Bulk Dirac Points in Distorted Spinels

Julia A. Steinberg, Steve M. Young, Saad Zaheer, C. L. Kane, E. J. Mele, and Andrew M. Rappe

Phys. Rev. Lett. 112, 036403 – Published 22 January 2014

Abstract

We report on a Dirac-like Fermi surface in three-dimensional bulk materials in a distorted spinel structure on the basis of density functional theory as well as tight-binding theory. The four examples we provide in this Letter are ${BiZnSiO}_{4}$ , ${BiCaSiO}_{4}$ , ${BiAlInO}_{4}$ , and ${BiMgSiO}_{4}$ . A necessary characteristic of these structures is that they contain a Bi lattice which forms a hierarchy of chainlike substructures, with consequences for both fundamental understanding and materials design.

Received 23 September 2013

DOI:https://doi.org/10.1103/PhysRevLett.112.036403

Authors & Affiliations

Julia A. Steinberg^1,2,*, Steve M. Young¹, Saad Zaheer², C. L. Kane², E. J. Mele², and Andrew M. Rappe¹

¹The Makineni Theoretical Laboratories, Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6323, USA
²Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6396, USA

^*Corresponding author. stjulia@sas.upenn.edu

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Vol. 112, Iss. 3 — 24 January 2014

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Images

Figure 1
The (a) spinel and (b) distorted spinel structures. Conventionally, the high-symmetry spinel unit cell is cubic; here we have chosen a reduced supercell allowing for direct comparison with the distorted structure. The $A$ sites, populated by Bi, are surrounded by a tetrahedral oxygen cage (not shown for clarity) and the $B$ sites are surrounded by octahedral oxygen cages. In the high-symmetry spinel all Bi neighbors are equidistant, and the Bi atoms form a diamond lattice. Upon distortion, bonds lying in the (0, 1, 0) plane elongate, and the structure can be described as coupled, close-packed, corrugated chains running parallel to [0, 1, 0].
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Figure 2
The distorted spinel structure with the Bi network annotated. $B^{″}$ cages are blue, $B^{'}$ cages are gray, and the Bi atoms are in purple. The chains (a) are composed of Bi atoms with alternating bond directions, labeled $α$ and $β$ , forming chains with distinct orientations from their nearest neighbors, signified by $+$ and $-$ . Running parallel to [0, 1, 0], the alternatingly oriented chains form their own chainlike structure in the [1, 0, 1] direction, so that each chain is now an object coupling to its neighbors in 1D. The resulting parallel planes of alternating sense (denoted by solid and dashed lines), constructed from the sheets of coupled chains, themselves couple with one another in the [1, 0, $- 1$ ] direction. Thus, at each level of structure the elements (atoms, chains, and planes) alternate in orientation along a particular direction, resulting in a fourfold representation for a point on the BZ surface corresponding to one half of a reciprocal lattice vector for each of those directions, and a Dirac point at the Fermi level for half-filling.
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Figure 3
The band structures of ${BiZnSiO}_{4}$ (a), ${BiCaSiO}_{4}$ (b), ${BiAlInO}_{4}$ (c), and ${BiMgSiO}_{4}$ (d) in the distorted spinel structure all contain a Dirac point at $T$ that is completely split along the line in the Brillouin zone from $T$ to $W$ due to the spin orbit interaction between the bismuth sites in the lattice.
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Figure 4
Projected density of states near the Dirac point in the ${ZnBiSiO}_{4}$ structure (a). States near the Dirac point are primarily Bi $p$ states, with a vanishing density at the Fermi energy. The angular portion of the wave functions (i.e., without the spin degree of freedom) of the ${ZnBiSiO}_{4}$ structure at $T$ is shown in (c), with a cartoon depiction in (b) emphasizing their $p$ -like nature. The neighboring bismuth sites interact in a way that can be depicted as chains of Bi $p$ orbitals running through an insulating structure, in which individual chains are analogous to polyacetylene; nonsymmorphic operation of “inversion followed by translation between $A$ sites” results in an equivalent but different state, creating a twofold degeneracy at the Dirac point, which then becomes fourfold due to the spin degree of freedom. Breaking inversion symmetry in this structure is analogous to having inequivalent coupling constants between Bi sites and single and double bonds in polyacetylene, and will lift the degeneracy, gapping the system.
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