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SCIENCE CHINA Physics, Mechanics & Astronomy, Volume 63 , Issue 10 : 107011(2020) https://doi.org/10.1007/s11433-019-1515-4

Layer construction of topological crystalline insulator LaSbTe

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  • ReceivedDec 14, 2019
  • AcceptedJan 20, 2020
  • PublishedMay 9, 2020
PACS numbers

Abstract


Acknowledgment

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11504117, 11674369, 11925408, 11921004, and 11974395). ZhiYun Tan acknowledges the support from the Foundation of Guizhou Science and Technology Department (Grant No. QKH-LHZ20177091). ZhiJun Wang acknowledges the support from the National Thousand-Young-Talents Program, the CAS Pioneer Hundred Talents Program, and the National Natural Science Foundation of China. HongMing Weng acknowledges the support from the National Key Research and Development Program of China (Grant Nos. 2016YFA0300600, 2016YFA0302400, and 2018YFA0305700), and the K. C. Wong Education Foundation (Grant No. GJTD-2018-01). We acknowledge helpful discussions with Yi Jiang and ZhiDa Song.


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Appendix

Appendix

Single-layer calculations of $o$-LaSbT

In this section, we determine whether one QL of tci is a 2D TI and two QLs of tci is a trivial insulator. First, we construct a corresponding single-layer structure for these two cases, as shown in Figure A1(a) and (e). Then, we compute the band structures without and with considering SOC. Without SOC, there are one band inversion and double band inversions along $\Gamma$-Y for one QL and two QLs, respectively. When SOC is considered, these two cases are fully gapped so that the TR $\mathbb~Z_2$ can be well defined. Since one QL keeps inversion symmetry, $\mathbb~Z_2$ can be simply derived by Fu-Kane formula. According to eq. (1), we obtain $\mathbb~Z_2=1$ for one QL. However, for two QLs, inversion symmetry is broken. Therefore, the Wilson-loop method is employed to calculate the Wannier-center flow. $\mathbb~Z_2$ equals zero because the number of crossings between the Wilson-loop bands and reference line is even. By calculating $\mathbb~Z_2$ index, we confirm each QL of tci is a well-defined 2D TI and stacking two weakly coupled 2D TIs can generate a normal insulator with trivial $\mathbb~Z_2$ index.

LC of $o$-LaSbTe

The LC method has been introduced to map the symmetry-based indicator and topological invariants [13]. For ti with SG. $P4/nmm$, our convention is the same as in ref. [13], so we can directly use the eLC of SG.129 as provided in S Table 5 of ref. [13]. However, for tci with SG. $Pmcn$ in this work, the convention is different from that in ref. [13]. Then, we derive the eLC of SG. $Pmcn$ and tabulate it in Table AA1. The second eLC (001; ${1}/{4}$) in Table AA1 represents the eLC of tci.

Appendix


References

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  • Figure 1

    (Color online) (a) Crystal structure of ti. (b) Layer construction (LC) of a weak TI as the stacking of 2D TI (yellow plane) in lattice space corresponding to ti. The red dots are inversion centers. (c) Bulk Brillouin zone (BZ) and high symmetrical crystal momenta for ti. The number in parentheses near high symmetrical momenta is the number of occupied Kramers pair bands with negative parity eigenvalues. (d)-(f) are for tci. In (e) the mirror plane, glide plane and screw axis are also shown. In (f) the blue plane indicates the mirror plane and the red lines inside of it represent the nodal rings calculated without spin-orbit coupling (SOC). The dark gray planes are the projected surface BZ for (100) and (1$\bar~1$0) surfaces of tci, where the red hourglass shapes represent the hourglass-like surface states along the two paths.

  • Figure 2

    (Color online) Calculated phonon spectra for (a) ti and (b) tci, respectively. (c) Two unit cells of ti along the $c$-axis with arrows on Sb atoms denoting the displacement in one of the soft phonon modes at $Z$. The view from $c$-axis of the (d) top and (e) bottom Sb layer for a $4~\times~4~\times~1$ tci supercell (black box). The blue box represents the actual selection of the primitive cell. The solid yellow circle represents Sb atom in tci due to the distortion from those in ti (dotted yellow circles) consistent with the soft modes in (c). (f) The side view of tci from $a$-axis shows the zigzag chain like Sb atoms along the $b$-axis.

  • Figure 3

    (Color online) Calculated band structures along the high-symmetry lines within GGA without ((a), (c)) and with ((b), (d)) SOC for ti ((a), (b)) and tci ((c), (d)). The inset in (c) shows the band inversion feature along $\Gamma$-Y without SOC for $o$-LaSbTe. The two highest valance bands and lowest conduction bands are marked in red and green, respectively.

  • Figure 4

    (Color online) The spectra of Wilson loop and SS band structures on surface BZ of (100) ((a)-(c)) and (1$\bar~1$0) ((d)-(f)) surfaces, respectively. (b), (e) The zoomed in image of the shadowed part in (a) and (d), respectively. Hourglass-like SSs in (b) and (c) but trivial SSs with full gap in (e) and (f) can be observed. The insets in (b) and (c) indicate the Hourglass-like SSs.

  • Table A1   Elementary layer construction for space group $Pmcn$
    $(hkl;~d)$ $\mathbb~Z_{2,2,2,4}$ weak $m^{100}_{(2)}$ $g^{010}_{0~\frac{1}{2}\frac{1}{2}~}$$g^{001}_{\frac{1}{2}~\frac{1}{2}~0~}$$i$$2^{100}_{1~}$ $2^{010}_{1}$ $2^{001}_{~1}$
    001; 0 0000 000 00 1 1 0 0 11
    001; $\frac{1}{4}$0002000 00 1 0 1 101
    100; $\frac{1}{4}$ 0000000 20 0 1 0 1 01
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