SUPPLEMENTARY INFORMATION Half-Heusler ternary compounds as new multifunctional platforms for topological quantum phenomena H. Lin, L.A. Wray, Y. Xia, S.-Y. Xu, S. Jia, R. J. Cava, A. Bansil, and M. Z. Hasan Topic: Topological equivalence of HgTe and LnPtSb (Ln=Lu, Y) via firstprinciple calculations. HgTe is a known topologically non-trivial semimetal with a single band inversion at the -point. On the other hand, CdTe is topologically trivial with no band inversion. We start with the comparison of band symmetries of (Hg,Cd)Te and half-heusler compounds. As shown in Fig. S0, the bands near the Fermi level (E F ) at the -point possess 6 (2-fold degenerate), 7 (2-fold degenerate), and 8 (4-fold degenerate) symmetry in all of these compounds. LuPtSb and HgTe have 8 states at E F. The 6 bands are below the 8 ones and are occupied, providing the band inversion that leads to the topological order. Although the order of 7 and 6 is different in LuPtSb and HgTe, it is not relevant in determining the topological nature since both states are occupied. In trivial insulators such as TiNiSn and CdTe, the 6 states are higher in energy and lie above the downward dispersing valence bands. These non-inverted 6 states lie above E F and are unoccupied. It is due to the occupancy of 6 bands at the -point in LuPtSb and HgTe that the Z 2 topological invariant picks up an extra factor of -1 in comparison to the bulk of TiNiSn and CdTe. a LuPtSb E F b 8 x4 6 x2 7 x2 TiNiSn E F 6 x2 8 x4 7 x2 c Non-trivial d Trivial E F 8 x4 7 x2 6 x2 K K E F Figure S0 Non-trivial and trivial band ordering. a-d, Band structures of LuPtSb (a), TiNiSn (b), HgTe (c) and CdTe (d). The symmetry of the states at the -point are labeled for each compound. For TiNiSn, the 6 states lie at very high energies above the Fermi level (not shown). 6 x2 8 x4 7 x2 nature materials www.nature.com/naturematerials 1
supplementary information Section 1. Introduction to adiabatic transformation method for determining Z 2 equivalents Two electronic materials are topologically equivalent if their Hamiltonians can be adiabatic transformed into each other without encountering a phase transition. In order to prove that zincblende HgTe is topologically equivalent to half-heusler LuPtSb and expanded YPtSb, we start out introducing an artificial material KrHgTe with half-heusler structure. The insertion of noble-gas atom Kr into HgTe makes no relevant change to the band structure near E F due to the chemically inert nature of Kr if the lattice constant is sufficiently large. The KrHgTe crystal can be continuously transformed into YPtSb by changing the atomic number Z's of the three atomic species. A specific pathway is found to guarantee that YPtSb is topologically equivalent to HgTe. Finally, substituting Y by Lu atoms we find that the topological nature remains intact. The DFT-GGA band structure evolution associated with the entire process shows no evidence for any intervening phase transition which is tantamount to the fact that YPtSb and LuPtSb are topologically equivalent to 3D-HgTe. The transformation process is carried out in three stages: Step 1. Expansion of the HgTe lattice and the insertion of Kr into the lattice Step 2. Changing of the atomic numbers Z s Step 3. Replacement of Y by Lu and restoration of the lattice constant to the experimental values The transformation details are discussed in the following sections, and DFT-GGA band structure evolution are illustrated in Figs. S1-S3. The sizes of the red dots in the figures indicate the amount of s-orbital character. The green arrows indicate the path of transformation. For the band structures along a path, the states at point with s-orbital character are occupied and lie below the 4-fold degenerate j=3/2 states at the Fermi level (E F ). This band inversion is the most important feature for the topological nature and should be kept track of throughout the entire process. We observed that no electronic gap opened along the path of transformation, and the system remained topologically nontrivial all the way from HgTe to LuPtSb. Section 2. Expansion of the HgTe lattice and the insertion of Kr In order to