Symmetry-broken three-dimensional (3D) topological Dirac semimetal systems with strong spin-orbit coupling can host many exotic Hall-like phenomena and Weyl fermion quantum transport. Here, using high-resolution angle-resolved photoemission spectroscopy, we performed systematic electronic structure studies on Cd3As2, which has been predicted to be the parent material, from which many unusual topological phases can be derived. We observe a highly linear bulk band crossing to form a 3D dispersive Dirac cone projected at the Brillouin zone centre by studying the (001)-cleaved surface. Remarkably, an unusually high in-plane Fermi velocity up to 1.5 × 106 ms−1 is observed in our samples, where the mobility is known up to 40,000 cm2 V−1s−1, suggesting that Cd3As2 can be a promising candidate as an anisotropic-hypercone (three-dimensional) high spin-orbit analogue of 3D graphene. Our discovery of the Dirac-like bulk topological semimetal phase in Cd3As2 opens the door for exploring higher dimensional spin-orbit Dirac physics in a real material.
Two-dimensional (2D) Dirac electron systems exhibiting many exotic quantum phenomena constitute one of the most active topics in condensed matter physics1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19. The notable examples are graphene and the surface states of topological insulators (TIs). Three-dimensional (3D) Dirac fermion metals, sometimes noted as the topological bulk Dirac semimetal (BDS) phases, are also of great interest if the material possesses 3D isotropic or anisotropic relativistic dispersion in the presence of strong spin-orbit coupling. It has been theoretically predicted that a topological (spin-orbit) 3D spin-orbit Dirac semimetal can be viewed as a composite of two sets of Weyl fermions where broken time-reversal or space inversion symmetry can lead to a surface Fermi-arc semimetal phase or a TI14. In the absence of spin-orbit coupling, topological phases cannot be derived from a 3D Dirac semimetal. Thus, the parent BDS phase with strong spin-orbit coupling is of great interest. Despite their predicted existence11,13,14, experimental studies on the massless BDS phase have been lacking as it has been difficult to realize this phase in real materials, especially in stoichiometric single crystalline non-metastable system with high mobility. It has also been noted that the BDS state can be achieved at the critical point of a topological phase transition20,21 between a normal insulator and a TI, which requires fine-tuning of the chemical doping/alloying composition thus by effectively varying the spin-orbit coupling strength. This approach also introduces chemical disorder into the system. In stoichiometric bulk materials, the known 3D Dirac fermions in bismuth are in fact of massive variety as there clearly exists a band gap in the bulk Dirac spectrum10. On the other hand, the bulk Dirac fermions in the Bi1−xSbx system coexist with additional Fermi surfaces5. Therefore, to this date, identification of a gapless BDS phase in stoichiometric materials remains experimentally elusive.
In this Article, we present the experimental identification of a gapless Dirac-like 3D topological (spin-orbit) semimetal phase in stoichiometric single crystalline system of Cd3As2, which is protected by the C4 crystalline (crystal structure) symmetry and spin-orbit coupling as predicted in theory14. Using high-resolution angle-resolved photoemission spectroscopy (ARPES), we show that Cd3As2 features a bulk band Dirac-like cone locating at the centre of the (001) surface projected Brillouin zone (BZ). Remarkably, we observe that the band velocity of the bulk Dirac spectrum is as high as ~10 eVÅ, which along with its massless character favourably contributes to its natural high mobility (~105 cm2 V−1s−1 (refs 22, 23)). We further compare and contrast the observed crystalline-symmetry-protected BDS phase in Cd3As2 with those of in the Bi-based 3D-TI systems such as in BiTl(S1−δSeδ)2 and (Bi1−δInδ)2Se3 systems. Our experimental identification and band-structure measurements of the Dirac-like bulk semimetal phase and its clear contrast with Bi2Se3 and 2D graphene discovered previously, opens the door for exploring higher dimensional spin-orbit Dirac physics in a stoichiometric material. These new directions are uniquely enabled by our observation of strongly spin-orbit coupled 3D massless Dirac semimetal phase protected by the C4 symmetry, which is not possible in the 2D Dirac fermions in graphene and the surfaces of TIs, or weak spin-orbit 3D Dirac fermions in other materials.
