Abstract
Symmetrybroken threedimensional (3D) topological Dirac semimetal systems with strong spinorbit coupling can host many exotic Halllike phenomena and Weyl fermion quantum transport. Here, using highresolution angleresolved photoemission spectroscopy, we performed systematic electronic structure studies on Cd_{3}As_{2}, 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 inplane Fermi velocity up to 1.5 × 10^{6} ms^{−1} is observed in our samples, where the mobility is known up to 40,000 cm^{2} V^{−1}s^{−1}, suggesting that Cd_{3}As_{2} can be a promising candidate as an anisotropichypercone (threedimensional) high spinorbit analogue of 3D graphene. Our discovery of the Diraclike bulk topological semimetal phase in Cd_{3}As_{2} opens the door for exploring higher dimensional spinorbit Dirac physics in a real material.
Introduction
Twodimensional (2D) Dirac electron systems exhibiting many exotic quantum phenomena constitute one of the most active topics in condensed matter physics^{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19}. The notable examples are graphene and the surface states of topological insulators (TIs). Threedimensional (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 spinorbit coupling. It has been theoretically predicted that a topological (spinorbit) 3D spinorbit Dirac semimetal can be viewed as a composite of two sets of Weyl fermions where broken timereversal or space inversion symmetry can lead to a surface Fermiarc semimetal phase or a TI^{14}. In the absence of spinorbit coupling, topological phases cannot be derived from a 3D Dirac semimetal. Thus, the parent BDS phase with strong spinorbit coupling is of great interest. Despite their predicted existence^{11,13,14}, 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 nonmetastable system with high mobility. It has also been noted that the BDS state can be achieved at the critical point of a topological phase transition^{20,21} between a normal insulator and a TI, which requires finetuning of the chemical doping/alloying composition thus by effectively varying the spinorbit 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 spectrum^{10}. On the other hand, the bulk Dirac fermions in the Bi_{1−x}Sb_{x} system coexist with additional Fermi surfaces^{5}. 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 Diraclike 3D topological (spinorbit) semimetal phase in stoichiometric single crystalline system of Cd_{3}As_{2}, which is protected by the C_{4} crystalline (crystal structure) symmetry and spinorbit coupling as predicted in theory^{14}. Using highresolution angleresolved photoemission spectroscopy (ARPES), we show that Cd_{3}As_{2} features a bulk band Diraclike 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 (~10^{5} cm^{2} V^{−1}s^{−1} (refs 22, 23)). We further compare and contrast the observed crystallinesymmetryprotected BDS phase in Cd_{3}As_{2} with those of in the Bibased 3DTI systems such as in BiTl(S_{1−δ}Se_{δ})_{2} and (Bi_{1−δ}In_{δ})_{2}Se_{3} systems. Our experimental identification and bandstructure measurements of the Diraclike bulk semimetal phase and its clear contrast with Bi_{2}Se_{3} and 2D graphene discovered previously, opens the door for exploring higher dimensional spinorbit Dirac physics in a stoichiometric material. These new directions are uniquely enabled by our observation of strongly spinorbit coupled 3D massless Dirac semimetal phase protected by the C_{4} symmetry, which is not possible in the 2D Dirac fermions in graphene and the surfaces of TIs, or weak spinorbit 3D Dirac fermions in other materials.
