Linear bands, zero-momentum Weyl semimetal, and topological transition in

skutterudite-structure pnictides

V. PardoDepartamento de Fısica Aplicada, Universidade Santiago de Compostela,

Spain, and Department of Physics, University of California Davis

J. C. Smith and and W. E. PickettDepartment of Physics, University of California Davis

(Dated: April 19, 2012)

It was reported earlier [Phys. Rev. Lett. 106, 056401 (2011)] that the skutterudite structurecompound CoSb3 displays a unique band structure with a topological transition versus a symmetry-preserving sublattice (Sb) displacement very near the structural ground state. The transition isthrough a massless Dirac-Weyl semimetal, point Fermi surface phase which is unique in that (1) itappears in a three dimensional crystal, (2) the band critical point occurs at k=0, and (3) linear bandsare degenerate with conventional (massive) bands at the critical point (before inclusion of spin-orbitcoupling). Further interest arises because the critical point separates a conventional (trivial) phasefrom a topological phase. In the native cubic structure this is a zero-gap topological semimetal; weshow how spin-orbit coupling and uniaxial strain converts the system to a topological insulator (TI).We also analyze the origin of the linear band in this class of materials, which is the characteristic thatmakes them potentially useful in thermoelectric applications or possibly as transparent conductors.We characterize the formal charge as Co+ d8, consistent with the gap, with its 3 site symmetry,and with its lack of moment. The Sb states are characterized as px (separately, py) σ-bonded Sb4

ring states occupied and the corresponding antibonding states empty. The remaining (locally) pz

orbitals form molecular orbitals with definite parity centered on the empty 2a site in the skutteruditestructure. Eight such orbitals must be occupied; the one giving the linear band is an odd orbitalsinglet A2u at the zone center. We observe that the provocative linearity of the band within thegap is a consequence of the aforementioned near-degeneracy, which is also responsible for the smallband gap.

PACS numbers:

I. MOTIVATION AND INTRODUCTION

Recent years have seen an explosion of interest intwo areas that, while distinct in themselves, havealso found some areas of overlap. One is the situa-tion where Dirac-Weyl linear bands emanate from apoint Fermi surface (FS), the so-called Dirac point.While linear bands around symmetry points are notuncommon in crystal structures, it is uncommonthat such points can determine the Fermi energy(EF ), which is what happens in the celebrated caseof graphene. It is necessary that there is a gapthroughout the Brillouin zone except for the touch-ing bands. Point FSs occur also in the case of con-ventional (massive) zero-gap semiconductors,1 andgive rise to properties that have been studied in somedetail. The case of the Dirac-Weyl semimetal (pointFermi surface) systems is quite different,2,3 leadingto a great deal of new phenomenology.

The other new and active area relevant to this pa-per is that of topological insulators (TIs), in whichthe Brillouin zone integral of a certain ‘gauge field’derived from the k-dependence of the periodic partof the Bloch states of the crystal is non-vanishing.4–6

These integer-valued topological invariants delineate

TIs from conventional (‘trivial’) insulators, with dis-tinctive properties that are manifested primarily inedge states.5,7–9 Identification of TIs, and study oftheir properties and the critical point that separatesthe phases, is highlighting new basic physics and sev-eral potential applications.

Recently we reported10 on the skutterudite struc-ture compound CoSb3, which combines and in-terrelates several of these properties (Dirac-Weylsemimetal at k=0; critical point of degeneracy withmassive bands; transition to topological insulator)in a unique way. The skutterudite structure, whichis illustrated in Fig. 1 and is discussed in Sec. II, iscritical to the electronic behavior in CoSb3 though itmay not be necessary for the unique type of criticalpoint that arises. The relevant portion of the bandstructure, in a very small volume of the Brillouinzone (BZ) centered at Γ, is unusually simple com-pared to most of the TIs that have been discovered,viz. the Bi2Se3 class,11, the HgTe class,12 and thethree dimensional system HgCr2Se4.

