Kondo Effect in the Presence of Itinerant-Electron Ferromagnetism Studied with the Numerical Renormalization Group Method

The Kondo effect in quantum dots (QDs)—artificial magnetic impurities—attached to ferromagnetic leads is studied with the numerical renormalization group method. It is shown that the QD level is spin split due to the presence of ferromagnetic electrodes, leading to a suppression of the Kondo effect. We find that the Kondo effect can be restored by compensating this splitting with a magnetic field. Although the resulting Kondo resonance then has an unusual spin asymmetry with a reduced Kondo temperature, the ground state is still a locally screened state, describable by Fermi liquid theory and a generalized Friedel sum rule, and transport at zero temperature is spin independent.

Figure 1

Spin-dependent occupation of the dot level at (a) $B=0$ and (b) $B=-0.1\Gamma$, as a function of spin polarization $P$. (a) For $B=0$, the condition $n_{\uparrow }=n_{\downarrow }$ holds only at $P=0$. (b) For finite $P$, it can be satisfied if a finite, fine-tuned magnetic field, $B_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}}(P)$, is applied, whose dependence on $P$ is shown in (c). As expected, it is approximately linear [12, 27]. Here $U=0.12D$, ${ϵ}_{\mathrm{d}}=-U/3$, $\Gamma =U/6$, and $T=0$.

Figure 2

(a) QD spectral function $A(\omega )=\sum _{\sigma }{A}_{\sigma }(\omega )$ for several values of spin polarization $P$; inset: expanded scale of $A(\omega )$ around ${ϵ}_{\mathrm{F}}$. (b) The spin-resolved spectral function for fixed $P$. For $P\to -P$, we have $A_{\sigma }\to A_{-\sigma }$. (c) Compensation of the spin splitting by fine-tuning an external magnetic field. Parameters $U$, ${ϵ}_{\mathrm{d}}$, $\Gamma$, and $T$ as in Fig. 1.

Figure 3

(a) The spin spectral function $\mathrm{I}\mathrm{m}\{{\chi }_s^z(\omega )\}$. (b) Dependence of $T_{\mathrm{K}}$ on spin polarization $P$. The dotted line shows the prediction from Ref. 12—namely, $T_{\mathrm{K}}(P)/T_{\mathrm{K}}(0)=\exp [C\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}(P)/P]$, where the best fit is obtained for $C=-5.98$. Equation (6) from Ref. 12 with $({\rho }_{\uparrow }+{\rho }_{\downarrow })J_0=(\Gamma /\pi )U/[|{ϵ}_{\mathrm{d}}|(U+{ϵ}_{\mathrm{d}})]$ would lead to $C=-4.19$. (c) QD spectral function for several values of $P$. Parameters $U$, ${ϵ}_{\mathrm{d}}$, $\Gamma$, and $T$ are as in Fig. 1, $B=B_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}}(P)$.

Figure 4

(a) Spin resolved QD spectral function amplitude $A_{\sigma }(\omega =0)$ at the Fermi level, and (b) the QD conductance $G_{\sigma }$, as functions of the spin polarization $P$, for $B=B_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}}(P)$ and symmetric couplings (${\Gamma }_{\mathrm{L}\sigma }={\Gamma }_{\mathrm{R}\sigma }$), with $U$, ${ϵ}_{\mathrm{d}}$, $\Gamma$, and $T$ as in Fig. 1, implying $n_{\sigma }\approx 0.5$. The dashed line in (a) is $1/(1\pm P)$ [Eq. (4) with $n_{\sigma }=0.5$]. As expected, we find $G_{\sigma }=e^2/h$, with a numerical error less than $1\%$.