1. IntroductionThe Dirac semimetals have isolated bulk Fermi nodes in the momentum space of the whole Brillouin zone, which are generated by a linear touching of the conduction band and the valence band.[1–32] In the vicinity of these Fermi nodes, the associated bands disperse linearly along all the three momentum directions and can be effectively described by massless Dirac equation. Thus, the isolated Fermi nodes are 3D Dirac nodes and Dirac semimetals can ideally mimic graphene[33,34] in three dimensions. Dirac semimetals are new topological states that are different from topological insulators[35,36] but closely related to Weyl semimetals.[1,9–14,37–45] The low-energy excitations of a Weyl semimetal is analogous to a Weyl fermion which can be described by Weyl equation,[37] which is a half of the massless Dirac equation. Therefore, a Dirac node in a 3D Dirac semimetal can be viewed as a pair of Weyl nodes with opposite chirality, which usually tend to annihilate each other when they overlap in momentum space. In crystal systems, however, such a Dirac node can survive under the protection of a certain crystallographic symmetry.[15] Recently, A3Bi (A = Na, K, Rb) are predicted to be Dirac semimetals, which have a pair of isolated 3D bulk Dirac nodes along the A–Γ–A direction at the Fermi level.[16] The robustness of the Dirac nodes are guaranteed by the crystal symmetry of A3Bi (A = Na, K, Rb). Moreover, topological non-trivial surface states are expected on the specific (100) crystalline surface of A3Bi (A = Na, K, Rb), showing up as fascinating Fermi arcs nesting the two bulk Dirac nodes at the Fermi level. A3Bi (A = Na, K, Rb) can be driven into various topologically distinct state such as Weyl semimetals, topological insulators, topological superconductors, and axion insulators.[16] Experimental evidence on the nature of 3D Dirac semimetal has been provided from angle-resolved photoemission (ARPES) measurements on either (001) surface[17–20] or the (100) surface of Na3Bi.[21,22] A clear chiral anomaly is also observed in Na3Bi.[23] Later on, Cd3As2 has been established as another Dirac semimetal both theoretically and experimentally.[26–32]
Here we report comprehensive high-resolution angle-resolved photoemission measurements on the two cleaved surfaces, (001) and (100), of Na3Bi. On the (001) surface, by comparison with theoretical calculations, we provide a proper assignment of the observed bands, and in particular, pinpoint the band that is responsible for the formation of the three-dimensional Dirac cones. By varying incident photon energies thus varying the out-of-plane kz values, a pair of separated 3D Dirac nodes are revealed, which locate symmetrically along A–Γ–A direction with respect to Γ. In addition to the single Dirac node at the zone center, we observed six faint point-like Fermi surface at zone corners that may be related with surface reconstruction. On the (100) surface, our ARPES measurements over a large momentum space raise an issue on the selection of the basic Brillouin zone in the (100) plane. We directly observe two isolated 3D Dirac nodes on the (100) surface. We observe the signature of the Fermi-arc surface states connecting the two 3D Dirac nodes that extend to a binding energy of ∼150 meV before merging into the bulk band. Our results constitute strong evidence on the observation of three-dimensional Dirac semimetal state in Na3Bi that are basically consistent with previous theoretical[16] and experimental work.[17,21,22] In addition, our results provide new information to clarify on the nature of the band that forms the 3D Dirac cones, on the possible formation of surface reconstruction of the (001) surface, and on the issue of basic Brillouin zone selection for the (100) surface. The electronic information is important in further study of the topological nature of the three-dimensional Dirac semimetal Na3Bi.
