Magneto-Optical Signature of Massless Kane Electrons in ${\mathrm{Cd}}_3{\mathrm{As}}_2$

We report on optical reflectivity experiments performed on ${\mathrm{Cd}}_3{\mathrm{As}}_2$ over a broad range of photon energies and magnetic fields. The observed response clearly indicates the presence of 3D massless charge carriers. The specific cyclotron resonance absorption in the quantum limit implies that we are probing massless Kane electrons rather than symmetry-protected 3D Dirac particles. The latter may appear at a smaller energy scale and are not directly observed in our infrared experiments.

Figure 1

(a) The reflectance of both studied samples at 5 K and $B=0$ dominated by sharp plasma edges and phonon response. For a purely conical band structure, the approximate ratio of plasma frequencies, ${\omega }_p^{001}/{\omega }_p^{112}\approx 2$, implies $2\times$ larger $E_F$, and therefore, $8\times$ higher carrier density ($n\propto E_F^3$) for the (001)-oriented sample. (b) The optical conductivity with a nearly linear dependence on the photon energy. Inset: optical conductivity in a wide energy range.

Figure 2

Relative magnetoreflectivity spectra of the (112)- and (001)-oriented sample measured at $T=1.8\mathrm{K}$ in (a) and (b), respectively. The observed resonances are marked by full (open) circles for dominant (weak, likely present) CR modes. Their $\sqrt{B}$ dependence is shown in the part (c), together with the theoretical curves based on the Dirac (dotted) and gapless Kane (dashed) models. The former curves were calculated for the velocity $10^6\mathrm{m}/\mathrm{s}$ and the zero-field Fermi energies $E_F^0=100$ and 200 meV; the kinks appear due to field dependence of $E_F$ [12].

Figure 3

The band structure calculated using the Bodnar model (at $k_y\equiv 0$) for various $E_g$ and $\delta$ chosen to approach (a) ab initio calculations [8], (b) the original Bodnar’s work [5] and (c) Kane electrons [11], and for $\mathrm{\Delta }=400\mathrm{meV}$ and $v=0.93\times 10^6\mathrm{m}/\mathrm{s}$. Two highly tilted Dirac cones (marked by black arrows) with the size comparable to $\delta$ appear in (a) and (b) at the crossing points of the upper ${\mathrm{\Gamma }}_7$ and ${\mathrm{\Gamma }}_6$ bands. The widely extended cone of Kane electrons is visible in (c) and partly also (b).

Figure 4

Idealized Landau levels of (a) Dirac and (b) Kane electrons as occupied in the quantum limit and their relative energy scale in the Bodnar model (central inset). The $n=0$ LLs typical of Dirac electrons, dispersing linearly with the momentum, are absent for Kane electrons. The solid arrows show CR modes active in the quantum limit, the dashed arrows those which vanish when this limit is reached. For details about the crossover from Dirac- to Kane-like LLs see Ref. [12].