Tunneling conductance of graphene ferromagnet-insulator-superconductor junctions

We study the transport properties of a graphene ferromagnet-insulator-superconductor (FIS) junction within the Blonder-Tinkham-Klapwijk formalism by solving spin-polarized Dirac-Bogoliubov-de-Gennes equation. In particular, we calculate the spin polarization of tunneling current at the I-S interface and investigate how the exchange splitting of the Dirac fermion bands influences the characteristic conductance oscillation of the graphene junctions. We find that the retro- and specular Andreev reflections in the graphene FIS junction are drastically modified in the presence of exchange interaction and that the spin polarization $\left(P_T\right)$ of tunneling current can be tuned from the positive to negative value by bias voltage $\left(V\right)$. In the thin-barrier limit, the conductance $G$ of a graphene FIS junction oscillates as a function of barrier strength $\chi$. Both the amplitude and phase of the conductance oscillation varies with the exchange energy $E_{ex}$. For $E_{ex}<E_F$ (Fermi energy), the amplitude of oscillation decreases with $E_{ex}$. For $E_{ex}^c>E_{ex}>E_F$, the amplitude of oscillation increases with $E_{ex}$, where $E_{ex}^c=2E_F+U_0$ ($U_0$ is the applied electrostatic potential on the superconducting segment of the junction). For $E_{ex}>E_{ex}^c$, the amplitude of oscillation decreases with $E_{ex}$ again. Interestingly, a universal phase difference of $\pi /2$ in $\chi$ exists between the $G\text{−}\chi$ curves for $E_{ex}>E_F$ and $E_{ex}<E_F$. Finally, we find that the transitions between retro- and specular Andreev reflections occur at $eV=\left|E_F-E_{ex}\right|$ and $eV=E_{ex}+E_F$, and hence the singular behavior of the conductance near these bias voltages results from the difference in transport properties between specular and retro-Andreev reflections.

Figure 1

(Color online) (a) A schematic plot of a graphene FIS junction. Here, F, I, and S denote the ferromagnetic, insulating, and superconducting regions, respectively. The F and S regions could be manufactured by contacting graphene with a ferromagnetic electrode and a superconducting electrode, respectively. The I region could be created by an external gate voltage $V_0$. An additional gate voltage $U_0$ may be applied on the S region. (b) A schematic plot of the electrostatic potential $U\left(\mathbf{r}\right)$ in the FIS junction of (a).

Figure 2

(Color online) The excitation spectra in the ferromagnetic graphene segment in (a) for $E_{ex}<E_F$ and in (b) for $E_{ex}>E_F$, calculated using Eq. (5). Light gray (pink) lines indicate electron excitations while dark gray (blue) lines indicate hole excitations. Solid and dotted lines denote the conduction and valence bands, respectively.

Figure 3

(Color online) Schematic plots: (a) the modified specular Andreev reflection, (b) the modified retro-Andreev reflection, and (c) the normal reflection.

Figure 4

(Color online) (a) The tunneling spin polarization for several values of the exchange energy $E_{ex}$ with $E_F=100{\Delta }_0$, $U_0=0$, and $\chi =\pi /2$. The inset: the $P_T\text{−}eV$ curve with the same parameters as (a) and $eV$ in the range of $0–500{\Delta }_0$. For $E_{ex}<E_F$, the spin polarization could be negative. (b) The effect of barrier on the curve of the tunneling spin polarization versus bias voltage for $E_F,E_{ex}⪢{\Delta }_0$ with $E_F=100{\Delta }_0$, $E_{ex}=50{\Delta }_0$, and $U_0=0$.

Figure 5

(Color online) (a) The conductance spectra for several values of the exchange energy $E_{ex}$ with $E_F=100{\Delta }_0$, $U_0=0$, and $\chi =\pi /2$. (b) The conductance spectra for several values of barrier strength $\chi$ with $E_F=100{\Delta }_0$, $E_{ex}=50{\Delta }_0$, and $U_0=0$. The curves for $\chi =0$ and $\chi =\pi$ are identical. Note that $G_0$ is not always constant. For $E_F$, $E_{ex}⪢eV$, $G_0$ is independent of $eV$. When $E_{ex}\le E_F$, $G_0=4\left(E_F/{\Delta }_0\right)\left(e^{2}W/h{\xi }_0\right)$, where ${\xi }_0=\pi \hslash v_F/{\Delta }_0$. Conversely, $G_0=4\left(E_{ex}/{\Delta }_0\right)\left(e^{2}W/h{\xi }_0\right)$.

Figure 6

(Color online) The zero-bias conductance as a function of barrier strength $\chi$ with $E_F=100{\Delta }_0$, $U_0=0$, and $E_{ex}=0,50{\Delta }_0,100{\Delta }_0,200{\Delta }_0,300{\Delta }_0$.

Figure 7

(Color online) (a) The transmission coefficient as a function of incident angle $\alpha$ in a NIS junction with $\epsilon =0$, $E_F=100{\Delta }_0$, and $U_0=0$ for $\chi =0,\pi /4,\pi /2$, respectively. (b) The spin-up transmission coefficient in $T_{\uparrow }$ in a FIS junction with $\epsilon =0$, $E_{ex}=50{\Delta }_0$, $E_F=100{\Delta }_0$, and $U_0=0$, where $T_{\uparrow }$ refers to $\left(1+A_{\uparrow }-B_{\uparrow }\right)$. (c) The spin-down transmission coefficient $T_{\downarrow }$ in a FIS junction with $\epsilon =0$, $E_{ex}=50{\Delta }_0$, $E_F=100{\Delta }_0$, and $U_0=0$, where $T_{\downarrow }$ refers to $\left(1+A_{\downarrow }-B_{\downarrow }\right)$.

Figure 8

(Color online) The conductance spectra (a) for several values of $E_{ex}$ with $E_F=0.5{\Delta }_0$ and $U_0=500{\Delta }_0$ and (b) for several values of $E_F$ with $E_{ex}=0.5{\Delta }_0$ and $U_0=500{\Delta }_0$. The curves for $\chi =0$ and $\chi =\pi$ are identical. Note that $G_0$ is not a constant and $\xi =\pi \hslash v_F/{\Delta }_0$. As $\epsilon \le \left(E_{ex}-E_F\right)$, $G_0=4\left(E_{ex}/{\Delta }_0\right)\left(e^{2}W/h{\xi }_0\right)$ and as $\epsilon >\left(E_{ex}-E_F\right)$, $G_0=4\left(\left(\epsilon +E_F\right)/{\Delta }_0\right)\left(e^{2}W/h{\xi }_0\right)$, where ${\xi }_0=\pi \hslash v_F/{\Delta }_0$.