13

The electronic structure of skutterudites was stud-ied early on by Singh and Pickett,14 who focused onthe peculiar valence band which was linear (exceptexceedingly near Γ) and whose linearity extended

surprisingly far out into the BZ. It became clearthat the linearity of the band was responsible forthe large thermopower15 that is potentially usefulin thermoelectric applications.16,17 The origin of thelinearity was however not identified, but it leads topeculiar consequences:14 the density of states variesas ε2 near the band edge rather than the usualthree dimensional (3D) form

√ε; as a consequence

the carrier density scales differently with Fermi en-ergy εF ; the inverse mass tensor ∇∇εk is entirelyoff-diagonal corresponding to an “infinite” transportmass; the cyclotron mass differs from conventional3D behavior, etc. As the Sb sublattice is shifted in asymmetry-preserving way,10 the small semiconduct-ing gap of CoSb3 closes at a critical point, givingrise to an unusual occurrence: a Dirac-Weyl pointin a 3D solid at the Γ point. This transition pointsets the stage for a conventional insulator to topo-logical insulator transition, although the specific be-havior of the system at this point was not spelledout in our earlier work. The only other topologicalinsulator phases in skutterudite materials that weare aware of are the examples by Jung et al.18 inCeOs4Pn12 filled skutterudites.

This paper is organized as follows. In SectionII the skutterudite structure and its relation to thesymmetric perovskite (ReO3) structure is reviewed.The computational methods are outlined in SectionIII. Section IV discusses the evolution of the bandstructure during a perovskite-to-skutterudite trans-formation, including the development of the gap andthe appearance of the linear band. Section V isdevoted to constructing an microscopic but trans-parent understanding of the electronic structure, in-cluding the development of the energy gap and thecharacter of the peculiar linear band and the lowerconduction band triplet that it interacts with near,at, and beyond the critical point. In Section VI weprovide details of the band inversion leading to thetopological transition, and analyze the anisotropy atthe critical point. A brief summary is provided inSec. VII.

II. STRUCTURE AND RELATION TO

PEROVSKITE

It is useful for the purposes of this paper to con-sider the skutterudite structure to be a strongly dis-torted perovskite ATPn3 structure in which the Aatom is missing (i.e. the ReO3 structure) and theinterconnected TPn6 octahedra are rotated substan-tially (T = transition metal; Pn = pnictogen). Theskutterudite structure,20–22 pictured in Fig. 1, hasspace group Im3 (#204) and a body-centered cubic

FIG. 1: (Color online) Crystal structure of skutteruditeminerals (viz. CoSb3) with space group Im3 (#204),which includes inversion. The Co site (small pink [darkgrey] sphere) is octahedrally coordinated to Sb atoms(small yellow [light gray] spheres). Each Sb atom con-nects two octahedra, as in the perovskite structure whichhas the same connectivity of octahedra. The large (blue)sphere denotes a large open site that is unoccupied inCoSb3 but is occupied in filled skutterudites (see for ex-ample Ref. [19]). The geometrical solid (center of figure)provides an indication of the volume and shape of theempty region.

(bcc) Bravais lattice, and is comprised of a bcc rep-etition of four formula units (f.u.) when expressedas TPn3. The pnictide (Pn) atoms form bondedunits (nearly square but commonly designated as“rings”) which are not required by local environmentor overall symmetry to be truly square. The threePn4 squares in the primitive cell are oriented per-pendicular to the coordinate axes. Transition metal(T ) atoms (usually in the Co column) lie in six ofthe subcubes of the large cube of lattice constant a;the other two subcubes (octants) are empty. Thestructure has inversion symmetry and is symmor-phic, with 24 point group operations. The cubic op-eration that is missing is reflection in (110) planes.The related filled skutterudites XT4Pn12 have anatom X incorporated into the large 2a site of 3msymmetry23 that remains empty in the compoundsthat we discuss.

