New Candidates for Topological Insulators : Pb-based chalcogenide series
Hosub Jin, Jung-Hwan Song, Arthur J. Freeman, Mercouri G. Kanatzidis
NNew Candidates for Topological Insulators : Pb-based chalcogenide series
Hosub Jin, Jung-Hwan Song, ∗ Arthur J. Freeman, and Mercouri G. Kanatzidis Department of Physics and Astronomy,Northwestern University, Evanston, Illinois 60208, USA Department of Chemistry, Northwestern University, Evanston, Illinois 60208, USA
Abstract
To date, Bi Se is known as the best three-dimensional high temperature topological insulator with alarge bulk band gap. Here, we theoretically predict that the series of Pb-based layered chalcogenides,Pb n Bi Se n +3 and Pb n Sb Te n +3 , are possible new candidates for topological insulators. As n increases,the phase transition from a topological insulator to a band insulator is found to occur between n = 2 and 3 for both series. Significantly, among the new topological insulators, we found a bulk band gap of0.40eV in PbBi Se which is one of the largest gap topological insulators, and that Pb Sb Te is locatedin the immediate vicinity of the topological phase boundary, making its topological phase easily tunableby changing external parameters such as lattice constants. Due to the three-dimensional Dirac cone at thephase boundary, massless Dirac fermions also may be easily accessible in Pb Sb Te . PACS numbers: ∗ [email protected] a r X i v : . [ c ond - m a t . m t r l - s c i ] J u l opological insulators, distinguished from normal band insulators by a nontrivial Z topolog-ical number and topologically protected surface states, have attracted great attention due to theirsignificance both for applications and for fundamental research on a new quantum state of matter[1, 2]. Since the suggestion of a quantum spin Hall effect in a honeycomb lattice [3, 4], which isa time-reversal pair of Haldane models [5] and induced by spin-orbit interactions, several topo-logical insulator materials were predicted and observed [6–12]. Efforts to find new topologicalinsulators are still being made [13, 14]. A large spin-orbit coupling (SOC) strength is essential torealizing non-trivial topological band structures.To classify the Z topological number, we adopted the method proposed in ref. [15], whichstudied the topological phase transition by making a variation of external parameters such as theSOC strength or lattice constants. In our work, we varied the SOC strength ( λ SO ) from 0 to the realvalues of the systems ( λ ), and investigated whether the topological phase transition occurs. Inthree-dimensional (3D) topological insulator materials, a band crossing between conduction andvalence bands should occur during this process: the 3D Dirac cone appears at the transition pointbetween the band insulator (BI) and topological insulator (TI).There are several advantages of this method. First, it is intuitive and only needs a direct obser-vation of the Dirac cone at the transition point. Second, it does not require a heavy computationalcost because of the simple unit-cell calculations. The surface calculations to see the topologicallyprotected surface state are also one of alternative ways to determine the topological phase. Thedegrees of complexity of such calculations are, however, grater than the method we employedhere. Third, this method can be applied to every system regardless of the existence of inversionsymmetry. Lastly, the critical SOC value at which the band crossing occurs shows how far thesystem is located from the phase boundary of BI and TI.By using this method, we found that the series of Pb-based chalcogenides, Pb n Bi Se n +3 andPb n Sb Te n +3 , are possible new candidates for TI materials and the topological phases are changedfrom TI to BI with increasing n . Among the above series, we focus on the material near the phaseboundary between BI and TI due to the possibility of tuning the topological phase by changingexternal parameters.To investigate the electronic structures and topological phases, first-principles calculations wereperformed using the full-potential linearized augmented plane wave method[16] with the gradient-corrected Perdew, Burke, and Ernzerhof form of the exchange-correlation functional [17]. Thecore states and the valence states were treated fully relativistically and scalar relativistically, re-2pectively. The calculations were carried out with the experimental lattice parameters for bulkPbBi Se , PbSb Te and Pb Bi Se , and with the fully optimized geometry for the others, becausecrystal structures of n = 1 compounds and Pb Bi Se were reported experimentally [18–20] andthose of other n ’s are designed here theoretically.PbBi Se and PbSb Te have the rhombohedral crystal structure, and a layered structurestacked along the c -axis of the hexagonal lattice, consisting of seven atoms in one septuple layer.The Pb atom is sandwiched by Se-Bi-Se or Te-Sb-Te layers and located at the inversion center,shown in Fig.1(a). There are van der Waals interactions between two septuple layers, and thisprovides a natural surface geometry which is appropriate for observing a topologically protectedsurface state.Upon increasing the SOC strength ratio, λ SO /λ , from 0 to 1, both materials show the phasetransition from BI to TI; in other words, there are band crossings during the process, cf., Fig1.(b)-(g). Critical values of the ratio are 0.46 for PbBi Se and 0.62 for PbSb Te , and the electronicband structures at each critical point are shown in Fig.1(c) and Fig.1(f). 3D Dirac cones are seenat the Z point, one of the time-reversal invariant momentum points where - k is equivalent to k ;they are doubly degenerate due to spatial inversion and time reversal symmetry.Without the spin-orbit interaction, the system should have a trivial Z topological number ( ν =0 ), the so called BI. The presence of band crossings between conduction and valence bands duringincreasing SOC represents the change of its Z topological number from trivial ( ν = 0 ) to non-trivial ( ν = 1 ) insulators. If the topological phase transition occurs before the SOC strengthreaches λ , the system is classified as TI. In these inversion symmetric systems, band crossing isequivalent to band inversion where conduction and valence bands exchange their parities at thecrossing point.From our calculations, the bulk gaps of PbBi Se and PbSb Te are 0.28eV and 0.13eV, respec-tively. The gap size of PbBi Se is comparable to that of Bi Se [11, 12]. The gap values fromthe calculations without SOC are 0.24eV (Fig.1(b)) and 0.26eV (Fig.1(e)), which reflects purelyhybridization effects. After increasing the SOC strength, gaps shrink to zero and then increaseagain, which shows the competition between hybridization and SOC strength in the topologicalphase transition. A non-trivial topological phase is a result of the predominance of SOC overthe hybridization strength. Compared to other TI chalcogenides, the large band gap in PbBi Se originates from weak hybridization and large SOC strength.To estimate the gap size of PbBi Se more precisely, we performed non-local screened-3xchange LDA (sX-LDA) calculations [21, 22] which overcome the well-known weakness ofgap underestimation in the LDA scheme and reproduces the experimental gap values in manysemiconducting materials. Significantly, the sX-LDA result yields a gap of 0.40eV in PbBi Se ,which confirms that PbBi Se is one of the largest gap TI, leading to the fact that PbBi Se maybe the best for high temperature applications among the known TI materials.One advantage of the method, used in this work to verify the topological phase, is that it de-scribes the exact distance from the phase boundary of BI and TI in terms of the SOC strength. Forinstance, the larger critical SOC strength of PbSb Te than that of PbBi Se means that the formeris closer to the phase boundary than the latter.In addition to PbBi Se and PbSb Te , we suggest the series of Pb n Bi Se n +3 and Pb n Sb Te n +3 structures where n is an integer larger than 1 and show that a phase transition from TI to BIoccurs as n increases. The crystal structure of Pb n Bi Se n +3 / Pb n Sb Te n +3 is composed of(PbSe) n /(PbTe) n core with Se-Bi-(Se)/Te-Sb-(Te) sandwich layers. As shown in Fig.2(a), thecritical SOC ratio ( λ cSO /λ ) exceeds 1 for n ≥ n = 2 (TI) and n = 3 (BI). The gap sizes decrease up to n = 2 and increase again in going beyond the phase boundary. (cf. Fig.2(b)) On the other hand, thecalculated gap values without SOC rise monotonically with n ; this shows that as n increases, thehybridization strength exceeds SOC, resulting in the topological phase transition.In Fig.2(c) and (d), electronic band structures from the slab geometry of the n = 1 and n = 3 composition of Pb n Bi Se n +3 , which consist of 6 septuple and 4 hendecuple layers with 78.4 ˚A and85.3 ˚A respectively, are shown. Consistent with the critical SOC ratio in Fig.2(a), PbBi Se hasa topologically protected surface state, whereas there is no such state connecting the valence andconduction bands in the Pb Bi Se slab. This is further evidence of the change in Z topologicalnumber from 1 to 0 as n increases. The 2D surface state Dirac cone in the PbBi Se slab is robustunder the presence