insert Kr atom into HgTe yet minimally modify the low energy Hamiltonian, we first expand the HgTe lattice to have more room to accommodate the Kr atoms. We start with HgTe in Zinc-blende structure with lattice constant a 0 HgTe =12.21 Bohr, and increase the lattice constant to 12.7 Bohr. The DFT-GGA bands are shown in Fig. S1. We also show the band structures of KrHgTe with half-heusler structures in the second row. As the lattice constants increase, the dispersion of the bands becomes smaller due to the decrease of the strength of interactions between orbitals on different atoms. By comparing the first row and the second row, it is observed that the local maximum of the valence bands at X point for KrHgTe goes away from E F and the band structures of KrHgTe get more and more similar to those of HgTe as the lattice constant increases. At a=12.6 Bohr, eight bands lie between -3 ev and 2 ev and they constitute the low-energy Hamiltonian. The similarities between Fig. S1 e and j evidently suggests that the low-energy Hamiltonians of HgTe and KrHgTe are essentially the same for the topological nature. The additional bands in KrHgTe lie below -4 ev and they cannot change the topological nature since they originate from complete-shell orbitals and remain fully occupied. Therefore, HgTe and KrHgTe are topologically equivalent. 2 nature MATERIALS www.nature.com/naturematerials
HgTe a b c d e f supplementary information a=a 0 HgTe a=12.3 a=12.4 a=12.5 a=12.6 a=12.7 L X W L X W L X W L X W L X W L X W KrHgTe g h i j k a=12.3 a=12.4 a=12.5 a=12.6 a=12.7 L X W L X W L X W L X W L X W Fig S1 Expansion of HgTe and insertion of Kr. The DFT-GGA band structures of HgTe with varying lattice constants are shown in a-f. The bands of KrHgTe are shown in g-k. The red dots indicate the strength of the s-orbital character. The green arrows indicate the path of transformation. Atomic unit Bohr is used for lattice constant a. The lattice constant of bulk HgTe is a 0 HgTe =12.21 Bohr. 3 nature materials www.nature.com/naturematerials 3
supplementary information Section 3. Change of the atomic number Z s KrHgTe can be continuously transformed into YPtSb by changing the atomic number Z's of the three atomic species. In the computations, the nuclear charge is changed according to the atomic number Z. For each step of changing the value of Z, self-consistency is achieved and the electrostatic potentials change accordingly. While Z is not an integer, there is no corresponding atom in reality. However, sometimes the fractional Z is physically meaningful. It can be considered as a doping effect in the virtual crystal approximation. Here, we treat Z as parameters of the Hamiltonian in the first-principle calculations. Both KrHgTe and YPtSb are half-heusler compounds MM X. We can define the atomic numbers of M, M, and X as Z M =36+2x+y, Z M =80-2x, and Z X =52-y respectively. This choice guarantees that the total charge is neutral during the adiabatic transformation. The band structures of various values of x and y with fixed lattice constant a=12.6 Bohr are shown in Fig. S2. While the starting point with x=0 and y=0 corresponds to KrHgTe (Fig. S2 a), the end point with x=1 and y = 1 corresponds to YPtSb (Fig. S2 p). In a-d, y is increased from 0 to 0.7. The value of y was not increased further in order to avoid the conduction bands that tends to dip below E F near the X-point. Then x is increased from 0 to 1 in d-m. Finally, y is increased from 0.7 to 1 in m-p. In Fig. S2 a-p, the band structures change continuously and the band inversion at -point persists. The systems remain semi-metal along this path. A complete adiabatic transformation is achieved without phase transition. Therefore, YPtSb with somewhat expanded lattice constant a=12.6 Bohr are proved to be topologically equivalent to KrHgTe. 4 nature MATERIALS www.nature.com/naturematerials
