Energy scale of Dirac electrons in ${\mathrm{Cd}}_{3}{\mathrm{As}}_{2}$

Energy scale of Dirac electrons in ${Cd}_{3} {As}_{2}$

M. Hakl, S. Tchoumakov, I. Crassee, A. Akrap, B. A. Piot, C. Faugeras, G. Martinez, A. Nateprov, E. Arushanov, F. Teppe, R. Sankar, Wei-li Lee, J. Debray, O. Caha, J. Novák, M. O. Goerbig, M. Potemski, and M. Orlita

Phys. Rev. B 97, 115206 – Published 26 March 2018

Abstract

Cadmium arsenide $({Cd}_{3} {As}_{2})$ has recently became conspicuous in solid-state physics due to several reports proposing that it hosts a pair of symmetry-protected 3D Dirac cones. Despite vast investigations, a solid experimental insight into the band structure of this material is still missing. Here we fill one of the existing gaps in our understanding of ${Cd}_{3} {As}_{2}$ , and based on our Landau-level spectroscopy study, we provide an estimate for the energy scale of 3D Dirac electrons in this system. We find that the appearance of such charge carriers is limited—contrary to a widespread belief in the solid-state community—to a relatively small energy scale (below 40 meV).

Received 4 January 2018
Revised 5 March 2018

DOI:https://doi.org/10.1103/PhysRevB.97.115206

Physics Subject Headings (PhySH)

Landau levels Semimetals

Dirac semimetal

Fourier transform infrared spectroscopy Infrared spectroscopy

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

M. Hakl^1,*, S. Tchoumakov², I. Crassee¹, A. Akrap³, B. A. Piot¹, C. Faugeras¹, G. Martinez¹, A. Nateprov⁴, E. Arushanov⁴, F. Teppe⁵, R. Sankar^6,7, Wei-li Lee⁶, J. Debray^8,9, O. Caha¹⁰, J. Novák¹⁰, M. O. Goerbig², M. Potemski¹, and M. Orlita^1,11,†

¹Laboratoire National des Champs Magnétiques Intenses, CNRS-UGA-UPS-INSA-EMFL, 25, Avenue des Martyrs, 38042 Grenoble, France
²LPS, Univ. Paris-Sud, Univ. Paris-Saclay, CNRS UMR 8502, 91405 Orsay, France
³DQMP, University of Geneva, 1211 Geneva 4, Switzerland
⁴Institute of Applied Physics, Academy of Sciences of Moldova, 2028 Chisinau, Moldova
⁵Laboratoire Charles Coulomb, CNRS, Université Montpellier, 34095 Montpellier, France
⁶Institute of Physics, Academia Sinica, Nankang, 11529 Taipei, Taiwan
⁷Center for Condensed Matter Sciences, National Taiwan University, Taipei 10617, Taiwan
⁸Université Grenoble Alpes, Institut NEEL, F-38000 Grenoble, France
⁹CNRS, Institut NEEL, F-38000 Grenoble, France
¹⁰CEITEC MU and Faculty of Science, Masaryk University, 61137 Brno, Czech Republic
¹¹Institute of Physics, Charles University, Ke Karlovu 5, 12116 Praha 2, Czech Republic

^*michaelhakl@email.cz
^†milan.orlita@lncmi.cnrs.fr

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Vol. 97, Iss. 11 — 15 March 2018

