Figure 1

(a) Top: scheme of the junction including the leads and a qualitative color plot of the barrier potential $V\left(\mathit{r}\right)$; $V_g$ is the value at the saddle-point. Bottom: qualitative two-mode picture of how the current through the barrier becomes spin polarized (left to right case). In the dispersion relations $E\left(k_x\right)$ at different positions, we indicate the relevant states using dots at the Fermi energy $E_{\mathrm{F}}$. Electrons from the left lead (at $x=x_L$) are injected in four states, $(n,{\sigma }_y)$, with $n=1,2$ and ${\sigma }_y=\pm$. The barrier allows only one transverse mode to be transmitted (first conductance plateau, $G=2e^2/h$). Electrons injected at $x=x_L$ in states $(1,+)$ and $(2,+)$ are transmitted to the right. They arrive at $x=x_R$ in states $(1,+)$ and $(2,+)$ and therefore the current is $up$ polarized along $y$ direction. Notice that—in the passage through the barrier—an electron injected in state $(1,+)$ does not see any avoided crossing (as in a zero-SOC barrier), whereas an electron injected in state $(2,+)$ suffers two SOC-induced avoided crossings: first it is mixed with $(1,-)$ and then it returns to $(2,+)$. (b) Summary of the left to right and right to left transport properties of the barrier. The spin polarization from right to left is $down$ as electrons injected (at $x=x_R$ from the right lead) in $(1,-)$ and $(2,-)$ are transmitted whereas electrons injected in $(1,+)$ and $(2,+)$ are reflected to $(2,-)$ and $(1,-)$, respectively.Reuse & Permissions

Figure 2

Conductance $G$ (gray line) and polarization $P$ (black line) as a function of $V_g$ for the quantum point contact ${\mathrm{QPC}}_2$ described by Eq. (A1) taking $(z;W_b;L_b)=(90;100;250)$ nm. The 2DEG parameters are for an InAs-based 2DEG, ($n={10}^{12}{\mathrm{cm}}^{-2}$ and $m^{*}=0.045m_e$ with $m_e$ the electron mass) $E_{\mathrm{F}}=53.3$ meV, $\alpha =5$ meVnm, and $L_N=x_R-x_L=1.2\mu$m. In the simulation, we take $W_y=180$ nm and $a_0=3$ nm. (a) Case I, 2DEG-QPC-2DEG: the leads are the same 2DEG and therefore the scattering at the interfaces is absent. As the Rashba strength is weak the polarization is small: less than $4\%$ in the first plateau. (b) Case II, M-QPC-M : the leads are a metallic material $M$ different from the 2DEG leading to a strong backscattering at the M-2DEG junctions; taking metallic Nb as a reference we use $m^{*}/m_s=0.045$ and the Fermi velocities ratio, $v_{\mathrm{F}}^{2\mathrm{DEG}}/v_{\mathrm{F}}^{\mathrm{Nb}}\approx 2.25$. Due to Fabry-Perot interference-like effects, both the conductance and the polarization show resonant behavior as a function of $V_g$; remarkably, there are values of $V_g$ in which the polarization can be much larger than the sub $4\%$ values observed when the device is in between 2DEG leads.Reuse & Permissions

Figure 3

Exact results for the full multichannel model of a Josephson junction with a SOC-based spin-polarizing barrier. The superconductors have ${\Delta }_0=1.5$ meV and are chosen to minimize normal reflection (see text). The normal region length is $L_N=300a_0=900$ nm and the QPC parameters are those given in Fig. 2. Here, the gate voltage is set in the middle of the first conductance plateau and the SOC is strong, $\alpha =40$ meVnm, so that the polarization is close to one. Color maps as a function of energy $\varepsilon$ and the superconducting phase difference $\varphi$ for (a) ${\rho }_{X_R}(\varepsilon ,\varphi )$, the local density of states at the last layer of the normal, $X_R$; (b) and (d) $J(\varepsilon ,\varphi )$, the current density. Notice that $J(\varepsilon ,\varphi )$, for $\left|\varepsilon \right|<{\Delta }_0$, images the Andreev states; those that contribute to the current (red and blue—in the online version—are positive and negative contributions, respectively, given by the sign of the slope $\frac{\partial {\varepsilon }_j}{\partial \varphi }$). (c) Total Josephson current $I$, i.e., the integral of $J(\varepsilon ,\varphi )$. (e) and (f) Magnetization $s_y(x,y)=\frac{\hslash }{2}\langle {\sigma }_y\rangle$ at ${\varphi }_a=0.3\pi$ for Andreev states $|A\rangle$ and $|B\rangle$ [see (d) and the discussion in Fig. 4]. The magnetizations, integrated over all the normal sample, are positive: $s_y^B=0.45s_y^A$.Reuse & Permissions

