Electronic transport in ferromagnetic conductors with inhomogeneous magnetic order parameter and domain-wall resistance

We microscopically derive transport equations for the conduction electrons in ferromagnetic materials with an inhomogeneous magnetization profile. Our quantum kinetic approach includes elastic scattering and anisotropic spin-flip scattering at magnetic impurities. In the diffusive limit, we calculate the resistance through a domain wall and find that the domain-wall resistance can be positive or negative. In the limit of long domain walls we derive analytical expressions and compare them with existing works, which used less general models or different theoretical frameworks.

Figure 1

Contact of length $L$ with a Bloch domain wall of characteristic size $w$ situated in between. The domain wall is described by the magnetization gradient $q\left(x\right)=\partial \theta \left(x\right)$. The lower plot shows $m_y\left(x\right)=-q\left(x\right)$ and $m_z\left(x\right)$ according to Eq. (38) with unity wall width $w=1$.

Figure 2

(Color online) Domain-wall resistance $\delta {\rho }_{\text{DW}}=\frac{1}{3}\frac{E_{\text{DW}}}{E_{\text{F}}}f\left(\tilde{\Delta },\gamma ,\xi ,\kappa \right)$ as a function of exchange splitting $\Delta$ for various values of $\xi \equiv \frac{{\gamma }_n^{\perp }}{{\gamma }_j^{\perp }}$, the ratio of spin-dephasing rate and momentum relaxation rate and denotes the transverse spin-dephasing strength. $E_{\text{DW}}=\frac{{\hslash }^2}{2M}\frac{2}{wL}$ is the domain-wall energy. $\kappa =1$ implies spin-isotropic magnetic scattering of strength ${\chi }_{\parallel }^{\left(m\right)}={\chi }_{\perp }^{\left(m\right)}$ (dotted lines), while $\kappa =2$ means ${\chi }_{\parallel }^{\left(m\right)}=0$ (normal lines) and $\kappa =0$ is the case of ${\chi }_{\perp }^{\left(m\right)}=0$ (dashed lines). $\gamma =P\left(1-\xi \kappa \right)$ is the scattering asymmetry parameter and is not an independent parameter in our model.

Figure 3

(Color online) Additional magnetization-gradient-induced flow of longitudinal spin-current inside the domain wall where $E_q\left(x\right)=\frac{q^2\left(x\right)}{2M}$ does not vanish, so that $j_{\text{s}}^{\left(2\right)}\left(x\right)=eE{D}_{\perp }{\nu }_0\frac{1}{3}\frac{E_q\left(x\right)}{E_{\text{F}}}g\left(\tilde{\Delta },\gamma ,\xi ,\kappa \right)$. Note that the divergence for $\tilde{\Delta }\to 0$ poses no problem, as discussed above. The magnetic scattering parameters are identical to Fig. 2.