3D Dirac semimetal \mbox{Cd}_{3}\mbox{As}_{2}: a review of material properties

3D Dirac semimetal : a review of material properties

I. Crassee Laboratoire National des Champs Magnétiques Intenses, CNRS-UGA-UPS-INSA, 25, avenue des Martyrs, 38042 Grenoble, France    R. Sankar Institute of Physics, Academia Sinica, Nankang, 11529 Taipei, Taiwan    W.-L. Lee Institute of Physics, Academia Sinica, Nankang, 11529 Taipei, Taiwan    A. Akrap University of Fribourg, Department of Physics, Chemin du Musée 3, CH-1700 Fribourg, Switzerland    M. Orlita milan.orlita@lncmi.cnrs.fr Laboratoire National des Champs Magnétiques Intenses, CNRS-UGA-UPS-INSA, 25, avenue des Martyrs, 38042 Grenoble, France Institute of Physics, Charles University, Ke Karlovu 5, 12116 Praha 2, Czech Republic
July 20, 2019
Abstract

Cadmium arsenide (CdAs) – 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 3D symmetry-protected Dirac electrons. Subsequent extended investigations involving spectroscopic and transport techniques have provided us with solid experimental evidence of conical bands in this system, and revealed a number of interesting properties and phenomena. 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. Here we review the basic material parameters and properties of CdAs.

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I Introduction

Cadmium arsenide (CdAs) is an old material for condensed-matter physics, with its very first investigations dating back to the thirties Stackelberg and Paulus (1935). Research on this material then continued extensively into the sixties and seventies, as reviewed in Ref. Zdanowicz and Zdanowicz (1975). This is when the physics of semiconductors, those with a sizeable, narrow, but also vanishing energy band gap, strongly developed.

In the early stages of research on CdAs, it was the extraordinarily high mobility of electrons, largely exceeding 10 cm/(V.s) at room temperature Rosenberg and Harman (1959); Turner et al. (1961); Rosenman (1969), that already attracted significant attention to this material. Even today, with a declared mobility well above 10 cm/(V.s) Liang et al. (2015) at low temperatures, cadmium arsenide belongs to the class of systems with the highest electronic mobilities, joining materials such as graphene, graphite, bismuth and 2D electron gases in GaAs/GaAlAs heterostructures Michenaud and Issi (1972); Brandt et al. (1988); Hwang and Das Sarma (2008); Bolotin et al. (2008); Neugebauer et al. (2009). Another interesting – and in view of recent developments, crucial – observation was the strong dependence of the effective mass of electrons on their concentration (see Fig. 1), implying a nearly conical conduction band Armitage and Goldsmid (1968); Rosenman (1969); Rogers et al. (1971).

To understand the electronic properties of CdAs, and its extraordinarily large mobility of electrons in particular, simple theoretical models for the electronic band structure have been proposed in the standard framework of semiconductors physics, and compared with optical and transport experiments. In the initial phase of research, cadmium arsenide was treated as an ordinary Kane-like semiconductor or semimetal Caron et al. (1977); Bodnar (1977). It was seen as a material with an electronic band structure that closely resembled zinc-blende semiconductors with a relatively narrow or vanishing band gap. However, no clear consensus was achieved concerning the particular band structure parameters. Most notably, there was disagreement about the ordering of electronic bands and the presence of another conduction band at higher energies Zdanowicz and Zdanowicz (1975).

Figure 1: One of the very first experimental indications of a conical band in CdAs presented by Rosenman Rosenman (1969): the square of the cyclotron mass , determined from the thermal damping of Shubnikov-de Haas oscillations, as a function of their inverse oscillation period . The theoretically expected dependence for a conical band, , allows us to estimate the velocity parameter  m/s from the slope indicated in the plot. Reprinted from Rosenman (1969) with permission by Elsevier, copyright (1969).

Renewed interest in the electronic properties of CdAs was provoked by a theoretical study where Wang et al. Wang et al. (2013) invoked the presence of a pair of symmetry-protected 3D Dirac cones. This theoretical prediction can be illustrated using a simple, currently widely accepted cartoon-like picture depicted in Fig. 2. It shows two 3D Dirac nodes located at the tetragonal axis of the crystal. Their protection is ensured by rotational symmetry Yang and Nagaosa (2014). With either increasing or decreasing energy, these two Dirac cones approach each other and merge into a single electronic band centered around the point, passing through a saddle-like Lifshitz point.

The prediction of a 3D Dirac phase in a material, which is not only well-known in solid-state physics but also relatively stable under ambient conditions, stimulated a considerable experimental effort. The first angular-resolved photoemission spectroscopy (ARPES) and scanning tunneling microscopy/spectroscopy (STM/STS) experiments Liu et al. (2014); Jeon et al. (2014); Borisenko et al. (2014); Neupane et al. (2014) confirmed the presence of widely extending conical features in the band structure, creating a large wave of interest. This wave gave rise to a number of experimental and theoretical studies, complementing those performed in the past, forming a rich knowledge of this material. As a result, cadmium arsenide now is among the most explored materials in current stolid-state physics.

In addition to the extraordinarily high electronic mobility, cadmium arsenide exhibits a very strong linear magneto-resistance  Liang et al. (2015), an anomalous Nernst effect Liang et al. (2017), and quantum Hall effect signatures when thinned down Uchida et al. (2017); Schumann et al. (2018). Furthermore, indications of the chiral anomaly, planar Hall effect, and electron transport through surface gates have been reported Li et al. (2015); Moll et al. (2016); Jia et al. (2016a); Wu et al. (). Currently, these effects are all considered to be at least indirectly related to the specific relativistic-like band structure of this material. Yet surprisingly, consensus about the complex electronic band structure of CdAs has not yet been fully established.

Figure 2: A schematic view of the 3D Dirac cones in CdAs. Two symmetry-protected cones emerge in the vicinity of the point with the apexes located at the momenta along the tetragonal axis of CdAs (often associated with the -axis). In general, the Dirac conical bands are characterized by an anisotropic velocity parameter () and possibly also by an electron-hole asymmetry. The energy scale of these cones (its upper bound) is given by the parameter, which refers to the energy distance between the upper and lower Lifshitz points.