Results
Crystalline symmetry protected topological Dirac phase
The crystal structure of Cd_{3}As_{2} has a tetragonal unit cell with a=12.67 Å; and c=25.48 Å; for Z=32 with symmetry of space group I4_{1}cd (see Fig. 1a,b). In this structure, arsenic ions are approximately cubic closepacked 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 CdAs_{4} 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 face^{24}. 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 highsymmetry points are also noted. Cd_{3}As_{2} has attracted attention in electrical transport because of its high mobility of 10^{5} cm^{2} V^{−1}s^{−1} reported in previous studies^{22,23}. The carrier density and mobility of our Cd_{3}As_{2} samples (shown in Figs. 1 and 2) are characterized to be of 5.2 × 10^{18} cm^{−3} and 42,850 cm^{2} V^{−1}s^{−1}, respectively, at temperature of 130 K, consistent with previous reports^{22,23}, which provide an evidence for the high quality of our single crystalline sample. In band theoretical calculations, Cd_{3}As_{2} is also of interest as it features an inverted band structure^{25}. More interestingly, a very recent theoretical prediction^{14} which motivated this work, has shown that the spin–orbit interaction in Cd_{3}As_{2} cannot open up a full energy gap between the inverted bulk conduction and valence bands due to the protection of an additional crystallographic symmetry^{12} (in the case of Cd_{3}As_{2} it is the C_{4} rotational symmetry along the k_{z} direction^{14}), which is in contrast to other bandinverted systems such as HgTe^{3}. This theory predicts^{14} that the C_{4} rotational symmetry protects two bulk (3D) Dirac band touching points at two special k points along the Γ−Z momentum space cutdirection, as shown by the red crossings in Fig. 1d. Therefore, Cd_{3}As_{2} serves a candidate for a spacegroup or crystal structure symmetry protected C_{4} BDS phase.
Observation of bulk Dirac cone
In order to experimentally identify such a BDS phase, we systematically study the electronic structure of Cd_{3}As_{2} on the cleaved (001) surface. Figure 1c shows momentumintegrated 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 3eV binding energy window, where the dispersion of several valence bands are identified. Moreover, a lowlying small feature that crosses the Fermi level is observed. In order to resolve it, highresolution 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 potential^{14}. Compared with the muchstudied 2D Dirac systems, the Fermi velocity of the 3D Dirac fermions in Cd_{3}As_{2} is thus about 3 times higher than that of in the topological surface states of Bi_{2}Se_{3} (ref. 6), 1.5 times higher than in graphene^{26} and 30 times higher than that in the topological Kondo insulator phase in SmB_{6} (refs 27, 28). The observed large Fermi velocity of the 3D Dirac band provides clues to understand unusually high mobility of Cd_{3}As_{2} reported in previous transport experiments^{22,23}. Therefore, one can expect to observe unusual magnetoelectrical and quantum Hall transport properties under highmagnetic field. It is wellknown that in graphene the capability to prepare highquality and highmobility 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 Cd_{3}As_{2} are among the important experimental criteria to explore the 3D relativistic physics in various Hall phenomena in tailored Cd_{3}As_{2}.
We compare ARPES observations with our theoretical calculations, which is qualitatively consistent with previous calculations^{14}. 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 cutdirection, 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 Cd_{3}As_{2}. 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 isoenergetic contours are likely due to matrix element effects associated with the standard ppolarization 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 lowenergy Diraclike bands in Cd_{3}As_{2}, we performed ARPES measurements as a function of incident photon energy to study the outofplane dispersion perpendicular to the (001) surface. Upon varying the photon energy, one can effectively probe the electronic structure at different outofplane momentum k_{z} values in a 3D BZ and compare with band calculations. In Cd_{3}As_{2}, the electronic structure or band dispersions in the vicinity of its 3D Diraclike node can be approximated as: , where k_{0} is the outofplane momentum value of the 3D Dirac point. Thus, at a fixed k_{z} value (which is determined by the incident photon energy value), the inplane electronic dispersion takes the form: . It can be seen that only at k_{z}=k_{0} the inplane dispersion is a gapless Dirac cone, whereas in the case for k_{z}≠k_{0} the nonzero k_{z}−k_{0} term acts as an effective mass term and opens up a gap in the inplane dispersion relation. Figure 3a shows the ARPES measured inplane electronic dispersion at various photon energies. At a photon energy of 102 eV, a gapless Diraclike cone is observed, which shows that photon energy hν=102 eV corresponds to a k_{z} value that is close to the outofplane momentum value of the 3D Dirac node k_{0}. 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 inplane dispersion. At photon energies sufficiently away from 102 eV, such as 90 eV or 114 eV in Fig. 3a, the inplane 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 inplane momenta at 0 and plot the ARPES data at k_{x}=k_{y}=0 as a function of incidence photon energy. As shown in Fig. 3b, a E−k_{z} dispersion is observed in the outofplane momentum space cut direction, which is in qualitative agreement with the theoretical calculations (Fig. 3c). The Fermi velocity in the zdirection can be estimated (only at the order of magnitude level) to be about 10^{5} ms^{−1}. We note that the sample we used for k_{z} dispersion measurements (Fig. 3a–c) is relatively ptype (Fermi velocity is about 80 meV from the Dirac point) as compared with the sample we used to measure the inplane 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 (ntype 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 energydependent measurements show that the observed Diraclike band disperses along both the inplane and the outofplane 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 Cd_{3}As_{2}. As shown in Fig. 3f, spinresolved ARPES measurements are performed on a relatively ptype sample. Two spinresolved energydispersive 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 Cd_{3}As_{2} 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 Cd_{3}As_{2}, whereas the predicted surface (resonance) states^{14} 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 Cd_{3}As_{2} is dominated by the spindegenerate bulk bands, which is very different from that of the 3D TIs.
