Quantum Renormalization of the Spin Hall Effect

By quantum Monte Carlo simulation of a realistic multiorbital Anderson impurity model, we study the spin-orbit interaction (SOI) of an Fe impurity in Au host metal. We show, for the first time, that the SOI is strongly renormalized by the quantum spin fluctuation. Based on this mechanism, we can explain why the gigantic spin Hall effect in Au with Fe impurities was observed in recent experiments, while it is not visible in the anomalous Hall effect. In addition, we show that the SOI is strongly renormalized by the Coulomb correlation $U$. Based on this picture, we can explain past discrepancies in the calculated orbital angular momenta for an Fe impurity in an Au host.

Figure 1

(a) Hybridization between the $\xi$ orbitals of an Fe impurity and an Au host. We show the LDA result including five $d$ orbitals. (b) Temperature dependence of the square of the magnetic moment $⟨(M_{\xi }^z{)}^2⟩$, (c) the temperature times susceptibility $T{\chi }_{\xi }$, and (d) the occupation number $⟨n_{\xi }⟩$ of the $\xi$ orbitals of the Fe impurity. (b)–(d) are the QMC results including three $d$ orbitals. See text for details.

Figure 2

Temperature dependence of spin-orbit interactions for an Fe impurity in an Au host. The parameter $\lambda$ in Eq. (2) is taken as a realistic 75 meV [18], and as 40 meV for comparison.

Figure 3

For an Fe impurity in an Au host, we show the spin-orbit interaction $⟨{\ell }^z{\sigma }^z⟩$ as a function of correlation energy $U$ and spin polarization $⟨(M_2^z+M_3^z)/2⟩$. The QMC results of $⟨{\ell }^z{\sigma }^z⟩$ are shown as squares with solid lines. In the phase diagram plane of $U$ to $⟨(M_2^z+M_3^z)/2⟩$, the different colors denote the different values of $⟨{\ell }^z{\sigma }^z⟩$. Here, $T=360\mathrm{K}$ and $\lambda =75\mathrm{meV}$ are fixed.