A detailed understanding of the basic material parameters of CdAs became essential for the correct interpretation of a wide range of observed, and yet to be discovered, physical phenomena. In this paper we review the the current knowledge of CdAs. We start with the properties of the crystal lattice and continue with the technological aspects of CdAs growth. This is followed by a discussion of the theoretical and experimental investigation results, providing different possible views of this material’s electronic bands. In particular, we focus on Dirac-like nodes, the most relevant aspect of the band structure when taking into consideration recent developments in this material’s field.

Ii Crystal lattice

Figure 3: The non-primitive tetragonal unit cell of CdAs is composed of weakly distorted antiflourite cells with two cadmium vacancies. This cell hosts 96 cadmium atoms and 64 arsenic atoms.

Cadmium arsenide has a relatively complex crystal structure, with 160 atoms in the unit cell. It has been the subject of several -ray studies, resulting in somewhat contradicting conclusions (see Refs. Stackelberg and Paulus (1935); Steigmann and Goodyear (1968); Pietraszko and Lukaszewicz (); Ali et al. (2014)). Most relevant for the fundamental and applied research conducted on this material, it is currently believed that at room or lower temperatures, CdAs is a tetragonal material with a unit cell of  nm and  nm. The particular space group remains a subject of discussion. More recent investigations favor a centrosymmetric group I4/acd (No. 142)  Pietraszko and Lukaszewicz (); Ali et al. (2014) over a non-centrosymmetric space group  I4cd (No. 110), as previously suggested Steigmann and Goodyear (1968). The correct space group assignment is of considerable importance for the complete understanding of CdAs. When space inversion symmetry is not present, spin degeneracy is lifted, and the Dirac nodes possibly split into pairs of Weyl nodes.

It is also important to note that the CdAs crystal lattice, though clearly tetragonal, remains nearly cubic (Ali et al. (2014). The lattice may therefore be seen, in the simplest approach, as being composed of antifluorite (cubic) cells with two missing cadmium cations (Fig. 3). Due to the cadmium vacancy ordering, a very large unit cell of CdAs is formed. Composed of of antifluorite cells, it is oriented along the tetragonal -axis, and contains 160 atoms (96 cadmium and 64 arsenic). Additionally, each single anti-fluorite cell is tetragonally distorted, with a small elongation along the -axis (Arushanov (1980); Ali et al. (2014). This simplified image of nearly cubic antifluourite cells, serving as building blocks for the entire CdAs lattice, appears as a useful starting point for simplified effective models and ab initio calculations.  Both are briefly discussed below.

Finally, let us mention that above room temperature the crystal lattice of CdAs undergoes a sequence of polymorphic phase transitions. The corresponding space group remains tetragonal but changes to P4/nbc (No. 133) around 220C and to P4/nmc (No. 137) around 470Pietraszko and Lukaszewicz (); Arushanov (1980). At temperatures above 600C, the symmetry of the crystal changes to a cubic one, characterized by the space group Fmm (No. 225) Ali et al. (2014). Each of these phase transitions is accompanied by an abrupt change in lattice constants, leading to potential microcracks in the crystal. Notably, crystals of CdAs can only be grown at temperatures above 425C, regardless of the growth method.

The complexity of the crystal lattice directly impacts the physical properties of CdAs, as well as how we understand them. For instance, the large number of unit cell atoms may partly complicate ab initio calculations of the electronic band structure. The complex crystal lattice, with a number of cadmium vacancies, may also be susceptible to small changes in ordering, possibly impacting distinct details in the electronic band structure. Some of the existing controversies about the electronic bands in CdAs may thus stem from differences in the investigated samples’ crystal structure, currently prepared using a wide range of crystal growth methods. The crystal lattice complexity is also directly reflected in the optical response of CdAs, characterised by a large number of infrared and Raman-active phonon modes Jandl et al. (1984); Neubauer et al. (2016); Houde et al. (1986).

Iii Technology of sample growth

The technology of CdAs growth encompasses a broad range of methods dating back over 50 years. These provide us with a multiplicity of sample forms: bulk or needle-like crystals, thin films, microplatelets, and nanowires. Not only mono- or polycrystalline samples exist, but also amorphous Zdanowicz and Zdanowicz (1975) samples, all exhibiting different quality and doping levels. In the past, various techniques have been used, such as the Bridgman Rogers et al. (1971) and Czochralski Silvey et al. (1961); Hiscocks and Elliott (1969) methods, sublimation in vacuum or in a specific atmosphere Zdanowicz (1964); Pawlikowski et al. (1975), pulsed-laser deposition Dubowski and Williams (1984), and directional crystallisation in a thermal gradient Rosenberg and Harman (1959). The results of these techniques were summarised in review articles dedicated to the growth of IIV materials Arushanov (1980, 1992).

Most recently, the fast developing field of CdAs has been largely dominated by experiments Liang et al. (2015); Jeon et al. (2014); Borisenko et al. (2014); Neupane et al. (2014); Liang et al. (2017); Moll et al. (2016); Akrap et al. (2016) performed on samples prepared by either growth from a Cd-rich melt Ali et al. (2014), or self-selecting vapor growth (SSVG) Sankar et al. (2015), which are below described in greater detail. With Cd-rich melt, CdAs is synthesized from a Cd-rich mixture of elements sealed in an evacuated quartz ampoule, heated to 825C, and kept there for 48 hours. After cooling to 425C at a rate of 6 C/h, CdAs single crystals with a characteristic pseudo-hexagonal (112)-oriented facets are centrifuged from the flux.

Figure 4: Schematic cross-sectional side view of an alumina furnace used for self-selecting vapor growth of a CdAs single crystal Sankar et al. (2015). The positions of the ampoule in which transport takes place and the temperature profile of the furnace for CdAs growth are shown. Reprinted from Sankar et al. (2015).

The SSVG method comprises several steps Sankar et al. (2015). In the first, the compound is synthesized from a stoichiometric mixture of elements in an evacuated sealed ampoule, and heated for 4 hours 50 C above the CdAs melting point. The resulting ingot is then purified using a sublimation process in an evacuated closed tube (kept around 800 C), refined with small amounts of excess metal or chalcogen elements (also at 50 C above the melting point), water quenched, annealed (around 700 C) and subsequently air cooled. Afterwards, the ingot is crushed and then sieved (targeting a particle size of 0.1-0.3 mm), with the obtained precursors then sealed in an evacuated ampoule. This is then inserted for about 10 days into a horizontal alumina furnace, whose temperature profile is shown in Fig. 4. The resulting crystals are plate-like or needle-like monocrystals, or polycrystals with large grains. The crystals have primarily (112)-oriented, but occasionally also (00)-oriented facets, with an even number, see Fig. 5.