Discussion
The distinct semimetal nature of Cd_{3}As_{2} is better understood from ARPES data if we compare our results with that of the prototype TI, Bi_{2}Se_{3}. In Bi_{2}Se_{3} 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 bandgap. In the case of Cd_{3}As_{2} (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 bandtouching 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 spinorbit 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(S_{1−δ}Se_{δ})_{2} and (Bi_{1−δ}In_{δ})_{2}Se_{3} 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 state^{20,21,29}. 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(S_{1−δ}Se_{δ})_{2} and (Bi_{1−δ}In_{δ})_{2}Se_{3} 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 Cd_{3}As_{2}. Currently, the origin of the filling behaviour is not fully understood irrespective of the bulk (outofplane 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(S_{1−δ}Se_{δ})_{2} and (Bi_{1−δ}In_{δ})_{2}Se_{3}, respectively, which is much lower than that of what we observe in Cd_{3}As_{2}, 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 achieve^{20}, especially while considering the chemical inhomogeneity introduced by the dopants. For example, although similarly high electron mobility on the order of 10^{5} cm^{2} V^{−1}s^{−1} has been reported in the bulk states of Pb_{1−x}Sn_{x}Se (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 Pb_{1−x}Sn_{x}Se. These issues do not arise in the stoichiometric Cd_{3}As_{2} 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 Cd_{3}As_{2} are nearly stoichiometric within the resolution of electron probe microanalyser (EPMA) and Xray diffraction analysis. The existence of some lowlevel 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 Cd_{3}As_{2} and the topological phase transition critical composition samples as discussed above, we also note that BDSs unrelated to the combination of C_{4} symmetry and bandinverted spinorbit coupling (combination of which has been termed ‘topological’ in theory^{14}) have been studied previously in pnictide BaFe_{2}As_{2} (ref. 31), heavy fermion LaRhIn_{5} (ref. 32) and organic compound α(BEDTTTF)_{2}I_{3} (ref. 33). The recent interest is actually focused on spinorbitbased 3D BDS phase as the spinorbit coupling can drive exotic topological phenomena and quantum transport in such materials as the Weyl phases, hightemperature linear quantum magnetoresistance and topological magnetic phases^{11,12,13,14,16,17,18,19}. Our observation of the bulk Dirac states in Cd_{3}As_{2} provides a unique combination of physical properties, including high spinorbit 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 physics^{11,12,13,14,16,17,18,19}.
In conclusion, we have experimentally identified the crystallinesymmetryprotected 3D spinorbit BDS phase in a stoichiometric system Cd_{3}As_{2} (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 C_{4} crystalline symmetry is broken, the 3D Dirac cone in Cd_{3}As_{2} can open up a gap and therefore a TI phase is realized in a highmobility setting (current Bibased TIs feature low carrier mobility). Furthermore, upon doping magnetic elements or fabricating superlattice heterostructures, the 3D Dirac node in Cd_{3}As_{2} can be split into two topologically protected Weyl nodes, realizing the much sought out Fermi arcs phases in solidstate setting.
