Strain driven onset of non-trivial topological insulating states in Zintl Sr 2 X compounds ( X =Pb, Sn)
Yan Sun, Xing-Qiu Chen, Dianzhong Li, Cesare Franchini, Seiji Yunoki, Yiyi Li, Zhong Fang
aa r X i v : . [ c ond - m a t . m t r l - s c i ] M a y Strain driven onset of non-trivial topological insulating states in Zintl Sr X compounds ( X =Pb, Sn) Yan Sun , Xing-Qiu Chen , ∗ Dianzhong Li , Cesare Franchini , , Seiji Yunoki , Yiyi Li , and Zhong Fang Shenyang National Laboratory for Materials Science, Institute of Metal Research,Chinese Academy of Sciences, Shenyang 110016, China Center for Computational Materials Science, University of Vienna, Sensengasse 8, A-1090 Vienna, Austria Computational Condensed Matter Physics Laboratory,RIKEN ASI, Saitama 351-0198, Japan, and CREST,Japan Science and Technology Agency (JST), Saitama 332-0012, Japan,Computational Materials Science Research Team, RIKEN AICS, Hyogo 650-0047 and Beijing National Laboratory for Condensed Matter Physics,Institute of Physics, Chinese Academy of Sciences, Beijing, 100081, China (Dated: September 11, 2018)We explore the topological behavior of the binary Zintl phase of the alkaline earth metals basedcompounds Sr Pb and Sr Sn using both standard and hybrid density functional theory. It is foundthat Sr Pb lies on the verge of a topological instability which can be suitably tuned through theapplication of a small uniaxial expansion strain ( > Pb(010) surface, whose evolution isstudied as a function of the film thickness.
PACS numbers: 71.20.-b, 73.43.-f, 73.20.-r
Topological insulators[1–6] (TI) are a new quantumstate of matter characterized by the existence of gap-less surface states sheltered against destructive scatteringeffects by time-reversal symmetry. Since the discoveryof two dimensional TI behaviors of HgTe-based quan-tum wells [9, 10], several families of topological materi-als have been theoretically predicted and experimentallyrealized afterwards.[1, 11] The variety of insulating ma-terials displaying topological features includes Bi x Sb − x alloys[5, 12], Bi Te , Bi Se and Sb Te binary com-pounds [6–8, 11, 13], ternary heavy-metals based com-pounds such as TlBiTe , TlBiSe [14–16], and PbBi Se [17, 18], and ternary rare-earth chalcogenides (LaBiTe )[19], and another honeycomb-lattice type of ternary com-pounds (LiAuSe) [20]. All these materials are character-ized by a layered structure stacked along the c -axis of thecentrosymmetric hexagonal lattice, similar to the struc-ture of Bi Se . In addition, other ternary TIs have beenrecently predicted such as the non-centrosymmetic cu-bic zinc-blende HgTe-like phase (half-Heusler compounds[21–23] and I-III-VI2 and II-IV-V2 chalcopyrite semicon-ductors such as AuTlTe [24, 25]), and Ca NBi with acentrosymmetric antiperovskite structure [26]. Althougha large number of ternary TI have been found[14–19, 21–25], up to now, binary TIs are only limited to threeclasses: Bi − x Sb x alloys and the family of Bi Te , Bi Se and Sb Te as well as Ag Te [27]. In searching for new TImaterials we have focused our attention on binary heavy-element-based small band gap semiconductors. One ofthe simplest way to realize a binary heavy-element-basedclosed-shell semiconductor is M X with an alkaline earthelement ( M = Mg, Ca, Sr, and Ba) which donates its s valence electron to a IV group element ( X = Si, Ge, Snand Pb). These are the so-called Zintl compounds. bc a ac (a) (b) (c) FIG. 1: (Color online) Structure representations of or-thorhombic (space group
