Anomalous Josephson effect induced by spin-orbit interaction and Zeeman effect in semiconductor nanowires

We investigate theoretically the Josephson junction of semiconductor nanowire with strong spin-orbit (SO) interaction in the presence of magnetic field. By using a tight-binding model, the energy levels $E_n$ of Andreev bound states are numerically calculated as a function of phase difference $\phi$ between two superconductors in the case of short junctions. The dc Josephson current is evaluated from the Andreev levels. In the absence of SO interaction, a 0-$\pi$ transition due to the magnetic field is clearly observed. In the presence of SO interaction, the coexistence of SO interaction and Zeeman effect results in $E_n(-\phi )\ne E_n\left(\phi \right)$, where the anomalous Josephson current flows even at $\phi =0$. In addition, the direction dependence of critical current is observed, in accordance with experimental results.

Figure 1

Our model for a quasi-one-dimensional semiconductor nanowire connected to two superconductors. The nanowire is along the $x$ axis. (a) Schematic view of the model. The pair potential is induced in the nanowire by the proximity effect, $\Delta \left(x\right)={\Delta }_0{e}^{i{\phi }_{\mathrm{L}}}$ at $x<0$ and ${\Delta }_0{e}^{i{\phi }_{\mathrm{R}}}$ at $L<x$, whereas $\Delta \left(x\right)=0$ at $0<x<L$. Several impurities are present in the nanowire. The spin-orbit interaction and Zeeman effect are taken into account only in the normal region. (b) The tight-binding model gives the scattering matrix ${\widehat{S}}_{\mathrm{e}}$ (${\widehat{S}}_{\mathrm{h}}$) for electrons (holes), which connects incoming ${\mathit{a}}_{\mathrm{e}}$ (${\mathit{a}}_{\mathrm{h}}$) and outgoing electrons ${\mathit{b}}_{\mathrm{e}}$ (holes ${\mathit{b}}_{\mathrm{h}}$). At $x=0$ and $L$, the electron ${\mathit{b}}_{\mathrm{e}}$ is reflected as the hole ${\mathit{a}}_{\mathrm{h}}$ by the Andreev reflection, whereas ${\mathit{b}}_{\mathrm{h}}$ is reflected as ${\mathit{a}}_{\mathrm{e}}$.

Figure 2

Calculated results for a sample when $N=1$ and $l_0/L=1$. The SO interaction is absent. (a) Andreev levels $E_n$ as a function of phase difference $\phi$ between two superconductors. Solid and broken lines indicate $E_{\uparrow i\pm }$ and $E_{\downarrow i\pm }$, respectively. The magnetic field is ${\theta }_B=0$ (left upper), $0.1\pi$ (left middle), $0.27\pi$ (left bottom), $0.53\pi$ (right upper), $0.79\pi$ (right middle), and $\pi$ (right bottom). At $B=0$, two lines are overlapped to each other, reflecting the Kramers' degeneracy. (b) Josephson current $I\left(\phi \right)$ through the nanowire when ${\theta }_B=0$ (solid), $0.27\pi$ (broken), $0.53\pi$ (dotted), and $\pi$ (dotted broken lines).

Figure 3

Calculated results for $N=1$ and $l_0/L=1$. The SO interaction is absent. (a) Phase difference ${\phi }_0$ at the minimum of ground-state energy as a function of magnetic field, ${\theta }_B$. (b) Critical current $I_{\mathrm{c},\pm }$. The current in the positive direction $I_{\mathrm{c},+}$ is identical with that in the negative direction $I_{\mathrm{c},-}$. The sample for (a) and (b) is same as that in Fig. 2. (c) Average of critical current, $\langle I_{\mathrm{c},\pm }\rangle$, with the average of fluctuation, $\sqrt{\langle {\left[\Delta I_{\mathrm{c},\pm }\right]}^2\rangle }$, as error bars, where $\Delta I_{\mathrm{c},\pm }\equiv I_{\mathrm{c},\pm }-\langle I_{\mathrm{c},\pm }\rangle$. The random average is taken for 400 samples.

Figure 4

Calculated results of random average of critical current $\langle I_{\mathrm{c},\pm }\rangle$ as a function of magnetic field ${\theta }_B$ when $N=2$ (a) and 4 (b). Error bars represent the average of fluctuation, $\sqrt{\langle {\left[\Delta I_{\mathrm{c},\pm }\right]}^2\rangle }$, where $\Delta I_{\mathrm{c},\pm }\equiv I_{\mathrm{c},\pm }-\langle I_{\mathrm{c},\pm }\rangle$. The SO interaction is absent. The mean free path is $l_0/L=1$ (upper) and $0.5$ (lower panels). The random average is taken for 400 samples.

Figure 5

Calculated results for a sample when $N=1$ and $l_0/L=1$. The SO interaction is $k_{\alpha }/k_{\mathrm{F}}=0.15$. The magnetic field is applied to the $y$ direction. (a) Andreev levels $E_n$ as a function of phase difference $\phi$ between two superconductors. The magnetic field is ${\theta }_B=0$ (left upper), $0.1\pi$ (left middle), $0.35\pi$ (left bottom), $0.7\pi$ (right upper), $1.05\pi$ (right middle), and $1.4\pi$ (right bottom). At $B=0$, two lines are overlapped with each other, reflecting the Kramers' degeneracy. (b) Josephson current $I\left(\phi \right)$ through the nanowire when ${\theta }_B=0$ (solid), $0.35\pi$ (broken), $0.7\pi$ (dotted), and $1.4\pi$ (dotted broken lines).

