3D Dirac semimetal ${\mathrm{Cd}}_3{\mathrm{As}}_2$: A review of material properties

Cadmium arsenide $\left({\mathrm{Cd}}_3{\mathrm{As}}_2\right)$, a time-honored and widely explored material in solid-state physics, has recently attracted considerable attention. This was triggered by a theoretical prediction concerning the presence of three-dimensional (3D) symmetry-protected massless Dirac electrons, which could turn ${\mathrm{Cd}}_3{\mathrm{As}}_2$ into a 3D analog of graphene. Subsequent extended experimental studies have provided us with compelling experimental evidence of conical bands in this system, and revealed a number of interesting properties and phenomena. At the same time, some of the material properties remain the subject of vast discussions despite recent intensive experimental and theoretical efforts, which may hinder the progress in understanding and applications of this appealing material. In this paper, we focus on the basic material parameters and properties of ${\mathrm{Cd}}_3{\mathrm{As}}_2$, in particular those which are directly related to the conical features in the electronic band structure of this material. The outcome of experimental investigations, performed on ${\mathrm{Cd}}_3{\mathrm{As}}_2$ using various spectroscopic and transport techniques within the past 60 years, is compared with theoretical studies. These theoretical works gave us not only simplified effective models, but more recently, also the electronic band structure calculated numerically using ab initio methods.

Figure 1

One of the very first experimental indications of a conical band in ${\mathrm{Cd}}_3{\mathrm{As}}_2$ presented by Rosenman [5]: the square of the cyclotron mass $m^{*}$, determined from the thermal damping of Shubnikov–de Haas oscillations, as a function of their inverse oscillation period $1/P$. The theoretically expected dependence for a conical band $m^2=2e\hslash /\left(P{v}^2\right)$ allows us to estimate the velocity parameter $v=(1.1\pm 0.1)\times {10}^6$ m/s from the slope indicated in the plot. Reprinted from [5] with permission by Elsevier, copyright (1969).

Figure 2

A schematic view of the 3D Dirac cones in ${\mathrm{Cd}}_3{\mathrm{As}}_2$. Two symmetry-protected cones emerge in the vicinity of the $\mathrm{\Gamma }$ point with the apexes located at the momenta $k=\pm k_D$ along the tetragonal axis of ${\mathrm{Cd}}_3{\mathrm{As}}_2$ (often associated with the $z$ axis). In general, the Dirac conical bands are characterized by an anisotropic velocity parameter $\left(v_x=v_y\ne v_z\right)$ and possibly also by an electron-hole asymmetry. The energy scale of these cones (its upper bound) is given by the $E_D$ parameter, which refers to the energy distance between the upper and lower Lifshitz points.

Figure 3

The nonprimitive tetragonal unit cell of ${\mathrm{Cd}}_3{\mathrm{As}}_2$ is composed of $2\times 2\times 4$ weakly distorted antiflourite cells with two cadmium vacancies. This cell hosts 96 cadmium atoms and 64 arsenic atoms.

Figure 4

Schematic cross-sectional side view of an alumina furnace used for self-selecting vapor growth of a ${\mathrm{Cd}}_3{\mathrm{As}}_2$ single crystal [46]. The positions of the ampoule in which transport takes place and the temperature profile of the furnace for ${\mathrm{Cd}}_3{\mathrm{As}}_2$ growth are shown. Reprinted from [46].

Figure 5

${\mathrm{Cd}}_3{\mathrm{As}}_2$ single crystals grown by the SSVG method. The larger facets are (112) and (even 00) planes, reaching an area up to 0.$7{\mathrm{cm}}^2$. Reprinted from [46].

Figure 6

The electronic band structure of a cubic semiconductor/semimetal implied by the Kane mode in the vicinity of the Brillouin zone center. At energies significantly larger than the band gap, both conduction and valence bands exhibit nearly conical shapes. For a negative band gap $E_g$ (chosen in the plotted case), the system is characterized by an inverted ordering of bands, and closely resembles the well-known semimetal HgTe [47]. A positive band gap would imply a band structure typical of narrow-gap semiconductors such as InSb [48].

Figure 7

The schematic view of the electronic band structure of ${\mathrm{Cd}}_3{\mathrm{As}}_2$ proposed by Bodnar in Ref. [15]. Three electronic bands form two types of 3D conical structures: a single cone hosting Kane electrons at the large energy scale, appearing due to the vanishing band gap, and two highly tilted and anisotropic 3D Dirac cones at low energies. These are marked by vertical gray arrows and emerge due to the partly avoided crossing of the flat band with the conduction band. The energy scale of massless Dirac electrons $E_D$ exactly matches the size of crystal-field splitting parameter $\delta$.

