Subgap states and quantum phase transitions in one-dimensional superconductor–ferromagnetic insulator heterostructures

We theoretically study the spectral properties of a one dimensional semiconductor-superconductor-ferromagnetic insulator (SE-SU-FMI) hybrid nanostructure, motivated by recent experiments where such devices have been fabricated using epitaxial growing techniques. We model the hybrid structure as a one-dimensional single-channel semiconductor nanowire under the simultaneous effect of two proximity-induced interactions: superconducting pairing and a (spatially inhomogeneous) Zeeman exchange field. The coexistence of these competing mechanisms generates a rich quantum phase diagram and a complex subgap Andreev bound state (ABS) spectrum. By exploiting the symmetries of the problem, we classify the solutions of the Bogoliubov-de Gennes equations into even and odd ABS with respect to the spatial inversion symmetry $x\to -x$. We find the ABS spectrum of the device as a function of the different parameters of the model: the length $L$ of the coexisting SU-FMI region, the induced Zeeman exchange field $h_0$, and the induced superconducting coherence length $\xi$. In particular we analyze the evolution of the subgap spectrum as a function of the length $L$. Interestingly, we generically find spin-polarized ABS emerging in the subgap region, which, depending on the ratio $h_0/\mathrm{\Delta }$, can eventually cross below the Fermi energy at certain critical values $L_c$, and induce spin- and fermion parity-changing quantum phase transitions. We argue that this type of device constitute a promising highly-tunable platform to engineer subgap ABS.

Figure 1

Schematic representation of the SC-FMI heterostructure.

Figure 2

Energy of the spin-polarized Andreev bound states (upper panel) and total spin $S_z$(lower panel) as a function of $k_{F}L$, for $k_F\xi =7.8$ and $h_0/\mathrm{\Delta }=3.0$ (left panel) and $k_F\xi =3.4$ and $h_0/\mathrm{\Delta }=2.1$ (right panel). Blue and red colors correspond to even and odd states respectively. Lines starting from the top gap edge at positive energy $E/\mathrm{\Delta }=1$ (bottom gap edge at negative energy $E/\mathrm{\Delta }=-1$) correspond to up (down) spin projections of the states. For smaller values of $k_{F}L$ (right panel), plateaus corresponding to regions of integer and half-integer spin are more separated and might become easier to observe in experiments.

Figure 3

Energy of the spin-polarized Andreev bound states as a function of $k_{F}L$ for the three different values of $h_0$ ($h_0/\mathrm{\Delta }=0.8,1.54,2.2$ for the lower, middle, and upper panels) and $k_F\xi =8.2$. Blue and red colors correspond to even and odd states respectively. Lines starting from negative (positive) energies correspond to down (up) spin projections of the state. Note that the value of $h_0/\mathrm{\Delta }$ sets the asymptotic limit for the Andreev states and is crucial to determine the overall subgap spectrum.