Figure 5: CdAs single crystals grown by the SSVG method. The larger facets are (112) and (even 00) planes, reaching an area up to 0.7 cm. Reprinted from Sankar et al. (2015).

Crystals prepared using the two methods discussed above, as well as past methods, display -type conductivity with a relatively high density of electrons. As-grown and without specific doping, the crystals rarely show an electron density below  cm. This translates into typical Fermi levels in the range of  meV measured form the charge-neutrality point. In literature this rather high doping is usually associated with the presence of arsenic vacancies Spitzer et al. (1966). The doping of CdAs has been reported to vary with conventional annealing Rambo and Aubin (1979). It seems to decrease after thermal cycling between room and helium temperatures Crassee et al. (2018). A combined optical and transport study also revealed a relatively thick (10-20 m) depleted layer on the surface of CdAs crystals Schleijpen et al. (1984). Optical studies also show relatively large inhomogeneities of up to 30% in the electron density, at the scale of hundred microns, in crystals prepared using different growth methods, with -ray studies indicating the presence of systematic twinning Crassee et al. (2018).

Recently, there has been progress in other growth methods. These include the CVD technique, employed to fabricate CdAs nanowires Wang et al. (2016). Other methods, such as pulsed laser deposition in combination with solid phase epitaxy Uchida et al. (2017) and molecular beam epitaxy Yuan et al. (2017); Schumann et al. (2018), have been successfully employed to prepare CdAs layers with a thickness down to the nanometer scale. Most importantly, when the thickness of CdAs films is reduced down to the tens of nanometers, ambipolar gating of CdAs becomes possible Galletti et al. (2018).

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 (chosen in the plotted case), the system is characterized by an inverted ordering of bands, and closely resembles the well-known semimetal HgTe Weiler (1981). A positive band gap would imply a band structure typical of narrow-gap semiconductors such as InSb Yu and Cardona (2010).

Iv Electronic band structure: theoretical views

Soon after the first experimental studies appeared Rosenberg and Harman (1959); Zdanowicz (1964); Haidemenakis et al. (1966); Sexer (1967); Rosenman (1969), the band structure of CdAs was approached theoretically. In this early phase, the electronic band structure was described using simple effective models, developed in the framework of the standard theory. Such models were driven by the similarity between the crystal lattice – and presumably also electronic states – in CdAs, and conventional binary semiconductors/semimetals such as GaAs, HgTe, CdTe or InAs Yu and Cardona (2010). In all of these materials, the electronic bands in the vicinity to the Fermi energy are overwhelmingly composed of cation -states and anion -states. In the case of CdAs, those are cadmium-like cations and arsenic-like anions.

In the first approach, the electronic band structure of CdAs has been described using the conventional Kane model Kane (1957), which is widely and successfully applied in the field of zinc-blende semiconductors. To the best of our knowledge, the very first attempt to interpret the experimental data collected on CdAs using the Kane model was presented by Armitage and Goldsmid Armitage and Goldsmid (1968) in 1968. Later on, similar studies appeared Rogers et al. (1971); Caron et al. (1977); Aubin et al. (1977); Jay-Gerin and Lakhani (1977) but with no clear consensus on the particular band structure parameters.

The size and nature of the band gap, describing the separation between the -like and -like states at the point, remained a main source of controversy Zdanowicz and Zdanowicz (1975). Most often, relatively small values of either inverted or non-inverted gaps have been reported, based on the analysis of different sets of experimental data Armitage and Goldsmid (1968); Rosenman (1969); Caron et al. (1977). Notably, when the band gap is significantly smaller than the overall energy scale of the considered band structure, the Kane model implies approximately conical conduction and valence bands, additionally crossed at the apex by a relatively flat band (Fig. 6Kacman and Zawadzki (1971); Orlita et al. (2014). This band is usually referred to as being heavy-hole-like in semiconductor physics, and may be considered as flat only in the vicinity of the point. At larger momenta, it disperses with a characteristic effective mass close to unity.

Figure 7: The schematic view of the electronic band structure of CdAs proposed by Bodnar in Ref. Bodnar (1977). 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 Dirac electrons exactly matches the size of crystal field splitting parameter .

Importantly, the presence of a 3D conical band in the Kane model is a result of an approximate accidental degeneracy of -like and -like states at the point, therefore making this cone not protected by any symmetry. Additionally, this cone is described by the Kane Hamiltonian Kacman and Zawadzki (1971), which is clearly different from the Dirac Hamiltonian. It should therefore not be confused with 3D Dirac cones subsequently predicted for CdAs Wang et al. (2013). The term massless Kane electron was recently introduced Orlita et al. (2014); Malcolm and Nicol (2015); Teppe et al. (2016); Akrap et al. (2016) to distinguish those two types of 3D massless charge carriers. Therefore, at the level of the strongly simplifying Kane model, CdAs does not host any 3D Dirac electrons. A nearly conical band, the presence of which was deduced from early transport experiments indicating energy-dependent effective mass Armitage and Goldsmid (1968); Rosenman (1969) (Fig. 1), was then interpreted in terms of the Kane model assuming a nearly vanishing band gap.

An improved effective model, which takes into account the tetragonal nature of CdAs, not included in the conventional Kane model for zinc-blende semiconductors, has been proposed by Bodnar Bodnar (1977). The tetragonal distortion of a nearly cubic lattice lifts the degeneracy of -type states (light and heavy hole bands) at the point; this degeneracy is typical of all zinc-blende semiconductors Yu and Cardona (2010). A closer inspection of the theoretical band structure reveals two specific points at the tetragonal axis. At these points the conduction and flat heavy-hole valence bands meet and form two highly anisotropic and tilted cones (Fig. 7). These may be associated with symmetry-protected 3D Dirac cones. Nevertheless, it was not until the work by Wang et al. Wang et al. (2013) when such a prediction appeared explicitly in the literature. The crystal-field splitting parameter  Kildal (1974) is employed by Bodnar Bodnar (1977) to quantify the impact of the tetragonal distortion of the cubic lattice. This parameter directly corresponds to the energy scale of the symmetry-protected Dirac electrons in CdAs (cf. Figs. 2 and 7). The work of Bodnar also inspired other theorists, who used the proposed model for electronic band structure calculations in quantizing magnetic fields Wallace (1979); Singh and Wallace (1983a); Singh et al. (1984).

