Yu-Shiba-Rusinov states in phase-biased superconductor–quantum dot–superconductor junctions

We study the effects of a phase difference on Yu-Shiba-Rusinov (YSR) states in a spinful Coulomb-blockaded quantum dot contacted by a superconducting loop. In the limit where charging energy is larger than the superconducting gap, we determine the subgap excitation spectrum, the corresponding supercurrent, and the differential conductance as measured by a normal-metal tunnel probe. In absence of a phase difference only one linear combination of the superconductor lead electrons couples to the spin, which gives a single YSR state. With finite phase difference, however, it is effectively a two-channel scattering problem and therefore an additional state emerges from the gap edge. The energy of the phase-dependent YSR states depend on the gate voltage and one state can cross zero energy twice inside the valley with odd occupancy. These crossings are shifted by the phase difference towards the charge degeneracy points, corresponding to larger exchange couplings. Moreover, the zero-energy crossings give rise to resonant peaks in the differential conductance with magnitude $4e^2/h$. Finally, we demonstrate that the quantum fluctuations of the dot spin do not alter qualitatively any of the results.

Figure 1

Sketch of the device studied here (following the experiments in Refs. [12, 13]), comprised of a Coulomb-blockaded QD (green) tunnel coupled to superconducting leads (blue) with an applied phase difference $\varphi$ controlled by a magnetic flux. A backgate voltage is adjusted to provide a single spin-1/2 on the QD, and the resulting phase-dependent YSR states are probed by a normal-metal tunnel probe (yellow).

Figure 2

Schematics of the two situations $E_C>\mathrm{\Delta }$ and $E_C<\mathrm{\Delta }$. In both cases there is a subgap state in the odd diamond. For large $\mathrm{\Delta }$ it appears because of hybridization of the $N=0$ and $N=2$ charge states by the superconductor, while for large $E_C$ it appears because of hybridization between the dot electron and quasiparticles in the leads, forming a bound state, which is the YSR state. In both cases the average occupation on the dot is equal to one and the subgap states have similar dependencies on gate voltage. In this paper we focus on the situation (a) with $E_C>\mathrm{\Delta }$.

Figure 3

The figure shows the YSR state energies $V=E_{\pm ,\sigma }$ given in Eq. (34) as dashed lines. The YSR energies are plotted as a function of dimensionless gate voltage $x=1+2{\varepsilon }_d/U$ in (a)–(d), and as function of phase difference in (e)–(h). $x=0$ corresponds to the middle of the odd diamond and $x=\pm 1$ are the charge degeneracy points, see Eqs. (11) and (12). The lines are overlayed on contour plots of the differential conductance [see Fig. 1 and Eq. (39)] using $g_{NN}=0.1$ and the parametrization of the couplings given in Eq. (A4). The arrows indicate the spin of the excited states. The exchange coupling at the middle of the diamond is set to $g(x=0)=0.5$ in (a), (c), (e), and (g) and to $g(x=0)=1.5$ in (b), (d), (f), and (h). In (a), the YSR states are seen to cross zero energy when the effective exchange coupling increases as $x$ is changed from 0 to 1 towards the charge degeneracy points. (b) With a stronger coupling, the eye-shaped feature in the middle becomes smaller, and for strong enough coupling it disappears. (c) With finite phase difference, two new bound states appear. (d) A finite phase difference reduces the effective exchange coupling and restores the eye-shaped crossings. The detailed phase evolution is shown in (e) and (f) for a cut in gate voltage corresponding to $x=0.5$. While (a)–(f) are for a symmetrically coupled junction ($\theta =\pi /4$), (g) and (h) show the phase dependence for an asymmetrically coupled junction ($\theta =\pi /3$). Note that the finite-bias degeneracy at $\varphi =\pi$ is lifted by the asymmetry.

Figure 4

Differential conductance $dI/dV$ in units of $2e^2/h$ between the normal electrode and the superconducting leads, see Fig. 1. The different lines correspond to constant phase cuts ($\varphi =0,\pi /2,\pi$) in Figs. 3–3, as indicated at the top of the figure. The left panels (a) and (c) are for weak coupling, $\left[g\right(x=0)=0.5]$, while the right panels (b) and (d) are for stronger coupling $\left[g\right(x=0)=1.5]$. The top panels (a) and (b) are for symmetric coupling $\theta =\pi /4$, while the bottom panels (c) and (d) are for asymmetric coupling $\theta =\pi /3$. In all cases the gate voltage is set to the particle-hole symmetric point $x=0$, and the coupling to the normal lead was chosen to be $g_{NN}=0.1$.

Figure 5

Phase diagram with couplings parametrized by Eq. (11), in terms of $\mathrm{\Gamma }={\nu }_F(t_L^2+t_R^2)$, and the dimensionless level position (gate voltage) $x=1+2{\varepsilon }_d/U$. The phase boundaries are independent of $\mathrm{\Delta }$, and their dependencies on $x$ and $\theta$ are indicated, with the function $f\left(\theta \right)$ defined as $f\left(\theta \right)={[1/2\left\{{sin}^2\left(2\theta \right)+\sqrt{4+{sin}^4\left(2\theta \right)}\right\}]}^{1/2}$. In this diagram the coupling asymmetry parameter was chosen to $\theta =\pi /3$, whereas for a symmetric junction with $\theta =\pi /4$, the $0^{\prime }-0$ boundary will never be reached, consistent with earlier results on the Anderson model [27, 50, 51].

Figure 6

Supercurrent vs phase difference at the particle-hole symmetric point $x=0$, for exemplary values of $g$ and $\theta$ (see inset) corresponding to the four different kinds of Josephson junctions. Similar results were first obtained in Refs. [27, 29] and experimentally demonstrated in Ref. [52].

Figure 7

Subgap excitation spectrum $E_{\mathrm{ex}}=E_{\delta }-E_{\downarrow }$, with respect to the ground-state doublet $|D_{\downarrow }\rangle$ for particular channel $\delta =1,2$. It is assumed that the magnetic field in the superconductor is screened $g_{\mathrm{ce}}\approx 0$. For antiferromagnetic coupling $g_{\delta }=0.05$, the excited state is singletlike, which for high magnetic fields becomes polarized-spin-like $|{\uparrow }_{\delta },\downarrow \rangle$.