Shiba states and zero-bias anomalies in the hybrid normal-superconductor Anderson model

Hybrid semiconductor-superconductor systems are interesting melting pots where various fundamental effects in condensed-matter physics coexist. For example, when a quantum dot is coupled to a superconducting electrode two very distinct phenomena, superconductivity and the Kondo effect, compete. As a result of this competition, the system undergoes a quantum phase transition when the superconducting gap $\Delta$ is of the order of the Kondo temperature $T_K$. The underlying physics behind such transition ultimately relies on the physics of the Anderson model where the standard metallic host is replaced by a superconducting one, namely the physics of a (quantum) magnetic impurity in a superconductor. A characteristic feature of this hybrid system is the emergence of subgap bound states, the so-called Yu-Shiba-Rusinov (YSR) states, which cross zero energy across the quantum phase transition, signaling a switching of the fermion parity and spin (doublet or singlet) of the ground state. Interestingly, similar hybrid devices based on semiconducting nanowires with spin-orbit coupling may host exotic zero-energy bound states with Majorana character. Both parity crossings and Majorana bound states (MBSs) are experimentally marked by zero-bias anomalies in transport, which are detected by coupling the hybrid device with an extra normal contact. We here demonstrate theoretically that this extra contact, usually considered as a nonperturbing tunneling weak probe, leads to nontrivial effects. This conclusion is supported by numerical renormalization-group calculations of the phase diagram of an Anderson impurity coupled to both superconducting and normal-state leads. We obtain this phase diagram for an arbitrary ratio $\frac{\Delta }{T_K}$, which allows us to analyze relevant experimental scenarios, such as parity crossings as well as Kondo features induced by the normal lead, as this ratio changes. Spectral functions at finite temperatures and magnetic fields, which can be directly linked to experimental tunneling transport characteristics, show zero-energy anomalies irrespective of whether the system is in the doublet or singlet regime. We also derive the analytical condition for the occurrence of Zeeman-induced fermion-parity switches in the presence of interactions which bears unexpected similarities with the condition for emergent MBSs in nanowires.

Figure 1

(a) Lowest energy many-particle eigenstates of an Anderson impurity coupled to a superconductor with the typical BCS density of states $\sim {[{(\omega /\Delta )}^2-1]}^{-1/2}$ for large on-site interaction $U\gg \Delta$. The magnetic impurity ground state develops singlet correlations with the quasiparticles in the superconducting leads and forms a Yu-Shiba-Rusinov-like (YSR) singlet eigenstate. This excited state gives rise to subgap spectral peaks at energies $E_b$ and $-E_b$. When these subgap excitations cross zero energy, the system undergoes a parity-changing quantum phase transition and the YSR singlet becomes the new ground state. At higher energies there are BCS-like excited singlet states resulting from the hybridization between the empty and doubly occupied states of the quantum impurity. These singlets occur at subgap energies in the opposite limit $U\ll \Delta$ (not shown). (b) Top: Schematics of a normal–quantum dot–superconducting hybrid system with all the relevant energies involved in the problem. In odd-occupancy Coulomb blockade valleys (charging energy $U$), the unpaired spin (green) mimics the physics of a magnetic impurity coupled to a superconductor (coupling ${\Gamma }_{SC}$) with a BCS density of states (purple) with gap $\Delta$. This physics can be considerably modified by the weak coupling $\left({\Gamma }_N\right)$ to a normal probe (orange-yellow), as we discuss in this work. Bottom: this hybrid system can be realized with, e.g., nanowires deposited on top of normal and superconducting electrodes. (c) Standard Kondo singlets that occur as quasiparticles in the normal metal (red) screen the magnetic doublet. (d) Typical spectral density of the hybridized quantum dot in the magnetic doublet ground-state regime showing the coexistence of YSR singlet subgap excitations and a Kondo resonance. The subgap excitations remove spectral weight from the BCS density of states.

