Fluxoid-induced pairing suppression and near-zero modes in quantum dots coupled to full-shell nanowires

We analyze the subgap excitations and phase diagram of a quantum dot (QD) coupled to a semiconducting nanowire fully wrapped by a superconducting (S) shell. We take into account how a Little-Parks (LP) pairing fluxoid (a winding in the S phase around the shell) influences the proximity effect on the dot. We find that under axially symmetric QD-S coupling, shell fluxoids cause the induced pairing to vanish, producing instead a level renormalization that pushes subgap levels closer to zero energy and flattens fermionic parity crossings as the coupling strength increases. This fluxoid-induced stabilization mechanism has analoges in symmetric S-QD-S Josephson junctions at phase $\pi$, and can naturally lead to patterns of near-zero modes weakly dispersing with parameters in all but the zeroth lobe of the LP spectrum.

Figure 1

QD/full-shell nanowire model. (a) Schematic lateral cut of a semiconducting nanowire (green) under a longitudinal magnetic field $B$. The wire is coupled to a normal (N) metal (yellow) to the left and fully wrapped by a superconducting (S) thin shell (red) to the right, with an uncovered segment in between. The junction traps quantum dot (QD) states that can be tuned with a back gate. A schematic wave function of the lowest energy QD state is shown in blue. It is mostly localized in the junction but penetrates into the covered region (dashed blue), where band-bending pushes it towards the superconductor-semiconductor interface. (b) Cross-sectional cut of the same wire inside the full-shell region. ${\mathit{r}}_0=(\langle y\rangle ,\langle z\rangle )$ represents the deviation of the wave-function mean position with respect to wire's axis, informing of its degree of axial symmetry. (c), (d) Generalized superconducting impurity Anderson (SIA) model introduced in this work to describe the above hybrid system. A spinful single impurity with energy ${\epsilon }_0$ and charging energy $U$ is coupled to N with coupling strength ${\mathrm{\Gamma }}_{\mathrm{N}}$ and to a hollow S cylinder of radius $R$ with a hopping amplitude $t\left(\theta \right)$ that depends on the polar angle $\theta$. A flux $\mathrm{\Phi }$ threads the full-shell wire inducing a winding in the phase of the S order parameter $\mathrm{\Delta }{e}^{in\theta }$, where the integer $n$ represents the fluxoid quantization and $\mathrm{\Delta }$ is the flux-dependent S gap.

Figure 2

QD spectral density. (Left half) Mean-field local density of states (LDOS) at the QD for a symmetric wave function distribution inside the covered region of the full-shell wire [${\mathit{r}}_0=0$, see Fig. 1]. The first, second and third columns correspond, respectively, to the regimes of weak (${\mathrm{\Gamma }}_{\mathrm{S}}/U=0.08$), intermediate (${\mathrm{\Gamma }}_{\mathrm{S}}/U=0.6$), and strong (${\mathrm{\Gamma }}_{\mathrm{S}}/U=2$) coupling between the QD and the superconductor. (Right half) The same but for an asymmetric wave function distribution ($|{\mathit{r}}_0|=R/2$, where $R$ is the shell inner radius). In the first row [(a)–(f)], the LDOS is plotted vs the magnetic field $B$ in the weak Little-Parks (LP) regime at half occupation (${\epsilon }_0/U=-0.5$), whereas the destructive LP regime is displayed in the second row [(g)–(l)]. The energy $\omega$ is measured with respect to the superconductor Fermi energy. In the third row [(m)–(r)], we show the LDOS vs ${\epsilon }_0/U$ at finite magnetic field ($B=115$ mT), corresponding to the center of the $n=1$ LP lobe of (g)–(l). Other parameters: $R=65$ nm, shell thickness $d=25$ nm, InAs g-factor $g=14$, Al gap at zero field ${\mathrm{\Delta }}_0=0.2$ meV, temperature $T=10$ mK, charging energy $U=1$ meV, and ${\mathrm{\Gamma }}_{\mathrm{N}}={10}^{-3}$ meV. The S coherence length is $\xi =185$ nm ($\xi =80$ nm) in the destructive (weak) LP regime.

Figure 3

QD phase diagram. [(a) and (b)] Spin polarization $P=\langle n_{\uparrow }\rangle -\langle n_{\downarrow }\rangle$ of the QD level vs ${\mathrm{\Gamma }}_{\mathrm{S}}/U$ and ${\epsilon }_0/U$ at the center of the $n=1$ destructive LP lobe. In the case (a) of a symmetric coupling (${\mathit{r}}_0=0$), the singlet-doublet phase transition boundary displays a chimneylike shape where the doublet ground state phase ($P\ne 0$) extends to arbitrarily large values of ${\mathrm{\Gamma }}_{\mathrm{S}}/U$. In the asymmetric case (b) ($|{\mathit{r}}_0|=R/2$ here), we recover a domelike shape for the singlet-doublet boundary, typical of the standard SIA, above which the ground state is a singlet ($P=0$). [(c) and (d)] Minimum excitation energy $E_{\min }$ normalized to the zero-field Al gap ${\mathrm{\Delta }}_0=\mathrm{\Delta }\left(0\right)$. The singlet-doublet boundary is associated to $E_{\min }=0$ fermionic parity crossings (black). Rest of parameters as in Fig. 2.

