5\mathrm{A}/\left(\mathrm{s}{\mathrm{m}}^2\right)$ and see that we get the right order.