Figure 2 illustrates two aspects of the structure.The top panel reveals that, although the Sb6 octa-hedra are rotated considerably (and strained) com-pared to the perovskite structure, a great deal of reg-ularity is retained. The bottom panel gives a pictureof the relationship between the Co atoms and thesix Sb4 rings that it is coordinated with (only threemutually orthogonal rings are pictured). A naturallocal coordinate system for describing an Sb4 unit isto let it lie in the x − y plane; then the pz orbitals

2

FIG. 2: (Color online) Top: view down a cubic axis ofCoSb3, illustrating the very regular appearing positionsof the Sb4 rings, drawn as connected by bonds. Notethat one of these rings looks like a dumbbell when viewededge on. The (dark) Co atom is pictured without bonds.Bottom: an indication of the environment of the Co atom(smaller sphere); only three of the six attached Sb4 ringsare pictured. In this structure, all Sb sites are equivalentas are all Co sites.

we will discuss will be oriented perpendicular to theplane of the Sb unit. The px, py orbitals are nearlysymmetry related and naturally lead to bonding andantibonding molecular orbitals, and hence are verysimilar in character, while the nonbonding pz or-bitals have distinctive character.

If the origin is chosen so the large empty sites arecentered at ( 1

4 , 14 , 1

4 ) and (34 , 3

4 , 34 ), then the rings are

centered in each of the other subcubes (14 , 3

4 , 34 ) etc.

A ring oriented perpendicular to z is neighbored byrings above and below (±z directions) oriented per-pendicular to (say) y, by rings in the ±x directionsperpendicular to x, and neighbored by the emptysites along the ±y directions. A symmetric com-bination of the four pz orbitals on a ring, call itPz, is orthogonal to the Pz orbitals on neighboringrings, so if dispersion is governed by inter-ring hop-ping (rather than through Co atoms) some ratherflat Sb pz bands should result. Such behavior is seen

in fatbands plots (see below).As mentioned, the skutterudite structure is re-

lated to the perovskite structure �TPn3 (� denotesan empty A site). Beginning from perovskite, a ro-tation of the octahedra keeping the Pn atoms alongthe cube faces results in the formation of the (nearlysquare) Pn4 rings, and the Pn octahedra become dis-torted and less identifiable as a structural feature.The transformation is, in terms of the internal coor-dinates u and v,

u′(s) =1

4+ s(u − 1

4); v′(s) =

1

4+ s(v − 1

4). (1)

The transformation path, from perovskite for s=0 tothe observed structure for s=1, is pictured in Fig. 1of Ref. 24. Below we make use of this transformationto follow the opening of the (pseudo)gap betweenoccupied and unoccupied states.

III. COMPUTATIONAL METHODS

Two all-electron full-potential codes, FPLO-725

and WIEN2k26 based on the augmented planewave+local orbitals (APW+lo) method,27 have beenused in these calculations, with consistent results.The Brillouin zone was sampled with regular 12 ×12×12 k-mesh during self-consistency. For WIEN2k,the basis size was determined by RmtKmax= 7.Atomic radii used were 2.50 a.u. for Co and 2.23 a.u.for Sb, and the Perdew-Wang form28 of exchange-correlation functional was used. The experimentallattice parameters and atomic positions were takenfrom Ref. 20–22.

IV. PEROVSKITE TO SKUTTERUDITE

TRANSFORMATION

The development of the electronic spectrum as thecrystal is distorted from perovskite to the skutteru-dite structure is pictured in the three panels of Fig.3. For the ideal perovskite the result is a highlymetallic state, with no gap or pseudogap near theFermi level. The important features only arise nearthe end of the distortion path. For s=0.75 (top panelof Fig. 3) the valence bands have just become dis-joint from the conduction band and gap formation isimminent. For s=0.90 the gap is well formed and theminimum direct gap is emerging at the Γ point. Theunusual high velocity band arising from the Γ point,already striking at s=0.75, remains unchanged atthis point. The valence bands are a mass of indeci-pherable spaghetti.

By s=0.95 (bottom panel) a similarly high ve-locity band is emerging from the valence bands to

3

the maximum at the Γ point. At s=1.0 it becomesclear that it is a partner of the high velocity conduc-tion band, as discussed in our earlier paper10 andto which we return in the next section. No doubtthe formation of the gap, i.e. formation of occupiedbonding bands and unoccupied antibonding bands,is behind the stability of the skutterudite structurein the Co pnictides class of materials. The fact thatthe gap only opens as the Sb px and py bonding-antibonding splitting becomes strong supports thepicture that this tppσ interaction takes precedenceover the Co-Sb interactions within the CoSb6 unit;however, these latter interactions (i.e. the presenceof the Co atom) are necessary for the gap formation.

