**3. Result and Discussion**

We suppose that the left and right leads are made from the same material and in the wide band limit, that which is of interest in the present investigation, the ferromagnetism of the leads can be accounted for by the polarization-dependent couplings Γ*L*1 = Γ*R*1 = (1 + *p*)<sup>Γ</sup>, Γ*L*2 = Γ*R*2 = (1 − *p*)<sup>Γ</sup> for the parallel (P) alignment, while Γ*L*1 = Γ*R*2 = (1 + *p*)<sup>Γ</sup>, Γ*L*2 = Γ*R*1 = (1 − *p*)<sup>Γ</sup> for the anti-parallel (AP) alignment. Here, Γ describes the tunneling coupling between the QD and the nonmagnetic leads, which is taken as the energy unit in the following calculations. In addition, *p* (0 ≤ *p* < 1) denotes the polarization strength of the leads. The Kondo temperature in the case of *p* = 0, given by *TK* = *U*√*D* exp(−*<sup>π</sup>*/*D*)/2*<sup>π</sup>* with *D* = −2*U*Γ/ *<sup>d</sup>*(*<sup>U</sup>* +  *<sup>d</sup>*), will be set as another dynamical energy scale of the nonlinear conductance.

In the following, we deal with the three-terminal QD system having a fixed finite Coulomb interaction *U* = 10 at zero temperature and consider the effects of changing the bare dot level  *d*, the spin polarization *p*, and the proximity strength Γ*<sup>s</sup>*, respectively.

### *3.1. Linear Local and Cross Conductances*

Firstly, we show the calculated linear conductances in Figure 2, including the local conductance *GL* and the nonlocal cross conductance *GC* as functions of the bare energy level  *d* of the QD at different superconducting coupling strengths, Γ*s* = 0, 0.2, 0.5, 1.0, 1.5, and 2.0, in the case of no spin-polarization *p* = 0. Without the superconducting coupling Γ*s* = 0, *GL* = *GC* and the linear conductance reaches

the unitary limit, *G*0 (*G*0 ≡ <sup>2</sup>*e*2/*h*), as expected in the Kondo regime. With increasing the coupling Γ*<sup>s</sup>*, the local conductance *GL* raises at the beginning, as seen in Figure 2a, since the AR channel starts to emerge and contribute to the electronic tunneling. A slightly bigger value of conductance, *GL* 1.1 *G*0, than the unitary limit of conductance of single-particle tunneling is reached at the coupling Γ*s* = 0.5 in the Kondo regime. On the other hand, it is known that the resonant AR leads to the unitary limit of conductance, 2 *G*0, of the Cooper pair tunneling in the two-terminal hybrid system, e.g., a normal metal-QD-superconductor system [24]. We can therefore deduce that such a larger value of the conductance is a signature indicating that the tunneling event in the present hybrid system is a mixture of the single-particle and Cooper pair tunnelings. Increasing the coupling Γ*s* further will, however, cause a decrease in the local conductance *GL*. The suppression of *GL* can be interpreted as follows: An electron coming from the left lead has much higher probability to form the Cooper pair breaking into the superconducting electrode due to the considerable strength of the coupling Γ*s* > 0.5, and as a result, the ET process is rapidly suppressed. Different from the local conductance, the nonlocal conductance *GC* decreases from the beginning and even becomes negative if the proximity-coupling is sufficiently strong. The negative cross conductance means that when the left lead is applied with a voltage which is bigger than the right lead, electrons will, instead of entering into the right lead from the QD, tunnel into the QD out of the right lead. Moreover, we find that when the QD leaves the Kondo regime, the cross conductance becomes positive again.

**Figure 2.** (Color online) (**a**) The local conductance and (**b**) the cross conductance vs. the bare dot level  *d* at zero temperature for different proximity-coupling strengths Γ*s* in the case of normal leads, i.e., *p* = 0.

