Two-orbital Kondo effect in a quantum dot coupled to ferromagnetic leads

We study the Kondo effect of a two-orbital vertical quantum dot (QD) coupled to two ferromagnetic leads by employing an equation of motion method. When the ferromagnetic leads are coupled with parallel spin polarization, we find three peaks in the single-particle excitation spectra. The middle one is the Kondo resonance caused by the orbital degrees of freedom. In magnetic fields, the Kondo effect vanishes. However, at a certain magnetic field new twofold degenerate states arise and the Kondo effect emerges there. In contrast, when the ferromagnetic leads are coupled with antiparallel spin polarization, the Kondo effect caused by the spin (orbital) degrees of freedom survives (is suppressed) in magnetic fields. We investigate the field dependence of the conductance in the parallel and antiparallel spin polarizations of the leads and find that the conductance changes noticeably in magnetic fields.

Figure 1

Schematic diagram of the magnetic configurations in our QD system. The relative orientations of the magnetic moments in the two ferromagnetic leads are (a) parallel and (b) antiparallel.

Figure 2

Single-particle excitation spectra (a) $\rho (\omega )={\rho }_{\uparrow }(\omega )+{\rho }_{\downarrow }(\omega )$, (b) ${\rho }_{\uparrow }(\omega )$, and (c) ${\rho }_{\downarrow }(\omega )$ for several $p$ values in the P configuration. The parameters are ${\Delta }_{\mathrm{orb}}/{\Gamma }_0=0$, $T/{\Gamma }_0=0.005$, and ${\varepsilon }_0/{\Gamma }_0=-2$. Inset: Expanded scale of $\rho (\omega )$.

Figure 3

(a) Energy splittings between up- and down-spin QD levels as functions of $p$ for the P configuration. $\Delta {\mathrm{\varepsilon ̃}}_{\mathrm{eom}}$ and $\Delta {\mathrm{\varepsilon ̃}}_{\mathrm{scal}}$ are obtained by the EOM method and the scaling approach, respectively. (b) Energy difference $\delta \omega /{\Gamma }_0$ between the peaks of $\rho (\omega )$ in the positively and negatively high energy region as a function of $p$ for the P configuration. For both (a) and (b), we set ${\Delta }_{\mathrm{orb}}/{\Gamma }_0=0$, $T/{\Gamma }_0=0.005$, and ${\varepsilon }_0/{\Gamma }_0=-2$.

Figure 4

QD electron numbers $\langle n_{1\uparrow }\rangle$, $\langle n_{2\uparrow }\rangle$, $\langle n_{1\downarrow }\rangle$, $\langle n_{2\downarrow }\rangle$, and $\langle n_{\mathrm{tot}}\rangle$ as functions of $p$ in the P configuration. $\langle n_{\mathrm{tot}}\rangle =\sum _{l\sigma }\langle n_{l\sigma }\rangle$. The parameters are ${\Delta }_{\mathrm{orb}}/{\Gamma }_0=0$, $T/{\Gamma }_0=0.005$, and ${\varepsilon }_0/{\Gamma }_0=-2$. Inset: $\langle n_{\mathrm{tot}}\rangle$ as a function of ${\varepsilon }_0/{\Gamma }_0$ with $p=0.2$.

Figure 5

Single-particle excitation spectra $\rho (\omega )$ for the several values of ${\Delta }_{\mathrm{orb}}/{\Gamma }_0$ in the P configuration. The parameters are $p=0.3$, $T/{\Gamma }_0=0.005$, and ${\varepsilon }_0/{\Gamma }_0=-2$.

Figure 6

Renormalized QD energy levels in the P configuration as functions of ${\Delta }_{\mathrm{orb}}/{\Gamma }_0$ for $p=0.3$. The vertical dashed lines are ${\Delta }_{\mathrm{orb}}/{\Gamma }_0=0.04$, $0.115$, and $0.16$, which correspond to the values used in Fig. 5. ${\mathrm{\varepsilon ̃}}_{l\uparrow }/{\Gamma }_0$ and ${\mathrm{\varepsilon ̃}}_{2\downarrow }/{\Gamma }_0$ cross at ${\Delta }_{\mathrm{orb}}/{\Gamma }_0=0.115$.

Figure 7

Single-particle excitation spectra $\rho (\omega )$ for the several values of ${\Delta }_{\mathrm{orb}}/{\Gamma }_0$ in the AP configuration. The parameters are ${\Delta }_{\mathrm{orb}}/{\Gamma }_0=0$, $p=0.3$, $T/{\Gamma }_0=0.005$, and ${\varepsilon }_0/{\Gamma }_0=-2$. Inset: Renormalized QD energy levels in the AP configuration as functions of ${\Delta }_{\mathrm{orb}}/{\Gamma }_0$ for $p=0.3$.

Figure 8

Conductance as functions of ${\Delta }_{\mathrm{orb}}/{\Gamma }_0$ for $p=0.3$ in the P and AP configurations. We set $T/{\Gamma }_0=0.005$. The local maximum appears at ${\Delta }_{\mathrm{orb}}/{\Gamma }_0~0.115$ in the P configuration.