Van Vleck Nature of Carrier-Free Ferromagnetic Order in Vanadium-Doped Three-Dimensional Topological Insulators

We experimentally demonstrate that the long-range ferromagnetic (FM) order in vanadium (V)-doped topological insulator Sb$_{\mathrm{2}}$Te$_{\mathrm{3}}$ has the nature of van Vleck-type ferromagnetism, using the state-of-art low-temperature Electron Energy Loss Spectroscopy (EELS). Contrary to the temperature-independent Te M$_{\mathrm{4,5\thinspace }}$peak, there is an unusual redshift of the V L$_{\mathrm{3}}$ and L$_{\mathrm{2}}$ peak positions and unambiguous change of the L$_{\mathrm{3}}$:L$_{\mathrm{2}}$ peak ratio at T$=$10K. Further high-order Green's function's EELS simulation and magnetotransport show that the shift of the peak position and change of the L$_{\mathrm{3}}$:L$_{\mathrm{2}}$ ratio are originated from the development of the core-level FM order, indicating that in V-doped Sb$_{\mathrm{2}}$Te$_{\mathrm{3}}$, partially filled core states will also contribute to FM order. Since van Vleck magnetism is a result of summing over all states, this magnetization of core level verifies the van Vleck-type ferromagnetism in a direct manner.