of non-magnetic perturbations. Again, the 0.35eV gap originating from bulkstates is useful for high temperature spintronics applications. In other words, a single isolatedsurface state Dirac cone is seen in the energy range from -0.03eV below the Fermi level to 0.32eVabove. In the case of the Pb Bi Se slab, due to the weak van der Waals type interlayer interactions,no distinct surface state appears from the bulk BI states.Among the Pb-based chalcogenides series, Pb Sb Te whose critical SOC ratio is 0.96, is lo-cated quite close to the phase boundary, that its Z topological number may be easily tuned bychanging external parameters. The 26meV gap in the fully optimized geometry of Pb Sb Te Sb Te with a . reduction of the ab -lattice constants is positioned exactly at the borderline of TIand BI, whose electronic band structure shows a gapless 3D Dirac cone at the Γ point, realizingmonopoles which generate an inverse-square type Berry curvature in the 3D Brillioun zone. A re-duction of the ab -lattice constants makes the hybridization stronger, resulting in equilibration withthe SOC strength. The band structure near the Dirac point shows anisotropic dispersions betweenin-plane and out-of-plane directions, described by a massless Dirac Hamiltonian H = v ⊥ (cid:126)σ ⊥ · (cid:126)k ⊥ + v (cid:107) (cid:126)σ (cid:107) · (cid:126)k (cid:107) (1)where (cid:126)σ are Pauli matrices, (cid:126)k ⊥ is a momentum vector along the c -axis and (cid:126)k (cid:107) is on the ab -plane.Nearly circular cross-section on the ( k x - k y ) plane and an ellipsoidal cross-sections on the ( k z - k x )plane are depicted in Fig.3(c) and (d), where the electronic band dispersion and the iso-energycontours of the 3D Dirac cone are shown.The substitution of Te by Se might be useful to control the topological phase of this materialin terms of varying both the SOC strength and lattice parameters. Considering the smaller ionicradius and SOC strength of Se than those of Te, the Se substitution can play a role in reducingnot only lattice constants but also the SOC strength which can both accelerate the phase transitionfrom TI to BI. Hence, Pb Sb (Te − x Se x ) might be appropriate for studying the topological phasetransition and the 3D Dirac cone.In this work, we predicted new TI materials of Pb-based chalcogenides by investigating thechange of the Z topological number during variation of the SOC strength. With increasing n in the Pb n Bi Se n +3 and Pb n Sb Te n +3 series, the TI to BI transition occurs between n = 2 and3. Among these compounds, Pb Sb Te has a critical ratio of SOC strength close to 1, whichmeans closeness to the phase boundary. Both topological phases and the gap at the 3D Diraccone are tunable by changing lattice constants or the SOC strength. Also, it will be interestingto investigate the physical properties of the bulk 3D Dirac cone manifested at the critical point.As an extension of the present work, the Pb n (Bi/Sb) m (Se/Te) n +3 m series, which are pseudo-binary [Pb(Se/Te)] n [(Bi/Sb) (Se/Te) ] m systems and combinations of Pb n (Bi/Sb) (Se/Te) n +3 and[(Bi/Sb) (Se/Te) ] m layered structures, are potential candidates for new TI materials which mightshow various bulk band gaps and topological phase transitions. The method used in our workcan be the most appropriate tool to determine the Z topological number of the systems withoutinversion symmetry, which show pair creation and annihilation of monopole and anti-monopoles5t the topological phase transition region [15]. ACKNOWLEDGMENTS
Support from the U.S. DOE under Grant No.DE-FG02-88ER45372 is gratefully acknowledged.
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FIG. 1. (a) Crystal structures of PbBi Se /PbSb Te . Electronic band structures of (b)-(d) PbBi Se and(e)-(f) PbSb Te . Calculations (b),(e) without spin-orbit interactions ( λ SO = 0 ), (d),(g) with real SOCstrength ( λ SO = λ ), and (c),(f) with critical SOC strength ( λ SO = λ cSO ). a) (b)(c) (d) FIG. 2. (a) Critical SOC ratio and (b) bulk band gap with respect to n for the Pb n Bi Se n +3 andPb n Sb Te n +3 series. Band structures of slab geometry of (c) 6 septuple layers of PbBi Se and (d) 4hendecuple layers of Pb Bi Se , which represent TI and BI, respectively. a)(b) (c)(d) E n er gy ( e V ) k x k z k x k y E n er gy ( e V ) FIG. 3. Band structures of Pb Sb Te with (a) fully optimized crystal structure and (b) 0.5 % reduction of ab -lattice constants. Three-dimensional plot of Dirac cone in (b) and its contours are shown in (c) and (d).Anisotropic Dirac cone dispersions on (c) k x - k y and (d) k z - k x plane.plane.