Nonlocal Andreev transport through a quantum dot in a magnetic field: Interplay between Kondo, Zeeman, and Cooper-pair correlations

TL;DR

The paper develops a comprehensive framework to analyze nonlocal transport through a strongly correlated quantum dot coupled to two normal and one superconducting leads, focusing on the interplay between CAR, Kondo physics, and Zeeman splitting in the large-gap limit. Using a Fermi-liquid description of interacting Bogoliubov quasiparticles and Wilson NRG, it expresses the nonlocal conductance in terms of Bogoliubov-phase shifts δ_σ and the Bogoliubov rotation angle Θ, with CAR governed by T_CP ∝ sin^2 Θ sin^2(δ_↑+δ_↓) and T_BG ∝ ∑_σ sin^2 δ_σ. At zero field CAR is enhanced near the crossover between Kondo-dominated and SC-proximity-dominated regimes, forming a crescent-shaped region in (ξ_d, Γ_S) (or E_A, Θ) space; under finite magnetic fields, a second crossover emerges between SC-dominated and Zeeman-dominated regimes, producing a flat negative valley in g_RL that tracks E_A ≈ U/2 + b. The results identify robust, experimentally accessible sweet spots around Θ ≈ π/2 where Cooper-pair tunneling is most entangled, offering clear signatures of Cooper-pair splitting in multiterminal quantum-dot devices and deepening understanding of the competition between many-body Kondo physics and superconducting proximity effects.

Abstract

We study the nonlocal magnetotransport through a strongly correlated quantum dot, connected to multiple terminals consisting of two normal and one superconducting (SC) leads. Specifically, we present a comprehensive view on the interplay between the crossed Andreev reflection (CAR), the Kondo effect, and the Zeeman splitting at zero temperature in the large SC gap limit. The ground state of this network shows an interesting variety, which varies continuously with the system parameters, such as the coupling strength $Γ_S^{}$ between the SC lead and the quantum dot, the Coulomb repulsion $U$, the impurity level $\varepsilon_d^{}$, and the magnetic field $b$. We show, using the many-body optical theorem which is derived from the Fermi-liquid theory, that the nonlocal conductance is determined by the transmission rate of the Cooper pairs $\mathcal{T}_{\mathrm{CP}}^{} = \frac{1}{4} \sin^2 Θ\, \sin^2 \bigl(δ_{\uparrow}+ δ_{\downarrow})$ and that of the Bogoliubov particles $\mathcal{T}_{\mathrm{BG}}^{}= \frac{1}{2}\sum_σ \sin^2 δ_σ^{}$. Here, $δ_σ^{}$ is the phase shift of the renormalized Bogoliubov particles, and $Θ\equiv \cot^{-1} (ξ_d^{}/ Γ_S^{})$ is the Bogoliubov-rotation angle in the Nambu pseudo spin space, with $ξ_d^{} =\varepsilon_d^{}+U/2$. It is also demonstrated, using Wilson's numerical renormalization group approach, that the CAR is enhanced in the crossover region between the Kondo regime and the SC-proximity-dominated regime at zero magnetic field. The magnetic fields induce another crossover between the Zeeman-dominated regime and the SC-dominated regime. We find that the CAR is enhanced and becomes less sensitive to magnetic fields in the SC-dominated regime close to the crossover region spreading over the angular range of $π/4 \lesssim Θ\lesssim 3π/4$.

Figures (12)