V. ELECTRONIC STRUCTURE

A. The Zintl Viewpoint

Although in the ‘parent’ perovskite system thestructure is that of vertex-connected CoSb6 octahe-dra, in the strongly distorted skutterudite structurethe Sb4 rings form a basic structural motif. Thesecompounds, viz. CoSb3, are sometimes character-ized as Zintl materials in which the Sb4 unit balancesthe charge of the Co unit. Within the Zintl picture,the charge of the Com+ ion must be balanced by a(Sb4)

(4m/3)− unit: there are only 3/4 as many Sb4

rings as Co atoms. Thus there is tension between theZintl picture, which does not lead to integral formalcharges on both of the primary units, and the pres-ence of the gap that specifies an integral number ofoccupied bonding states. Several papers14,23,24,29–35

have presented results and some analysis for emptyand for a few filled skutterudites. We have benefitedfrom the previous work as we proceed on a deeperanalysis.

We first follow the commonly held line of reason-ing that the short Sb-Sb distance in the Sb4 ringis the most fundamental aspect of the electronicstructure.35 The px − px bonding along an x-axisis characterized by a hopping amplitude tppσ ≈ 3eV, so the corresponding bonding and antibondingstates lie at εp ± tppσ. The separation of Sb atomsand the σ bonding and antibonding between py or-bitals along the y-direction is indistinguishable fromthat of the px σ case. Indeed, the correspondingpσ bonding and antibonding character (not shownin figures) is centered roughly ±3-4 eV below andabove the gap. The on-site energies εpx

≈ εpy= εp

therefore lie in the vicinity of the gap. This pictureleads to half-filled px and py orbitals.

The pz orbitals within an Sb4 unit are orthogonal

Γ H N Γ P N P

s=0.75

−2.0

−1.0

0.0

1.0

2.0

Ene

rgy

εn(

k) [

eV]

Γ H N Γ P N P

s=0.90

−2.0

−1.0

0.0

1.0

2.0

Ene

rgy

εn(

k) [

eV]

Γ H N Γ P N P

s=0.95

−2.0

−1.0

0.0

1.0

2.0

Ene

rgy

εn(

k) [

eV]

FIG. 3: Transformation of the band structure frommetallic in the perovskite to semiconducting in the skut-terudite, following the (linear) transformation describedin the text. From the top, the panels show s = 0.75,0.90, 0.95. The gap only begins to emerge around s=0.75, where the conduction bands are already disjointfrom the valence bands. The linear band near Γ emergesfrom the valence bands around s=0.95.

4

FIG. 4: (Color online) Band structure of CoSb3 alonghigh symmetry directions, with fatbands highlighting theSb pz character. The character is substantial in the -3 to-6 eV region, in the 3 to 5 eV region, but most notablyin the linear band extending through the gap, which isprimarily pz in character.

to the px, py orbitals and couple by a tppπ hoppingamplitude. This coupling leads to doubly degeneratelevels at ±tppπ relative to εp. The evidence is thatthere is negligible crystal field splitting of the threeon-site p orbitals. However, inter-unit pz − pz inter-action will be stronger than this intra-unit coupling,and the pz orbitals also couple to the Co orbitals.The pz character, provided by the fatbands repre-sentation in Fig. 4, is in fact distributed throughoutthe -6 eV to +5 eV region and is not reproducibleby a simple model due to other couplings. The pz

projected density of states (DOS) indicates the pz

orbitals are at least half-filled, perhaps slightly more.

B. Complications with the Zintl Viewpoint

It is readily demonstrated that analysis focusingprimarily on the Sb4 rings has strong limitations.We have performed calculations with the Co atomremoved, �Sb3. The Sb p band complex is broad (10eV) and featureless. There is no hint of a gap or evenpseudogap, indicating that the px and py bonding-

FIG. 5: (Color online) Band structure of CoSb3 alonghigh symmetry directions, with fatbands highlighting theCo 3d character. This plot reflects a narrow mass of d

bands throughout the -1 to -2 eV range, more around -3eV, and ∼4 unoccupied bands in the 0-1 eV range. Theamount of 3d occupation is discussed in the text.

antibonding feature described above is far from adominant feature. Only when Co is re-inserted intothe structure does the remarkably clean 1 eV gapbecome carved out around the Fermi level, exceptof course for the peculiar linear band near the zonecenter. The gap therefore arises from the Co 3d mix-ing (evidently strongly) with Sb 5p orbitals, with thelinear band emerging from the valence bands beingof strong Sb character.