Figure 2

Phase diagram for fixed $U=0.01$ and typical spectra for doublet (D), singlet (S), and doublet-singlet crossover (DS) regimes. Shading indicates the estimated width of the crossover region.

Figure 3

(a) Spectra for $\Delta$ ranging from $0.5$ (bottom) to 0 (top). Offsets are added for clarity. (b) Anomalous spectral function $B\left(\omega \right)$ for $\Delta =0.004$ (doublet), $\Delta =0.002$ (doublet-singlet crossover, dashed line), and $\Delta =0.001$ (singlet). Inset: $B\left(\omega \right)$ for $\omega >0$ on the logarithmic frequency scale. The arrow indicates the peak with negative weight in the doublet regime, which is associated with the Kondo effect and the ultimate spin-singlet ground state.

Figure 4

Phase diagram [according to criterion (a)] for the $\text{DS}$ crossover when ${\Gamma }_{SC}$ and $U$ are tuned for different values of ${\Gamma }_N$. The gap is fixed to $\Delta =0.01$. Compared to Fig. 2, here $\Delta$ is fixed rather than $U$. For this reason, the behavior near the origin is different. In this figure, the origin corresponds to the noninteracting $U\to 0$ limit, while in Fig. 2 the origin corresponds to the large-gap $\Delta \to \infty$ limit.

Figure 5

(a) Phase diagram for fixed $U=0.05$ and two values of ${\Gamma }_{SC}$ as a function of ${\Gamma }_N$ and $\Delta$. The colored areas denote doublet regions. (b) Spectral function (curves offset) and (c) anomalous spectrum as we increase ${\Gamma }_N$ along the direction of the arrow in panel (a) ($\Delta =0.0002$ and ${\Gamma }_{SC}=0.004$).

Figure 6

Impurity spectral function $A\left(\omega \right)$ vs coupling to the superconducting lead ${\Gamma }_{SC}$. Calculations are performed at fixed $U=0.01$ and $U/\Delta =5$, and plotted for a range of increasing coupling to the normal-state lead ${\Gamma }_N$: (a) ${\Gamma }_N=0.0001$, (b) ${\Gamma }_N=0.0002$, (c) ${\Gamma }_N=0.0003$, and (d) ${\Gamma }_N=0.0004$. Note the progressively wide range of ${\Gamma }_{SC}$ where a zero-bias resonance exists as ${\Gamma }_N$ increases.

Figure 7

Large $U$ case, $U=0.05$. (a) Spectral densities for decreasing $\Delta$ (curves offset) from $\Delta =0.05$ (bottom) to $\Delta =0.001$ (top). Other parameters: ${\Gamma }_{SC}=0.004,{\Gamma }_N=0.0004$. (b) $T_K^N$ vs $\Delta$ for the same parameters. (c) Spectral functions at finite temperature $T=0.002U$ (shaded curves) as the gap decreases (for a typical charging energy of $U\sim 1\text{meV}$, the temperature used in the calculations would correspond to $T\sim 2\mu \text{eV}\sim 23\text{mK}$). The corresponding zero-temperature results are shown as thin lines.

Figure 8

Effect of the magnetic field on the spectral functions. We plot the spin-averaged spectral function for a range of magnetic fields $B$ in (a) doublet $\left({\Gamma }_{SC}=0.001\right)$, (b) DS crossover $\left({\Gamma }_{SC}=0.00225\right)$, and (c) singlet $\left({\Gamma }_{SC}=0.003\right)$ regimes (curves vertically offset for clarity). Other parameters are $U=0.01,{\Gamma }_N=0.0002,\Delta =0.003$. (d) Position of the parity crossing in magnetic field vs ${\Gamma }_{SC}$.

Figure 9

Impurity spectral function $A\left(\omega \right)$ as a function of the magnetic field $B$. The calculations are performed at fixed $U=0.01,{\Gamma }_{SC}=0.003,\Delta /U=0.3$, and plotted for a range of increasing coupling to the normal-state lead ${\Gamma }_N$: (a) ${\Gamma }_N=0.0001$, (b) ${\Gamma }_N=0.0002$, (c) ${\Gamma }_N=0.0003$ and (d) ${\Gamma }_N=0.0004$. Note that ${\Gamma }_N\ll {\Gamma }_{SC}$ for all cases considered.