Crystalline symmetry protected topological Dirac phase
The crystal structure of Cd3As2 has a tetragonal unit cell with a=12.67 Å; and c=25.48 Å; for Z=32 with symmetry of space group I41cd (see Fig. 1a,b). In this structure, arsenic ions are approximately cubic close-packed and Cd ions are tetrahedrally coordinated, which can be described in parallel to a fluorite structure of systematic Cd/As vacancies. There are four layers per unit and the missing Cd-As4 tetrahedra are arranged without the central symmetry as shown with the (001) projection view in Fig. 1b, with the two vacant sites being at diagonally opposite corners of a cube face24. The corresponding BZ is shown in Fig. 1d, where the centre of the BZ is the Γ point, the centres of the top and bottom square surfaces are the Z points, and other high-symmetry points are also noted. Cd3As2 has attracted attention in electrical transport because of its high mobility of 105 cm2 V−1s−1 reported in previous studies22,23. The carrier density and mobility of our Cd3As2 samples (shown in Figs. 1 and 2) are characterized to be of 5.2 × 1018 cm−3 and 42,850 cm2 V−1s−1, respectively, at temperature of 130 K, consistent with previous reports22,23, which provide an evidence for the high quality of our single crystalline sample. In band theoretical calculations, Cd3As2 is also of interest as it features an inverted band structure25. More interestingly, a very recent theoretical prediction14 which motivated this work, has shown that the spin–orbit interaction in Cd3As2 cannot open up a full energy gap between the inverted bulk conduction and valence bands due to the protection of an additional crystallographic symmetry12 (in the case of Cd3As2 it is the C4 rotational symmetry along the kz direction14), which is in contrast to other band-inverted systems such as HgTe3. This theory predicts14 that the C4 rotational symmetry protects two bulk (3D) Dirac band touching points at two special k points along the Γ−Z momentum space cut-direction, as shown by the red crossings in Fig. 1d. Therefore, Cd3As2 serves a candidate for a spacegroup or crystal structure symmetry protected C4 BDS phase.
Observation of bulk Dirac cone
In order to experimentally identify such a BDS phase, we systematically study the electronic structure of Cd3As2 on the cleaved (001) surface. Figure 1c shows momentum-integrated ARPES spectral intensity over a wide energy window. Sharp ARPES intensity peaks at binding energies of and 41 eV that correspond to the cadmium 4d and the arsenic 3d core levels are observed, confirming the chemical composition of our samples. We study the overall electronic structure of the valence band. Figure 1e shows the second derivative image of an ARPES dispersion map in a 3-eV binding energy window, where the dispersion of several valence bands are identified. Moreover, a low-lying small feature that crosses the Fermi level is observed. In order to resolve it, high-resolution ARPES dispersion measurements are performed in the close vicinity of the Fermi level as shown in Fig. 1f. Remarkably, a linearly dispersive upper Dirac cone is observed at the surface BZ centre point, whose Dirac node is found to locate at a binding energy of . At the Fermi level, only the upper Dirac band but no other electronic states are observed. On the other hand, the linearly dispersive lower Dirac cone is found to coexist with another parabolic bulk valence band, which can be seen from Fig. 1e. From the observed steep Dirac dispersion (Fig. 1f), we obtain a surprisingly high Fermi velocity of about 9.8 eVÅ; (). This is more than tenfold larger than the theoretical prediction of 0.15 eVÅ; at the corresponding location of the chemical potential14. Compared with the much-studied 2D Dirac systems, the Fermi velocity of the 3D Dirac fermions in Cd3As2 is thus about 3 times higher than that of in the topological surface states of Bi2Se3 (ref. 6), 1.5 times higher than in graphene26 and 30 times higher than that in the topological Kondo insulator phase in SmB6 (refs 27, 28). The observed large Fermi velocity of the 3D Dirac band provides clues to understand unusually high mobility of Cd3As2 reported in previous transport experiments22,23. Therefore, one can expect to observe unusual magneto-electrical and quantum Hall transport properties under high-magnetic field. It is well-known that in graphene the capability to prepare high-quality and high-mobility samples has enabled the experimental observations of many interesting phenomena that arises from its 2D Dirac fermions. The large Fermi velocity and high mobility in Cd3As2 are among the important experimental criteria to explore the 3D relativistic physics in various Hall phenomena in tailored Cd3As2.
