Electronic structure and bonding in skutterudite-type phosphides

The electronic structures of the skutterudite-type phosphides CoP 3 and NiP3 have been investigated by means of first-principles linear muffin-tin orbital–atomic sphere approximation band-structure calculations. The presence of P 4 rings in the skutterudite structure is of great importance in determining the nature of the electronic bands around the Fermi level, composed mainly of p-type molecular orbitals of these units. The metallic character found for NiP 3 should be ascribed to the phosphorus framework rather than to the metal atoms.

Although these compounds show interesting physical properties, relatively little is known about their electronic structure, probably because of the complex crystal structure with a relatively open unit cell.So far, only two bandstructure calculations have been reported for the skutteruditetype compounds.In the first study, 21 the electronic structures of LaFe 4 P 12 and related compounds were analyzed by using the extended Hu ¨ckel tight-binding ͑EHTB͒ method.Recently, self-consistent band-structure calculations, using the linear augmented plane-wave method for IrSb 3 , CoAs 3 , and CoSb 3 , have been published by Singh and Pickett. 22n this contribution, we report first-principles bandstructure calculations using the linear muffin-tin orbital method in the atomic sphere approximation ͑LMTO-ASA͒ ͑Refs.23-25͒ to explore the band structure of CoP 3 and NiP 3 .

CRYSTAL STRUCTURE
The nature of the highest occupied bands, responsible for the electronic properties of a material, is strongly dependent on its crystal structure.A detailed knowledge of the crystal structure is, therefore, indispensable before analyzing the details of the electronic structure.In the case of the skutterudite structure ͑space-group Im3 ¯, n. 204͒, 26 the conventional body-centered-cubic unit cell contains eight M X 3 units.Metal atoms are located on 8c sites, while the nonmetal ones occupy 24g sites.8][29][30] A simple way to describe the skutterudite structure is to deduce it as a distortion of the more symmetric ReO 3 structure, 31,32 formed by a simple cubic array of metal atoms octahedrally surrounded by six oxygen atoms.This structure can also be considered as an array of vertex-sharing octahedra ͓Fig.1͑a͔͒.
The characteristic X 4 rings in skutterudites can be obtained by displacing four of the nonmetal atoms located on parallel edges of a metal cube to its center, as shown in Fig. 1͑b͒.Since each cube has 12 nonmetal atoms on its edges, the displacement must be done simultaneously in eight neighboring cubes.The new unit cell is thus eight times larger than the original ReO 3 unit cell.Since there are not enough nonmetal atoms to make X 4 rings at the center of each cube, the distortion leaves one out of each four metal cubes empty, as shown in Fig. 2͑a͒.In the resulting skutterudite structure, each metal atom is in an octahedral environ- The American Physical Society ment ͓Fig.2͑b͔͒ and each M X 6 octahedron is sharing vertices with six neighboring octahedra as in ReO 3 .The main differences between both structures are that, in the case of skutterudite, the M X 6 octahedra are distorted ͑local D 3d symmetry͒ with their relative orientations in the threedimensional array tilted.The X 4 rings in the skutterudite structure are arranged in mutually orthogonal linear arrays that run parallel to each crystallographic direction ͓Fig.2͑c͔͒.