Figure 10

(a) Slope of the real part of ${\Sigma }_B\left(B\right)$ self-energy function. This quantity can be interpreted as the renormalization of the effective $g$ factor due to interactions. (b) Zero-frequency value of the real part of the anomalous self-energy, $\mathrm{Re}{\Sigma }_a(\omega =0)$ in the singlet regime, $\Gamma >{\Gamma }_{DS}$.

Figure 11

(a) Phase diagram in the $(B,\Delta )$ plane for ${\Gamma }_N=0$. At ${\Delta }_c\sim 0.0012$, the ground state of the system at $B=0$ changes from singlet to doublet. (b),(c) Zero-frequency spectral function $A(\omega =0)$ plotted as a function of the gap $\Delta$ and the external magnetic field $B$, revealing the behavior of the zero-bias anomaly in the $(\Delta ,B)$ plane. The coupling to the normal-state lead is (b) ${\Gamma }_N=0.0002$, and (c) ${\Gamma }_N=0.0004$. In (b) we also plot (in blue) two possible lines for the evolution of the gap for increasing magnetic fields. We use the function $\Delta \left(B\right)\sim \Delta [1-0.32B-0.1B^2]$, based on a fitting of the experimental data from Ref. [25]. Both curves correspond to gap values $\Delta =0.0011$ and $0.0013$, respectively, which are located on either side of the ${\Gamma }_N=0$ transition around ${\Delta }_c$. The rest of the parameters are ${\Gamma }_{SC}=0.002$ and $U=0.01$.

Figure 12

Left panel: Impurity spectral functions for a range of hybridization strengths to the superconducting lead ($U=0.05$ and $\Delta /U=0.01$). Normal lead is nearly decoupled. The doublet-singlet transition occurs for $\Delta =3.6T_K$ or $T_K=0.3\Delta$, where $T_K$ is Wilson's Kondo temperature. Right panel: Expectation values as a function of ${\Gamma }_{SC}$. All the criteria show a DS transition at ${\Gamma }_{SC}\approx 5.3\times {10}^{-3}$ (arrows).

Figure 13

Expectation values as a function of ${\Gamma }_N$ for a fixed ${\Gamma }_{SC}=0.004$ (the rest of parameters are the same as in Fig. 12). The pairing term (top panel) changes sign at ${\Gamma }_N\approx 1.25\times {10}^{-3}$ whereas the magnetization (bottom panel) is non-zero for a slightly smaller value ${\Gamma }_N={10}^{-3}$.

Figure 14

(a) Characterization of the singlet phase by the anomalous spectral function $B\left(\omega \right)$ for a range of ${\Gamma }_N$. (b) Spectral function $A\left(\omega \right)$. The model parameters are $U=0.05,{\Gamma }_{SC}=0.004,\Delta =0.0002$.

Figure 15

From top to bottom: zero-bias peak in the spectral function $A\left(0\right)$, width of the peak at zero frequency, ${\Omega }^{\pm }\equiv {\int }_0^{\Delta }d\omega B^{\pm }\left(\omega \right)$ [with $B^{\pm }\left(\omega \right)$ being the positive and negative parts of $B\left(\omega \right)$], and positive and negative peak positions of $B\left(\omega \right)$.

Figure 16

NRG results for the Kondo temperature $T_K^N$ of the needle resonance as a function of the exchange coupling to the superconducting lead, ${\Gamma }_{SC}$, for several values of the BCS gap $\Delta$, both in the small $\Delta$ and large $\Delta$ limits.

Figure 17

NRG results for the Kondo temperature $T_K^N$ of the needle resonance as a function of the exchange coupling to the normal lead, ${\Gamma }_N$.