Figure 8: The band structure of CdAs deduced theoretically using ab initio approach by Ali et al. Ali et al. (2014) and Conte et al. Conte et al. (2017), 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 point, due the symmetry-allowed crossing of bands along the -Z line. The energy scale of the Dirac electrons in these two cases may be estimated to be 45 and 20 meV, respectively. Part (a) reprinted with permission from Ali et al. (2014), copyright (2014) American Chemical Society. Part (b) reprinted from Conte et al. (2017).

The above described effective models have been applied with some success to explain experimental data available those times. However, these models should be confronted with the much broader recently acquired experimental knowledge. Similar to all other models based on the expansion, their validity is limited to the near vicinity of the Brillouin zone center, and to the number of spin-degenerate bands taken into account (restricted to 4 in the Kane/Bodnar model). Moreover, the number of band structure parameters is strongly restricted in these models. This ensures their relative simplicity, but at the same time, limits the potential to describe the band structure in greater detail, even in the immediate vicinity of the point.

It is therefore important to reconcile the implications of such effective models with other theoretical approaches. Such band structure calculations already appeared in the early stages of research on CdAs. These were based on either pseudopotential calculations Lin-Chung (1969, 1971); Dowgiałło-Plenkiewicz and Plenkiewicz (1979) or the semiempirical tight-binding method Sierański et al. (1994). More recently, ab initio calculations have been performed, with their main focus on the Dirac-like states. To the best of our knowledge, the first ab initio study of CdAs predicting the presence of 3D Dirac electrons was presented by Wang et al. Wang et al. (2013). Nevertheless, the considered space groups did not comprise the most probable one, I4/acd (No. 110) Pietraszko and Lukaszewicz (); Ali et al. (2014). Other ab initio calculations may be found in Refs. Liu et al. (2014); Neupane et al. (2014); Borisenko et al. (2014); Conte et al. (2017), often carried out in support of experimental findings.

Even though the results of ab initio calculations may differ in some details – most likely related to the particular approximation of exchange and correlation functionals and the size of the unit cell considered – they provide us with a rather consistent theoretical picture of the electronic bands in CdAs. They predict that CdAs is a semimetal, with a pair of well-defined 3D Dirac cones, emerging at relatively low energies. In line with symmetry arguments, the Dirac nodes are found to be located at the tetragonal axis, with an exception of Ref. Liu et al. (2014).

When the non-centrosymmetric I4cd space group is considered, the pair of Dirac nodes’ double degeneracy due to spin is lifted Wang et al. (2013). The loss of inversion symmetry then transforms the 3D Dirac semimetal into a 3D Weyl semimetal with two pairs of Weyl nodes. The Dirac cones are dominantly formed from -type arsenic states, and are well separated from the bands lying at higher or lower energies. Notably, the -like cadmium states are found well below the Fermi energy. The band structure is therefore inverted and may be formally described by a negative band gap ( eV). To certain extent, the electronic bands in CdAs resemble those in HgTe, which is another semimetal with an inverted band structure Weiler (1981).

To illustrate the typical results of ab initio calculations, two examples have been selected from Refs. Ali et al., 2014 and Conte et al., 2017 and plotted in Figs. 8a and b, respectively. The Dirac cones are located in close vicinity to the point, with the charge neutrality (Dirac) points at the -Z line and the Fermi level. The parameters of the Dirac cones derived in selected ab initio studies are presented in Tab. 1. For instance, Conte et al. Conte et al. (2017) deduced, using the GGA approximation, that the energy scale of Dirac electrons reaches  meV, the strongly anisotropic velocity parameter is of the order of  m/s, and two Dirac nodes are located at  nm.

Let us now explore the experiments carried out on CdAs using various techniques. These provided us with unambiguous evidence for the conical features in this materials. However, let us clearly note from the beginning that the scale of the conical bands deduced – using ARPES Liu et al. (2014); Neupane et al. (2014); Borisenko et al. (2014) and STM/STS Jeon et al. (2014) techniques, or from the optical response Ali et al. (2014); Neubauer et al. (2016); Akrap et al. (2016) – is not consistent with energy scale predictions for Dirac electrons based on ab initio calculations. While the theoretically expected scale of Dirac electrons rarely exceeds  meV (Tab. 1), the experimentally observed cones extend over a significantly broader interval of energies.

Technique Scale (meV) Location (nm) Orientation
ab initio, GGA, I4cd Wang et al. (2013) 40 0.32 [001]
ab initio, GGA, I4/acd Ali et al. (2014) 45 0.4 [001]
ab initio, GGA, I4/acd Conte et al. (2017) 20 0.23 [001]
ab initio, GGA, I4cd Liu et al. (2014) 200 1.2 [112]111equivalent to the [111] direction when the cubic cell approximation is considered as in Ref. Liu et al. (2014)
ARPES Liu et al. (2014) several hundred 1.6 [112]11footnotemark: 1
ARPES Neupane et al. (2014) several hundred [001]
ARPES Borisenko et al. (2014) several hundred [001]
STM/STS Jeon et al. (2014) 20 0.04 [001]
Magneto-optics Hakl et al. (2018); Akrap et al. (2016) 40 0.05 [001]
Magneto-transport & Bodnar model Rosenman (1969); Bodnar (1977) 85 0.15 [001]
Magneto-transport Zhao et al. (2015) 200 [001]
Table 1: The band structure parameters of CdAs: the energy scale (defined in Fig. 2), the Dirac point position and the crystallographic axis at which the Dirac nodes are located, as deduced using different experimental techniques or theoretical calculations/analysis. The velocity parameter is usually found to be around, or slightly below,  m/s in the direction perpendicular to the line connecting Dirac nodes, and reduced down to the  m/s range along this direction.

V Angular-resolved photoemission spectroscopy

The ARPES technique provided us with a solid piece of evidence for conical features in the electronic band structure of CdAs, soon after predictions of Dirac-like states by Wang et al. Wang et al. (2013). This largely contributed to the renewed interest in the electronic properties of this material. Such initial observations were made by several groups Liu et al. (2014); Borisenko et al. (2014); Neupane et al. (2014); Yi et al. (2014), and elaborated further later on Roth et al. (2018).

