Controllable nonlocal transport of Majorana fermions with the aid of two quantum dots

Majorana fermions (MFs) are appearing in pairs and nonlocally distributed at the two ends of the wire. The two MFs can combine together to form a nonlocal fermion state. Due to the nonlocal nature of Majorana fermions in the topological superconductor, there are two types of nonlocal transport processes supported by MFs: crossed Andreev reflection and electron transmission. In this paper, we study electron transport through a normal lead-quantum dot-topological superconductor-quantum dot-normal lead (N-QD-TS-QD-N) junction. We show that these two nonlocal processes can be directly controlled by gating the energy level of QDs. For instance, when both energies of QDs (labeled as ${\epsilon }_1$ and ${\epsilon }_2)$ are equal to the coupling energy of MFs (denoted as $E_M)$, the electron in the left lead can transport to the right lead via MFs while when ${\epsilon }_1=-{\epsilon }_2=E_M$ is satisfied, the electron in the left lead can combine one electron in the right lead to form a Cooper pair and tunnel into the topological superconductor via MFs.

Figure 1

(a) The schematic plot of experimental setup: it can be used to detect the nonlocality of MFs. (b) Local Andreev reflection process: an electron from one lead is reflected as a hole in the same lead. (c) Crossed Andreev reflection: an electron from one lead tunnels to the other lead as a hole and a Cooper pair is injected into the superconductor. (d) Electron transmission: an electron from one lead transmit to the other lead via MFs.

Figure 2

The transmission coefficients vs incident energy $E$ in various situations. (a) $E_M=0$: in this case two MFs are well separated and only the local AR can occur. (b) $E_M=0.2$: in this case, two MFs are coupled to each other and the nonlocal process can happen. In both (a) and (b), the MFs are well coupled with the QDs with $t_1=t_2=0.1$. (c) $E_M=0$: the local AR still survives though $t_1=t_2=0.01$. (d) For $E_M=0.2$ and $t_1=t_2=0.01$, all the processes are suppressed due to the weak coupling between QDs and MFs except at $E=\pm E_M$. The energies of QDs are in line with the Fermi level in all the cases: ${\varepsilon }_1={\varepsilon }_2=0$.

Figure 3

Contour plot of (a) $T_h$ and (b) $T_e$, as functions of right QD's energy level ${\varepsilon }_2$ and incident energy E. In both cases, $t_1=t_2=0.01$, and ${\varepsilon }_1=E_M=0.2$. (c) $T_A$, $T_e$, $T_h$ as a function of incident energy $E$ with the QD's energies fixed at ${\varepsilon }_1=-{\varepsilon }_2=E_M=0.2$ denoted by the dashed lines in (a). In this case, only the CAR survives. (d) $T_A$, $T_e$, $T_h$ as a function of incident energy $E$ with the QD's energy levels fixed at ${\varepsilon }_1={\varepsilon }_2=E_M=0.2$ denoted by the dashed lines in (b). The ET is in the resonant regime.

Figure 4

Contour plot of (a) $T_e$ and (b) $T_h$ as functions of MFs' coupling energy $E_M$ and incident energy $E$. We fixed the QD's energy level equal to $E_M$: ${\varepsilon }_1={\varepsilon }_2=E_M$. In this case, only the ET process is allowed. In panels (c) and (d), contour plot of (c) $T_e$ and (d) $T_h$ as functions of MFs' coupling energy $E_M$ and incident energy $E$ are shown with the parameter ${\varepsilon }_1=-{\varepsilon }_2=E_M$. The CAR is allowed in this case.

Figure 5

(a) The energy eigenvalues of a short 1D wire versus chemical potential, the lowest-energy states are indicated in red. The topological region is marked in the figure. (b) and (c) show the contour plot of $T_h$ and $T_e$, respectively, as functions of right QD's energy level ${\varepsilon }_2$ and incident energy $E$. The TS wire's chemical potential $\mu =-48.7$ is denoted by the dashed lines in (a). We fix the left QD's energy level to be $E_M$: ${\varepsilon }_1=E_M=0.11\Delta$. We can see that the CAR is in resonant regime when ${\varepsilon }_1=-{\varepsilon }_2=E_M$, while the ET is in resonant regime when ${\varepsilon }_1={\varepsilon }_2=E_M$.

Figure 6

Contour plot of (a) $T_A$ and (b) $T_e$ and (c) $T_h$ as functions of chemical potential $\mu$ and incident energy $E$. We fixed the QD's energy level: ${\varepsilon }_1=-{\varepsilon }_2=0.08$. In this case, the CAR process is dominant.

Figure 7

Contour plot of (a) $T_A$ and (b) $T_e$ and (c) $T_h$ as functions of chemical potential $\mu$ and incident energy $E$. We fixed the QD's energy level equal to $E_M$: ${\varepsilon }_1={\varepsilon }_2=0.08$. The ET process is dominant in this case.

Figure 8

The disorder has little effect on the nonlocal process. Contour plot of (a) $T_A$ and (b) $T_e$ and (c) $T_h$ as functions of chemical potential $\mu$ and incident energy $E$. We fixed the QD's energy level equal to $E_M$: ${\varepsilon }_1=-{\varepsilon }_2=0.08$. We can see that the CAR process is still dominant despite the influence of disorder. The disorder strength is $\omega =\Delta$.