Kondo effect in a quantum dot coupled to ferromagnetic leads and side-coupled to a nonmagnetic reservoir

Equilibrium transport properties of a single-level quantum dot tunnel coupled to ferromagnetic leads and exchange coupled to a side nonmagnetic reservoir are analyzed theoretically in the Kondo regime. The equilibrium spectral functions and conductance through the dot are calculated using the numerical renormalization-group method. It is shown that in the antiparallel magnetic configuration, the system undergoes a quantum phase transition with increasing exchange coupling $J$, where the conductance drops from its maximum value to zero. In the parallel configuration, on the other hand, the conductance is generally suppressed due to an effective spin splitting of the dot level caused by the presence of ferromagnetic leads, irrespective of the strength of exchange constant. However, for $J$ ranging from $J=0$ up to the corresponding critical value, the Kondo effect and quantum critical behavior can be restored by applying properly tuned compensating magnetic field.

Figure 1

(Color online) The schematic of a quantum dot (QD) tunnel coupled to external ferromagnetic leads and exchange coupled to a nonmagnetic electron reservoir. The spin-dependent coupling to the left (right) lead is described by ${\Gamma }_{\text{L}\sigma }$ $\left({\Gamma }_{\text{R}\sigma }\right)$, while $J$ denotes the exchange coupling constant. The magnetizations of the leads can form either parallel or antiparallel magnetic configuration, as indicated.

Figure 2

(Color online) The spectral function of the $d$-level operator in the (a) antiparallel and [(b) and (c)] parallel magnetic configurations for the symmetric Anderson model and for different values of the exchange coupling $J$. The parameters are: ${\epsilon }_{\text{d}}=-0.05$, $U=0.1$, $\Gamma =0.0077$, $p=0.4$, and $T=0$. The Kondo temperature is defined as a half-width of the spectral function for $J=0$ and $p=0$, $T_K=2.5\times {10}^{-4}$, while $A_0={\sum }_{\sigma }{A}_{\sigma }\left(\omega =0\right)$ for $J=0$ and $p=0$. All the parameters are given in the units of $D\equiv 1$.

Figure 3

The linear conductance as a function of exchange coupling constant $J$ for the antiparallel magnetic configuration. The conductance was determined from the spectral function shown in Fig. 2a. The parameters are the same as in Fig. 2 and $J$ is in units of $D=1$.

Figure 4

(Color online) The dependence of the critical exchange coupling $J_{\text{c}}^{\text{P}}$ (in units of $D=1$) on the spin polarization of the leads $p$ in the case of symmetric Anderson model and parallel magnetic configuration for three different values of the tunnel coupling $\Gamma$, as indicated in the figure. The other parameters are the same as in Fig. 2.

Figure 5

(Color online) The spectral function of the $d$-level operator in the antiparallel magnetic configuration for the asymmetric Anderson model, ${\epsilon }_{\text{d}}=-0.05$, $U=0.2$, and for indicated values of the exchange coupling $J$. The Kondo temperature for assumed parameters (and for $J=0$ and $p=0$) is $T_K=3.4\times {10}^{-5}$. The other parameters are the same as in Fig. 2.

Figure 6

(Color online) The spectral function of the $d$-level operator in the parallel magnetic configuration for the asymmetric Anderson model. The other parameters are the same as in Fig. 5.

Figure 7

(Color online) The spectral function of the $d$-level operator in the parallel magnetic configuration for the asymmetric Anderson model in the presence of external magnetic field $B$ applied along the $z\text{th}$ direction for different values of exchange coupling constant [(a) and (b)] $J=0$, [(c) and (d)] $J=0.17$, and [(e) and (f)] $J=0.17875$. The other parameters are the same as in Fig. 5.

Figure 8

The dependence of the compensating magnetic field $B_{\text{c}}$ on the exchange coupling constant $J$ in the case of the parallel configuration and asymmetric Anderson model. The other parameters are the same as in Fig. 5. $J$ and $B_{\text{c}}$ are in units of $D=1$.