We compare ARPES observations with our theoretical calculations, which is qualitatively consistent with previous calculations14. The reason for the use of our calculations is twofold: first, our calculations are fine tuned based on the characterization of samples used in the present ARPES study; second, sufficiently detailed cuts are not readily available from ref. 14 which is necessary for a detailed comparison of ARPES data with theory. In theory, there are two 3D Dirac nodes that are expected at two special k points along the Γ−Z momentum space cut-direction, as shown by the red crossings in Fig. 1d. At the (001) surface, these two k points along the Γ−Z axis project on to the point of the (001) surface BZ (Fig. 1d). Therefore, at the (001) surface, theory predicts one 3D Dirac cone at the BZ centre point, as shown in Fig. 2a. These results are in qualitative agreement with our data, which supports our experimental observation of the 3D BDS phase in Cd3As2. We also study the ARPES measured constant energy contour maps (Fig. 2c and d). At the Fermi level, the constant energy contour consists of a single pocket centred at the point. With increasing binding energy, the size of the pocket decreases and eventually shrinks to a point (the 3D Dirac point) near . The observed anisotropies in the iso-energetic contours are likely due to matrix element effects associated with the standard p-polarization geometry used in our measurements.
3D dispersive nature
A 3D Dirac semimetal is expected to feature nearly linear dispersion along all three momentum space directions close to the crossing point, even though the Fermi/Dirac velocity can vary significantly along different directions. It is well known that in real materials, such as pure Bi, graphene or TIs, the Dirac cones are never perfectly linear over a large energy window yet they can be approximated to be so within a narrow energy window and in comparison to the large effective mass of conventional band electrons in many other materials. In order to probe the 3D nature of the observed low-energy Dirac-like bands in Cd3As2, we performed ARPES measurements as a function of incident photon energy to study the out-of-plane dispersion perpendicular to the (001) surface. Upon varying the photon energy, one can effectively probe the electronic structure at different out-of-plane momentum kz values in a 3D BZ and compare with band calculations. In Cd3As2, the electronic structure or band dispersions in the vicinity of its 3D Dirac-like node can be approximated as: , where k0 is the out-of-plane momentum value of the 3D Dirac point. Thus, at a fixed kz value (which is determined by the incident photon energy value), the in-plane electronic dispersion takes the form: . It can be seen that only at kz=k0 the in-plane dispersion is a gapless Dirac cone, whereas in the case for kz≠k0 the nonzero kz−k0 term acts as an effective mass term and opens up a gap in the in-plane dispersion relation. Figure 3a shows the ARPES measured in-plane electronic dispersion at various photon energies. At a photon energy of 102 eV, a gapless Dirac-like cone is observed, which shows that photon energy hν=102 eV corresponds to a kz value that is close to the out-of-plane momentum value of the 3D Dirac node k0. As photon energy is changed away from 102 eV in either direction, the bulk conduction and valence bands are observed within experimental resolution to be separated along the energy axis and a gap opens in the in-plane dispersion. At photon energies sufficiently away from 102 eV, such as 90 eV or 114 eV in Fig. 3a, the in-plane gap is large enough so that the bottom of the upper Dirac cone (bulk conduction band) is moved above the Fermi level, and therefore only the lower Dirac cone is observed. We now fix the in-plane momenta at 0 and plot the ARPES data at kx=ky=0 as a function of incidence photon energy. As shown in Fig. 3b, a E−kz dispersion is observed in the out-of-plane momentum space cut direction, which is in qualitative agreement with the theoretical calculations (Fig. 3c). The Fermi velocity in the z-direction can be estimated (only at the order of magnitude level) to be about 105 ms−1. We note that the sample we used for kz dispersion measurements (Fig. 3a–c) is relatively p-type (Fermi velocity is about 80 meV from the Dirac point) as compared with the sample we used to measure the in-plane dispersion and Fermi surfaces (Figs 1 and 2) where chemical potential is about 200 meV from the Dirac point. It is important to note that the magnitude of Fermi velocity anisotropy strongly depends on the position of the sample chemical potential (n-type sample leads to weaker anisotropy), and therefore the direct comparison between our results and previous transport data in terms of this anisotropy is not applicable. These systematic incident photon energy-dependent measurements show that the observed Dirac-like band disperses along both the in-plane and the out-of-plane directions, suggesting its 3D or bulk nature consistent with theory.
In order to further understand the nature of the observed Dirac band, we studied the spin polarization or spin texture properties of Cd3As2. As shown in Fig. 3f, spin-resolved ARPES measurements are performed on a relatively p-type sample. Two spin-resolved energy-dispersive curve cuts are shown at momenta of ±0.1 Å−1 on the opposite sides of the Fermi surface. The obtained spin data shown in Fig. 3g,h show no observable net spin polarization or texture behaviour within our experimental resolution, which is in remarkable contrast with the clear spin texture in 2D Dirac fermions on the surfaces of TIs. The absence of spin texture in our observed Dirac fermion in Cd3As2 bands is consistent with their bulk origin, which agrees with the theoretical prediction. It also provides a strong evidence that our ARPES signal is mainly due to the bulk Dirac bands on the surface of Cd3As2, whereas the predicted surface (resonance) states14 that lie along the boundary of the bulk Dirac cone projection has a small spectral weight (intensity) contribution to the photoemission signal. In other words, according to our experimental data, the surface electronic structure of Cd3As2 is dominated by the spin-degenerate bulk bands, which is very different from that of the 3D TIs.