[b]

Figure 9: The valence bands in CdAs visualized by low-temperature ARPES technique by Liu et al. Liu et al. (2014). The data was collected on the (112)-terminated surface. The widely extending 3D conical band – characterized by a velocity parameter of  m/s and interpreted in terms of 3D Dirac electrons – coexists with another hole-like weakly dispersing parabolic band, also observed in Ref. Neupane et al. (2014). Reprinted by permission from Springer Nature: Nature Materials Liu et al. (2014), copyright (2014).

Characteristic data collected in ARPES experiments on CdAs Liu et al. (2014); Neupane et al. (2014) are plotted in Figs. 9 and 10, and respectively show well-defined conical features for both valence and conduction bands. In addition, the valence conical band has been found to coexist with another weakly dispersing hole-like parabolic band Neupane et al. (2014) which has a characteristic effective mass close to unity. The conical features in Fig. 9 and 10 were interpreted in terms of bulk states. Liu et al. Liu et al. (2014) and Neupane et al. Neupane et al. (2014) concluded the presence of a pair of 3D Dirac nodes at the [112] and [001] axes, respectively. However, it is worth noting that the orientation along the [112] axis – or alternatively, along the [111] axis when an approximately cubic unit cell is considered like it is the case in Ref. Liu et al. (2014) – is not consistent with expectations based on symmetry arguments Yang and Nagaosa (2014), which only allow the Dirac nodes to be present at the tetragonal axis (the [001] direction). The velocity parameter was found to be close to  m/s in the plane perpendicular to the axis connecting the Dirac nodes, and reduced down to  m/s Liu et al. (2014) along this axis. The ARPES data in Refs. Liu et al. (2014); Neupane et al. (2014) do not directly show any signatures of Dirac cones merging via the corresponding Lifshitz points. Nevertheless, the indicated velocity parameters, and the position of the cones (), allow us to estimate the scale of Dirac electrons to be several hundred meV or more.

ARPES data similar to Refs. Liu et al. (2014); Neupane et al. (2014), obtained on the [112]-terminated surface of CdAs, were also presented by Borisenko et al. Borisenko et al. (2014), who primarily focused on the conical feature in the conduction band. The presence of a pair of symmetry-protected 3D Dirac cones, with the corresponding nodes at the [001] axis axis, has been concluded, and the electron velocity parameter  m/s deduced. The Fermi energy in the studied -doped sample exceeded  meV, and may serve as a lower bound for the parameter. Since the shape of the observed conical band does not provide any signature of the approaching upper Lifshitz point, one may conclude that .

More recently, other ARPES experiments on CdAs have been performed by Roth et al. Roth et al. (2018). They provided experimental data similar to previous studies, but differed in their interpretation. From their experiment realized on a sample with a (112)-terminated surface, they concluded that a part of observed conical features does not come from bulk, but instead originate from the surface states (cf. Ref. Yi et al. (2014)). Let us note that, since each Dirac node is composed of two Weyl nodes with opposite chiralities Potter et al. (2014), such surface states may have the form of the Fermi arcs in a 3D symmetry-protected Dirac semimetal.

Figure 10: The conduction band of CdAs visualized by ARPES by Neupane et al. Neupane et al. (2014). The data were collected on the (001)-terminated surface of CdAs at 14 K and show the dispersion in the direction perpendicular to the -Z line. The observed 3D conical band was interpreted in terms of 3D Dirac electrons, implying a velocity parameter of  m/s. Reprinted by permission from Springer Nature: Nature Communications Neupane et al. (2014), copyright (2014).

Vi Scanning tunneling spectroscopy and microscopy

Similar to ARPES, the STS/STM technique also played an important role in the recent revival of CdAs. The data collected in STS experiments performed in magnetic fields Jeon et al. (2014) revealed, via the characteristic -dependence of Landau levels, the presence of a single widely extending conical band located at the center of the Brillouin zone. In the literature, this observation is often taken as experimental evidence for the symmetry-protected 3D Dirac electrons in CdAs. However, according to Jeon et al. Jeon et al. (2014), the existence of “this extended linearity is not guaranteed by the Dirac physics around the band inversion”.

Such a conclusion agrees with theoretical expectations based on symmetry arguments. These exclude the existence of a symmetry-protected Dirac cone located at the center of the Brillouin zone Yang and Nagaosa (2014). In addition, this observation is in line with the Kane/Bodnar models used to explain the band structure of CdAs in the past. Nevertheless, Jeon et al. Jeon et al. (2014) also conclude – by extrapolating their Landau level spectroscopy data to vanishing magnetic fields – that a pair of symmetry-protected Dirac cones emerges at low energies. They give a rough estimate of  meV for the characteristic Dirac energy scale.

Beyond insights into the bulk electronic states of CdAs, the natural sensitivity of the STM/STS technique to the surface of explored systems may provide us with deeper knowledge about their surface states. A 3D symmetry-protected Dirac semimetal like CdAs can be viewed as two copies of a 3D Weyl semimetal with nodes having opposite chiralities; two sets of Fermi arcs are expected on the surface. The recent STS/STM study Butler et al. (2017) dedicated to the surface reconstruction of cadmium vacancies in CdAs may be considered as the initial step in such investigations.

Figure 11: The approximately conical band centered at the point deduced from Landau-level spectroscopy (open circles) and quasi-particle interference (QPI) pattern (closed circles with error bars) in the STS/STM experiments performed by Jeon et al. Jeon et al. (2014). The slope of the conical band corresponds to a velocity parameter of  m/s. Reprinted by permission from Springer Nature: Nature Materials Jeon et al. (2014), copyright (2014).

Vii Optical properties

The optical and magnetic optical properties of CdAs have been a topic of study for more than 50 years, and resulted in a series of works Turner et al. (1961); Haidemenakis et al. (1966); Zdanowicz (1967). These were often interpreted in terms of Kane/Bodnar models Wagner et al. (1971); Rogers et al. (1971); Radoff and Bishop (1972); Aubin et al. (1977); Gelten et al. (1980); Aubin et al. (1981); Jay-Gerin et al. (1983); Singh and Wallace (1983b, a); Houde et al. (1986); Lamrani and Aubin (1987); Akrap et al. (2016); Hakl et al. (2018); Crassee et al. (2018), and more recently using the picture of 3D Dirac electrons Neubauer et al. (2016); Jenkins et al. (2016); Yuan et al. (2017); Uykur et al. (2018).

