Interplay of spin orbit interaction and Andreev reflection in proximized quantum dots

TL;DR

This work analyzes a three-terminal Cooper pair splitter formed by two quantum dots proximized by an s-wave superconductor, focusing on how spin-orbit coupling and crossed Andreev reflection shape molecular Andreev bound states. Using an exact Keldysh Green's function approach in the Δ→∞ subgap limit, the authors show a duality between CAR and SOI and identify a sweet-spot condition $\Delta_{12}=t_{so}$ that yields zero-energy, spin-polarized Majorana-like bound states localized on separate dots. The zero-energy states manifest as four Dirac-like modes whose transport signatures include near-perfect transmissions $T_{LR}(0)\approx T_S(0)\approx1$ under symmetric bias, revealing a robust entangled-electron transport that probes the molecular quasiparticles. Dissipation from the normal electrodes can suppress these features, suggesting that conductance measurements in both bias configurations can empirically detect the Majorana-like states and the SOC/CAR interplay in this minimal Kitaev-like setup.

Abstract

We investigate a hybrid device, consisting of two quantum dots proximized by a BCS superconductor and coupled to two external normal electrodes. Assuming charge tunneling between quantum dots through the spin-flip processes, we study the molecular Andreev bound states appearing in the proximized quantum dots. We show that the spin-orbit coupling induces four quasiparticle states. For the appropriate set of model parameters, two of these internal quasiparticles merge, forming the zero-energy state. Under such circumstances, we obtain fully spin-polarized versions of the Majorana quasiparticles, localized on different quantum dots. This situation occurs solely when the spin-orbit interaction is equally strong to the magnitude of crossed Andreev reflections, i.e. in the sweet spot. Otherwise, these processes are competitive, as indicated in expectation values of the corresponding order parameters. We analyze signatures of such competition manifested under the nonequilibrium conditions, for various configurations of bias voltage. In particular, for the symmetric bias voltage between the normal electrodes and the Cooper pair splitter bias configuration we reveal duality in the transport properties. Charge transport through the zero-energy state at the sweet spot is contributed by perfectly entangled electrons with an (almost) ideal transmission. Transport studies would thus enable empirical detection of the molecular quasiparticle states and the efficiency of dissipation processes caused by the external normal electrodes.

Figures (5)

• Figure 1: Scheme of the Andreev molecule hybridized by a spin-orbit (SO) coupling. The inter-dot (CAR) electron-hole processes are denoted in blue and the inter-dot SO processes in green color, respectively. These processes couple particles with the opposite spins. Additionally the molecule is coupled to the external metallic electrodes, which serve as the electron and hole reservoirs characterized by the Fermi distributions: $f_{Le}$, $f_{Lh}$, $f_{Re}$ and $f_{Rh}$. Notice, that by ignoring the local Andreev reflections and the spin-conserved hopping we obtain two separated subspaces.
• Figure 2: The spectral function $\rho_{1\uparrow,1\uparrow}=(-1/\pi)\Im \ll c_{1\uparrow}|c^{\dag}_{1\uparrow}\gg^r_{\omega}$ for $\Delta_{12}=1$ and several values of the spin-orbit coupling $t_{so}=0$, $1/3$, $2/3$, $1$ and $4/3$. The horizontal lines show the positions of the poles: $\omega=\pm (\Delta_{12} \pm t_{so})$. Results are obtained for the particle-hole symmetric case $\epsilon_1=0=\epsilon_2$, assuming small symmetric couplings to the external normal electrods $\gamma=0.1$.
• Figure 3: Plots of the thermal averages $\langle c^{\dag}_{1\uparrow} c_{1\uparrow}\rangle$ - green dashed, $\langle c^{\dag}_{2\downarrow} c_{2\downarrow}\rangle$ - cyan, $|\langle c^{\dag}_{2\downarrow} c_{1\uparrow}\rangle|$ - blue and $|\langle c_{1\uparrow} c_{2\downarrow}\rangle|$ - red as a function of $\epsilon$ for $\delta=0$, $\Delta_{12}=1$ and $t_{so}=0.5$, 1, 1.5 (the left column); and as a function of $\delta$ for $\epsilon=0$, $t_{so}=1$ and $\Delta_{12}=0.5$, 1, 1.5 (the right column). Results are obtained for a small coupling $\gamma=0.01$ at zero temperature $T=0$.
• Figure 4: Evolution of the transmission $T_{LR}(\omega)$ (blue) and $T_{S}(\omega)$ (dashed red) for $t_{so}=1$ and several $\Delta_{12}=0$, 0.5, 1 and 1.5 at $\epsilon_1=0$, $\epsilon_2=0$ and a small symmetric coupling $\gamma=0.1$ to the reservoirs. Compare with the spectral density in Fig.\ref{['Fspectral']}.
• Figure 5: Left-column (figure a,b and c) presents transmission $T_{LR}(\omega)$ (blue) and $T_{S}(\omega)$ (dashed-red) for various couplings $\gamma=0.1$, $\gamma=0.3$ and $\gamma=0.5$ at $\epsilon=0$ and $\delta=0$. Right column (figure d,e,and f) presents how $T_{LR}(\omega)$ and $T_S(\omega)$ change for various $\epsilon=0$, 1 and 1.5 at $\gamma=0.3$ and $\delta=0$. The other parameters are $t_{so}=1$ and $\Delta_{12}=0.5$