The distinct semimetal nature of Cd3As2 is better understood from ARPES data if we compare our results with that of the prototype TI, Bi2Se3. In Bi2Se3 as shown in Fig. 4b, the bulk conduction and valence bands are fully separated (gapped), and a linearly dispersive topological surface state is observed that connect across the bulk band-gap. In the case of Cd3As2 (Fig. 4a), there does not exist a full bulk energy gap. On the other hand, the bulk conduction and valence bands ‘touch’ (and only ‘touch’) at one specific location in the momentum space, which is the 3D band-touching node, thus realizing a 3D BDS. For comparison, we further show that a similar BDS state is also realized by tuning the chemical composition δ (effectively the spin-orbit coupling strength) to the critical point of a topological phase transition between a normal insulator and a TI. Figure 4c,d presents the surface electronic structure of two other BDS phases in the BiTl(S1−δSeδ)2 and (Bi1−δInδ)2Se3 systems. In both systems, it has been shown that tuning the chemical composition δ can drive the system from a normal insulator state to a TI state20,21,29. The critical compositions for the two topological phase transitions are approximately near δ=0.5 and δ=0.04, respectively. Figure 4c,d shows the ARPES measured surface electronic structure of the critical compositions for both BiTl(S1−δSeδ)2 and (Bi1−δInδ)2Se3 systems, which are expected to exhibit the BDS phase. Indeed, the bulk critical compositions where bulk and surface Dirac bands collapse also show Dirac cones with intensities filled inside the cones, which is qualitatively similar to the case in Cd3As2. Currently, the origin of the filling behaviour is not fully understood irrespective of the bulk (out-of-plane dispersive behaviour) nature of the overall band dispersion interpreted in connection to band calculations (see Fig. 2). Based on the ARPES data in Fig. 4c,d, the Fermi velocity is estimated to be ~4 eV Å and ~2 eV Å for the 3D Dirac fermions in BiTl(S1−δSeδ)2 and (Bi1−δInδ)2Se3, respectively, which is much lower than that of what we observe in Cd3As2, thus likely limiting the carrier mobility. The mobility is also limited by the disorder due to the strong chemical alloying. More importantly, the fine control of doping/alloying δ value and keeping the composition exactly at the bulk critical composition is difficult to achieve20, especially while considering the chemical inhomogeneity introduced by the dopants. For example, although similarly high electron mobility on the order of 105 cm2 V−1s−1 has been reported in the bulk states of Pb1−xSnxSe (x=0.23) (ref. 30), the bulk Dirac fermions are in fact massive because of the difficulty of controlling the composition exactly at the critical point. These facts taken together exclude the possibility of realizing proposed topological physics including the Weyl semimetal and quantum spin Hall phases using the bulk Dirac states in the Pb1−xSnxSe. These issues do not arise in the stoichiometric Cd3As2 system as its BDS phase is protected by the crystal symmetry, which does not require chemical doping and therefore the natural high electron mobility is retained (not diminished). We note that our crystals of Cd3As2 are nearly stoichiometric within the resolution of electron probe micro-analyser (EPMA) and X-ray diffraction analysis. The existence of some low-level defects is not ruled out. However, these defects do not affect the main conclusion regarding the 3D Dirac band structure ground state of this compound. Beside Cd3As2 and the topological phase transition critical composition samples as discussed above, we also note that BDSs unrelated to the combination of C4 symmetry and band-inverted spin-orbit coupling (combination of which has been termed ‘topological’ in theory14) have been studied previously in pnictide BaFe2As2 (ref. 31), heavy fermion LaRhIn5 (ref. 32) and organic compound α-(BEDT-TTF)2I3 (ref. 33). The recent interest is actually focused on spin-orbit-based 3D BDS phase as the spin-orbit coupling can drive exotic topological phenomena and quantum transport in such materials as the Weyl phases, high-temperature linear quantum magnetoresistance and topological magnetic phases11,12,13,14,16,17,18,19. Our observation of the bulk Dirac states in Cd3As2 provides a unique combination of physical properties, including high spin-orbit coupling strength, high electron mobility, massless nature guaranteed by the crystal symmetry protection without compositional tuning, making it an ideal and unique platform to realize many of the proposed exciting new topological physics11,12,13,14,16,17,18,19.
