Three-body Fermi-liquid corrections for Andreev transport through quantum dots

Abstract

We study crossed Andreev reflection occurring in quantum dots connected to one superconducting lead and two normal leads at low temperatures $T$. Specifically, we derive an exact formula for the conductance up to order $T^2$ in the large superconducting gap limit, which is expressed in terms of the transmission probabilities of Cooper pairs and interacting Bogoliubov quasiparticles. Our formulation is based on the latest version of Fermi-liquid theory for the Anderson impurity model, which has clarified the quasiparticle energy shifts of order $ω^2$ and $T^2$ -- that is, corrections of the same order as those arising from the finite lifetime of quasiparticles -- can be exactly taken into account through three-body correlations of impurity electrons. We also demonstrate how the three-body contributions evolve and affect the Cooper-pair tunneling as the Andreev level moves away from the Fermi level, using the numerical renormalization group approach. The results demonstrate that the Cooper-pair contribution to the $T^2$ term of the nonlocal conductance becomes comparable to the Bogoliubov-quasiparticle contribution in the parameter region where superconducting proximity effects dominate over the Kondo effect.

Figures (11)

• Figure 1: Anderson impurity ($\bullet$) coupled to one superconducting (SC) lead and two normal leads on the left and right. $\Delta_{S}$ is the SC gap, and $\epsilon_d^{}$ and $U$ are the level position and Coulomb interaction of the impurity electrons. $\Gamma_L^{}$, $\Gamma_R^{}$, and $\Gamma_S^{}$ represent the tunneling couplings between the impurity site and the left ($L$), right ($R$), and SC leads, respectively.
• Figure 2: Parameter space of $\mathcal{H}_\mathrm{eff}^{}$ at zero magnetic field $b=0$, defined in Eq. \ref{['eq:Heff_single']}. Here, $\theta \equiv \cot^{-1}(\xi_d^{}/\Gamma_S^{})$ is the Bogoliubov rotation angle, with $\xi_d^{} \equiv \epsilon_d^{} +U/2$. The semicircle corresponds to the line along which the energy of the Andreev level, $E_A^{} \equiv \sqrt{\xi_d^2+\Gamma_S^2}$, coincides with $U/2$. In the atomic limit $\Gamma_N ^{} =0$, the ground state is a magnetic spin doublet ($S=1/2$) inside the semicircle, which is eventually screened by conduction electrons to form the Kondo singlet when the tunnel coupling $\Gamma_N^{}$ is switched on, whereas outside the semicircle the ground state is a spin singlet ($S=0$) due to Cooper pairing.
• Figure 3: Coefficients $C_{T}^{\mathrm{BG}}$ and $C_{T}^{\mathrm{CP}}$ for noninteracting electrons ($H_d^{U}=0$), plotted as functions of $E_{A}^{}/\Gamma_{N}^{}$ for the Bogoliubov angle $\theta=\pi/2$.
• Figure 4: Phase shift and transmission probabilities at $T=0$ plotted as functions of $E_A^{}/U$. (a): Occupation number of Bogoliubov quasiparticles, $\langle q_{d}^{} \rangle =2\delta/\pi$. (b): Transmission probability of Bogoliubov quasiparticles, $\mathcal{T}_{\mathrm{BG}}(0)=\sin^2 \delta$ and the Cooper-pair contributions, $-2\mathcal{T}_{\mathrm{CP}}^{}(0)$ with $\mathcal{T}_{\mathrm{CP}}(0)=(1/4) \sin \theta\, \sin^2 2\delta$. (c) and (d): Nonlocal conductance, $g_{RL}^{T=0} / (2 g_0^{}) = \mathcal{T}_{\mathrm{BG}}^{}(0) - 2\mathcal{T}_{\mathrm{CP}}^{}(0)$ with $g_0^{} = \frac{e^2}{h}{4\Gamma_R^{}\Gamma_L^{}}/{\Gamma_N^{2}}$. The results plotted in panels (a)--(c) are obtained for $U/(\pi \Gamma_{N})=1.0$, $2.0$, $3.0$, $4.0$, and $5.0$. Specifically, the Bogoliubov angle is fixed at $\theta = \pi/2$ for $-2\mathcal{T}_{\mathrm{CP}}^{}(0)$ and $g_{RL}^{T=0}$. For comparison, panel (d) shows the results for $g_{RL}^{T=0}$ at $\theta =0$, $\pi/8$, $\pi/4$, $3\pi/8$, and $\pi/2$, with $U/(\pi \Gamma_{N})=5.0$.
• Figure 5: Inverse of the Fermi-liquid energy scale $1/T^{*}$ and the Wilson ratio $R-1$, defined in Eqs. \ref{['eq:T*_def']} and \ref{['eq:Wilson_ratio_def']}, plotted as functions of $E_A/U$ for $U/(\pi \Gamma_{N})= 1.0$, $2.0$, $3.0$, $4.0$, and $5.0$. Here, $T_{K}^{} \equiv T^{*}|_{E_A^{}\to 0}^{}$ corresponds to the Kondo temperature, which takes the values $T_{K}^{}/\Gamma_{N}^{} =0.494$, $0.188$, $0.063$, $0.020$, and $0.006$, respectively, for these five values of $U$.