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

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 ${\mathrm{\Gamma }}_S^{}$ 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 ${\mathcal{T}}_{\mathrm{CP}}^{}=\frac{1}{4}{sin}^2\mathrm{\Theta }{sin}^2({\delta }_{\uparrow }+{\delta }_{\downarrow })$ and that of the Bogoliubov particles ${\mathcal{T}}_{\mathrm{BG}}^{}=\frac{1}{2}{\sum }_{\sigma }{sin}^2{\delta }_{\sigma }^{}$. Here, ${\delta }_{\sigma }^{}$ is the phase shift of the renormalized Bogoliubov particles, and $\mathrm{\Theta }\equiv {cot}^{-1}({\xi }_d^{}/{\mathrm{\Gamma }}_S^{})$ is the Bogoliubov-rotation angle in the Nambu pseudospin space, with ${\xi }_d^{}={\varepsilon }_d^{}+U/2$. 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, which occurs when the renormalized Andreev resonance level of majority spin crosses the Fermi level. 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 $\pi /4\lesssim \mathrm{\Theta }\lesssim 3\pi /4$. At the level crossing point, a spin-polarized current flows between the two normal leads, and it is significantly enhanced in the directions of $\mathrm{\Theta }\simeq 0$ and $\mathrm{\Theta }\simeq \pi$ where the SC proximity effect is suppressed.

Figure 1

Single quantum dot (QD) connected to two normal leads (N) and one superconductor lead (SC). ${\mathrm{\Gamma }}_L^{}$, ${\mathrm{\Gamma }}_R^{}$, and ${\mathrm{\Gamma }}_S^{}$ 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 ${\mathrm{\Gamma }}_N^{}={\mathrm{\Gamma }}_L^{}+{\mathrm{\Gamma }}_R^{}$.

Figure 2

Parameter space of $H_{\mathrm{eff}}^{}$ at zero magnetic field $b=0$, defined in Eqs. (A5) and (2.8). The semicircle represents the line along which the energy of the Andreev level $E_A^{}\equiv \sqrt{{\xi }_d^2+{\mathrm{\Gamma }}_S^2}$ coincides with one-half of the Coulomb interaction $U/2$, where ${\xi }_d^{}\equiv {\varepsilon }_d^{}+U/2$. In the atomic limit ${\mathrm{\Gamma }}_N^{}=0$, 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 ${\mathrm{\Gamma }}_N^{}$ is switched on, whereas the ground state is a spin singlet due to the Cooper paring outside the circle. $\mathrm{\Theta }$ 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 $U/\left(\pi {\mathrm{\Gamma }}_N\right)=1.0$, 2.0, 3.0, 5.0 at $b=0$. (a) Occupation number of Bogoliubov particles $Q$ ($=2\delta /\pi$). (b) Pair correlation $|〈d_{\uparrow }^{†}{d}_{\downarrow }^{†}+d_{\downarrow }^{}{d}_{\uparrow }^{}〉|$. (c) Renormalized Andreev-resonance energy ${\tilde{E}}_A$. (d) Renormalization factor $Z={\tilde{\mathrm{\Gamma }}}_N/{\mathrm{\Gamma }}_N$, 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 $|〈d_{\uparrow }^{†}{d}_{\downarrow }^{†}+d_{\downarrow }^{}{d}_{\uparrow }^{}〉|=1-Q$, at $\mathrm{\Theta }=\pi /2$ along the ${\mathrm{\Gamma }}_S^{}$ axis of Fig. 2. The dashed line in (c) denotes the Hartree-Fock energy shift $E_A^{\mathrm{HF}}\stackrel{E_A^{}\gg U/2}{\to }{E}_A^{}-U/2$, given in Eq. (4.6).

Figure 4

Transport coefficients plotted vs $E_A/U$ for $U/\left(\pi {\mathrm{\Gamma }}_N\right)=1.0$, 2.0, 3.0, 5.0 at $b=0$, keeping the Bogoliubov angle fixed at $\mathrm{\Theta }=\pi /2$ (i.e., ${\xi }_d=0$). (a) Single-electron transmission ${\mathcal{T}}_{\mathrm{ET}}^{}$, defined in Eq. (3.3). (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. (3.4) and (3.5), respectively. (c) Nonlocal conductance $g_{\mathrm{RL}}^{}/g_0^{}=2({\mathcal{T}}_{\mathrm{BG}}-2{\mathcal{T}}_{\mathrm{CP}})$ with $g_0^{}=\frac{e^2}{h}4{\mathrm{\Gamma }}_R^{}{\mathrm{\Gamma }}_L^{}/{\mathrm{\Gamma }}_N^2$. (d) CAR efficiency ${\eta }_{\text{CAR}}^{}$ defined in Eq. (3.7).