Such effects of Γ*s* are clearly manifested in Figure 3, in which the local and nonlocal conductances, and their three respective parts, *<sup>G</sup>ET*, *<sup>G</sup>DAR*, and *<sup>G</sup>CAR*, are illustrated as functions of the coupling Γ*s* for the specific system which has bare dot level,  *d* = − *U*/2 = −5. It is observed that a maximum value of the local conductance, *GL* = 1.125 *G*0, is arrived at, Γ*s* = 0.58. After this point of Γ*<sup>s</sup>*, the AR process becomes the predominate tunneling mechanism over the ET process. When the proximity-coupling is equal to the tunnel-coupling, i.e., Γ*s* = 1.0, a new resonance is reached, originating from interplay between the Kondo effect and AR. Consequently, *GDAR* = *G*0/2 and *GCAR* = *GET* = *G*0/4, and the

local conductance arrive at the unitary value, *GL* = *G*0 once more. At the same time, the nonlocal conductance completely vanishes, *GC* = 0, which indicates no current response in the right lead to the bias voltage applied to the left lead.

**Figure 3.** (Color online) The zero temperature local conductance (black-solid line) and the cross conductance (black-dotted line) vs. the proximity coupling Γ*s* for the system with a bare dot level at the particle-hole symmetric point,  *d* = −*U*/2 = −5 in the case of normal leads (*p* = 0). The three parts of the conductance are also plotted for illustration purposes.

Secondly, in Figure 4, we investigate the cross conductance *GC* as a function of the bare energy level  *d* of the QD at different proximity couplings Γ*s* in the AP configuration with a large spin polarization *p* = 0.5. In the AP configuration, similar with the case of zero spin polarization *p* = 0, electrons with up-spin and down-spin are equally available in the whole system, favoring the formation of the Kondo-correlated state within a wide dot level range centered at  *d* = −*U*/2 = −5. Meanwhile, since there is no splitting of the renormalized dot levels,  *d* + *<sup>λ</sup>*2*σ*, for different spins, the usual tunneling and charging peaks, around  *d* = 0 and <sup>−</sup>*U*, respectively, are relatively narrow. The local conductance *GL* vs.  *d* curves show a similar behavior as the case of zero spin polarization even in the presence of superconducting coupling Γ*<sup>s</sup>*. Furthermore, since no spin-flip scattering exists in the tunneling processes, in the AP configuration, the majority-spin (e.g., up-spin) states in the left lead increase but the available up-spin (minority-spin) states in the right lead decrease with increasing spin polarization strength, and as a consequence, the transfer of the majority-spin (up-spin) electrons through the QD is suppressed, such that the local conductance goes down and eventually vanishes at *p* = 1 as expected. On the contrary, the available down-spin states in the right lead increase in the AP configuration, which just facilitates the occurrence of the CAR process [30]. Therefore, one can observe that *GC* becomes negative in almost the whole region of dot levels, from the mixed-valence regime to the empty orbital regime, even when Γ*s* < 1, and nearly arrives at a considerably bigger negative value, *GC* −*G*0/5, at the Kondo regime at *p* = 0.5. It is interesting to consider the extreme case of *p* = 1. As mentioned above, in the AP configuration electrons with up-spin and down-spin are identical to each other, preferring the formation of the Kondo-correlated state for all values of *p*. However, since the up-spin states are almost unavailable in the right lead in the case of large polarization, the ET process for the left lead to the right lead is completely damaged (implying an exactly vanishing conductance in the usual QD system), but the CAR process survives here as a unique tunneling mechanism, exclusively making a contribution to electronic tunneling. It is anticipated that in this case, *GET* = *GDAR* = 0 and *GL* = −*GC* = *GCAR* = *G*0/2 (this is the unitary limit of conductance of the single channel).

**Figure 4.** (Colour online) (**a**) The local conductance and (**b**) the cross conductance versus the bare dot level  *d*for different proximity-coupling strengths Γ*s* in the AP configuration with *p* = 0.5.

