**1. Introduction**

Recently, electron transport through a hybrid nanodevice, for instance, a quantum dot (QD), connected to normal and superconducting electrodes, has attracted much attention in many experimental [1–20] and theoretical studies [21–29] due to the associated physical challenges and potential applications in spintronics and quantum information. When a QD is connected to a superconductor, superconducting order can leak into it to give rise to pairing correlations and an induced superconducting gap, known as the superconducting proximity effect; this privileges the tunneling of Cooper pairs of electrons with opposite spin, and thereby favors QD states with even numbers of electrons and a zero total spin. At the same time, the local Coulomb repulsion enforces a one-by-one filling of the QD, and thereby induces the Coulomb blockade and even the Kondo effect at very low temperatures, which exhibits the zero-bias anomaly in the differential conductance with odd numbers of electrons residing in the QD. In this case, the superconducting proximity effect competes with the on-site Coulomb correlation [1,6,10,21,24,25,28,29].

It is even more intriguing when the QD additionally connects to a ferromagnetic lead [30,31]. It is known that the effective exchange field induced by the ferromagnetic correlation can cause a spin imbalance inside the QD, and as a result, suppress and/or even split the Kondo peak in the differential conductance [32–37]. Furthermore, spin polarization of the QD, on the one hand, is disadvantageous to the formation of on-dot superconducting pairing. However, the spin polarization in the antiparallel configuration, on the other hand, is favorable to the Andreev reflection (AR) and Cooper pair splitting [30,38]. It is, therefore, very interesting to study how the interplay of the Kondo, superconducting pairing, and ferromagnetic correlations affects the electron tunneling through a QD [39]. In a recent paper, Futterer et al. present a theoretical analysis of the subgap transport of such a three-terminal hybrid system, which consists of n interacting QD attached to two ferromagnetic leads and one superconducting lead [40,41]. They focused on the first-order sequential tunneling by using a master equation and found that the strong on-dot electron–electron interaction, rather than the nonlocal AR, leads to negative values of the nonlocal current response at an appropriately large bias voltage. Moreover, the bias-dependent supercurrent in the superconducting electrode was proposed as a sensitive detector to probe the exchange field of the QD induced by ferromagnetic leads [42]. Thereafter, the tunneling magnetoresistance was calculated for the same system to display a nontrivial dependence on the bias voltage and the level detuning caused by the AR [43]. Very recently, it has been reported, in contrast to [40], that the cross AR is indeed the dominant nonlocal transport channel at a low bias voltage and leads to a negative value of the cross conductance in the three-terminal hybrid nanodevice with two normal electrodes instead [44,45].

In the present work, we extend the finite-*U* slave boson mean field (SBMF) approach of Kotliar and Ruckenstein [46] with the help of the nonequilibrium Green function (NGF) method to investigate the subgap transport for the same three-terminal hybrid QD as in [40]. This kind of SBMF approach is generally believed to be reliable in describing not only spin fluctuations rigorously but also charge fluctuations to a certain degree in the Kondo regime at zero temperature [46–49]. This nonperturbative approach has been successfully utilized to calculate the linear and nonlinear conductance within a relatively wide dot-level range from the mixed valence to the empty orbital regimes, in which the major characteristics induced by the external magnetic field and the magnetization in Kondo transport arise [49–52]. Furthermore, this approach has been applied to analyze the *π*-phase transition in a double-QDs Josephson junction caused by competition between Kondo and interdot antiferromagnetic coupling [53]. The main purpose of this paper is to analyze in detail the interplay of the Kondo, superconducting proximity induced on-dot pairing, and ferromagnetic correlations and their influence on electronic tunneling.

The rest of the paper is organized as follows. In Section 2, we introduce our model of the three-terminal hybrid system, and the equivalent slave-boson field Hamiltonian. Then, we present the self-consistent equations of the expectation values of slave-boson operators within the SBMF approach and NGF method. Moreover, the formulas for current and linear conductance, including the local and cross conductances, are given. In Section 3, we present and analyze our numerical calculations for the linear conductance and nonlinear conductance in detail. Finally, a brief summary is given in Section 4.