Early optical studies were focused on the basic character of the electronic band structure in CdAs. They aimed at clarifying the existence of a band gap and at determining its size. Using infrared reflectivity and transmission techniques, Turner et al. Turner et al. (1961) concluded that CdAs should be classified as a narrow-gap semiconductor with a direct band gap of 0.16 eV. Similarly, an indirect band gap around 0.2 eV, was concluded by Zdanowicz et al. Zdanowicz (1967) based on transmission experiments. Much lower values were found in magneto-optical studies. Haidemenakis et al. Haidemenakis et al. (1966) concluded  meV, suggesting a semimetallic nature of CdAs. The difference in conclusions between optical and magneto-optical studies may be related to the large doping of the explored samples. In that case, the onset of interband absorption, often referred to as the optical band gap, appears due to the Moss-Burstein shift (Pauli blocking) Burstein (1954), at photon energies significantly exceeding the size of the energy band gap.

Further series of optical and magneto-optical CdAs studies were performed on various mono- or polycrystalline samples during the seventies and eighties. The collected data was primarily analyzed within the framework of the Kane model Wagner et al. (1971); Rogers et al. (1971); Radoff and Bishop (1972); Aubin et al. (1977); Gelten et al. (1980), and later on the Bodnar model Aubin et al. (1981); Jay-Gerin et al. (1983); Lamrani and Aubin (1987). The authors of these works concluded that the electronic band structure at the point is fairly well described using these models: it is inverted, with the -type arsenic states above the -like cadmium states, and characterized by a relatively small negative band gap. A schematic view of such a band structure is plotted in Figs. 6 and 7. The deduced value of the band gap reached  eV Wagner et al. (1971); Radoff and Bishop (1972); Aubin et al. (1977) at room or low temperatures, but also values as low as  eV have been reported Aubin et al. (1981); Lamrani and Aubin (1987); Jay-Gerin et al. (1983). Particular attention has been paid to the profile and position of the weakly dispersing band, which is nearly flat in the vicinity of the point. Several times, the possibility of the band maxima located at non-zero momenta has been discussed Aubin et al. (1977); Gelten et al. (1980). At the same time, Lamrani and Aubin concluded a surprisingly flat heavy-hole band, when the theoretically expected Landau-quantized Bodnar band structure Singh and Wallace (1983b, a) was confronted with the experimentally determined energies of interband inter-Landau level excitations Lamrani and Aubin (1987).

Figure 12: The real part of the optical conductivity of CdAs obtained by Neubauer et al. Neubauer et al. (2016). 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  cm, the optical conductivity increases roughly linearly with the photon energy, which is behavior expected for massless charge carries in 3D. Reprinted with permission from Neubauer et al. (2016), copyright (2016) by the American Physical Society.

The renewed interest in CdAs motivated several groups to take a fresh look at the optical, magneto-optical, and ultra fast optical properties of this material Weber et al. (2015); Neubauer et al. (2016); Jenkins et al. (2016); Akrap et al. (2016); Yuan et al. (2017); Sharafeev et al. (2017); Hakl et al. (2018); Crassee et al. (2018); Weber et al. (2015); Zhu et al. (2017); Lu et al. (2017, 2018). Thanks to this, to the best of our knowledge, the optical conductivity of CdAs (Fig. 12) has been extracted from the experimental data for the very first time Neubauer et al. (2016). At low energies, the optical conductivity is characterized by a pronounced Drude peak due to the presence of free charge carriers, and a rich set of phonon excitations. Similar to the Raman response Jandl et al. (1984); Sharafeev et al. (2017), the complexity of the phonon-related response directly reflects the relatively high number of atoms in the CdAs unit cell. Above the onset of interband absorption, optical conductivity increases with a slightly superlinear dependence on photon energy. Such behavior is not far from the expectation for 3D massless charge carriers,  Goswami and Chakravarty (2011); Timusk et al. (2013). The observed optical response was thus interpreted in terms of 3D Dirac electrons, with a low anisotropy and a velocity parameter lying in the range of  m/s. No clear indications of Dirac cones merging at the Lifshitz points were found. Reflectivity data similar to Ref. Neubauer et al. (2016) were also presented by Jenkins et al. Jenkins et al. (2016), with basically the same conclusions.

The Dirac model was similarly used to interpret the classical-to-quantum crossover of cyclotron resonance observed in magneto-transmission data collected on thin MBE-grown CdAs layers Yuan et al. (2017) as well as pump-probe experiments in the visible spectral range, which revealed the transient reflection Weber et al. (2015) and transmission Zhu et al. (2017). These latter experiments show that a hot carrier distribution is obtained after 400–500 fs, after which the charge carrier relax by two processes. Subsequent pump-probe experiments using mid-infrared Lu et al. (2017) and THz probe Lu et al. (2018) confirm this two process relaxation, which can be qualitatively reproduced using a two temperature model.

A different view was proposed in a recent magneto-reflectivity study of CdAs by Akrap et al. Akrap et al. (2016) (see Fig. 13). In high magnetic fields, when the samples were pushed into their corresponding quantum limits with all electrons in the lowest Landau level, the dependence of the observed cyclotron mode was found to be inconsistent with 3D Dirac electrons. The data was interpreted in terms of the Kane/Bodnar model, with a vanishing band gap. This model also leads to the appearance of 3D massless electrons, and consequently, to a magneto-optical response linear in . In this model, the dependence of the cyclotron mode is also expected in the quantum limit. This is in contrast to 3D Dirac electrons, which host characteristic so-called zero-mode Landau levels. These are independent of magnetic field, disperse linearly with momentum along the direction of the applied field, and imply a more complex cyclotron resonance dependence in the quantum limit. An approximately isotropic velocity parameter was found for the nearly conical conduction band:  m/s Akrap et al. (2016). Estimates of the band gap and crystal field splitting parameter at low temperatures were obtained in a subsequent magneto-transmission study performed on thin CdAs slabs Hakl et al. (2018): meV and  meV.