The situation is quite different in the case of the P configuration, as demonstrated in Figure 5, in which the two conductances are plotted as functions of bare dot level with spin polarization *p* = 0.5. In the P configuration, finite spin polarization splits the dot level for up- and down-spins and thus broadens the usual resonance peaks around  *d* = 0 and  *d* = −*U* [32–36]. On one hand, since minority-spin electrons are still available in the two electrodes to build the Kondo screening correlation to a certain degree, the central Kondo peak can still be reached at the unitary limit *G*0 at the large polarization *p* = 0.5 in the case of Γ*s* = 0. On the other hand, the number of minority-spin electrons is too small to construct the Kondo-correlated state at *p* = 0.5, and thus Kondo-induced conductance enhancement disappears rapidly when the QD moves away from the particle-hole symmetric point  *d* = −*U*/2. These two factors cause the appearance of kinks or splitting peaks in both conductance vs.  *d* curves. Besides, it can be observed from Figure 5a that the central Kondo peak in the local conductance is progressively splitting with increasing proximity coupling Γ*s* ≥ 1.0 in this P configuration. Furthermore, a decrease in minority-spin states in both leads in the P configuration hinders the emergence of AR processes, which leads to weakly negative cross conductance in the Kondo regime, e.g., *GC* ≥ −0.1*G*0, and even causes CAR to totally vanish, thus *GC GL* at the two usual resonance peaks, as shown in Figure 5b. This states that strong ferromagnetism destroys proximitized superconductivity in this three-terminal hybrid nanosystem.

**Figure 5.** (Color online) (**a**) The local conductance and (**b**) the cross conductance vs. the bare dot level  *d* with *U* = 10 at zero temperature for different proximity-coupling strengths Γ*s* in the P configuration with *p* = 0.5.

### *3.2. Nonlinear Local and Cross Conductances*

Now, we turn to the investigation of nonlinear tunneling, since the nonlinear differential conductance *dIL*/*dV* is believed to be a very useful tool in experiments aimed at detecting the formation of the Kondo-correlated state due to its proportionality to the transmission spectrum, supposing that the total transmission is unchanged subject to the external bias voltage. In the present three-terminal hybrid device, one can define the local and cross differential conductances, *gL* = *∂IL*/*∂V* and *gC* = *∂IR*/*∂V*, if the bias voltage *V* is applied to the left lead and while the superconducting and the right leads are kept grounded. From the Equations (26)–(29), we can obtain that the two diffenertial conductances are both proportional to the normal transmission spectrum *TN*(*ω*) and the AR spectrum *TA*(*ω*) at *ω* = *V* at zero temperature, *gL* ∝ *TN*(*V*) + *aTA*(*V*) and *gC* ∝ *TN*(*V*) − *bTA*(*V*) (*a* and *b* are constants).

Figure 6 shows the local and cross differential conductances as functions of bias voltage at various proximity couplings Γ*s* for the system with a single dot level  *d* = −5 (*TK* 0.03) at the Kondo regime. These curves for weak proximity coupling Γ*s* < 1.0 present a single zero-bias anomaly, which is the signature of the Kondo effect. Nevertheless, there appears non-zero-bias peak with increasing proximity coupling Γ*s* ≥ 1.0. It is announced that the Kondo correlation enhances not only the normal ET, but also the AR; nonetheless. the increasing superconducting proximity coupling induces splitting of the Kondo peaks in the normal transmission spectrum as well as the AR spectrum. This peak splitting is the reason that the three parts of the linear conductance are all suppressed when Γ*s* > 1.0, as shown in Figure 3. Finally, one can observe that the negative cross differential conductance becomes positive in the case of large bias voltage. External bias voltage plays a role in dissipation so as to destroy not only the Kondo correlation but the negative nonlocal current response as well.

**Figure 6.** (Color online) The zero-temperature local (**a**) and cross (**b**) differential conductances vs. bias voltage *V* for various couplings Γ*s* for the system with bare dot level  *d* = −5 and *U* = 10 in the case of normal leads (*p* = 0).
