Theory of charge transport in ferromagnetic semiconductor/$s$-wave superconductor junction

We study tunneling conductance in ferromagnetic semiconductor/insulator/$s$-wave superconductor junction where Rashba spin-orbit interaction (RSOI) and exchange field are taken into account in the ferromagnetic semiconductor. We show that normalized conductance at zero voltage has a maximum as a function of RSOI for high-transparent interface and finite exchange field. This is because Andreev reflection probability shows a nonmonotonic dependence on RSOI in the presence of the exchange field. On the other hand, for intermediate transparent interface, normalized conductance at zero voltage has a reentrant shape at zero or small exchange field with increasing RSOI but is monotonically increasing by RSOI at large exchange field.

Figure 1

(Color online) The energy spectrum of the ferromagnetic semiconductor $\left(\beta =0.5,\gamma =0.1\right)$ $E_1=\frac{{\hslash }^2}{2m}{k}^2+\sqrt{H^2+{\left(\lambda k\right)}^2}$ and $E_2=\frac{{\hslash }^2}{2m}{k}^2-\sqrt{H^2+{\left(\lambda k\right)}^2}$. $E_F$ is Fermi energy and $k_F$ is Fermi wave number.

Figure 2

(Color online) Schematic illustration of scattering processes. ${\theta }_{1\left(2\right)}$ is the angle of the wave number $k_{1\left(2\right)}$ with respect to the interface normal. $\theta$ denotes the direction of motions of quasiparticles in the S measured from the interface normal.

Figure 3

(Color online) (a) Normalized tunneling conductance $\left({\sigma }_{\text{T}}=\frac{{\sigma }_{\text{S}}}{{\sigma }_{\text{N}}\left(\beta =0\right)}\right)$ as a function of $\beta$ at $Z=0$ and zero voltage with $\gamma =0,0.1,0.5$. (b) ${\sigma }_1$ (injection with $k_1$) and ${\sigma }_2$ (injection with $k_2$) at zero voltage for $\gamma =0.5$.(c) $N_1$ and $N_2$ are the densities of states normalized by those with $\lambda =0$ and $H=0$ for wave numbers $k_1$ and $k_2$, respectively. (d) Normalized tunneling conductance at nonzero voltage and zero voltage for $\gamma =0.5$ with $eV/\Delta =0$, 0.5,1.0, and 1.5.

Figure 4

(Color online) The angular averaged AR probabilities as a function of $\beta$ at zero voltage for $Z=0$ with $\gamma =0,0.1,0.5$. At $\gamma =0$, $A_2=0,B_1=0$.

Figure 5

(Color online) Normalized tunneling conductance as a function of $\beta \left[\frac{{\sigma }_{\text{S}}}{{\sigma }_{\text{N}}\left(\beta =0\right)}\right]$ at $Z=1.0$ with $\gamma =0,0.1,0.5$ at zero voltage.(a)Normalized tunneling conductance with $\gamma =0,0.1,0.5$. (b)$\gamma =0$ and 0.1(c)$\gamma =0.5$.

Figure 6

(Color online) The angular averaged AR probability at zero voltage for $Z=1.0$ with $\gamma =0,0.1,0.5$. At $\gamma =0$, $A_2=0,B_1=0$.

Figure 7

(Color online) Normalized tunneling conductance as a function of $\beta \left[\frac{{\sigma }_{\text{S}}}{{\sigma }_{\text{N}}\left(\beta =0\right)}\right]$ at $\gamma =0.5$ with $Z=0,0.1,0.3,0.5,0.7,1.0$ at zero voltage.

Figure 8

(Color online) Normalized tunneling conductance as a function of $\gamma \left[\frac{{\sigma }_{\text{S}}}{{\sigma }_{\text{N}}\left(\gamma =0\right)}\right]$ at zero voltage with $\beta =0,0.1,0.5$. (a) $Z=0$. (b) $Z=1.0$.

Figure 9

(Color online) The angular averaged AR probabilities as a function of $\gamma$ at zero voltage for $Z=0$ with $\beta =0,0.1,0.5$. At $\beta =0$, $A_1=0,B_2=0$. At $\gamma =1.0$, $A_2=0,B_1=0$.

Figure 10

(Color online) The angular averaged AR probabilities as a function $\gamma$ at zero voltage for $Z=1.0$ with $\beta =0,0.1,0.5$. At $\beta =0$, $A_1=0,B_2=0$. At $\gamma =1.0$, $A_2=0,B_1=0$.