• Figure 1: Single quantum dot (QD) connected to two normal leads (N) and one superconductor lead (SC). $\Gamma_L^{}$, $\Gamma_R^{}$, and $\Gamma_S^{}$ represent the coupling strengths of the QD with the left ($L$), the right ($R$), and the SC leads, respectively. The contributions of the normal tunnelings are given by $\Gamma_N^{}=\Gamma_L^{}+\Gamma_R^{}$.
• Figure 2: Parameter space of $H_\mathrm{eff}^{}$ at zero magnetic field $b=0$, defined in Eqs. \ref{['eq:Heff_single']} and \ref{['eq:SingleDot_new']}. The semicircle represents the line along which the energy of the Andreev level $E_A^{} \equiv \sqrt{\xi_d^2+\Gamma_S^2}$ coincides with one-half of the Coulomb interaction $U/2$, where $\xi_d^{} \equiv \varepsilon_d^{} +U/2$. In the atomic limit $\Gamma_N ^{} =0$, the ground state is a magnetic spin doublet inside the semicircle, which eventually is screened by conduction electrons to form the Kondo singlet when the tunnel coupling $\Gamma_N^{}$ is switched on, whereas the ground state is a spin singlet due to the Cooper paring outside the circle. $\Theta$ is the Bogoliubov-rotation angle, which parametrizes the contributions of the Andreev scattering on the transport coefficients.
• Figure 3: Renormalized parameters plotted vs $E_A^{}/U$ for $U/(\pi \Gamma_{N})=1.0$, $2.0$, $3.0$, $5.0$ at $b=0$. (a): Occupation number of Bogoliubov particles $Q$ ($=2\delta/\pi$). (b): Pair correlation $|\bigl \langle d^\dagger_\uparrow d^\dagger_\downarrow + d^{}_\downarrow d^{}_\uparrow \bigr \rangle |$. (c): Renormalized Andreev-resonance energy $\widetilde{E}_A$. (d): Renormalization factor $Z = \widetilde{\Gamma}_{N}/\Gamma_N$, which at $E_A^{}=0$ takes the values $0.629$, $0.239$, $0.080$, $0.008$, respectively, for the above four values of $U$. Note that $\langle n_d^{} \rangle = 1.0$ and $|\bigl \langle d^\dagger_\uparrow d^\dagger_\downarrow + d^{}_\downarrow d^{}_\uparrow \bigr \rangle |=1-Q$, at $\Theta = \pi/2$ along the $\Gamma_S^{}$ axis of Fig. \ref{['fig:SingleDotPhase']}. The dashed line in (c) denotes the Hartree-Fock energy shift $E_{A}^\mathrm{HF} \xrightarrow{E_A^{} \gg U/2} E_{A}^{} -U/2$, given in Eq. \ref{['eq:HF_energy_shift']}.
• Figure 4: Transport coefficients plotted vs $E_A/U$ for $U/(\pi \Gamma_{N})= 1.0$, $2.0$, $3.0$, $5.0$ at $b=0$, keeping the Bogoliubov angle fixed at $\Theta = \pi/2$ (i.e., $\xi_d=0$). (a): Single-electron transmission $\mathcal{T}_{\mathrm{ET}}^{}$, defined in Eq. \ref{['eq:T_ET_single']}. (b): Cooper-pair contributions $-2\mathcal{T}_{\mathrm{CP}}^{}$ ($< 0$) and the Bogoliubov-particle transmission $\mathcal{T}_{\mathrm{BG}}=\sin^2 \delta$, defined in Eqs. \ref{['eq:T_CP_single']} and \ref{['eq:T_BG_single']}, respectively. (c): Nonlocal conductance $g_{\mathrm{RL}}^{} / g_0^{} = 2(\mathcal{T}_{\mathrm{BG}} - 2\mathcal{T}_{\mathrm{CP}})$ with $g_0^{} = \frac{e^2}{h}{4\Gamma_R^{}\Gamma_L^{}}/{\Gamma_N^{2}}$. (d): CAR efficiency $\eta_{\text{CAR}}^{}$ defined in Eq. \ref{['eq:CARefficiencyDef']}.
• Figure 5: Three-dimensional plots of (a) $|\bigl\langle d^\dagger_\uparrow d^\dagger_\downarrow + d^{}_\downarrow d^{}_\uparrow \bigr\rangle|$ and (b) $\left\langle n_d^{}\right\rangle$, described as functions of $\xi_d^{}$ and $\Gamma_S^{}$, choosing $U/(\pi\Gamma_N) = 5.0$. Mesh lines are drawn along the polar coordinate ($E_A^{}$, $\Theta$), with $E_A^{}=\sqrt{\xi_d^2+\Gamma_S^2}$ and $\Theta = \tan^{-1} (\Gamma_S^{}/\xi_d^{})$.