Deep-lying semi-Dirac fermions in hexagonal close-packed cadmium

TL;DR

Problem: identify deep-lying semi-Dirac fermions in a real metal and explain their origin. Approach: perform first-principles LDA calculations on hexagonal Cd to reveal linearly dispersing out-of-plane bands near $-3$ eV, with hybridization between $s$ and $p_z$ orbitals; orbital analysis shows direction-dependent mixing. Key findings: a lens-shaped electron pocket with cross-section $A(k_z)$ linearly dependent on $k_z$ and slope $dA/dk_z oughly 8.73$ Å$^{-1}$, matching Sondheimer oscillation data; comparison with hypothetical hcp Ag shows chemistry controls the semi-Dirac dispersion and tensile strain can tune the inversion. Significance: demonstrates a concrete solid-state realization of semi-Dirac fermions governed by orientation-dependent hybridization and strain engineering, with observable signatures in quantum oscillations including the Sondheimer effect.

Abstract

Semi-Dirac fermions are massless in one direction and massive in the perpendicular directions. Such quasiparticles have been proposed in various contexts in condensed matter. Using first principles calculations, we identify a pair of semi-Dirac bands anti-crossing at $-3$ eV below the Fermi level in the electronic structure of hexagonal close-packed cadmium. The linear out-of-plane dispersion is kept up to the Fermi level. We demonstrate that the dichotomy between the linear and quadratic dispersions is driven by an orientation-sensitive hybridization between the $s$ and $p_z$ orbitals. The upper semi-Dirac band produces a lens-shaped nonellipsoidal Fermi sheet whose cross-section area has a $k$-dependence that is in excellent agreement with the experimentally measured period of Sondheimer oscillations.

Figures (7)

• Figure 1: Hexagonal closed packed structure of cadmium. The inter- and intra-layer nearest-neighbor Cd-Cd distances are 3.293 and 2.979 Å, respectively.
• Figure 2: a) Calculated LDA band structure of hcp Cd. The band structure features a pair of bands that disperse linearly along the out-of-plane direction $\Gamma A$ and quadratically along the inplane directions $\Gamma M$ and $\Gamma K$. These bands appear to converge near $-3$ eV at $\Gamma$, but they are in fact separated by 50 meV. The upper and lower semi-Dirac bands are shown in blue and red, respectively. Calculated Fermi surface of hcp Cd consisting of b) hole-type "cap", c) hole-type "monster", and d) electron-type "lens" sheets. The lens sheet derives from the upper semi-Dirac band.
• Figure 3: Calculated LDA band structure of a hypothetical hcp Ag using the lattice parameters of hcp Cd. The pair of bands that linearly disperse along $\Gamma A$ in hcp Cd disperse quadratically in hypothetical hcp Ag. The gap at $\Gamma$ between these bands also increases to 1.4 eV.
• Figure 4: Band structures of hcp Cd and hypothetical hcp Ag isostructural to hcp Cd as a function of respective $s$ and $p_z$ orbital characters along out-of-plane $\Gamma A$ and inplane $\Gamma M$ and $\Gamma K$ directions. The semi-Dirac bands in hcp Cd show a mixed $s$ and $p_z$ character in the out-of-plane direction. In hcp Ag, the bands disperse quadratically in all directions, and the upper band is predominantly $s$-like, while the lower band is mainly $p_z$-like.
• Figure 5: a) The electron-like Fermi surface generated by the upper semi-Dirac band projected in the $k_x, k_z$ plane (empty circles). The blue solid line represents a regular ellipsoid whose axes are equally long. Note the difference. b) The cross section area as a function of $k_z$. Over a long distance, near the poles (that is $k_{z,min}$ and $k_{z,max}$), the area is linear in $k_z$, consistent with the linear dispersion of the semi-Dirac band from which this Fermi sheet derives.