P nma , No. 62) M X compounds( M = Mg, Ca, Sr and Ba; X = Si, Ge, Sn and Pb): (a) theunit cell (u.c.), (b) the projection of the unit cell in the ac plane, perpendicular to the b -axis and (c) the local environ-ment around the X atom. The large (gray) and small (red)balls denote the M and X atoms, respectively. In addition,along the b -axis (y=1/4 and y=3/4) the atoms can be ar-ranged in two parallel planes shown in the dashed and solidballs, respectively. M occupied two inequivalent 4 c sites, M ( x , 1/4, z ) and M ( x , 1/4, z ) whereas X occupies the 4 c site ( x , 1/4, z ). For Sr Pb the optimized atomic positionsare Sr (0.0214, 1/4, 0.683), Sr (0.1580, 1/4, 0.073) and Pb(0.2496, 1/4, 0.3929). The Zintl compounds M X crystallize in a simple or-thorhombic crystal structure [28, 29], as illustrated inFig. 1. The X atoms surround the M atoms in a slightlydistorted trigonal-prismatic coordination. The coordi-nation sphere is augmented by the three X atoms situ-ated on the rectangular faces of these prisms (tri-cappedtrigonal prism). Along the a direction the prisms are FIG. 2: (Color online) DFT and HSE electronic structure and band inversion strength. Panels (a,b) and (c,d) show DFTcalculated electronic band structures for Sr Sn and Sr Pb, respectively. Panels (e,f) and (g,h) reports HSE band structuresaround the Γ point along the Z-Γ-T directions for Sr Pb at zero strain and under 5% tensile strain in the ac -plane, respectively.The upper and lower panels show the results calculated without (a,c,e,g) and with (b,d,f,h) SOC, respectively. The solid (red)circles denote the states with a predominant s -like character. Panel (i) illustrates the comparison of the band inversion strengthbetween p x - and s -orbitals at the Γ point as a function of strain. Negative values denote the occurrence of band inversion. condensed via common edges while they share commontriangular faces along the b direction. The semiconduct-ing character of these compounds originates from theirclosed-shell nature, with 6 valence electrons (2 × s + p )per formula unit [(2 M ) X − ].We have performed band structure calculations usingthe Perdew-Burke-Ernzerhof[30] (PBE) based standardand hybrid (HSE[31]) density functional theory (DFT) asimplemented in the Vienna Ab initio Simulation Package (VASP) [32, 33], with the inclusion of relativistic spin-orbit coupling (SOC) effects. Within the HSE method,the many body exchange and correlation functional isconstructed by mixing 25% exact Hartree-Fock exchangewith 75% PBE, and the long-range Coulomb interactionis suitably screened according to the parameter µ (here µ = 0.3 ˚A − ). By using the experimental lattice constantswe have relaxed all atomic positions using a very densek-points mesh (up to 4200 kpoints).Our preliminary PBE-based research of possible topo-logical features in this class of materials revealed that twomembers of the Zintl family (Sr Pb and Ba Pb) exhibitpromising fingerprints of TI behaviors whereas the re-maining compounds are either trivial insulator (Ca Ge,Sr Ge,Ca Sn and Sr Sn) or semimetal (Ba Ge, Ca Pband Ba Sn). Therefore in the following we focus ouranalysis on these two specific Zintl compounds: Sr Pband Sr Sn.The DFT band structures with and without SOC ef-fects are compared in Fig. 2(a-d). The inclusion of SOC effects does not affect the overall electronic char-acter of Sr Sn [Fig. 2 (a,b)] which remains semiconduc-tor with a small bandgap at Γ ( ≈ p x states and highly dispersive Sr d -like empty orbitals (mostly d x − y ). By replacing Sn withthe isoelectronic heavier element Pb the band structurechanges dramatically, as schematically depicted in Fig.3(a). Without SOC, Sr Pb displays a metallic (gapless)state [Fig. 2(c)] whereas the inclusion of SOC [Fig. 2(d)]opens a band gap of about 100 meV at Γ (the indirect gapis about 50 meV), as a consequence of the anti-crossingbetween the conduction band minimum (CBM) and va-lence band maximum (VBM), a typical fingerprint of theSOC-induced formation of topological insulating states.The major differences between the electronic structure ofSr Sn and Sr Pb resides in the orbital character of thevalence and conduction bands near Γ as highlighted bythe (red) solid circles in the band plot of Fig.2(a-d): inSr Sn the states with a predominant s -like character lieabout 0.5 eV above the Fermi Level [see Fig. 2(a,b)],whereas in Sr Pb these states are