Dirac Semimetal in Three Dimensions

We show that the pseudorelativistic physics of graphene near the Fermi level can be extended to three dimensional (3D) materials. Unlike in phase transitions from inversion symmetric topological to normal insulators, we show that particular space groups also allow 3D Dirac points as symmetry protected degeneracies. We provide criteria necessary to identify these groups and, as an example, present ab initio calculations of $\beta$-cristobalite ${\mathrm{BiO}}_2$ which exhibits three Dirac points at the Fermi level. We find that $\beta$-cristobalite ${\mathrm{BiO}}_2$ is metastable, so it can be physically realized as a 3D analog to graphene.

Figure 1

3D Dirac semimetal in $\beta$-cristobalite ${\mathrm{BiO}}_2$. (a) Brillouin zone (BZ) of the FCC lattice. The plane highlighted in gray joins the three symmetry related $X$ points. Other high symmetry points are also indicated. (b) Conduction and valence bands of $\beta$-cristobalite ${\mathrm{BiO}}_2$ are plotted as functions of momentum on the plane highlighted in gray on the left. Each band is twofold degenerate due to inversion symmetry. Dirac points appear at the center of the three zone faces of the BZ. (c) Dirac, Weyl, and insulating phases in the diamond lattice. (1) The states at the Dirac point at $X$ span a four dimensional projective representation of the little group at $X$ which contains a fourfold rotation accompanied by a sublattice exchange operation. (2) Four Weyl points on the zone face due to a small inversion breaking perturbation. The Chern number of each Weyl point is indicated. (3) Two Weyl points appear on the line from $X$ to $W$ for a $T$-breaking Zeeman field $\mathbf{B}$ oriented along that direction. $\mathbf{B}$ oriented along other directions gaps all the Dirac points by breaking enough rotational symmetry that no two-dimensional representations are allowed. (4) Gapped phase obtained by breaking the fourfold rotation symmetry or by applying a magnetic field in any direction except along $\widehat{x}$, $\widehat{y}$, or $\widehat{z}$. The insulating phase can be a normal, strong, or a weak topological insulator [11].

Figure 2

(a) Band structure of $\beta$-cristobalite ${\mathrm{SiO}}_2$. Energy bands are plotted relative to the Fermi level. Each band is twofold degenerate due to inversion symmetry. The (highlighted) FDIR at $-4.5\mathrm{eV}$ is split into two linearly dispersing bands between $X$ and $\Gamma$ while the two degenerate bands along $X$ and $W$ are weakly split. This FDIR is buried deep below the Fermi level. (b) The $\beta$-cristobalite structure of ${\mathrm{SiO}}_2$ (${\mathrm{BiO}}_2$). Silicon (bismuth) atoms (light gray) are arranged on a diamond lattice, with oxygen atoms (dark gray) sitting midway between pairs of silicon (bismuth).

Figure 3

Band structures of (a) ${\mathrm{AsO}}_2$, (b) ${\mathrm{SbO}}_2$, and (c) ${\mathrm{BiO}}_2$ in the $\beta$-cristobalite structure, and (d) $s$ states on a diamond lattice in the tight-binding model of Ref. 11. Energy bands are plotted relative to the Fermi level. Each band is twofold degenerate due to inversion symmetry. Insets: with increasing atomic number of the cation, spin-orbit coupling widens the gap along the line $V$ from $X$ to $W$. In ${\mathrm{BiO}}_2$ and ${\mathrm{SbO}}_2$, the dispersion around the $X$ point is linear in all directions indicating the existence of Dirac points at $X$. ${\mathrm{BiO}}_2$ and ${\mathrm{SbO}}_2$ are Dirac semimetals because their Fermi surface consists entirely of Dirac points.

Figure 4

Linear splitting of fourfold degenerate irreducible representations (FDIRs). If the symmetric Kronecker product of a FDIR with itself contains the vector representation of the group to which the FDIR belongs, it will split in one of the four possible ways displayed above. (a) The FDIR splits into two twofold degenerate bands. This situation is realized at the $X$ point of the FCC Brillouin zone in a diamond lattice. (b) The FDIR splits into four nondegenerate bands. This situation arises at the $\Gamma$ point in zinc blende if mirror symmetry is broken (although the FDIR in zinc blende develops a nonzero Chern number due to threefold rotation symmetry at $\Gamma$). (c) The FDIR splits into two nondegenerate and one twofold degenerate band with linear dispersion. (d) The splitting of the FDIR at $\Gamma$ in zinc blende. The twofold degenerate band is constrained to be flat implying quadratic dispersion along that direction. The Chern number of this representation is zero in spite of a threefold rotation symmetry because the conduction and valence bands are degenerate away from $\Gamma$.