Proximity-induced global time-reversal symmetry (TRS) breaking and enhanced surface ferromagnetism mediated by Dirac fermions in bilayers of magnetic topological insulators (TIs)

Proximity-induced magnetic effects on the surface Dirac spectra of TIs are investigated by scanning tunneling spectroscopic (STS) studies of bilayer structures consisting of an undoped TI layer Bi$_{\mathrm{2}}$Se$_{\mathrm{3}}$ and (Bi$_{\mathrm{1-x}}$Sb$_{\mathrm{x}})_{\mathrm{2}}$Te$_{\mathrm{3}}$ on top of a Cr-doped, magnetic TI of 6 quintuple-layer (QL) thickness.$^{\mathrm{1}}$ For all samples with the top layer thinner than 4-QL, a surface gap $\Delta $ opens up below $T_{c}^{2D} $, much higher than the bulk Curie temperature $T_{c}^{3D} $ derived from the anomalous Hall resistance. The temperature ($T)$ evolution of $\Delta $ shows an initial increase below $T_{c}^{2D} $, followed by a `dip' near $T_{X}$, and then rises again, reaching maximum at $T\ll T_{c}^{3D} $. The gap is spatially inhomogeneous, and its average value and spatial homogeneity at low $T$ increases with applied magnetic field $H$ and Cr-doping level $x$. The appearance of massive Dirac spectra below $T_{c}^{2D} $is the result of global TRS breaking in the surface state of TIs. The non-monotonic $T$-dependence of $\Delta $ and the finding of $T_{c}^{2D} \gg T_{c}^{3D} _{\mathrm{\thinspace }}$may be attributed to proximity magnetism induced by a 3D contribution from the bulk magnetism that dominates at low $T$, and a 2D contribution from the RKKY interaction mediated by surface Dirac fermions, which dominates at $T_{c}^{3D} \ll T_{X}