Figure 5 illustrates the Co 3d character. It liesmostly below the gap, in the very narrow -1 to -2 eV range. A much smaller amount of character,roughly something like four bands, lies immediatelyabove the gap in the 0-1 eV range. These two partsof the 3d projected DOS are neatly and impressivelysplit by the 1 eV gap. The “charge state” (or for-mal valence, or oxidation state) of the Co atom is aquestion that can be asked, since this is an insulat-ing material. If there are four unoccupied d bands(though this cannot be claimed very conclusively dueto the strong hybridization) then considering thereare four Co atoms in the cell, the occupation wouldbe d8 Co1+. A nonmagnetic ion with this filling isnatural for the Co 3 site symmetry. As the CoSb6

5

octahedron is distorted strongly in progressing fromthe cubic perovskite structure to the skutteruditeone, the crystal field levels reduce as t2g → t1 + t2,the latter being the doublet, and eg → e1 + e2. Oneunoccupied orbital singlet (both spins) gives the d8

occupation. The magnetic quantum number m = 0orbital with respect to the local three-fold axis is thenatural one for the 3d holes to occupy.

Charge balance will then leave the rings as(Sb4)

−4/3, a very unsatisfactory result for account-ing for the gap: each Sb4 ring would not have aninteger number of (fully) occupied bands (or molec-ular orbitals). Within the picture that the px and py

orbitals are half-filled due to the large tppσ bonding-antibonding splitting, this Co charge state wouldsuggest pz orbitals to be 1/6 more than half-filled,a peculiar result and one that would only be com-patible with a metallic, rather than semiconducting,state.

The integrated DOS for the Co 3d states in factgives a clear d8 occupation. However, with near-neutral atoms there might be significant Co 4s, 4poccupation, which could bring Co nearer to a neu-tral d8s1 configuration. Some Co sp character is infact found in the upper valence bands, though it ispossible that this represents tails of Sb p orbitals.The Co d occupation d8 is clear, however.

C. Sb 6p - Co 3d Mixing

It was noted above that when Co is removed fromthe structure, there is no gap nor any indication ofone, nor any candidate for a potential linear band.The most direct interaction of the Sb pz orbitals onthe ring is between the pz orbital on the nearest Sb inthe ring with the rotated xz±yz orbitals on the eightCo neighbors. Sb6 molecular orbitals (MOs) formedby linear combinations of pz orbitals can be sortedaccording to whether they are even or odd paritywith respect to the center of inversion at the Co site.The even parity MOs mix with the Co 3d orbitals,forming bonding and antibonding pairs tending toopen a gap; the odd parity MOs do not mix withthe 3d orbitals. As a result of this coupling, four ofthe five 3d orbitals are lowered and occupied, whileone is raised and unoccupied, giving the 1 eV gapand the d8 occupation.

D. Band Character near the Gap

The fatbands representation of Sb pz characterin Fig. 4 shows the substantial pz character of thelinear band in the upper valence bands near thezone center. This pz character is in fact the dom-

inant character, although various other small con-tributions from other valence orbitals are mixed inby the low Sb site symmetry. The character of the3-fold degenerate conduction band minimum stateis more complex. These bands have primarily Co“t2g” character; the quotes arise from the fact thatthe degeneracies of the cubic t2g and eg subshellsare broken by the distortion of the octahedra in theskutterudite structure. There is a strongly dz2 flatband just above (+0.2 eV) the gap, and additionalstrong dx2

−y2 character in the 0.2-0.6 eV region.The band figures we show below demonstrate that

the linear band couples to only one of the triplet ofconduction bands at and near Γ, i.e. couples to onlyone linear combination of the triplet states. It doesnot couple to the Co 3d orbitals.

