Persistent Ferromagnetism and Topological Phase Transition at the Interface of a Superconductor and a Topological Insulator

At the interface of an $s$-wave superconductor and a three-dimensional topological insulator, Majorana zero modes and Majorana helical states have been proposed to exist respectively around magnetic vortices and geometrical edges. Here we first show that randomly distributed magnetic impurities at such an interface will induce bound states that broaden into impurity bands inside (but near the edges of) the superconducting gap, which remains open unless the impurity concentration is too high. Next we find that an increase in the superconducting gap suppresses both the oscillation magnitude and the period of the Ruderman-Kittel-Kasuya-Yosida interaction between two magnetic impurities. Within a mean-field approximation, the ferromagnetic Curie temperature is found to be essentially independent of the superconducting gap, an intriguing phenomenon due to a compensation effect between the short-range ferromagnetic and long-range antiferromagnetic interactions. The existence of robust superconductivity and persistent ferromagnetism at the interface allows realization of a novel topological phase transition from a nonchiral to a chiral superconducting state at sufficiently low temperatures, providing a new platform for topological quantum computation.

Figure 1

(a) Schematic of randomly distributed magnetic impurities at the interface of a 3D TI and a superconductor. (b) Side view of the TI-SC heterostructure exhibiting the positions of the magnetic impurities.

Figure 2

(a) DOS as a function of the electron energy $E$ at different $x$. (b) Renormalized superconducting gap as a function of $x$ (black), and the critical concentration $x_c$ for systems with different superconducting gaps (blue).

Figure 3

RKKY interaction between two magnetic impurities as a function of the separation $R$ and the Fermi energy $E_f$ calculated with $J=0.5\mathrm{eV}$, $E_f=100\mathrm{meV}$ for (a) and $R=10\mathrm{nm}$ for (b). The insert in (a) shows the case for the Fermi surface located at the Dirac point.

Figure 4

(a) The effective exchange field $V_{\text{ex}}$ (black line) and the superconducting gap $\mathrm{\Delta }$ (red line) as a function of the temperature. $T_s$ indicates the topological phase transition temperature, which is below the ferromagnetic Curie temperature $T_c$. (b) The Chern number as a function of $V_{\text{ex}}$. The insets illustrate the bulk band spectra (black solid lines) and edge states (red dashed lines) in the helical and chiral phases, calculated with $V_{\text{ex}}=5$ and 25 meV, respectively. Other parameters include $\mathrm{\Delta }=15\mathrm{meV}$, $J=0.5\mathrm{eV}$, $E_f=0\mathrm{meV}$, and $x=3\%$.