In conclusion, we have experimentally identified the crystalline-symmetry-protected 3D spin-orbit BDS phase in a stoichiometric system Cd3As2 (see Fig. 5). The combination of a large Fermi velocity and very high electron mobility of the 3D carriers with nearly linear dispersion at the crossing point makes it a promising platform to explore novel 3D relativistic physics in various types of quantum Hall phenomena. Our band structure study of the predicted 3D BDS phase also paves the way for designing and realizing a number of related exotic topological phenomena in future experiments. For example, if the C4 crystalline symmetry is broken, the 3D Dirac cone in Cd3As2 can open up a gap and therefore a TI phase is realized in a high-mobility setting (current Bi-based TIs feature low carrier mobility). Furthermore, upon doping magnetic elements or fabricating superlattice hetero-structures, the 3D Dirac node in Cd3As2 can be split into two topologically protected Weyl nodes, realizing the much sought out Fermi arcs phases in solid-state setting.
During the preparation of this manuscript, we became aware of another study34 reporting ARPES studies of experimental realization of 3D Dirac semimetal phase in Cd3As2; however, many of the experimental details and interpretations of the data differ from ours. Two other studies35,36 also reported experimental realization of the 3D Dirac phase in a metastable low-mobility compound, Na3Bi.
Sample growth and characterization
Single crystalline samples of Cd3As2 were grown using the standard method, which is described elsewhere24. The Cd3As2 samples used for our ARPES studies show carrier density of 5.2 × 1018 cm−3 and mobility up to 42,850 cm2 V−1s−1 at temperature of 130 K, which is consistent with the mobility of 104–105 cm2 V−1s−1 reported elsewhere22,23. A slight variation of the value of carrier density and mobility is observed for different growth batch samples. We note that our samples show different chemical potential position (measured by ARPES) and different carrier density (measured by transport) depending on the detailed growth conditions. Moreover, our crystals of Cd3As2 are nearly stoichiometric within the resolution of electron probe micro-analyser and X-ray diffraction analysis. The existence of some low-level defects is not ruled out.
ARPES measurements for the low-energy electronic structure were performed at the PGM beamline in Synchrotron Radiation Center (SRC) in Wisconsin, and at the beamlines 4.0.3, 10.0.1 and 12.0.1 at the Advanced Light Source in Berkeley, California, equipped with high-efficiency VG-Scienta R4000 or R8000 electron analysers. Spin-resolved ARPES measurements were performed at the ESPRESSO endstation at HiSOR. Photoelectrons are excited by an unpolarized He-Iα light (21.21 eV). The spin polarization is detected by state-of-the-art very-low-energy electron diffraction spin detectors using preoxidized Fe(001)-p(1 × 1)-O targets37. The two spin detectors are placed at an angle of 90° and are directly attached to a VG-Scienta R4000 hemispheric analyser, enabling simultaneous spin-resolved ARPES measurements for all three spin components as well as high-resolution spin-integrated ARPES experiments. The energy and momentum resolution were better than 40 meV and 1% of the surface BZ for spin-integrated ARPES measurements at the SRC and the Advanced Light Source, and 80 meV and 3% of the surface BZ for spin-resolved ARPES measurements at ESPRESSO endstation at HiSOR. Samples were cleaved in situ and measured at 10–80 K in a vacuum better than 1 × 10−10 torr. They were found to be very stable and without degradation for the typical measurement period of 20 h.
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The work at Princeton University and Princeton-led synchrotron X-ray-based measurements and the related theory at Northeastern University are supported by the Office of Basic Energy Sciences, US Department of Energy (grants DE-FG-02-05ER46200, AC03-76SF00098 and DE-FG02-07ER46352). We thank J. Denlinger, S.-K. Mo and A. Fedorov for beamline assistance at the DOE supported Advanced Light Source (ALS-LBNL) in Berkeley. We also thank M. Bissen and M. Severson for beamline assistance at SRC, WI. M.Z.H. acknowledges Visiting Scientist support from LBNL, Princeton University, and the A.P. Sloan Foundation.
The authors declare no competing financial interests.
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Neupane, M., Xu, S., Sankar, R. et al. Observation of a three-dimensional topological Dirac semimetal phase in high-mobility Cd3As2. Nat Commun 5, 3786 (2014). https://doi.org/10.1038/ncomms4786
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