Figure 4

Sketch for the WKB interpretation of the Andreev states $|A\rangle$ and $|B\rangle$ shown in Figs. 3d–3(f). Both states have positive velocity, they transfer Cooper pairs from the left ($L$) to the right ($R$) superconductor. Figure 3 shows that the state $|A\rangle$ transports Cooper pairs at a faster rate (since $\frac{\partial {\varepsilon }_{\mathrm{A}}}{\partial \varphi }>\frac{\partial {\varepsilon }_{\mathrm{B}}}{\partial \varphi }$) and its $s_y$ profile is always positive with a single maximum along the $y$ direction, whereas $s_y$ for the Andreev state $|B\rangle$ shows two positive maxima except from the center of the barrier where there exists a negative contribution to $s_y$ having a single maximum. This is in full agreement with the WKB picture presented here. For the state $|A\rangle$ (left), the electron occupies the $(1,+)$ transverse mode all along the barrier as much as the reflected hole occupies the $(1,-)$ transverse mode. On the other hand for $|B\rangle$ (right), because of the smoothness of the barrier and the avoided crossings presented in Fig. 1, the electron (hole) is on the $(2,+)$ [($2,-$ )] mode away from the barrier and it mixes with the $(1,-)$ [$(1,+)$] close to the barrier. This explains the different patterns of local magnetization along the $y$ direction, and why the Andreev state $|B\rangle$ is slower than the $|A\rangle$ one.Reuse & Permissions

Figure 5

Phase shifts due to external magnetic fields. The N-S interfaces are transparent. (a) Andreev levels in the long junction limit. At zero field, due to the symmetry under $\mathcal{T}$ of the BdG Hamiltonian, for every state $|j\rangle$, at ${\varphi }_x$, there is an associated state $|\overline{j}\rangle$, at $2\pi -{\varphi }_x$ ($\equiv -{\varphi }_x$) that (i) has the same energy, (ii) has opposite local spins properties (see text), and (iii) has opposite velocity. A weak in-plane magnetic field $B_y$—if applied along a direction in which the total spin of the Andreev state is nonzero—generates an increment (decrement) of the energy of the state $|\overline{j}\rangle$ ($|j\rangle$) as shown with vertical arrows. The new pattern of Andreev levels (thick lines) are phase-shifted (horizontal arrow) from the $B=0$ levels (thin lines). (b) Shows the Andreev levels of (a) restricting $\varphi$ to the $(0,2\pi )$ region.Reuse & Permissions

Figure 6

Josephson current at a fixed finite Zeeman energy, $E_Z={\mu }_{\mathrm{eff}}{B}_y$, and a fixed barrier profile—$V_g$ is set to allow transmission of a single transverse mode—as $L_N$ is enlarged. For convenience, we plot it as a function of the effective length, $L_{\mathrm{eff}}={\delta }_j{L}_N$ [see Eq. (31c) for ${\xi }_N\ll L_N$]. (a) and (b) correspond to $\alpha =0$, whereas (c) and (d) to $\alpha \ne 0$. Qualitatively similar results are obtained if $L_N$ is fixed and $E_Z$ is varied. For $\alpha =0$, the normal device behaves as a ferromagnetic polarizer; the current at $\varphi =0$ is zero, while changing the length produce 0-$\pi$ junction transitions. On the other hand, for the case shown for $\alpha \ne 0$, the QPC behaves as an efficient polarizer in the normal state. The magnetic field $B_y$ is applied along the SOC polarization axis. Note that the current at $\varphi =0$ is nonzero and that the phase-shift changes with the length of the junction.Reuse & Permissions