FIG. 6: Surface plot of the density arising from the linearband in CoSb3 near k=0. The closed surface surroundsthe unoccupied site in the skutterudite lattice and en-closes a region of lower density than outside of the sur-face.

E. A consistent viewpoint: an empty-site

molecular orbital

A generalized viewpoint of the most physical or-bitals in CoSb3 is that of four Co1+ d8 atoms (asdiscussed above), three Sb4 rings with px- and py-σbonding states filled and antibonding states unoccu-pied, and molecular orbitals formed from the twelvepz orbitals and centered on the empty 2a site. Thissite is an inversion center, so the MOs can be clas-

6

sified as parity-even or -odd. (Note that these aredifferent MOs than were discussed in Sec. V.C.)Twelve MOs can be formed, which can be classifiedalso by the tetrahedral 2a site symmetry. Chargecounting requires that eight of these MOs must beoccupied (with both spins).

We will not attempt to identify the positions anddispersions of these MOs, except to note that linearband is one of these MOs. Jung et al.35 identifiedthis MO as A2u, an orbitally nondegenerate statewith odd parity at k=0, and therefore not mixingwith the Co 3d orbitals at (or near) Γ. The densityarising from this orbital is of interest. We had earlierinvestigated the possibility that an electron may re-side in the large empty 2a site in the skutteruditestructure. This type of “electride” configuration,in which an interstitial electron without a nucleusbecomes an anion, is well established in molecularsolids.36,37 However, the density at the 2a site is verysmall, ruling out this possibility.

An isosurface plot of the density arising from thelinear band near Γ is shown in Fig. 6. The closedsurface in the density encloses a region of low density

centered on the unoccupied site. Exploring the den-sity with isosurfaces at various values of density re-veals no local maximum at this large interstitial site,thus no electride-like character. Lefebvre-Devos et

al. have presented30 complementary isosurface den-sity plots of this band, with the high density regionsof two types. One region surrounds the empty site(with rather complex shape) consistent with its ori-gin as a MO comprised of pz Sb4 ring states, withthe tetrahedral symmetry of the 2a site. The secondregion is within each CoSb6 octahedron, centered oneither side of the Co atom along the local 3 axis, ap-pearing as a three-bladed propeller that arises fromthe lobes of the pz orbitals that lie closer to the Coatom.

Interaction of the pz states with Co 3d orbitalsappears to cause a majority of the density of thisstate to border the large empty site. The result is notan electride state but rather a large molecular orbital(one of twelve in total) centered on the empty site,with one of them (A2u) per primitive cell. We havepreviously reported that this state at Γ is parity-odd.10

Such an orbital has the same hopping amplitudet along each nearest neighbor direction on a bcc lat-tice, hence its dispersion looks like that of an s or-bital, Ek = 8t cos(kxa/2) cos(kya/2) cos(kza/2), anddoes not lead to a linear band at k=0. This shouldn’tbe surprising, as the true linearity at Γ is acciden-tal, due to a degeneracy that has to be tuned. Inour previous paper,10 this tuning was accomplishedby making two on-site energies in the tight-bindingmodel identical.

The “linear band” aspect of the skutterudites mayhave been over-interpreted, because it arose in themost common member (the mineral skutteruditeCoAs3) which was studied most heavily initially. Ina general skutterudite, for example the filled oneLaFe4P12, the same band rises out of the valenceband complex but is not unduly “linear.” The iso-valent members XPn3, with X = Co, Rh, Ir, andpnictides Pn = As and Sb, all seem to lie not farfrom the critical degeneracy that extends the bandlinearly to k=0; however, we know of nothing ingeneral about the skutterudite crystal structure thatotherwise promotes a linear band.

