Chern insulator and Chern half-metal states in the two-dimensional. spin-gapless semiconductor Mn 2 C 6 S 12

Transcription

1 Supporting Information for Chern insulator and Chern half-metal states in the two-dimensional spin-gapless semiconductor Mn 2 C 6 S 12 Aizhu Wang 1,2, Xiaoming Zhang 1, Yuanping Feng 3 * and Mingwen Zhao 1 * 1 School of Physics and State Key Laboratory of Crystal Materials, Shandong University, Jinan, , Shandong, China. 2 Departmen of Electrical and Computer Engineering & Department of Physics, National University of Singapore, Singapore, , Singapore 3 Department of Physics & Centre for Advanced Two-dimensional Materials, National University of Singapore, Singapore, , Singapore This SI is meant to support the explanations described in the main text entitled Chern insulator and Chern half-metal states in the two-dimensional spin-gapless semiconductor Mn2C6S12. Part I The common 2D materials with Dirac cones. Part II The comparison of band structures based on DFT and Wannier90 Part III The spin-polarized electron density (Δρ) at FM and AFM ordering Part IV The mutation of the Chern number and the Berry curvature 1

2 Part I The common 2D materials with Dirac cones. Figure S1 The common 2D materials with Dirac cones, which contain various elementary substance and compounds. 2

3 Part II The comparison of band structures based on DFT and Wannier90 Figure S2 The comparison of band structures without SOC around the Fermi level, obtained from DFT calculations and Wannier method without and with SOC. Red and blue lines indicate the spin up and spin down channels. The energy at the Fermi level was set to zero 3

4 Part III The spin-polarized electron density (Δρ) at FM and AFM ordering Figure S3 Schematic representations of the ferromagnetic (FM) and antiferromagnetic (AFM) ordering in the Mn2C6S12 framework. Yellow and blue bubbles represent the spin-polarized electron density Δρ calculated from Δρ=ρ - ρ. The arrows indicate the two spin directions. The total magnetic moments (m) and relative energies (E) in one unit cell are presented 4

5 Part IV The mutation of the Chern number and the Berry curvature Figure S4. (a) The band structures of Mn2C6S12 lattice with spin-orbit coupling (SOC). (b) The calculated anomalous Hall conductivity (AHC) and (c) the enlarge AHC near the Fermi level. (d) The distribution of the Berry curvature of the band II between Δ1 and Δ2 in the reciprocal space. The energy at the Fermi level was set to zero. The anomalous Hall conductivity (AHC) or the Chern number is related to the integral of Berry curvature of the occupied states in the BZ. As the Fermi level locates in the band gap Δ1 or Δ3, the Mn2C6S12 lattice act as a topological insulator with an integral Chern number of -1, as shown in Figure S4(a). When the Fermi level resides in the band gap Δ2 or Δ4, however, the Mn2C6S12 lattice is not a topological insulator but a half metal, i.e., an insulating spin channel with a band (bands II and III) of the opposite spin direction crossing the band gap, as shown 5

6 Figure S4(a). In this case, the Berry curvature of the conducting band contributes to the AHC and Chern number. Usually, Berry curvature is mainly localized around the K points where the band gap-opening or inversion occurring induced by SOC, which make the distribution of Berry curvature being not uniform. For example, the distribution of the Berry curvature of the band II is highly localized at the corners the BZ, as shown in Figure S4(d), corresponding to the bottom the band. Therefore, as the Fermi level is in the band gap Δ2 and cross the band II, the Chern number undergoes a drastic change from -1 to almost +1, due to contribution of the band II. When the Fermi level crosses the band I in a large energy region (e.g. from -0.5 ev to ev), the AHC remains zero, because the Berry curvature of the band in this energy region is zero. This is also the reason why the Chern number is almost -1 when the Fermi level is in the band gap Δ4. It is noteworthy that as the Fermi level is in the band gap Δ2 (or Δ4), the Mn2C6S12 nanoribbon will carry quantized spin up (or down) current along the edges characterized by AHC, together with the current in the opposite spin direction in the interior region arising from the conducting bands crossing the band gap. 6