Experimental observation of dual magnetic states in topological insulators

The dual magnetic states in topological insulators reveal the interplay between two ordering phenomena: magnetism and topology.


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Section S1. Sample preparation Section S2. XAS/XMCD measurement Section S3. Multiplet calculations Section S4. Sum-rules analysis Fig. S1. Schematic diagram of the experimental setup for XAS and XMCD measurement. Fig. S2. Deconvolution of the mixed Cr valences. Fig. S3. The sum-rules analysis. Table S1. Summary of the XMCD-derived m spin for the global-, surf-, and mid-doped Cr-doped Bi 2 Se 3 , respectively, at 3 K. References (37-43) Section S1. Sample preparation The 10 nm 3% Cr-doped Bi 2 Se 3 or Bi 1.94 Cr 0.06 Se 3 thin films used in this study were grown in an ultra-high vacuum (UHV) chamber using a Perkin-Elmer molecular beam epitaxy (MBE) system on Si (111) substrates using the well-established recipe (37,38,39).  substrates are first cleaned with standard RCA cleaning method. The substrates are then treated by hydrofluoric acid wet etching so that the surface dangling bonds are saturated by a layer of hydrogen atoms and the (111) surface is 1 x 1 reconstructed. The substrates were kept at 200°C in UHV until the two-dimensional streak pattern appeared as monitored by the real-time reflection high-energy electron diffraction (RHEED).
High-purity Bi (99.9999%) was evaporated from a conventional effusion cell at 470°C, while Se (99.99%) was created from a cracker cell (SVTA) at 270°C. The desired doping concentrations of Cr was obtained by evaporating Cr (99.99%) from the effusion cell at different temperatures and then calibrated by energy dispersive X-ray analysis. Immediately after the growth of the Bi 1.94 Cr 0.06 Se 3 thin films, 2 nm Al was insitu evaporated onto the sample to protect it from oxidation and environmental doping during the transport to the synchrotron facility.
All the global-and the modulation-doped Bi 2 Se 3 thin films used in this study have identical thicknesses (10 nm). In the global-doped samples, the Cr dopants are uniformally distributed throughout the sample, wheares in the modulation-doped Bi 2 Se 3 , the dopants were only introduced into the topmost (referred to as surf-doped) or in the middle (referred to as mid-doped) one quintuple layer (QL), respectively, as described in the main text. This was achieved by the slow deposition rate (i.e., ~1 QL per minute as calibrated from the RHEED oscillation patterns) and the layer-by-layer growth geometry of Bi 2 Se 3 matrix. 40,41,42 It allows us to accurately control the Cr-dopant distribution profiles along the epitaxial growth direction (deviation < ±0.1 nm) (40). Note that the 'surface doping' in this work has a different physical meaning as in ref. (41), where metallic adatoms are deposited onto the surface of a bare TI. Here we discuss the as-grown Bi 2 Se 3 with the topmost 1 QL doped with Cr ions, in order to make it comparable with the mid-doped Bi 2 Se 3 (both contain only 1 QL of magnetic impurities). The merit of this design is to eliminate any influence of the thickness of the magnetically doped layer.

Section S4. Sum-rules analysis
The spin (m spin ) and orbital (m orb ) magnetic moments of the Cr-doped Bi 2 Se 3 epitaxial thin films were calculated by applying the sum rules (42,43) to the integrated XMCD and summed XAS spectra of the Cr L 2,3 edges, based on where E, n h , SC, and <T z > represent the photon energy, the number of d holes, the spin correction (SC) factor, and the magnetic dipole term, respectively. In order to exclude the non-magnetic contribution of the XAS spectra an arctangent-based step function is used to fit the threshold. The spectral overlap or j-j mixing 33 had to be taken into account because of the relatively small spin-orbit coupling in the Cr 2p level.
The value of SC, i.e. 2.0 ± 0.2 for Cr, was estimated by calculating the L 2,3 multiplet structure for a given ground state, applying the sum rule on the calculated XMCD spectrum, and comparing the result with the spin moment calculated directly for this ground state. 39 Furthermore, m spin needs to be corrected for the magnetic dipole term <T z >, however, its contribution is small for a Cr t 2g 3 configuration, giving an error < 5%. Using n h = 6.30 for the Cr  antiparallel alignment of the spin and orbital moment in Cr. This agrees with the Hund's rule for Cr, whose 3d shell is less-than-half full. The octahedral crystal-field interaction quenches m orb , since the 3d electrons occupy the threefold degenerate majority-spin t 2g orbitals, leading to a nearly vanishing m orb . In reality the octahedral symmetry is trigonally distorted to C 3v with three anions above and below the Cr ion, which can alter the m orb but hardly affects the m spin (29). Figure S3 presents typical pairs of XAS for opposite circular polarizations and the corresponding XMCD of the global-, surf-, and mid-doped Bi 2 Se 3 , respectively, obtained at 3 K. Noting that the integrated total XAS and XMCD spectra well saturate at ∼ 590 eV, the selected integration range, i.e. up to 600 eV, is sufficient. posite circular mid-doped Cr spectra aver r polarization r-doped Bi 2 Se raged over the s and e 3 obtained at e two Cr