Figure 13: The energies of cyclotron resonance excitations observed in Ref. Akrap et al. (2016) as a function of applied magnetic field, obtained on (112)- and (001)-oriented CdAs samples. The dashed and dotted lines show theoretically expected positions within the Kane and Dirac models, see Ref. Akrap et al. (2016) for details. Adapted with permission from Akrap et al. (2016), copyright (2016) by the American Physical Society.

Lately, spatially resolved infrared reflectivity has been used to characterize the homogeneity of CdAs crystals Crassee et al. (2018). In all the studied samples, independently of how they were prepared and how they were treated before the optical experiments, conspicuous fluctuations in the carrier density up to 30% have been found. These charge puddles have a characteristic scale of 100 m. They become more pronounced at low temperatures, and possibly, they become enhanced by the presence of crystal twinning. Such an inhomogeneous distribution of electrons may be a generic property of all CdAs crystals, and should be considered when interpreting experimental data collected using other techniques.

Viii Magneto-transport properties and quantum oscillations

Renewed interest in CdAs brought upon an explosion of interest in magneto-transport studies. Many of these studies focused on the previously reported very high carrier mobility Rosenberg and Harman (1959), and its possibilities for device application. Often, newer transport results are interpreted within the scenario of two 3D Dirac cones, which are well separated in -space and with the Fermi level lying below the Lifshitz transition.

The first detailed magneto-transport study of CdAs dates back to the 1960s Rosenman (1966); Armitage and Goldsmid (1968); Rosenman (1969). The geometry of the Fermi surface was for the first time addressed by Shubnikov-de Haas (SdH) measurements. Rosenman Rosenman (1966) explored such quantum oscillations on a series of -doped samples, concluding that the Fermi surface is a simple ellipsoid symmetric around the -axis, and inferred a low anisotropy factor of 1.2. These very first papers also show nearly conical shape of the conduction band (Fig. 1). A decade later, Zdanowicz et al. Zdanowicz et al. (1979) studied SdH oscillations in thin films and single crystals of CdAs, confirming the ellipsoidal geometry of the Fermi surface. In addition, they reported a striking linear magnetoresistance (MR).

Figure 14: (a) Weyl orbit in a thin slab of thickness in a magnetic field . 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  K measured on a thin CdAs sample ( nm) which was cut from a single crystal by focused ion beam. The field was oriented parallel (90) and perpendicular (0) to the surface. Only one frequency is observed for parallel field, and two frequencies for perpendicular field. Adapted by permission from Springer Nature: Nature Moll et al. (2016), copyright (2016).

In a new bout of activity, several groups confirmed that the Fermi surface consists of a simple, nearly spherical ellipsoid, with an almost isotropic Fermi velocity Liang et al. (2015). Such a simple Fermi surface was questioned by Zhao et al.. They found that, for particular directions of the magnetic field with respect to the main crystal axes, the magnetoresistance (MR) shows two oscillation periods which mutually differ by 10-25% Zhao et al. (2015), pointing to a dumbbell-shaped Fermi surface. This was interpreted to originate from two nested Fermi ellipsoids arising from the two separated Dirac cones, where is placed just above the Lifshitz transition. Another picture was proposed by Narayanan et al. Narayanan et al. (2015), who found single-frequency SdH oscillations, and concluded two nearly isotropic Dirac-like Fermi surfaces. In subsequent studies, Desrat et al. followed the SdH oscillations as a function of the field orientation, and always found two weakly separated frequencies (5-10%). Their results were interpreted within the picture of two ellipsoids that are separated in -space due to the possible absence of inversion symmetry Desrat et al. (2018). Clear beating patterns, indicating a multiple frequency in SdH oscillations, were also reported in Nernst measurements Liang et al. (2017), and magnetoresistance Guo et al. (2016) on single crystals of CdAs. Such beating patterns were attributed to the lifting of spin degeneracy due to inversion symmetry breaking either by an intense magnetic field Xiang et al. (2015); Liang et al. (2017), or by Cd-antisite defects Guo et al. (2016), which may turn a Dirac node into two Weyl nodes.

Several transport studies reported on strikingly linear nonsaturating MR in CdAs Narayanan et al. (2015); Feng et al. (2015); Liang et al. (2015) being more pronounced in samples with lower mobilities. Since linear MR is observed at magnetic fields far below the quantum limit, the standard Abrikosov’s theory Abrikosov (2003) – referring to the transport in the lowest Landau level – cannot be applied, and a different explanation was needed. Liang et al. Liang et al. (2015) therefore suggest there is a mechanism that protects from backscattering in zero field. This protection is then rapidly removed in field, leading to a very large magnetoresistance. They propose an unconventional mechanism, caused by the Fermi surface splitting into two Weyl pockets in an applied magnetic field. Similarly, Feng et al. Feng et al. (2015) assigned the large non-saturating MR to a lifting of the protection against backscattering, caused by a field-induced change in the Fermi surface. The authors judged that linear MR cannot be due to disorder, as the CdAs samples are high-quality single crystals. They instead conclude that it is due to the Dirac node splitting into two Weyl nodes. In contrast, Narayanan et al. Narayanan et al. (2015) find that the Fermi surface does not significantly change up to 65 T, except for Zeeman splitting caused by a large -factor. Through comparing quantum and transport relaxation times, they conclude that transport in CdAs is dominated by small-angle scattering, which they trace back to electrons scattered on arsenic vacancies, and that the linear MR is linked to mobility fluctuations.

Figure 15: (a) Angular dependence of the transverse magnetoresistance with magnetic field rotated in the (010) plane. The tilt angle is the angle between and the [100] direction. (b) Landau index plots vs at different ’s. (Inset) Angular dependence of the oscillation frequency and the total phase , extracted from the linear fitting from the first Landau level to the fifth Landau level. Adapted with permission from Xiang et al. (2015), copyright (2015) by the American Physical Society.

The Potter et al. theory Potter et al. (2014) predicts the existence of specific closed cyclotron orbits in a Dirac or Weyl semimetal (Fig.14a). These orbits are composed of two Fermi arcs located on opposite surfaces of the sample, which are then interconnected via zero-mode Landau levels. Moll et al. Moll et al. (2016) report on the SdH oscillations measured in mesoscopic devices. These were prepared using the focused ion beam technique, allowing CdAs crystals to be cut into sub-micron platelets. They report two series of oscillations, and via specific angle dependence, they associate them with surface-related and bulk-related orbits (Fig. 14b).