Figure 6

Calculated results for $N=1$ and $l_0/L=1$. The SO interaction is $k_{\alpha }/k_{\mathrm{F}}=0.15$. The magnetic field is applied to the $y$ direction. (a) Phase difference ${\phi }_0$ at the minimum of ground-state energy as a function of magnetic field ${\theta }_B$. (b) Anomalous Josephson current $I(\phi =0)$. (c) Critical current $I_{\mathrm{c},\pm }$.The current in the positive direction $I_{\mathrm{c},+}$ is identical with that in the negative direction $I_{\mathrm{c},-}$. The sample for (a)--(c) is same as that in Fig. 5. (d) Average of critical current $\langle I_{\mathrm{c},\pm }\rangle$ with the average of fluctuation, $\sqrt{\langle {\left[\Delta I_{\mathrm{c},\pm }\right]}^2\rangle }$, as error bars, where $\Delta I_{\mathrm{c},\pm }\equiv I_{\mathrm{c},\pm }-\langle I_{\mathrm{c},\pm }\rangle$. The random average is taken for 400 samples.

Figure 7

Calculated results for a sample when $N=4$ and $l_0/L=1$. The SO interaction is $k_{\alpha }/k_{\mathrm{F}}=0.15$. The magnetic field is applied to the $y$ direction. (a) Andreev levels $E_n$ as a function of phase difference $\phi$ between two superconductors. The magnetic field is ${\theta }_B=0$ (left upper), $0.1\pi$ (left middle), $0.4\pi$ (left bottom), $0.8\pi$ (right upper), $1.2\pi$ (right middle), and $1.6\pi$ (right bottom). At $B=0$, two lines are overlapped with each other, reflecting the Kramers degeneracy. (b) Josephson current $I\left(\phi \right)$ through the nanowire when ${\theta }_B=0$ (solid), $0.4\pi$ (broken), $0.8\pi$ (dotted), and $1.6\pi$ (dotted broken lines).

Figure 8

Calculated results for $N=4$ and $l_0/L=1$. The SO interaction is $k_{\alpha }/k_{\mathrm{F}}=0.15$. The magnetic field is applied to the $y$ direction. The sample is same as that in Fig. 7. (a) Phase difference ${\phi }_0$ at the minimum of ground-state energy as a function of magnetic field ${\theta }_B$. (b) Anomalous Josephson current $I(\phi =0)$. (c) Critical current in the positive $I_{\mathrm{c},+}$ (solid) and in the negative direction $I_{\mathrm{c},-}$ (broken lines).

Figure 9

Calculated results of random average when $N=4$ and $l_0/L=1$. The SO interaction is $k_{\alpha }/k_{\mathrm{F}}=0.15$. The magnetic field is applied to the $y$ direction. The random average is taken for 400 samples. (a) Average of anomalous Josephson current $\langle I(\phi =0)\rangle$ as a function of magnetic field, ${\theta }_B$. Error bars represent the average of fluctuation, $\sqrt{\langle {\left[\Delta I\left(0\right)\right]}^2\rangle }$, where $\Delta A\equiv A-\langle A\rangle$. (b) Average of difference of critical current $\langle \delta I_{\mathrm{c}}\rangle$ where $\delta I_{\mathrm{c}}\equiv |I_{\mathrm{c},+}-I_{\mathrm{c},-}|$. Error bars are $\sqrt{\langle {\left[\Delta \left\{\delta I_{\mathrm{c}}\right\}\right]}^2\rangle }$.

Figure 10

Schematic views of dispersion relation with SO interaction. (a) Spin-splitting due to the $p_x{\sigma }_y$ term in Eq. (2). Solid and broken lines indicate the branches with spin $\sigma =+$ and $-$, respectively. (b) Dispersion relation mixed by $p_y{\sigma }_x$ term (solid line). The broken line indicate the case without $p_y{\sigma }_x$ term. (c) Shift of wave number due to the Zeeman effect in the vicinity of Fermi level.

Figure 11

Grayscale plot of the phase difference ${\phi }_0$ at the minimum of ground-state energy in the plane of magnetic field ${\theta }_B$ and SO interaction $k_{\alpha }/k_{\mathrm{F}}$ when $N=1$ and $l_0/L=1$. The sample is the same as that in Fig. 5. The magnetic field is applied in the $y$ (a) and $x$ directions (b).

Figure 12

Calculated results for $N=1$ and $l_0/L=1$. The magnetic field is applied to the $x$ direction. The sample is the same as that in Fig. 5. (a) Phase difference ${\phi }_0$ at the minimum of ground-state energy as a function of magnetic field ${\theta }_B$. (b) Critical current $I_{\mathrm{c},\pm }$. The current in the positive direction $I_{\mathrm{c},+}$ is identical to that in the negative direction $I_{\mathrm{c},-}$. Solid and broken lines in each panel indicates $k_{\alpha }/k_{\mathrm{F}}=0.01$ and $0.005$, respectively.