pushed down in energyand eventually hybridize the VBM at Γ thus inducingthe anti-crossing feature responsible for the opening ofthe gap and the creation of a TI state. The downwardshift of the s -states is accompanied by a upward shift ofthe heavy metal p bands which ultimately intermix withthe Sr d states as schematized in Fig. 3 (a).As recently reported by Zunger and coworkers[34], theidentification of band-inverted TI[9, 10] on the basis ofconventional DFT may lead to false-positive assignmentif the band inversion strength is not large enough, due tothe well documented bandgap underestimation problem.For this reason we have revisited the electronic dispersionof Sr Pb by HSE, which indeed yields a substantially dif-ferent physical picture, as illustrated in Fig. 2(e,f). HSEfinds a much larger bandgap (0.25 eV) which is reducedto 0.13 eV with the inclusion of SOC effects, and, mostimportantly, prevent the occurrence of band inversion be-tween the p x -like and s -like states as schematically illus-trated in Fig. 3(b). The s states remain well localized onthe bottom of the conduction band [Fig. 2(e,f)]: Sr Pbis not a TI in its native phase. DFT wrongly stabilizes aspurious TI solution because of the relatively small bandinversion strength [0.12 eV, see Fig.2(a)]. Though HSE isexpected to provide a generally more accurate descriptionof band dispersion in small bandgap insulators, future op-tical experiment will be necessary in order to validate ourfirst principles findings. no-SOC no-SOC SOC SOC
Sr-s/Sn-sSr-dx -y
Sn-p x Sr-s/Pb-s Sr-s/Pb-sPb-p x Sr-dx -y
Pb-p /Sr-dx -y x 2 2
Pb-p /Sr-dx -y x 2 2
Sr Sn Sr Pb Sr Pb ( ) DFT a Sr-dx -y
Sr-s/Pb-s
Sr Pb TI no-SOC no-SOC SOC SOC Sr-s/Sn-sSr-dx -y
Sn-p x Sr-s/Pb-s Sr-s/Pb-sSr-dx -y
Sr Sn Sr Pb ( ) HSE b Pb-p ;p x y
Sr-dx -y
Sr-s/Pb-sSr-dx -y
Sr Pb Sr Pb TITI (Strain>3%)(Strain) E f E f +++ ++++ + G ZUX S YR T b * a*c* +++ ++ + - + G ZUX S YR T b * a*c* ( ) c ( ) d Pb-p ;p x y
Sn-p y Pb-p y Pb-p ;p ;p y z x
Pb-p y Sn-p y Pb-p ;p x y
Pb-p x Pb-p y Pb-p ;p x y
Sr-p /Pb-p ;p y z y
FIG. 3: (Color online) Evolution of atomic orbitals at the Γpoint from Sr Sn to Sr Pb with and without SOC effects in-cluded for both DFT (a) and HSE (b) calculations. The bandinversion can be observed by Pb (or Sr) s and Pb p x orbitals.In panels (c) and (d) we show the product of wave functionparities of the occupied bands for eight time-reversal invariantmomenta (TRIM) in the Brillouin zone [Γ (0,0,0), X ( π ,0,0),Y (0, π ,0), Z (0,0, π ), S ( π , π ,0), T (0, π , π ), U ( π ,0, π ), and R( π , π , π )] of Sr Pb obtained before and after band inversion,respectively.
To explore possible routes for designing a TI phasein the native phase of Sr Pb, we carried out a seriesof calculations applying strain, a gap-engineering tech-nique which was successful when applied to zero-gap semiconductors such as ternary Heusler compounds[22]and Ca NBi. [26] We have chosen to study the effect ofuniaxial strain ( ǫ ) in the ac -plane, by leaving the b -axisunconstrained (free to relax) in order to simulate at bestthe experimental condition for thin film growth. The re-sults, obtained by HSE and shown in Fig. 2(g and h),indicate that for relatively small uniaxial strain largerthan 3% Sr Pb can be tuned towards a TI phase. Therole of SOC effects is essential to open a small gap aroundΓ and to induce an inverted band order. The s -like statesshift downward below the Fermi level and becomes occu-pied and, simultaneously, the p x -like states become un-occupied and promoted at higher energy. This kind ofinverted band behavior can be ascribed to the fact thatthe strain-induced expansion in the ac -plane reduces thecrystal field effect, resulting in the less hybridization be-tween s - and p -like states and stabilizing the s -like stateat lower energy, as evidenced in Fig. 2(e) and (g). In sucha way, the spin-orbit coupling strength is now enough toinvert the band order between s -like and p -like states