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Figure 1
A schematic view of electronic bands in ${Cd}_{3} {As}_{2}$ at the Brillouin zone center. Three electronic bands form two types of 3D conical structures: a single cone hosting Kane electrons at the larger energy scale, appearing due to the vanishing band gap, and two highly tilted and anisotropic 3D Dirac cones at low energies. The Dirac cones emerge at the crossing points of $p$ -type bands, which in semiconductors with noninverted ordering of bands are referred to as heavy (HH) and light (LH) hole bands [13]. In materials with an inverted band gap (e.g., in HgTe or ${Cd}_{3} {As}_{2})$ , the LH band becomes the conduction band.
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Figure 2
(a) Transmission and reflectivity spectra of the investigated thin layer of ${Cd}_{3} {As}_{2}$ in the far and middle infrared spectral ranges. The upper and lower limits of the transmission window are given by the plasma energy $ℏ ω_{p}$ and by the onset of interband absorption, defined by the particular position of the Fermi energy $E_{F}$ (by Pauli blocking). This is shown in the inset of (a), where the band structure expected within the (gapped) Kane model is schematically plotted. Spectrally sharp features in the reflectivity spectrum visible at low energies are due to infrared active optical phonon modes. The optical conductivity spectrum deduced using Kramers-Kronig relations from the reflectivity curve is plotted in (b). The observed broadening of the onset of interband absorption indicates somewhat inhomogeneous distribution of electrons across the sample, with the mean value of the Fermi energy $E_{F} \approx 160$ meV. The optical conductivity in a broader spectral range is plotted in the inset of (b).
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Figure 3
(a) Relative magnetotransmission spectra $T_{B} / T_{B = 0}$ of the explored ${Cd}_{3} {As}_{2}$ sample plotted for selected values of the magnetic field $B$ applied. Individual inter-LL resonances are denoted by triangles. (b) The false color-map of magneto-absorbance spectra. The gray circles show the cyclotron energy (fundamental mode) obtained in the previous high-field magnetoreflectivity experiments [6]. (c) The experimentally extracted energies of inter-LL resonances versus $\sqrt{B}$ . To exclude effects related to the coupling of interband (single-particle) excitations with plasmon (so-called Bernstein modes), only resonances at photon energies above $ℏ ω = 100$ meV are plotted $(ℏ ω_{p} \approx 50$ meV). The inset shows the linear energy-momentum dependence (the separation of the conduction and valence band) obtained using a simple semiclassical model based on the Bohr-Sommerfeld quantization; see the text.
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Figure 4
The electronic band structure of ${Cd}_{3} {As}_{2}$ calculated within the Bodnar/Kane model for the fixed values of $E_{g} = - 70$ meV and $v = 0.94 \times 10^{6}$ m/s, for three different strengths of crystal field splitting: $δ = 30, 0$ and $- 30$ meV, in parts (a), (b) and (c) respectively. The Dirac cones are expected to appear for positive $δ$ , while for the negative one, the system should enter a topologically insulating phase. The electronic bands have been plotted along ( $1 - 1 0$ ) and $(- 1 - 1 1$ ) directions, which are perpendicular to the applied magnetic field, $B ∥ (112)$ , as well as to each other. The nonzero $δ$ parameter mainly impacts the HH band, which becomes modulated, anisotropic and shifted in energy by the average amount of $δ / 3$ . As shown by the corresponding asymptotes, the upper cone thus becomes shifted with respect to the flatband, implying thus the offset of $E_{g} / 2 + δ / 3$ for interband absorption [and interband inter-LL excitations in Fig. 3]. The vertical arrows denote interband excitations allowed by the occupation of the upper conical band, which dominantly contribute to interband absorption.
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Figure 5
(a)–(c) The experimentally determined fan chart of observed inter-LL excitations compared to theoretically calculated excitation energies within the Bodnar (Kane) model for three different combinations of $E_{g}$ and $δ$ , which fulfill the condition $E_{g} / 2 + δ / 3 \approx - 35$ meV implied by the offset of interband excitations. The individual excitations are labeled by the corresponding final-state LL and also by particular color-coding in (d), where the field-dependence of the LL spectrum $(n = 0, 1, 2 ... 6)$ , calculated for $E_{g} = - 60$ meV and $δ = 15$ meV, has been plotted. The velocity parameter was fixed at $v = 0.94 \times 10^{6}$ m/s and the strength of the spin-orbit coupling at $Δ = 400$ meV. The model does not include any other free parameters. The CR data are taken from our previous magnetoreflectivity experiments [6]. The dashed line in (d) denotes an approximate position of the Fermi energy as a function of $B$ . In (a)–(c), the lowest in energy interband transition (light blue) may be identified as the excitation from the flat band to $2_{↓}$ level, but its energy also coincides with the intraband $1_{↓} \to 3_{↓}$ excitation, which may become electric-dipole active due to the lack of full isotropy in the studied system.
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Physical Review B

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Energy scale of Dirac electrons in ${Cd}_{3} {As}_{2}$

M. Hakl, S. Tchoumakov, I. Crassee, A. Akrap, B. A. Piot, C. Faugeras, G. Martinez, A. Nateprov, E. Arushanov, F. Teppe, R. Sankar, Wei-li Lee, J. Debray, O. Caha, J. Novák, M. O. Goerbig, M. Potemski, and M. Orlita

Phys. Rev. B 97, 115206 – Published 26 March 2018

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Physical Review B

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Energy scale of Dirac electrons in Cd3As2

M. Hakl, S. Tchoumakov, I. Crassee, A. Akrap, B. A. Piot, C. Faugeras, G. Martinez, A. Nateprov, E. Arushanov, F. Teppe, R. Sankar, Wei-li Lee, J. Debray, O. Caha, J. Novák, M. O. Goerbig, M. Potemski, and M. Orlita

Phys. Rev. B 97, 115206 – Published 26 March 2018

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Energy scale of Dirac electrons in ${Cd}_{3} {As}_{2}$