Figure 7

Anomalous current, $I(\varphi =0)\ne 0$, phase-asymmetric CPRs and current-asymmetric CPRs, due to in-plane magnetic field (parallel to the SOC polarization axis). The barrier is the quantum point contact, ${\mathrm{QPC}}_1$, described by Eq. (A1) taking $(z;W_b;L_b)=(30;100;250)$ nm. The junction length is $L_N=$ 1.2 $\mu$m and the S-N interfaces are transparent. The left (L) panels show CPRs (La) and Andreev levels (Lc) for a fixed magnetic field, $E_Z=g{\mu }_B{B}_y=\frac{{\Delta }_0}{10}=150\mu$eV, and different values of the Rashba strength $\alpha$. The gate voltage, $V_g=46.5$ meV, sets the number of open transverse channels, $N_{\mathrm{oc}}$, in 1. The larger $\alpha$ the bigger is the current $I(\varphi =0)$ and the less distorted is the Josephson current from the one at $B_y=0$: it looks as a phase-shifted version of the CPR at zero field [see (Ra) and (Rc)]. (Lc) Deduced relative magnetic moments, ${\eta }_1$ and ${\eta }_2$, of the two Andreev bound states (see text) as a function of $\alpha$. We show the simplest estimation for ${\eta }_2$, $(2P_{\mathrm{av}}-1)$, with $P_{\mathrm{av}}$ the average polarization induced by the device in the energy range of the superconducting gap. The right (R) panels show how the opening of more than one transverse mode in the QPC can lead to asymmetric CPRs when $B_y$ is nonzero. We fix $\alpha =40$ meVnm and show the Andreev levels, for $(V_g,N_{\mathrm{oc}})=(46.5\mathrm{meV},1)$ [ $(39.4\mathrm{meV},2)$] in (Ra) [(Rd)] for $B_y=0$ and in (Rb) [(Re)] for $B_y\ne 0$: $E_Z=150\mu$eV. In (Rf), the CPR for $B_y\ne 0$ and $N_{\mathrm{oc}}=2$ does show current-asymmetry, i.e., it fulfills Eq. (17). On the other hand, for the case of $N_{\mathrm{oc}}=1$ shown in (Rc), the CPR gets phase shifted and distorted for $B_y\ne 0$ but no strong current-asymmetry is observed.Reuse & Permissions

Figure 8

Results for a Josephson junction, S-QPC-S, based in QPC${}_2$ with realistic 2DEG-S junctions: an In-based 2DEG and Nb superconductors are simulated [parameters given in Fig. 2b]. We take $\alpha =5$ meVnm, $L_N=1.2\mu$m; in (a) and (b) for the JJ results the Zeeman energy in the 2DEG is $E_Z=-150\mu$eV. We plot the dependence with the gate voltage $V_g$ of: (a) the average critical current ${\overline{I}}_c$ and (b) the anomalous Josephson current $I_a$. In (a) [(b)], we include the conductance $G$ [the polarization $P$] as a function of $V_g$ for the same device between normal contacts (dashed line, 2DEG-QPC-2DEG junction), such that the leads do not generate backscattering and the leads are metallic (solid black line, M-QPC-M junction), we take the parameters of Nb with ${\Delta }_0=0$. We observe correlation between ${\overline{I}}_c$($I_a$) for the S-QPC-S and $G$($P$) for the M-QPC-M. (c) We fix $V_g$ at a peak for $I_a$ and present, as a function of $E_Z$, ${\overline{I}}_c$ and the critical current asymmetry $I_c^{+}/I_c^{-}$. For such value of $V_g$, the departure from the $I_c^{+}=I_c^{-}$ situation is strong and controllable via magnetic field (and gate voltage as shown in Ref. 22). (d) We plot the full current-phase relations for some situations labeled in (c).Reuse & Permissions

Figure 9

Results for a junction with potential $V_{\mathrm{wide}}(x,y)$ with $L_b=360$ nm, i.e., a plain barrier (B) without a lateral constriction. We take $\alpha =20$ meVnm and the 2DEG and superconductor parameters given in Figs. 2 and 8. (a) Spin polarizing properties of the barrier when the leads are normal, we show the dependence on gate voltage $V_g$ for the conductance $G$ and the polarization $P$. Case $Z=0$ is for a 2DEG-B-2DEG junction, i.e., the interfaces are ideal as the leads are the same In-based 2DEG. The other two cases are for the M-B-M junction, the leads are metallic Nb and a there is mismatch of Fermi velocities and effective masses; moreover, away from $A=1$, see Eq. (6), the transparency of the junction is decreased. (b) The normal systems of (a) are turned into Josephson junctions due to a superconducting pairing ${\Delta }_0=1.5$ meV in the contacts and we also include a Zeeman field along the $y$ axis with energy $E_Z=-150\mu$eV. Current-phase relations showing both current for $\varphi =0$ and/or CPR asymmetry $I_c^{+}\ne I_c^{-}$, as those appearing in QPCs, can also be obtained for wide barriers.Reuse & Permissions