Figure 4

Microscopic simulations. (a) Energy spectrum at zero applied magnetic field of a finite-length device like the one in Fig. 1 of the main text, as a function of back-gate voltage $V_{\mathrm{bg}}$ (which is effective only in the uncovered segment of the wire). The level quantization is produced by lateral and longitudinal confinement. The presence of the electrostatic environment is taken into account at a self-consistent mean-field level. (b) Wave-function weight $W_{\mathrm{S}}$ inside the shell-covered region of the wire vs $V_{\mathrm{bg}}$ for the lowest-energy eigenstate. (c) Expectation value of the $z$ position of the wave function inside the full-shell region $\langle z\rangle$ (red curve) and standard deviation ${\sigma }_z^2$ (blue curve) normalized to the wire's radius $R$ as a function of $V_{\mathrm{bg}}$ for the lowest-energy eigenstate. [(d) and (e)] Wave-function profiles (probability density) inside the wire for the lowest energy level marked with a red/blue circle in (a). (f) Electrostatic profile inside the wire at $V_{\mathrm{bg}}=0.4$ V. Parameters are provided in Table 1.

Figure 5

QD phase diagrams for $n=0$ and 2. [(a) and (b)] Mean-field dot-level spin polarization $P=\langle n_{\uparrow }\rangle -\langle n_{\downarrow }\rangle$ vs ${\mathrm{\Gamma }}_{\mathrm{S}}/U$ and ${\epsilon }_0/U$ at the center of the $n=0$ LP lobe ($B=0$ mT) in the destructive LP regime: (a) a symmetric coupling (${\mathit{r}}_0=0$) and (b) an asymmetric one ($|{\mathit{r}}_0|=R/2$). In both cases, we have the usual dome shape for the singlet-doublet transition boundary, where the system has a singlet ground state ($P\approx 0$) in the limit of large coupling (${\mathrm{\Gamma }}_{\mathrm{S}}/U\gtrsim 1$). [(c) and (d)] Plot of the minimum excitation energy normalized to the zero-field Al gap. The singlet-doublet boundary is associated to a zero energy parity crossing (black). [(e)–(h)] Same but at the center of the $n=2$ LP lobe ($B=230$ mT), in the destructive LP regime as well. Notice that the phase diagram for $n=2$ is very similar to the $n=1$ case, shown in Fig. 3 of the main text.

Figure 6

QD phase diagram vs QD-S coupling asymmetry ${\mathbf{r}}_{\mathbf{0}}$. Mean-field dot-level spin polarization (a) and minimum excitation energy (b) vs $r_0/R$ and ${\epsilon }_0/U$ at the center of the $n=1$ lobe ($B=115$ mT) in the destructive LP regime, for a weak coupling to the superconductor (${\mathrm{\Gamma }}_{\mathrm{S}}=0.08$ meV). In (c), we show the LDOS for three different values of $r_0$: the symmetric case $r_0=0$, $r_0/R=0.2$, and $r_0/R=0.5$. In (d)–(f) and (g)–(i), we show the same simulations for intermediate (${\mathrm{\Gamma }}_{\mathrm{S}}=0.2$ meV) and a strong (${\mathrm{\Gamma }}_{\mathrm{S}}=2$ meV) coupling to the superconductor, respectively.

Figure 7

Evolution of the near-zero-energy anomaly with QD-S coupling asymmetry $r_0/R$. LDOS at strong coupling to the superconductor (${\mathrm{\Gamma }}_{\mathrm{S}}=2$ meV) and within the first lobe ($B=115$ mT) vs QD-S coupling asymmetry $r_0/R$ and (a) vs energy $\omega$ at half-filling (${\epsilon }_0/U=-0.5$) or (b) vs the QD energy level ${\epsilon }_0/U$ at $\omega =0$. These figures show that the ABSs remain robustly close to zero energy as long as the degree of QD-S coupling asymmetry is $r_0\lesssim 0.1R$.

Figure 8

QD phase diagram for different QD's inverse widths. Minimum excitation energy vs $r_0/R$ and ${\epsilon }_0/U$ at the center of the $n=1$ lobe ($B=115$ mT) in the destructive LP regime, for a weak coupling to the superconductor (${\mathrm{\Gamma }}_{\mathrm{S}}=0.08$ meV) [(a)–(c)] and for a strong coupling (${\mathrm{\Gamma }}_{\mathrm{S}}=2$ meV) [(d)–(f)].