KrHgTe -> YPtSb a x=0, y=0 b supplementary information a=12.6 Bohr c x=0, y=0.2 x=0, y=0.5 x=0, y=0.7 d L X W L X W L X W L X W e x=0.1, y=0.7 f g x=0.25, y=0.7 x=0.4, y=0.7 x=0.5, y=0.7 h L X W L X W L X W L X W i x=0.6, y=0.7 j k x=0.7, y=0.7 x=0.8, y=0.7 x=0.9, y=0.7 l L X W L X W L X W L X W m x=1, y=0.7 n o x=1, y=0.8 x=1, y=0.9 x=1, y=1 p L X W L X W L X W L X W Fig S2 The transformation from KrHgTe to YPtSb. The DFT-GGA band structures of half-heusler MM X with Z M =36+2x+y, Z M =80-2x, and Z X =52-y are shown with various values of x and y. The green arrows indicate the path of transformation from KrHgTe to YPtSb with fixed lattice constant a=12.6 Bohr. The path begins with KrHgTe, x=0 and y=0 in a. By increasing y in a-d, followed by increasing x in d-m, then by increasing only y in m-p, the end point YPtSb with x=1 and y=1 is reached in p. The red dots indicate the strength of the 5 s-orbital character. nature materials www.nature.com/naturematerials 5
supplementary information Section 4. Replacement of Y by Lu and the restoration of the lattice constant to the experimentally observed value The elements Y and Lu have similar chemical properties due to the similarity of their outermost electron configurations. The electronic configurations of Y 3+ and Lu 3+ differ by 5s 2 4d 10 5p 6 4f 14 which is a complete shell. One then expects that replacing Y by Lu in a material the low-energy band structure should remain very similar to the original one. In Fig. S3, we present that DFT-GGA bands for YPtSb in the first row and those for LuPtSb in the second row with various lattice constants. The panels are arranged such that each panel in the first row has the same topological nature in the corresponding panel in the second row. The lattice constant is slightly different in each corresponding panel due to the difference in the atomic size of Y and Lu. In panel a and d, gaps are obtained in bulk YPtSb and compressed LuPtSb. They are both topologically trivial. While a and d show semiconducting band structures, b and e are semi-metallic. There exists a phase transition between a and b (d and e) for YPtSb (LuPtSb). The topological nature of bulk YPtSb in a is different from the expanded YPtSb in b. We now focus on panel b and e. By invoking arguments presented Section 2 for inserting Kr atom, we show that the topological character is the same for YPtSb and LuPtSb. There are eight bands between -3 ev and 1 ev and they form the low-energy Hamiltonian. The similarities between b and e speaks for themselves that the low-energy Hamiltonians of YPtSb and LuPtSb are essentially the same as far as the topological nature is concerned. The additional bands in LuPtSb around -4 ev are originate from 4f 14 and there are more such bands in higher binding energies. These bands do not change the topological nature since they are all complete-shell orbitals and fully occupied. Therefore, YPtSb and LuPtSb are topologically equivalent. 6 nature MATERIALS www.nature.com/naturematerials
YPtSb a a=a 0 YPtSb =12.34 b supplementary information a=12.6 c a=12.7 L X W L X W L X W LuPtSb d a=12.0 e a=a LuPtSb 0 =12.2 f a=12.3 L X W L X W L X W Fig S3 Replacement of Y by Lu. The DFT-GGA band structures with varying lattice constants are shown in a-c for YPtSb and d-f for LuPtSb. The red dots indicate the strength of the s-orbital character. Atomic units are given in Bohr. The green arrows indicate the final step of transformation. The end point of the transformation is the bulk crystal of LuPtSb with a LuPtSb 0 =12.2 Bohr shown in e. Note that the bulk YPtSb with a YPtSb 0 =12.34 Bohr in a and compressed LuPtSb in d are semiconductors with gaps near E F. 7 nature materials www.nature.com/naturematerials 7
supplementary information Section 5. Summary We have shown that HgTe and KrHgTe are topologically equivalent (section 2), KrHgTe and YPtSb are topologically equivalent (section 3), and finally YPtSb and LuPtSb are topologically equivalent (section 4) leading to the conclusion that HgTe and LuPtSb are topologically equivalent. By tracing through the green arrows in Figs. S1-S3, it is seen that the most relevant 8 bands form the topologically non-trivial groundstate (band structure) near E F through the entire adiabatic transformation process. Our method presented here can be applied to determine the topological character of other half-heusler compounds. 8 nature MATERIALS www.nature.com/naturematerials