Optical signatures of Dirac nodal lines in NbAs2

Abstract

Using polarized optical and magneto-optical spectroscopy, we have demonstrated universal aspects of electrodynamics associated with Dirac nodal lines that are found in several classes of unconventional intermetallic compounds. We investigated anisotropic electrodynamics of [Formula: see text] where the spin-orbit coupling (SOC) triggers energy gaps along the nodal lines. These gaps manifest as sharp steps in the optical conductivity spectra [Formula: see text] This behavior is followed by the linear power-law scaling of [Formula: see text] at higher frequencies, consistent with our theoretical analysis for dispersive Dirac nodal lines. Magneto-optics data affirm the dominant role of nodal lines in the electrodynamics of [Formula: see text].

Keywords: Dirac fermions; magneto-optics; nodal-line semimetal; optical conductivity.

Fig. 1.

Band structure schematic (Left) and corresponding optical conductivity (Right) for 3D Weyl cones (A), flat nodal line (B), and dispersive nodal line (C). Red color in the band structure schematic indicates filled electronic states. Gray lines and dashed red lines in the optical conductivity represent gapless and gapped responses, respectively. In A and B, the optical gap comes from finite doping ($E_F\ne 0$), while the electronic structure is gapless. Note that the conductivity spectra in A apply to Dirac nodes as well. For C, the gap originates from SOC, and the bands are dispersive along the line direction ${\mathrm{k}}_{line}$. The orange line in C is the nodal line projected in momentum space.

Fig. 2.

(A) Ab initio calculations of the nodal lines (orange) in momentum space of ${\mathrm{N}\mathrm{b}\mathrm{A}\mathrm{s}}_2$. Red symbols are the high symmetry points in the BZ near the nodal lines. Blue arrows label the crystallographic axes in the (001) plane. (B) Band structure for ${\mathrm{N}\mathrm{b}\mathrm{A}\mathrm{s}}_2$ calculated along high symmetry points in A, with (solid lines) and without (dotted lines) SOC. Gray arrows indicate possible optical transitions.

Fig. 3.

(A) Anisotropic reflectance for the ${\mathrm{N}\mathrm{b}\mathrm{A}\mathrm{s}}_2$ (001) surface. A, Inset is a schematic of a unit cell of ${\mathrm{N}\mathrm{b}\mathrm{A}\mathrm{s}}_2$. (B) Optical conductivity from experiment (Left) and DFT calculations (Right). Blue shaded regions highlight the low-energy part where the response is dominated by the massive Dirac bands. Green arrows indicate positions of steps in ${\sigma }_1\left(\omega \right)$. Solid and dashed lines indicate the $b$-axis and $a$-axis response, respectively.

Fig. 4.

(A) Optical conductivity for E$\left|\right|b$. Blue dotted lines are fitted ${\sigma }_1^b$ curves with nodal-line structure parameters. Gray dashed and solid lines denote contributions from the nodal lines near ${\mathrm{I}}_1$-Z (C) and near ${\mathrm{X}}_1$-Y (D), respectively. The linear increase of ${\sigma }_1^b\left(\omega \right)$ saturates at $E_{max}\sim$0.3 eV. A, Inset shows the ratio ${\sigma }_1^b$/${\sigma }_1^a$ above the gap region. B displays the results of band structure calculations along high-symmetry points near the nodal-line regions. The 3D version of the band structure calculation is shown in C and D. Green arrows illustrate onsets of interband transitions for dispersive (C) and energy-flat (D) segments of the nodal line. (E) The energy dispersion of the gapped nodal line displayed as a function of the line length $k_0$, calculated by using DFT. Dirac-cone schematics indicate different fillings of the Dirac bands along the line. The gray dotted line is the Fermi energy $E_F$. Vertical arrows show different onsets of interband transition, and horizontal arrows are the effective line length.

Fig. 5.

(A) Magneto-reflectance spectra normalized by zero-field reflectance, showing a series of LL transitions systematically changing with increasing $B$. We also observe only a weakly field-dependent mode at $\sim$85 meV. (B) Derivative contour ($d$R/$d$B) for E$\left|\right|b$. The energies of peaks extracted from A are displayed as green dots. Gray dashed lines indicate the subtle in-gap states. (C) $d$R/$d$B for E$\left|\right|a$ and peak energies extracted from A. Green dashed lines in B and C are fits using Eq. 4 with $\overline{v}\sim 2.53eV\cdot$Å and $2{\mathrm{\Delta }}_2$ = 114 meV. C, Inset shows gapped Dirac bands and the LL dispersion with magnetic field $B$. Arrows indicate allowed interband LL transitions across the gap.