Figure 5

Three-dimensional plots of (a) $|〈d_{\uparrow }^{†}{d}_{\downarrow }^{†}+d_{\downarrow }^{}{d}_{\uparrow }^{}〉|$ and (b) $\langle n_d^{}\rangle$, described as functions of ${\xi }_d^{}$ and ${\mathrm{\Gamma }}_S^{}$, choosing $U/\left(\pi {\mathrm{\Gamma }}_N\right)=5.0$. Mesh lines are drawn along the polar coordinate ($E_A^{}$, $\mathrm{\Theta }$), with $E_A^{}=\sqrt{{\xi }_d^2+{\mathrm{\Gamma }}_S^2}$ and $\mathrm{\Theta }={tan}^{-1}({\mathrm{\Gamma }}_S^{}/{\xi }_d^{})$.

Figure 6

Transport coefficients (a) ${\mathcal{T}}_{\mathrm{BG}}^{}$, (b) ${\mathcal{T}}_{\mathrm{CP}}^{}$, (c) $g_{\mathrm{RL}}^{}$, and (e) the CAR efficiency ${\eta }_{\text{CAR}}^{}$ plotted as functions of ${\xi }_d^{}$ and ${\mathrm{\Gamma }}_S^{}$, for $U/\left(\pi {\mathrm{\Gamma }}_N\right)=5.0$ and $b=0$. For these three-dimensional plots, mesh lines are drawn along the polar coordinate ($E_A^{}$, $\mathrm{\Theta }$). Two-dimensional plot (d) is a contour map for the region where the nonlocal conductance $g_{\mathrm{RL}}^{}$ becomes negative: contour lines are drawn with 0.05 increments. The CAR dominates $g_{\mathrm{RL}}^{}$ over the parameter region ${\mathrm{\Gamma }}_S^{}\gtrsim U/2$ and $\pi /4<\mathrm{\Theta }<3\pi /4$, i.e., ${\mathrm{\Gamma }}_S>\left|{\xi }_d\right|$.

Figure 7

Parameter space of $H_{\mathrm{eff}}^{}$ at finite magnetic fields. Near the semicircle of radius $E_A^{}=U/2+b$ with $E_A^{}=\sqrt{{\xi }_d^2+{\mathrm{\Gamma }}_S^2}$, the occupation number of the Bogoliubov particles in the Andreev level varies rapidly from $Q\simeq 1.0$ to $Q\simeq 0.0$ for large $U$. Specifically, in the atomic limit ${\mathrm{\Gamma }}_N^{}\to 0$, the ground state is spin polarized $Q_{\uparrow }^{}\to 1.0$ inside the semicircle at finite fields $b\ne 0$. The Andreev scattering can dominate the transport in the range of $\pi /4<\mathrm{\Theta }<3\pi /4$ outside the semicircle which evolves with $b$.

Figure 8

Magnetic-field dependence of the renormalized parameters calculated at different Andreev-level positions $E_A/U=0.0$, 0.2, 0.4, 0.5, 0.6, and 0.8, for a fixed interaction $U/\left(\pi {\mathrm{\Gamma }}_N\right)=5.0$. (a), (b) Occupation number of Bogoliubov particles $Q_{\uparrow }^{}$ and $Q_{\downarrow }^{}$. Inset in (b) is an enlarged plot of $Q_{\downarrow }$ vs $b/T^{*}$, with $T^{*}$ the characteristic energy scale of the Kondo regime defined at $b=0$ in Eq. (4.5). For $E_A^{}=0.0$, $0.2U$, and $0.4U$, it takes the value $T^{*}/\left(\pi {\mathrm{\Gamma }}_N^{}\right)=0.002$, 0.005, and 0.097, respectively (or $T^{*}/U=0.0004$, 0.001, and 0.019). The values of $Z={\tilde{\mathrm{\Gamma }}}_N^{}/{\mathrm{\Gamma }}_N^{}$ at these three points of $E_A^{}$ are given by $Z=0.008$, 0.02, 0.31, respectively. (c) Magnetization $m_d^{}=\langle n_{d\uparrow }^{}\rangle -\langle n_{d\downarrow }^{}\rangle$, which does not depend on the Bogoliubov angle $\mathrm{\Theta }$. (d) Enlarged view of $m_d^{}$ for $E_A^{}=0.0$, $0.2U$, and $0.4U$ plotted vs $b/T^{*}$ in the Kondo regime. (e) Pair correlation $|〈d_{\uparrow }^{†}{d}_{\downarrow }^{†}+d_{\downarrow }^{}{d}_{\uparrow }^{}〉|$ at $\mathrm{\Theta }=\pi /2$, which in this case is given by $1-Q$ and varies with $b$ and $E_A^{}$, in contrast to the electron filling $\langle n_d^{}\rangle \equiv 1.0$ that remains unchanged along the ${\mathrm{\Gamma }}_S^{}$ axis at ${\xi }_d^{}=0$. (f) Renormalized Andreev levels ${\tilde{E}}_{A,\uparrow }$. The dashed straight lines represent the Hartree-Fock (HF) result $E_A^{\mathrm{HF}}\stackrel{E_A^{}\gg U/2}{\to }{E}_A^{}-U/2-b$.