COMPUTATIONAL METHODOLOGY
4][25] In this method, the one-electron potential entering the Schro ¨dinger equation is a superposition of overlapping spherical potential wells centered at the atomic positions.A spherical charge density inside the atomic spheres is also assumed.These approximations are specially well suited for highly compact crystal structures.In the case of CoP 3 and NiP 3 with the relatively open skutterudite structure, the use of only atom-centered spheres would result in substantial errors.To avoid this problem, it has been necessary to introduce interstitial spheres in order to choose sufficiently small radii for the spheres that result in small overlaps between them and at the same time fulfill the usual condition of maximizing the volume of the crystal occupied by the atomic spheres.The location and size of the atomic and interstitial spheres has been determined, using the superposition of Hartree atomic potentials. 34Application of this procedure to CoP 3 and NiP 3 results in the sphere positions and radii shown in Table I.Together with the real atoms, our calculations included also 19 empty spheres on each primitive unit cell that, according to symmetry, belong to three nonequivalent classes.The same internal parameters and sphere radii have been used for both compounds.For the cell parameter a, the experimentally determined value ͑7.073 Å for CoP 3 and 7.8192 Å for NiP 3 ͒ has been used in each case.
All calculations were performed within the tight-binding ͑TB͒ minimal basis set representation of the LMTO-ASA   25 The structure constants were first generated for the short-range TB basis set, but afterwards the transformation to the standard orthonormal basis set has been performed.The down-folding technique 25 based on Lo ¨wdin's perturbation theory 35 has been applied to this latter set.Integrations in k space were performed using the tetrahedron method 36,37 with a mesh of 512 points in the irreducible wedge of the first Brillouin zone.
9][40] The parameters used in these calculations are indicated in Ref. 41.

RESULTS AND DISCUSSION
The calculated ͑LMTO-ASA͒ band structures of CoP 3 and NiP 3 are shown in Figs.3͑a͒ and 3͑b͒, respectively.The general trends observed in both band-structure diagrams are qualitatively similar to those found by Singh and Pickett 22 for IrSb 3 , CoSb 3 , and CoAs 3 .The most important feature in the dispersion diagrams is the more or less well-defined pseudogap that appears just below the Fermi level ͑see Fig. 4͒.In CoP 3 , this pseudogap is crossed by a totally occupied single band that almost reaches the conduction band at the center of the first Brillouin zone.The calculated width of the indirect pseudogap is 1.26 eV, that of the indirect band gap ͑from ⌫ to a point on the ⌫-H line͒ is 0.07 eV and that of the direct gap at ⌫, 0.28 eV.Our calculated direct band gap is somewhat smaller than the value given by Ackermann and Wold 7 for the optical gap in CoP 3 ͑0.45eV͒, probably due to the well-known tendency of the local-density-approximationbased methods to underestimate this property.As already pointed by Singh and Pickett 22 for IrSb 3 , CoSb 3 , the dispersion of the band in the pseudogap region of CoP 3 is remarkable: although it has a parabolic shape in a small region around the ⌫ point, it rapidly becomes linear in energy when leaving the center of the Brillouin zone.This effect is most evident in the ⌫-H direction, where the crossover between parabolic and linear dispersion occurs for wave vectors of about 5% the distance to the zone boundary.The transport properties of the hole-doped material should thus be modified by this linear dispersion from those expected for standard semiconductor behavior.The dispersion diagram for NiP 3 , although similar in its rough features to that found for CoP 3 , shows some distinct features.The larger dispersion of the bands reduces the pseudogap region to approximately 0.57 eV.In this case, the band in the pseudogap region penetrates the bottom of the conduction band.In this compound, the metal atoms provide one more electron per formula unit than in CoP 3 .These electrons occupy the bottom of the conduction band and are responsible for the metallic behavior of NiP 3 .
Figures 4͑a͒ and 4͑b͒ show the calculated total and projected densities of states ͑DOS͒ for both compounds.The pseudogap region with low values for the DOS is clearly visible just below the Fermi level.The phosphorus projection can be roughly divided in three separate regions: the first one ranging from approximately Ϫ15 to Ϫ9 eV, the second one from Ϫ8 eV to the lower edge of the pseudogap, and the last one starting from the upper edge of the pseudogap.Since these results agree in their main features qualitatively well with those obtained from EHTB calculations ͑not shown in the figures͒, this method will be used as a qualitative tool in the analysis of bonding in these materials.
Our EHTB calculations indicate that the lowest region is due basically to the -bonding orbitals of the P 4 rings with large phosphorus 3s contributions.The second region is formed mainly by both the nonbonding and the bonding -type molecular orbitals of the P 4 rings.The third region is due mainly to ring antibonding molecular orbitals, both of -type ͑lowest part͒ and -type ͑region above 5 eV͒.It is also very important to note that a large contribution of the nonbonding and -type bonding orbitals that have the proper spatial orientation to interact strongly with the metal d orbitals is found at the lowest part of this third region.EHTB calculations show for these states an important positive ͑bonding͒ overlap population for the P-P bonds.
The projection of the DOS for the metal atoms shows that for both CoP 3 and NiP 3 the main peak is found in the region directly below the pseudogap.According to these results, the d orbitals of the metal atoms seem to be almost completely filled.3][44][45] The most relevant conclusion that can be drawn from these facts is that the bands responsible for the electrical conduction in NiP 3 are mainly centered on the phosphorous sublattice.This is clearly visible in our LMTO-ASA calculations, where, the contributions of phosphorous and nickel states to the DOS at the Fermi level are equal to 51.7 and 13.7 states/eV per unit cell, respectively.A more detailed picture of the nature of these states shows that they are mostly formed by nonbonding and -type bonding orbitals of the P 4 rings.Coupling of these orbitals to neighboring rings in all three crystallographic directions is achieved through interaction with the s and p orbitals of the metal atoms. 21r CoP 3 , it is also interesting to describe the nature of the single band in the pseudogap region.Our calculations confirm the result obtained earlier by other authors, 21 indicating that it is basically formed by the -type antibonding orbital of the P 4 rings shown in Fig. 5͑a͒.The important dispersion of this band is due to mixing of these orbitals belonging to neighboring rings with the metal p orbitals ͓Fig.5͑b͔͒.