atΓ (c.f., Fig. 2(f) and (h)). For larger strains up to 7% theTI state is preserved and further stabilized as inferred bythe evolution of the band inversion strength as a func-tion of strain reported in Fig. 2(i) at both, DFT andHSE levels.Beside the band inversion, an alternative way to iden-tify TI states is the parity criteria proposed by Fu andKane [12]. Considering that orthorhombic Pnma possesthe inversion symmetry this criteria can be applied andwill serve as a further support for our analysis. The prod-uct of the parities of the Bloch wave function for theoccupied bands at all eight time reversal invariant mo-menta (TRIM), illustrated in Fig. 3(c,d), suggest thatat six TRIMs (X, Y, Z, S, T and U) all bands share thesame doubly degenerate character, whereas the TRIM Ris found to be fourfold degenerate. Therefore all theseseven TRIMs display a positive (”+”) global parity. AtΓ the situation is different: the product of the parities is”+” and ”-” depending on whether or not the band in-version occurs, in consistency with the HSE band struc-ture interpretation [Fig. 3(c,d)]. We can therefore trust-fully conclude that distorted orthorhombic Sr Pb is atopological non-trivial insulator with Z index (1; 000).Strained-driven gap-engineering on Sr Sn dose not re-sults in any topological transition: Sr Sn remains a con-ventional semiconductor.After discussing the onset of topological features inthe bulk phase of distorted Sr Pb we turn our atten-tion to the surface properties focusing on the non-polar(010) termination. Considering that both DFT and HSElead to an essentially identical TI state in strained Sr Pband that SOC-HSE calculations for thick slabs are com-putationally very demanding (if not prohibitive at all)we study the surface band structure at DFT level only.The results on the Sr Pb(010) surface are summarizedin Fig. 4, which shows that very thick slabs have topo-logically protected surface metallic states which remainrobust with increasing the film thickness, thus corrobo-rating the conclusion of the bulk parity analysis and theinverted band order. The evolution of the band structureas a function of the film thickness show a very peculiarbehavior. Already at low film thickness surface relatedstates emerge in a small energy window ( ±
200 meV)around the Fermi level. This leads to a quenching ofthe bandgap as compared to the bulk value (130 meV).The bandgap is then progressively reduced by increasingthe slab thickness. Though for a thickness of 22 unit cell(u.c.) [Fig. 4(f)] the gap is almost completely suppressed(10 meV), it is necessary to increase the thickness up to100 u.c. in order to fabricate a well-defined metallic film[Fig. 4(g)]. At the critical thickness of 100 u.c. the low-est conduction band and the highest valence band meetat the Fermi level at Γ and establishes a single-Dirac-cone-like metallic surface state.
FIG. 4: (Color online) DFT Surface properties of strainedSr Pb(010) ( ǫ =5%). (a–d) Evolution of the band structureswith slab thickness (e) structural model of the symmetric slabadopted to simulate the Sr Pb(010) surface (the image cor-respond to the 3 unit cells case, i.e. 6 layers per side. Weadopted a vacuum region of 30 ˚A). (f) Band structure for athickness of 22 u.c. Shaded areas refer to bulk bands. (g) Pro-gressive closing of the gap at Γ (E Γ g ) as a function of the slabthickness. Note that, due to the prohibitive computationalcost, for slab thickness larger than 34 u.c. ( >
400 atoms) theband gaps were calculated at Γ only.
In conclusion, our computational study has disclosedthe non-trivial topological nature of the binary com-pound Sr Pb. The detailed analysis of the bulk and sur-face structural and electronic properties is of relevancefor the design principles of TI and provides helpful in-sight for tunability of TI states in trivial insulator by gap-engineering techniques. We believe that our findingswill encourage immediate experimental investigations.
