Transport properties of a hybrid Majorana wire-quantum dot system with ferromagnetic contacts

The transport properties of a quantum dot coupled to ferromagnetic contacts and attached to a topological superconductor wire hosting Majorana zero-energy modes at its ends are studied theoretically in the Kondo regime. By using the numerical renormalization group method, the temperature and dot level dependence of the spectral function, the conductance, and its spin polarization are studied for different coupling strengths to a topological superconductor. It is shown that the transport characteristics are generally determined by the interplay of three competing energy scales, resulting from Kondo correlations, a ferromagnetic-lead-induced exchange field, and topological-wire-induced splitting. These two splittings are found to have opposite signs. Moreover, they can compensate for each other in an appropriate parameter regime.

Figure 1

The schematic of the considered system. It consists of a quantum dot (QD), with energy level $\varepsilon$ and Coulomb correlations $U$, coupled to two ferromagnetic (FM) leads with coupling strengths ${\mathrm{\Gamma }}_L^{\sigma }$ and ${\mathrm{\Gamma }}_R^{\sigma }$ for the left and right junction, and side-coupled (with matrix element $V_M)$ to a topological superconductor (TS) wire hosting two Majorana zero-energy modes described by the operators ${\gamma }_1$ and ${\gamma }_2$.

Figure 2

Different contributions to the ground-state splitting $\mathrm{\Delta }{\varepsilon }_{\mathrm{exch}}^{\mathrm{tot}}$ as a function of the quantum-dot energy level $\varepsilon$ for different couplings to the Majorana wire $V_M$. The other parameters are $U=5\mathrm{\Gamma },\mathrm{\Gamma }=0.015$ in units of band half-width, $p=0.5$, and ${\varepsilon }_M=0$ (left panel) or ${\varepsilon }_M=0.05U$ (right panel). The first row presents the splitting for an isolated Majorana quantum-dot system, $\delta {\varepsilon }_M$, the second row shows the splitting due to the interaction with nonmagnetic leads, $\mathrm{\Delta }{\varepsilon }_{\mathrm{exch}}^{\mathrm{Maj}}$, while the third row presents the contribution coming from finite spin polarization of the leads. Note that the curves in the right column overlap for small values of $V_M$.

Figure 3

The zero-temperature normalized spectral function, $\mathcal{A}\equiv {\sum }_{\sigma }\pi {\mathrm{\Gamma }}_{\sigma }{A}_{\sigma }\left(\omega \right)$, calculated for different couplings to Majorana wire $V_M$, as indicated. The left (right) column corresponds to the case of ${\varepsilon }_M=0$ $\left({\varepsilon }_M/U=0.05\right)$, while the consecutive rows show the results for $\varepsilon =-0.5U,\varepsilon =-0.49U,\varepsilon =-0.48U$, and $\varepsilon =-0.33U$, respectively. The vertical arrows indicate the magnitude of the exchange field. The curves for finite ${\varepsilon }_M$ start to overlap when $V_M/U\lesssim 0.005$. The other parameters are the same as in Fig. 2.

Figure 4

The energy and dot-level position dependence of the zero-temperature normalized spectral function in the case of NM, ${\mathcal{A}}_{\mathrm{NM}}$ (left column), and FM, ${\mathcal{A}}_{\mathrm{FM}}$ (right column), leads calculated for different strengths of coupling to Majorana wire, as indicated. The parameters are the same as in Fig. 3. This figure corresponds to the case of infinite Majorana wire, ${\varepsilon }_M=0$. The dashed lines indicate the splitting of the ground state, $\mathrm{\Delta }{\varepsilon }_{\mathrm{exch}}^{\mathrm{tot}}$, estimated according to Eq. (11).

Figure 5

The same as in Fig. 4 calculated in the case of ${\varepsilon }_M/U=0.05$. The dashed lines indicate the splitting of the ground state, as described by Eq. (11).

Figure 6

The zero-temperature linear conductance (first row) and the corresponding conductance spin polarization (second row) as a function of the dot-level position calculated in the case of nonmagnetic leads $\left(p=0\right)$ for different couplings to topological superconductor $V_M$. The left (right) column corresponds to the case of ${\varepsilon }_M=0$ $\left({\varepsilon }_M/U=0.05\right)$. Note that the curves for finite ${\varepsilon }_M$ start to overlap when $V_M/U\lesssim 0.01$. The other parameters are the same as in Fig. 3.

Figure 7

The same as in Fig. 6 calculated in the case of ferromagnetic leads with $p=0.5$. For ${\varepsilon }_M/U=0.05$, the curves overlap when $V_M/U\lesssim 0.01$.

Figure 8

The linear conductance as a function of the dot-level position $\varepsilon$ and the overlap between the two Majorana fermions ${\varepsilon }_M$ calculated for different couplings $V_M$, as indicated. The left (right) column corresponds to the nonmagnetic (ferromagnetic) lead case. The parameters are the same as in Fig. 3.

Figure 9

The spin polarization of the conductance in the case of nonmagnetic $({\mathcal{P}}_{\mathrm{NM}}$, left column) and ferromagnetic $({\mathcal{P}}_{\mathrm{FM}}$, right column) leads as a function of the dot-level position and the overlap between the two Majorana modes calculated for different couplings $V_M$. The parameters are the same as in Fig. 3.

Figure 10

The temperature dependence of the linear conductance $G_{\mathrm{NM}}$ in the case of nonmagnetic leads calculated for different couplings to the Majorana mode $V_M$, as indicated. The first (second) row shows the results for $\varepsilon =-0.5U$ $\left(\varepsilon =-0.33U\right)$, while the first (second) column presents the data for ${\varepsilon }_M=0$ $\left({\varepsilon }_M/U=0.05\right)$. Note that in the case of finite ${\varepsilon }_M$, the curves overlap when $V_M/U\lesssim 0.01$. The other parameters are the same as in Fig. 3.

Figure 11

The temperature dependence of the linear conductance $G_{\mathrm{FM}}$ in the case of ferromagnetic leads calculated for different couplings to the Majorana mode $V_M$, as indicated. The left (right) column corresponds to the case of ${\varepsilon }_M=0$ $\left({\varepsilon }_M/U=0.05\right)$, while the consecutive rows show the data for $\varepsilon =-0.5U,-0.48U$, and $-0.33U$, respectively. Note that in the case of finite ${\varepsilon }_M$, the curves overlap when $V_M/U\lesssim 0.01$. The other parameters are the same as in Fig. 3.