VI. NONANALYTICITY, ANISOTROPY,

AND THE TOPOLOGICAL INSULATOR

PHASE

A. Band Inversion: Internal strain, spin-orbit

coupling, and tetragonal strain

In previous work10 we demonstrated that it is pos-sible, with very small Sb sublattice displacement (seeSec. II for the linear distortion, characterized by theparameter s), for this band to remain linear all theway to k=0, marking a critical point. In that work,the progression of the band crossing at the criticalpoint was plotted. Due to the 3-fold degeneracy ofthe conducting band edge, the progression versus swas from semiconductor for s ≤ 1.19, to the criticalpoint at s=1.020 with linear valence and conduc-

tion bands degenerate with two quadratic bands, toa zero-gap semiconductor for s ≥ 1.021 due to asymmetry-determined degeneracy, even when spin-orbit coupling (SOC) is included. Due to the bandinversion and the character of the states at Γ, thetransition is from trivial insulator to one with topo-logical character. Because the degeneracy at thepoint Fermi surface (at Γ) remains, the latter stateis a topological zero-gap semiconductor rather thana topological insulator.

We now demonstrate that a simple tetragonaldistortion, such as might result from growth on asubstrate with small lattice mismatch, does pro-duce a topological insulator (TI) but with an un-expected placement of the gap. The progression ofthe band structure very near Γ is shown in Figure7 for strain c/a=1.01. The gap before the criticalpoint is reached always has quadratic bands for suf-ficiently small k. At the critical point s = 1.020(when the strain is imposed) leads, unexpectedly, tolinear bands only in certain of the high symmetry di-rections (such as the < 111 > direction, toward theP point). In other high symmetry directions (from

7

H/50 Γ P/50

s=1.019

-20

0

20

40

60

80

100

Ene

rgy ε

n(k)

[m

eV]

H/50 Γ P/50

s=1.020

-40

-20

0

20

40

60

80

100

Ene

rgy ε

n(k)

[m

eV]

H/50 Γ P/50

s=1.022

-20

0

20

40

60

80

100

Ene

rgy ε

n(k)

[m

eV]

FIG. 7: Transformation of the band structure verynear k=0 from insulating at s=1.019 (top) through thesemimetallic critical point at s=1.020 to the invertedband insulator at s=1.022 (bottom). H/50 and P/50denotes the point 2% of the way from Γ to H and P, re-spectively. The horizontal line lies in the gap (or at theFermi level) in each panel. In the bottom panel the gaphas moved out along the Γ-H line but is non-vanishing.The much larger band repulsion along Γ-P produces theunique band dispersion at the critical point.

Γ toward H is pictured) the bands are quadratic andthe upper one is quite flat (heavy mass). We returnto this point in the next subsection.

Beyond the critical point, the four bands becomenon-degenerate once more; the lower (linear) bandhas crossed the lowest of the three conduction bandsand acquired mixed character. The coupling be-tween bands is clearly much smaller along the axes(Γ-H direction) than along the other symmetry lines,and the gap occurs along this direction. Because of

the heavy mass of the lower conduction bands, thegap is quite small, being of the order of 1 meV forthe tetragonal strain that is pictured in Fig. 7.

B. Anisotropy at the Critical Point

Inspecting the bands along the three cubic highsymmetry directions, a second peculiar feature be-comes evident. At the critical point, the “linearband” disperses linearly along the < 111 > di-rections, while they are quadratic (with distinctmasses) along both the < 100 > and < 110 > di-rections. The directional dependence as well as themagnitude dependence of the wavevector is anoma-

lous. The form of the dispersion for small |~k| canbe constructed. A polynomial in the direction of~k (k) with cubic symmetry that vanishes along the< 100 > directions can be constructed as

P100(k) =

√

√

√

√

6∏

j=1

(k − kj)2

=6

∏

j=1

|k − kj |, (2)

where {kj} are the six < 100 > unit vectors. The de-nominator normalizes the polynomial to unity along

each of the < 111 > directions; k111 lies alongthe < 111 > direction (any of them). The anal-

ogous polynomial P110(k) that vanishes along each< 110 > direction and is normalized to unity alongeach < 111 > direction can be constructed. Thenthe peculiar band pair is given by

ε(~k) = ±v|~k|P100(k)P110(k) +k2

2ms± k2

2ma, (3)

where ms,ma serve as symmetric and antisymmetricmasses.

C. Isovalent Skutterudite Antimonides

We have studied the band structures of anti-monide sisters of CoSb3, based on the transitionmetal atoms Rh and Ir. Band structures, or some-times only gap values, of these compounds have beenreported earlier.23,31,38 The band structures aroundthe gap at k=0 differ somewhat in the literature,due to sensitivity to structure (and that some userelaxed coordinates while others use the measuredstructure), to the level of convergence of the calcu-lation, and the exchange-correlation functional.