Figure 9

Magnetic-field dependence of the transport coefficients at $\mathrm{\Theta }=\pi /2$ for different positions of $E_A/U=0$, 0.2, 0.4, 0.5, 0.6, and 0.8, for a strong interaction $U/\left(\pi {\mathrm{\Gamma }}_N\right)=5.0$. Top panel describes ${\mathcal{T}}_{\mathrm{BG}}^{}={\sum }_{\sigma }{sin}^2{\delta }_{\sigma }^{}/2$, and $\mathrm{\Theta }$-dependent part $-2{\mathcal{T}}_{\mathrm{CP}}^{}$ for (a) small $E_A^{}\lesssim 0.5U$ and (b) large $E_A^{}\gtrsim 0.5U$. (c), (d) Nonlocal conductance $g_{\mathrm{RL}}^{}/g_0^{}$. Inset in (d) is an enlarged view in the region around $g_{\mathrm{RL}}^{}/g_0\approx 0.0$: the dashed lines represent the perturbation results obtained with Eq. (5.2). The characteristic energy is given by $T^{*}/U=0.0004$, 0.001, 0.019, and 0.097 for $E_A^{}/U=0.0$, 0.2, 0.4, and 0.5, respectively. (e) CAR efficiency ${\eta }_{\text{CAR}}^{}$. (f) $g_{\mathrm{RL}}^{}/g_0^{}$ at a fixed Andreev level position $E_A^{}=0.8U$ for several different angles $\mathrm{\Theta }=0$, $\pi /8$, $\pi /4$, $3\pi /8$, $\pi /2$: the dashed lines here also represent the perturbation results.

Figure 10

Contour maps of $|〈d_{\uparrow }^{†}{d}_{\downarrow }^{†}+d_{\downarrow }^{}{d}_{\uparrow }^{}〉|$ at finite magnetic fields: (a) $b=T_K$ and (b) $b=0.1U$, for $U/\left(\pi {\mathrm{\Gamma }}_N\right)=5.0$. Here, $T_K=0.0004U$ is defined as the value of $T^{*}$ at $E_A=b=0$. The contours are drawn with 0.1 increments, and the dashed line represents the semicircle of radius $E_A=U/2+b$.

Figure 11

Bogoliubov-angle dependence of transport coefficients in a strong interaction case $U/\left(\pi {\mathrm{\Gamma }}_N\right)=5.0$ at small $b=T_K$ and large $b=0.1U$ fields, with $T_K/U=0.0004$ defined as the value of $T^{*}$ at $E_A=b=0$. In the top panels for (a) $b=0$, (b) $b=T_K^{}$, and (c) $b=0.1U$, the coefficients ${\mathcal{T}}_{\mathrm{BG}}^{}$, ${\mathcal{T}}_{\mathrm{CP}}^{}$, and CAR efficiency ${\eta }_{\mathrm{CAR}}^{}$ at $\mathrm{\Theta }=\pi /2$ are plotted vs $E_A^{}/U$. (d), (e) Contour maps of ${\mathcal{T}}_{\mathrm{CP}}^{}$, drawn with 0.05 increments. (f), (g) Contour maps of ${\eta }_{\mathrm{CAR}}^{}$, drawn with 0.1 increments. (h)–(k) Describe the nonlocal conductance $g_{\mathrm{RL}}^{}/g_0^{}$. In particular, (j) and (k) are the contour maps for negative conductance region $g_{\mathrm{RL}}^{}<0$, for which contours are drawn with 0.05 increments. The dashed semicircle of radius $E_A=b+U/2$ in (d)–(g) and (j)–(k) corresponds to the one in Fig. 7.

Figure 12

Spin-dependent transport coefficients at finite magnetic field $b=0.1U$, for $U/\left(\pi {\mathrm{\Gamma }}_N\right)=5.0$. Top panel (a) shows ${sin}^2{\delta }_{\uparrow }$ and ${sin}^2{\delta }_{\downarrow }$, plotted vs $E_A^{}$. Three-dimensional figures represent (b) spin transmission coefficient ${\mathcal{T}}_{\mathrm{spin}}$ and (c) current polarization $P_R$, plotted as functions of ${\xi }_d^{}$ and ${\mathrm{\Gamma }}_S^{}$, for ${\mathrm{\Gamma }}_L={\mathrm{\Gamma }}_R$.