Magnetic impurities in spin-split superconductors

Hybrid semiconductor-superconductor quantum dot devices are tunable physical realizations of quantum impurity models for a magnetic impurity in a superconducting host. The binding energy of the localized subgap Shiba states is set by the gate voltages and external magnetic field. In this work we discuss the effects of the Zeeman spin splitting, which is generically present both in the quantum dot and in the (thin-film) superconductor. The unequal $g$ factors in semiconductor and superconductor materials result in respective Zeeman splittings of different magnitude. We consider both classical and quantum impurities. In the first case we analytically study the spectral function and the subgap states. The energy of bound states depends on the spin-splitting of the Bogoliubov quasiparticle bands as a simple rigid shift. For the case of collinear magnetization of impurity and host, the Shiba resonance of a given spin polarization remains unperturbed when it overlaps with the branch of the quasiparticle excitations of the opposite spin polarization. In the quantum case, we employ numerical renormalization group calculations to study the effect of the Zeeman field for different values of the $g$ factors of the impurity and of the superconductor. We find that in general the critical magnetic field for the singlet-doublet transition changes nonmonotonically as a function of the superconducting gap, demonstrating the existence of two different transition mechanisms: Zeeman splitting of Shiba states or gap closure due to Zeeman splitting of Bogoliubov states. We also study how in the presence of spin-orbit coupling, modeled as an additional noncollinear component of the magnetic field at the impurity site, the Shiba resonance overlapping with the quasiparticle continuum of the opposite spin gradually broadens and then merges with the continuum.

Figure 1

Spin-projected spectral functions (blue for spin-up, red for spin-down) for a range of the dimensionless impurity coupling $\alpha =\pi \rho JS/2$ in a Zeeman-split superconductor with $b_{\mathrm{bulk}}/\mathrm{\Delta }=0.4$.

Figure 2

(a) Schematic phase diagram for $B=0$. (b) Subgap splitting for finite field $B$.

Figure 3

Spectral function of the impurity for the spin singlet (a) and spin doublet ground state (b). The parameters are $b_{\mathrm{imp}}/U=0.005$ and $\mathrm{\Delta }/U=0.02$. For the singlet ground state $\mathrm{\Gamma }/U=0.2$ and for the doublet $\mathrm{\Gamma }/U=0.075$. The spectrum for $r=g_{\mathrm{bulk}}/g_{\mathrm{imp}}=0$ is shown in central panels, the adjoining panels show the evolution of the position of the Shiba resonances as $\left|r\right|$ increases, and the top/bottom panels correspond to $r=1$ and $r=-1$, respectively.

Figure 4

(a) Phase diagram in the $(B,\mathrm{\Delta })$ plane for several values of $r=g_{\mathrm{bulk}}/g_{\mathrm{imp}}$. Here $\mathrm{\Gamma }/U=0.2, {\mathrm{\Delta }}_c/U\approx 0.13$. (b) For small $\mathrm{\Delta }$ the singlet-doublet transition coincides with the closure of the SC gap for $b_{\mathrm{bulk}}\approx 2\mathrm{\Delta }$. (c) The expectation value $\langle S_z\rangle$ and (d) the pairing amplitude $\langle d_{\uparrow }{d}_{\downarrow }\rangle$ abruptly change across the phase transition. Here $\mathrm{\Delta }=0.385{\mathrm{\Delta }}_c$.

Figure 5

Spin up in-gap resonances and continuum of excitations for several values of $r$. Here $b_{\mathrm{bulk}}/U=0.01$ was kept constant. The finite width of the Shiba resonances is a broadening artifact: These resonances are true $\delta$ peaks at zero temperature.

Figure 6

Spectral function of the spin-up Shiba resonance and the quasiparticle continuum for several values of the spin-flipping transverse magnetic field. As $b_x$ increases, the Shiba resonance broadens.