Figure 8

The band structure of ${\mathrm{Cd}}_3{\mathrm{As}}_2$ deduced theoretically using ab initio approach by Ali et al. [20] and Conte et al. [67], using GGA approximations, in parts (a) and (b), respectively. In both cases, the 3D Dirac cones appear at the Fermi energy in close vicinity to the $\mathrm{\Gamma }$ point, due the symmetry-allowed crossing of bands along the $\mathrm{\Gamma }\text{−}Z$ line. The energy scale of the massless Dirac electrons $E_D$ in these two cases may be estimated to be 45 and 20 meV, respectively. Part (a) reprinted with permission from [20], copyright (2014) American Chemical Society. Part (b) reprinted from [67].

Figure 9

The valence bands in ${\mathrm{Cd}}_3{\mathrm{As}}_2$ visualized by low-temperature ARPES technique by Liu et al. [23]. The data were collected on the (112)-terminated surface. The widely extending 3D conical band, characterized by a velocity parameter of $1.3\times {10}^6$ m/s and interpreted in terms of 3D massless Dirac electrons, coexists with another holelike weakly dispersing parabolic band, also observed in Ref. [26]. Reprinted by permission from Springer Nature: Nature Materials [23], copyright (2014).

Figure 10

The conduction band of ${\mathrm{Cd}}_3{\mathrm{As}}_2$ visualized by ARPES by Neupane et al. [26]. The data were collected on the (001)-terminated surface of ${\mathrm{Cd}}_3{\mathrm{As}}_2$ at 14 K and show the dispersion in the direction perpendicular to the $\mathrm{\Gamma }\text{−}Z$ line. The observed 3D conical band was interpreted in terms of 3D massless Dirac electrons, implying a velocity parameter of $1.5\times {10}^6$ m/s. Reprinted by permission from Springer Nature: Nature Communications [26], copyright (2014).

Figure 11

The approximately conical band centered at the $\mathrm{\Gamma }$ point deduced from Landau-level spectroscopy (open circles) and quasiparticle interference (QPI) pattern (closed circles with error bars) in the STS/STM experiments performed by Jeon et al. [24]. The slope of the conical band corresponds to a velocity parameter of $0.94\times {10}^6$ m/s. Reprinted by permission from Springer Nature: Nature Materials [24], copyright (2014).

Figure 12

The real part of the optical conductivity of ${\mathrm{Cd}}_3{\mathrm{As}}_2$ obtained by Neubauer et al. [37]. At low energies, the conductivity is dominated by Drude (free-carrier) response accompanied by a series of infrared-active phonon modes. Above the onset of the interband absorption around $1500–2000{\mathrm{cm}}^{-1}$, the optical conductivity increases roughly linearly with the photon energy, which is behavior expected for massless charge carries in 3D. Reprinted with permission from [37], copyright (2016) by the American Physical Society.

Figure 13

The energies of cyclotron resonance excitations observed in Ref. [45] as a function of applied magnetic field, obtained on (112)- and (001)-oriented ${\mathrm{Cd}}_3{\mathrm{As}}_2$ samples. The dashed and dotted lines show theoretically expected positions within the Kane and Dirac models (see Ref. [45] for details). Adapted with permission from [45], copyright (2016) by the American Physical Society.

Figure 14

(a) Weyl orbit in a thin slab of thickness $L$ in a magnetic field $B$. The orbit involves the Fermi-arc surface states connecting the Weyl nodes of opposite chirality, and the bulk states of fixed chirality. (b) The Fourier transform of magnetoresistance at $T=2$ K measured on a thin ${\mathrm{Cd}}_3{\mathrm{As}}_2$ sample $(L=150$ nm) which was cut from a single crystal by focused ion beam. The field was oriented parallel $\left({90}^{\circ }\right)$ and perpendicular $\left(0^{\circ }\right)$ to the surface. Only one frequency is observed for parallel field, and two frequencies for perpendicular field. Adapted by permission from Springer Nature: Nature [31], copyright (2016).

Figure 15

(a) Angular dependence of the transverse magnetoresistance with magnetic field $B$ rotated in the (010) plane. The tilt angle $\theta$ is the angle between $B$ and the [100] direction. (b) Landau index plots $n$ vs $1/B$ at different $\theta$'s. (Inset) Angular dependence of the oscillation frequency $F$ and the total phase $\gamma -\delta$, extracted from the linear fitting from the first Landau level to the fifth Landau level. Adapted with permission from [106], copyright (2015) by the American Physical Society.

Figure 16

The quantum Hall effect in a 20-nm-thick epitaxial ${\mathrm{Cd}}_3{\mathrm{As}}_2$ film grown by molecular beam epitaxy, showing the Hall $\left(R_{xy}\right)$ and longitudinal $\left(R_{xx}\right)$ resistances as a function of magnetic field measured at 1 K. Reprinted with permission from [29], copyright (2018) by the American Physical Society.