PHYSICAL REVIEW B 86, 075316 (2012) Large insulating gap in topological insulators induced by negative spin-orbit splitting Julien Vidal, 1 Xiuwen Zhang, 2 Vladan Stevanovi´ c, 1 Jun-Wei Luo, 1 and Alex Zunger 3,* 1 National Renewable Energy Laboratory, Golden, Colorado 80401, USA 2 Department of Physics, Colorado School of Mines, Golden, Colorado 80401, USA 3 University of Colorado, Boulder, Colorado 80309, USA (Received 7 February 2012; revised manuscript received 16 May 2012; published 24 August 2012) In a cubic topological insulator (TI), there is a band inversion whereby the s -like Ŵ 6c conduction band is below the p-like Ŵ 7v + Ŵ 8v valence bands by the “inversion energy”  i < 0. In TIs based on the zinc-blende structure such as HgTe, the Fermi energy intersects the degenerate Ŵ 8v state so the insulating gap E g between occupied and unoccupied bands vanishes. To achieve an insulating gap E g > 0 critical for TI applications, one often needs to resort to structural manipulations such as structural symmetry lowering (e.g., Bi 2 Se 3 ), strain, or quantum confinement. However, these methods have thus far opened an insulating gap of only <0.1 eV. Here we point out that there is an electronic rather than structural way to affect an insulating gap in a TI: if one can invert the spin-orbit levels and place Ŵ 8v below Ŵ 7v (“negative spin-orbit splitting”), one can realize band inversion ( i < 0) with a large insulating gap (E g up to 0.5 eV). We outline design principles to create negative spin-orbit splitting: hybridization of d orbitals into p-like states. This general principle is illustrated in the “filled tetrahedral structures” (FTS) demonstrating via GW and density functional theory (DFT) calculations E g > 0 with  i < 0, albeit in a metastable form of FTS. DOI: 10.1103/PhysRevB.86.075316 PACS number(s): 73.43.−f, 31.15.A−, 72.25.Hg, 73.20.−r I. INTRODUCTION In conventional cubic insulators such as GaAs or CdTe [Fig. 1(a)] the s -like conduction band Ŵ (2) 6c lies above the p-like valence band Ŵ (4) 8v + Ŵ (2) 7v , thus defining a positive “inversion energy”  i = E[Ŵ (2) 6c ] − max(E[Ŵ (4) 8v ],E[Ŵ (2) 7v ]) (superscript denotes degeneracy). It has been known for a long time 1 that as the elements making up common cubic insulators become progressively heavier, e.g., in the sequence ZnTe → CdTe → HgTe or Si → Ge → Sn, the s -like Ŵ (2) 6c moves down in energy (a relativistic Mass-Darwin effect 2 ) and eventually (e.g., HgTe or α-Sn) dive below p-like Ŵ (4) 8v leading to  i < 0, i.e., band inversion. Recently 3,4 it was pointed out that when such band inversion occurs at time-reversal invariant wave vectors it produces interesting electronic properties, e.g., Dirac cones inside the insulating bulk band gap. The observation 3,5 and usefulness of such effects necessitate that in addition to a negative inversion energy  i < 0 there must be a positive “insulating gap” E g ; i.e., the lowest-unoccupied states lie above the highest occupied state. Small band-gap TIs, on the other hand, lead to two obstacles to realize TI-based devices at room temperature. First, the omnipresent carrier-producing defects have a significant effect in small gap materials as they impede the tuning of the Fermi level. 6 Second, in narrow gap materials, the generally present band bending has a particularly large effect on narrowing the gap. 7 In a normal insulator such as CdTe [Fig. 1(a)], the positive inversion energy  i equals the insulating gap E g [Fig. 1(a)], but in a cubic TI such as HgTe [Fig. 1(c)]  i < 0 is associated with E g = 0. As is evident from Fig. 1(c), this results from the fact that the Ŵ (2) 6c state which dives below the Ŵ (4) 8v state is twofold degenerate (including spin) whereas, in HgTe, Ŵ (4) 8v state is fourfold degenerate (including spin), so the Fermi level dissects Ŵ (4) 8v and results in E g = 0. The opening of an insulating band gap can be accomplished in conventional materials by splitting the degeneracy of the Ŵ 8v -like state via application of external nonuniform strain (as was proposed in LuPtBi 8 ) or by selecting noncubic materials with sufficiently low symmetry to create a natural (“crystal-field”) splitting (e.g., chalcopyrite 9 or in rhombohedral Bi 2 Se 3 ) or by quantum confinement pushing the highest occupied state down and the lowest unoccupied state up. 10,11 However, finding materials with band inversion ( i < 0) yet with large positive insulating gap E g has not been easy with conventional TI materials. To date, many of them displayed semimetallic or very small insulating gap E g < 0.1 eV, with very few materials such as Bi 2 Se 3 being above 0.2 eV. In this paper we suggest an electronic mechanism to create a significant insulating gap (larger than 0.5 eV in some cases), yet with band inversion. It is based on the observation that the order of Ŵ (2) 7v and Ŵ (4) 8v in cubic materials can be switched between “Ŵ 8v -above-Ŵ 7v ” [“positive spin-orbit (SO) splitting”  so > 0, Figs. 1(a) and 1(c)] and “Ŵ 8v -below-Ŵ 7v ” [“negative spin-orbit splitting”  so < 0, Figs. 1(b) and 1(d)]. In the latter case, if in addition  i is significantly negative then there is a natural insulating gap between the unoccupied Ŵ (2) 7v and the occupied Ŵ (4) 8v . We show that  so < 0 can be designed by increasing d character (not the usual p character) in Ŵ (2) 7v + Ŵ (4) 8v , and we point to a group of materials likely to have this property. The existence of a topological Mott insulator for some negative spin-orbit systems 12–15 has been recently proposed. In these cases, an insulating gap would result from a complex interplay between electron-electron correlation, spin-orbit coupling, and crystal-field splitting. 16 Thus, a mechanism based purely on negative spin orbit has yet to find a stable candidate material. Nevertheless, we propose specific design principles which might help in identifying TI systems having large insulating gaps without the use of nanostructuring or application of external nonuniform strain. 075316-1 1098-0121/2012/86(7)/075316(5) ©2012 American Physical Society