The difference in lattice constant is compensatedvery closely by the effect of difference in interatomic

8

distances on hopping amplitudes, and the resultingbands are very similar. The single feature of inter-est is the magnitude of the gap (and whether posi-tive or negative, the latter giving an inverted bandstructure) and the strength of SOC effects. As men-tioned above, band inversion alone does not in it-self give rise to a topological insulator state in thesecompounds because even after SOC included, a two-fold degeneracy at Γ is lowest and the system is apoint Fermi surface, zero-gap semiconductor. Suchcases are referred to as topological semimetals. How-ever as shown above, the topological semimetal canbe driven into the TI phase by tetragonal distortionthat lifts the final degeneracy.

RhSb3. For a range of volumes around the ex-perimental one (a= 9.23 A), this compound has aninverted band structure, hence it is a point Fermisurface, zero-gap topological semimetal. In the in-verted band structure before considering SOC, thesinglet lies 23 meV above the triplet that pins theFermi level. Inclusion of SOC leaves the Fermi en-ergy pinned to a doublet. A tetragonal distortionraises the degeneracy; a topological insulator is ob-tained for c/a > 1 in the same way that happensfor CoSb3 although no sublattice strain is requiredin RhSb3.

Generally, reducing volume (applying pressure) inthese Sb-based skutterudites leads to a larger gap.In the case of RhSb3, our calculations indicate a gapnever actually opens within cubic symmetry, for pe-ripheral reasons. When the band inversion at Γ isundone and a gap at Γ appears at sufficiently highpressure (a= 9.06 A), the material is metallic dueto other (Rh 4d) bands having become lowered inenergy, crossing the Fermi level along the Γ-N direc-tion.

IrSb3. The larger SOC for heavier atoms led usto consider this 5d compound. However, the linearband and several others have practically no metalatom character and are not affected by its spin-orbitstrength. We obtain, using the same computationalmethods as for the other two compounds, that likeCoSb3, IrSb3 has a ∼80 meV gap and therefore notopological character at the experimental equilib-rium structure. Like CoSb3, it can be driven to aninverted band structure and hence topological char-acter by internal and tetragonal strains.

VII. SUMMARY

One goal of this work was to provide a simple butfaithful picture of the electronic structure of CoSb3:the origin of the gap and the character of the linearband being the most basic. The picture is this. Cois in a d8 configuration, leaving four electrons in theCo4Sb12 unit cell to go into other bands. The px

and py orbitals on the Sb4 ring form strongly bond-ing (occupied) and antibonding (unoccupied) states,accounting for two of the three 5p electrons of eachSb atom, for a total of twelve occupied σ-bondingstates (of each spin). The twelve π-oriented pz or-bitals couple between units as well as with the Co3d orbitals, and form “molecular orbitals” centeredon the vacant 2a sites in the skutterudites lattice.Eight of these MOs are occupied by 16 electrons,one from each Sb and one from each Co. The linearband is best pictured as arising from the couplingand dispersion of these MOs.

We have also justified our earlier statement10 thatCoSb3 is very near a conventional-to-topologicaltransition. A small symmetry-preserving internalstrain, a small applied tetragonal strain, and spin-orbit coupling drives CoSb3 into a topological insu-lator phase. This critical point also marks a pointof accidental (but tunable, by the strains) degener-acy that gives rise to emergence of linear Dirac-Weylbands emanating from k=0, where they are degen-erate with massive bands. At this critical point, thedispersion is non-analytic at k=0 and anisotropy isextreme, with the mass varying from normal to van-ishing. In this respect CoSb3 at its critical pointbears several similarities to the semi-Dirac (Dirac-Weyl) behavior encountered39–41 in ultrathin layersof VO2.

VIII. ACKNOWLEDGMENTS

We acknowledge illuminating discussions with S.Banerjee, R. R. P. Singh, C. Felser, and L. Muchler.Work at UC Davis was supported by DOE GrantDE-FG02-04ER46111. V.P. acknowledges supportfrom the Spanish Government through the Ramony Cajal Program.

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