The chiral anomaly is yet another theoretically expected signature of the field-induced splitting of a Dirac point into a pair of Weyl nodes. The parallel application of an electric and magnetic field is predicted to transfer electrons between nodes with opposite chiralities. Such a transfer should be associated with lowering resistivity (negative MR). For CdAs, there are indeed several reports of a negative MR and a suppression of thermopower for particular magnetic-field directions Li et al. (2015); Jia et al. (2016b); Li et al. (2016). Typically, such behavior was observed in microdevices (platelets or ribbons). It should be noted that a negative MR can also emerge in a topologically trivial case due to so-called current jetting dos Reis et al. (2016); Liang et al. (2018). This is a simple consequence of a high transport anisotropy when the magnetic field is applied. In such a case, the current is jetting through a very narrow part of the sample. The measured MR strongly then depends on the distance of the voltage contacts from the current path.

The analysis of the quantum oscillation phase represents a unique way to identify the nature of probed charge carriers. For conventional Schrödinger electrons, one expects so-called Berry phase , whereas Dirac electrons should give rise to . Indeed, several reports based on SdH oscillations in CdAs indicate that  Zdanowicz et al. (1979); He et al. (2014); Desrat et al. (2015); Pariari et al. (2015). A weak deviation from the ideal non-trivial Berry phase, was reported by Narayanan Narayanan et al. (2015), who also discussed the influence of Zeeman splitting on the phase of quantum oscillations Mikitik and Sharlai (2003). Notably, the electron -factor in CdAs is relatively large and anisotropic, and it implies Zeeman splitting comparable to cyclotron energy Blom et al. (1980).

Contrasting results were obtained by Xiang et al. Xiang et al. (2015), who found the non-trivial Berry phase only when the field is applied along the tetragonal -axis of CdAs. When the magnetic field is rotated to be parallel with the or axis, the measured Berry phase becomes nearly trivial (see Fig. 15 and discussion in Ref. He and Li (2016)). This result is interpreted in terms of 3D Dirac-phase symmetry-breaking effects, when the magnetic field is tilted away from the -axis. A certain angle dependence of the Berry phase has been also reported by Zhao et al. Zhao et al. (2015), reporting the Berry phase in between 0 and values. Another study by Cheng et al. Cheng et al. (2016) was dedicated to the thickness-dependence of the Berry phase in MBE-grown thin films of CdAs, implying a non-trivial to trivial transition of the Berry phase with changing the layer thickness.

Figure 16: The quantum Hall effect in a 20-nm-thick epitaxial CdAs film grown by molecular beam epitaxy, showing the Hall () and longitudinal () resistances as a function of magnetic field measured at 1 K. Reprinted with permission from Schumann et al. (2018), copyright (2018) by the American Physical Society.

The partly contradicting results in determining the Berry phase illustrate that even though the phase of quantum oscillation in principle identifies the nature unambiguously, the practical analysis of this phase is straightforward only in well-defined system such as graphene Novoselov et al. (2005). In more complex materials such as CdAs, where quantum oscillations are superimposed on the much stronger effect of linear magneto-resistance Liang et al. (2015), the precise and reliable determination of the phase may represent a more challenging task. This may be illustrated on bulk graphite, which is another high-mobility system with a strong and approximately linear magneto-resistance. There, the Dirac-like or normal massive nature of hole-type carriers has been a subject of intensive discussion in literature Luk’yanchuk and Kopelevich (2004, 2006); Schneider et al. (2009); Luk’yanchuk and Kopelevich (2010); Schneider et al. (2010).

More recently, the very first experiments showing the quantum Hall effect (QHE) in thin films of CdAs appeared. Zhang et al. measured SdH oscillations on a series of CdAs nanoplates Zhang et al. (2017). They report multiple cyclotron orbits, distinguishing both 3D and 2D Fermi surfaces. They also observe a quantized Hall effect (QHE), which they attribute to the surface states of CdAs, linked to the Weyl orbits. Schumann et al. Schumann et al. (2018) studied MBE-grown, 20 nm thick films of CdAs, and observed the QHE at low temperatures (Fig. 16). Similarly, they attribute the QHE to surface states, and conclude that the bulk states are weakly gapped at low temperatures. In continuation of this work, Galletti et al. Galletti et al. (2018) tuned the carrier concentration in thin films across the charge neutrality point using a gate voltage and concluded that the observed magneto-transport response is in line with expectations for a 2D electron gas of massless Dirac electrons.

The planar Hall effect has also been reported in CdAs, an effect in which a longitudinal current and an in-plane magnetic field give rise to a transverse current or voltage. Li et al. investigated microribbons of CdAs, and find an anisotropic MR and planar Hall effect, which they attribute to the physics of Berry curvature Li et al. (2018). Guo et al. also uncovered unusually large transverse Hall currents in needle-like single crystals of CdAs Guo et al. (2016). Wu et al. carried out similar measurements on CdAs nanoplatelets, finding a large negative longitudinal MR, and a planar Hall effect with non-zero transverse voltage when the magnetic field is tilted away from the electric field. These observations are interpreted as transport evidence for the chiral anomaly Wu et al. ().

Ix Summary

Cadmium arsenide is a prominent member of the topological materials class, widely explored using both theoretical and experimental methods. At present, there is no doubt that CdAs hosts well-defined 3D massless charge carriers. These are often associated with the symmetry-protected 3D Dirac phase, but may also be interpreted using alternative approaches developed in the past. This calls for further investigations of CdAs, preferably using a combination of different experimental techniques. Such investigation may resolve the currently existing uncertainties about the electronic band structure, which slow down the overall progress in physics of CdAs, and may contribute to our understanding of rich phenomena associated with this appealing material.

Acknowledgements

The authors acknowledge discussions with F. Bechstedt, S. Borisenko, C. C. Homes, N. Miller, B. A. Piot, A. V. Pronin, O. Pulci, A. Soluyanov, and S. Stemmer. This work was supported by ANR DIRAC3D projects and MoST-CNRS exchange programme (DIRAC3D). A. A. acknowledges funding from The Ambizione Fellowship of the Swiss National Science Foundation. I. C. acknowledges support from the postdoc mobility programme of the Suisse National Science Foundation. A. A. acknowledges the support SNF through project PP00P2_170544.

References

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