Spatial trends of noncollinear exchange coupling mediated by itinerant carriers with different Fermi surfaces

We study the exchange coupling mediated by itinerant carriers with spin-orbit interaction by both analytic and numeric approaches. The mediated exchange coupling is noncollinear and its spatial trends depend on the Fermi-surface topology of the itinerant carriers. Taking Rashba interaction as an example, the exchange coupling is similar to the conventional Ruderman-Kittel-Kasuya-Yosida type in weak coupling. On the other hand, in the strong coupling, the spiral interaction dominates. In addition, inclusion of finite spin relaxation always makes the noncollinear spiral exchange interaction dominant. Potential applications of our findings are explained and discussed.

Figure 1

(Color online) Noncollinear exchange coupling mediated by itinerant carriers with Rashba interaction and finite spin relaxation. The hard magnet on the left is pined while the direction of the soft magnet on the right is determined by the effective exchange coupling.

Figure 2

(Color online) The noncollinear angle ${\varphi }_x$ in the weak Reshba regime with $k_R/k_F=0.02⪡1$, $\eta \to 0$, and $L_F$ is defined as $\frac{\pi }{k_F}$. The green dotted line is represented the upward tendency resulted from the Rashba effect. Notice that the lumps at discontinuous points are come from numerical error.

Figure 3

(Color online) The noncollinear angle ${\varphi }_x$ in the dilute density regime with $k_R/k_F=3.5$ and $\eta \to 0$. The period of the spiral angle is $L_s^{+}=\pi /\left(k_R+k_F\right)$.

Figure 4

(Color online) The noncollinear angle ${\varphi }_x$ in the critical regime with $k_R/k_F=1$ and $\eta \to 0$. The spiral length at the critical point is $\pi /2k_R=L_s/2$, only half of that in the strong-coupling regime.

Figure 5

(Color online) Suppression of RKKY oscillations in the presence of spin relaxation. The RKKY phase with $k_R/k_F=0.02$ and $k_{\eta }/k_F=1$.

Figure 6

(Color online) Suppression of RKKY oscillations in the presence of spin relaxation with closer to the phase transition with $k_R/k_F=0.2$ and $k_{\eta }/k_F=0.5$.

Figure 7

(Color online) Contours in the complex plane. The black points denote the Matsubara frequencies $z=i\frac{\left(2n+1\right)\pi }{\beta }$ for fermions.