CONCLUDING REMARKS
First-principles band-structure calculations for CoP 3 and NiP 3 show that both materials have similar electronic structures with predominantly covalent or metallic bonding.As a consequence, the highest occupied bands are mainly composed of phosphorous-centered orbitals.The electrical properties of phosphorous-rich late transition metal phosphides seem thus to be determined principally by the phosphorous sublattice rather than by the metallic atoms present in the structure.This finding is specially surprising for NiP 3 , since it has been experimentally found to be a metallic conductor with Pauli paramagnetism.For d 6 metal skutterudites like CoP 3 , electrical properties should be determined by the existence of a pseudogap separating the valence and conduction bands.The low density of states in this pseudogap region is provided by a single occupied band that exhibits quasilinear dispersion relations in large parts of the symmetry lines joining ⌫ and the zone boundaries.To confirm whether the highest occupied bands originate largely from the phosphorous or the metal sublattices, it would be interesting to perform a spectroscopic investigation of the valence-band region for this family of compounds.
the Eleventh International Conference on Thermoelectrics, edited by K. R. Rao ͑University of Texas at Arlington Press, Arlington, TX, 1993͒.

FIG. 1 .
FIG. 1. ReO 3 crystal structure ͑a͒ and distortion that leads to the skutterudite structure ͑b͒.Black and gray balls represent O and Re atoms, respectively.

FIG. 2 .
FIG.2.Skutterudite structure: Unit cell ͑a͒, coordination of the metal atoms and array of vertex-sharing octahedra ͑b͒, and arrangement of the X 4 rings in chains ͑c͒.Black and gray balls represent nonmetal and metal atoms, respectively.In ͑c͒, three mutually orthogonal chains have been highlighted for the sake of clarity.

FIG. 4 .
FIG. 4. Total and projected LMTO-ASA electronic density of states ͑in states/eV cell͒ for CoP 3 ͑a͒ and NiP 3 ͑b͒.The dashed line indicates the Fermi level.

FIG. 5 .
FIG. 5. Phosphorous contribution ͑a͒ and interaction between phosphorous and metal p orbitals ͑b͒ in the highest occupied band for CoP 3 .

TABLE I .
Geometrical parameters used for the LMTO-ASAcalculations on CoP 3 and NiP 3 .S R indicates the radius chosen for each atomic sphere.E represent empty spheres.