Acknowledgement . The authors are grateful forsupports from the “Hundred Talents Project” of ChineseAcademy of Sciences and from the project of the NSFCof China (Grand Numbers: 51074151 and 51050110444)as well as Supercomputing Center of Chinese Academyof Sciences (including Shenyang branch) and a local HPCcluster in the Materials Process Simulation Division,IMR of CAS as well as the Vienna Scientific Cluster(VSC) in Austria. F.Z. acknowledges the supports fromthe 973 program of China (No. 2007CB925000). ∗ The corresponding author should be addressed:[email protected][1] M. Z. Hasan, C. L. Kane, Rev. Mod. Phys. , 3045(2010).[2] J. E. Moore, Nature , 194 (2010).[3] Zhang, S. -C. Physics , 6 (2008)[4] D. Hsieh, D. Hsieh, Y. Xia, L. Wray, D. Qian, A. Pal,J. H. Dil, F. Meier, J. Osterwalder, C. L. Kane, G.Bihlmayer, Y. S. Hor, R. J. Cava and M. Z. Hasan. Sci-ence, , 5916, (2009).[5] D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J.Cava, and M. Z. Hasan, Nature , 970 (2008).[6] Y. Xia, D. Qian, D. Hsieh, L.Wray, A. Pal, H. Lin, A.Bansil, D. Grauer, Y. S. Hor, R. J. Cava and M. Z. Hasan,Nature Physics , 398 (2009).[7] D. Hsieh, J.W. McIver, D. H. Torchinsky, D. R. Gardner,Y. S. Lee, and N. Gedik, Phys. Rev. Lett, , 057401(2011).[8] Y. Xia, D. Qian, L. Wray, D. Hsieh, G. F. Chen, J. L.Luo, N. L. Wang, and M. Z. Hasan, Phys. Rev. Lett, , 037002 (2009).[9] B. A. Bernevig, T. L.Tughes and S.-C. Zhang, Science, , 1757 (2006).[10] M. K¨onig, S. Wiedmann, C. Br¨une, A. Roth, H. Buh-mann, L. Molenkamp, X. -L. Qi, and S. -C. Zhang, Sci-ence, , 766 (2007).[11] Y. L. Chen, J. G. Analytis, J. -H. Chu, Z. K. Liu, S.-K.Mo, X. -L. Qi, H. J. Zhang, D. H. Lu, X. Dai, Z. Fang,and S. -C. Zhang, Science, , 178, (2009).[12] L. Fu and C.L. Kane, Phys. Rev. B, , 045302 (2007).[13] H. J. Zhang, C.-X. Liu, X.-L. Qi, X. Dai, Z. Fang, andS. -C. Zhang, Nature Physics, , 438 (2009).[14] B. Yan, C.-X. Liu, H.-J. Zhang, C.-Y. Yam, X.-L. Qi, T.Frauenheim, and S.-C. Zhang, Europhys. Lett. , 37002(2010).[15] Y. Chen, Z. Liu, J. Analytis, J. Chu, H. Zhang, S. Mo,R. Moore, D. Lu, I. Fisher, S. Zhang, arXiv:1006.3843v1(2010).[16] T. Sato, K. Segawa, H. Guo, K. Sugawara, S. Souma, T.Takahashi, Y. Ando, Phys. Rev. Lett. 105, 136802 (2010).[17] S.-Y. Xu, L.A. Wray, Y. Xia, R. Shankar, A. Petersen,A. Fedorov, H. Lin, A. Bansil, Y. S. Hor, R. J. Cava, andM. Z. Hasan, arXiv: 1007.5111 (2010).[18] H. Jin, J.-H. Song, A. J. Freeman and M. G. Kanatzidis,Phys. Rev. B, , 041202(R) (2011).[19] B. H. Yan, H. -J. Zhang, C. -X. Liu, X. -L. Qi, T. Frauen- heim, and S.-C. Zhang, Phys. Rev. B, , 161108(R)(2010)[20] H.-J. Zhang, S. Chadov, L. Muchler, B. Yan, X.-L. Qi,J. K¨ubler, S.-C. Zhang, and C. Felser, Phys. Rev. Lett., , 156402 (2011).[21] H. Lin, L. A. Wray, Y. Xia, S. Xu, S. Jia, R. J. Cava, A.Bansil, and M. Z. Hasan, Nature Mater. , 546 (2010).[22] S. Chadov, X. L. Qi, and et.al., Nature Mater. , 541(2010);[23] D. Xiao, Y. Y. Yao, W. Feng, J. Wen, W. Zhu, X.-Q.Chen, G. M. Stocks, and Z. Zhang, Phys. Rev. Lett. ,096404 (2010).[24] M. Klintenberg, arXiv:1007.4838 (2010).[25] W. Feng, D. Xiao, J. Ding, and Y. Yao, Phys. Rev. Lett., , 016402 (2011).[26] Y. Sun, X.-Q. Chen, S. Yunoki, D. Z. Li and Y. Y. Li,Phys. Rev. Lett., , 216406 (2010). [27] W. Zhang, R. Yu, W. Feng, Y. Yao, H. Weng, X. Dai,and Z. Fang, Phys. Rev. Lett., , 156808 (2011).[28] G. Bruzzone and E. Franceschi, J. Less-Comm Met., ,201-208 (1978).[29] P.Eckerlin and E. Wolfel, Z. Anorg. Chem. , 321(1955).[30] J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev.Lett. , 3865 (1996).[31] J. Heyd, G. E. Scuseria, and M. Ernzerhof, J. Chem.Phys. , 8207 (2003).[32] G. Kresse and J. Hafner, Phys. Rev. B , 13115 (1993).[33] G. Kresse and J. Furthmuller, Comput. Mater. Sci.6