During the preparation of this manuscript, we became aware of another study^{34} reporting ARPES studies of experimental realization of 3D Dirac semimetal phase in Cd_{3}As_{2}; however, many of the experimental details and interpretations of the data differ from ours. Two other studies^{35,36} also reported experimental realization of the 3D Dirac phase in a metastable lowmobility compound, Na_{3}Bi.
Methods
Sample growth and characterization
Single crystalline samples of Cd_{3}As_{2} were grown using the standard method, which is described elsewhere^{24}. The Cd_{3}As_{2} samples used for our ARPES studies show carrier density of 5.2 × 10^{18} cm^{−3} and mobility up to 42,850 cm^{2} V^{−1}s^{−1} at temperature of 130 K, which is consistent with the mobility of 10^{4}–10^{5} cm^{2} V^{−1}s^{−1} reported elsewhere^{22,23}. 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 Cd_{3}As_{2} are nearly stoichiometric within the resolution of electron probe microanalyser and Xray diffraction analysis. The existence of some lowlevel defects is not ruled out.
Spectroscopic measurements
ARPES measurements for the lowenergy 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 highefficiency VGScienta R4000 or R8000 electron analysers. Spinresolved ARPES measurements were performed at the ESPRESSO endstation at HiSOR. Photoelectrons are excited by an unpolarized HeIα light (21.21 eV). The spin polarization is detected by stateoftheart verylowenergy electron diffraction spin detectors using preoxidized Fe(001)p(1 × 1)O targets^{37}. The two spin detectors are placed at an angle of 90° and are directly attached to a VGScienta R4000 hemispheric analyser, enabling simultaneous spinresolved ARPES measurements for all three spin components as well as highresolution spinintegrated ARPES experiments. The energy and momentum resolution were better than 40 meV and 1% of the surface BZ for spinintegrated ARPES measurements at the SRC and the Advanced Light Source, and 80 meV and 3% of the surface BZ for spinresolved 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.
Theoretical calculations
The firstprinciples calculations are based on the generalized gradient approximation^{38} using the projector augmented wave method^{39,40} as implemented in the VASP package^{41,42}. The experimental crystal structure was used^{24}. The electronic structure calculations were performed over 4 × 4 × 2 MonkhorstPack kmesh with the spinorbit coupling included selfconsistently.
Additional information
How to cite this article: Neupane, M. et al. Observation of a threedimensional topological Dirac semimetal phase in highmobility Cd_{3}As_{2}. Nat. Commun. 5:3786 doi: 10.1038/ncomms4786 (2014).
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Acknowledgements
The work at Princeton University and Princetonled synchrotron Xraybased measurements and the related theory at Northeastern University are supported by the Office of Basic Energy Sciences, US Department of Energy (grants DEFG0205ER46200, AC0376SF00098 and DEFG0207ER46352). We thank J. Denlinger, S.K. Mo and A. Fedorov for beamline assistance at the DOE supported Advanced Light Source (ALSLBNL) 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.
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M.N. and S.Y.X. performed the experiments with assistance from N.A., G.B., C.L., I.B. and M.Z.H.; M.N. and M.Z.H. performed data analysis, figure planning and draft preparation; R.S. and F.C. provided the singlecrystal samples and performed sample characterization; T.R.C., H.T.J., H.L. and A.B. carried out calculations; M.Z.H. was responsible for the conception and the overall direction, planning and integration among different research units.
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Correspondence to M. Zahid Hasan.
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Neupane, M., Xu, S., Sankar, R. et al. Observation of a threedimensional topological Dirac semimetal phase in highmobility Cd_{3}As_{2}. Nat Commun 5, 3786 (2014). https://doi.org/10.1038/ncomms4786
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Journal of Advanced Research (2020)
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