2. Experimental methodsAngle-resolved photoemission measurements were carried out on our lab ARPES system equipped with the Scienta R4000 electron energy analyzer[46] and by ARPES systems using synchrotron light source. For our Lab ARPES measurements, light sources including vacuum ultraviolet (VUV) laser (hυ = 6.994 eV)[46] and He Iα (hυ = 21.2 eV, He Iα hereafter) were used, with the corresponding energy resolution set at 1.5 meV and 20 meV, respectively. The angular resolution is ∼0.3 degree. Synchrotron radiation ARPES measurements were performed at beamlines BL-1 and BL-7 in Hiroshima Synchrotron Radiation Center (HiSOR), Hiroshima University, Japan. The incident photon energy at BL-1 was varied between 48 and 64 eV and the energy resolution was set at ∼15 meV. The incident photon energy at BL-7 was varied from 48 to 116 eV and the energy resolution was set at ∼50 meV. The Fermi level is referenced by measuring on a clean polycrystalline gold that is electrically connected to the sample. High-quality single crystals of Na3Bi were grown by the self-flux method. The atomic ratio of Na to Bi of the grown crystals were measured by Inductively Coupled Plasma Atomic Emission Spectroscopy (ICP-AES) which corresponds to 1 to 0.33 that is consistent with the stoichiometry of Na3Bi. Since Na3Bi is extremely sensitive to air, all the sample preparations are carried out in a glove box filled with Argon. Specifically, in order to obtain two cleaving surfaces, i.e., (001) and (100) surfaces, the Na3Bi samples were cut and pasted along two orthogonal directions. For ARPES measurements, all the samples were cleaved in situ in vacuum with a base pressure better than 8.0 × 10−11 mbar and at a temperature of ∼25 K.
The electronic structure calculations were performed by using the full-potential augmented plane-wave and Perdew–Burke–Ernzerhof parametrization of the generalized gradient approximation (GGA-PBE) exchange-correlation function[47] as implemented in the WIEN2k code.[48] The spin-orbital interaction is included by using a second variational procedure. The muffin-tin radii RMT is set to be 2.50 bohrs for both Na and Bi. The plane-wave cutoff (Kmax) is determined by RminKmax = 7.0, where Rmin is the minimal RMT. The local density of states (LDOS), from which we can obtain the dispersion of the surface states, is conveniently obtained through the maximally localized Wannier functions derived from the ab initio calculations.
3. Results and discussion3.1. Preparation of two cleaved surfaces (001) and (100) of Na3Bi and their characterizationsFigure 1 shows basic information and characterization of Na3Bi. Figure 1(a) shows the crystal structure of Na3Bi with a space group of P63/mmc and the lattice constants a = b = 5.448 Å and c = 9.655 Å. Figure 1(b) shows the corresponding bulk Brillouin zone, projected (001) surface Brillouin zone (hexagon above the bulk Brillouin zone), and (100) surface Brillouin zone (rectangle in front of the bulk Brillouin zone) of Na3Bi.
We performed x-ray diffraction (XRD) analysis on the crystal structure of Na3Bi on both the powder and crystal samples. Since Na3Bi is very reactive to air, the Na3Bi powder was put inside a small ultra-thin quartz capillary (mark-tubes made of special glass No. 10, Hilgenberg GmbH, see Fig. 1(e)) for x-ray diffraction analysis. Figure 1(c) shows the x-ray diffraction patterns on an empty quartz capillary (blue curve) and a quartz capillary with Na3Bi powder sealed inside (red curve). Their difference represents the XRD diffraction signal from the Na3Bi sample that is shown in Fig. 1(d). The observed peaks can be well identified as the diffraction peaks from Na3Bi. Figure 1(f) shows XRD diffraction pattern for Na3Bi single crystal. In this case, the sample was covered with a piece of Mylar thin film (right inset of Fig. 1(f)) to protect it from air. Two groups of peaks can be observed, with the main contribution coming from the (001) surface. The weak peaks of (100) surface may be due to a small deviation of sample orientation or a presence of a small piece of crystal with (100) orientation. We also measured photoemission spectra of Na3Bi over a wide energy range (Fig. 1(g)). The integrated energy distribution curve (EDC, red curve in Fig. 1(g))) shows the location of the Na 2p core level and the doublet of Bi 5d core level.
Since the confirmation of three-dimensional Dirac cones needs to have access to all three-dimensional Brillouin zone, it is ideal to have ARPES measurements on multiple cleaved surfaces. In Fig. 2, we show our approach to obtain two cleaved surfaces (001) and (100) of Na3Bi in situ in vacuum. We first chose a piece of high-quality Na3Bi single crystal and cut it along two orthogonal directions as sketched in Fig. 2(a). As marked by white arrows in Figs. 2(b) and 2(c), the obtained cleaved (001) surface (Fig. 2(b)) and (100) surface (Fig. 2(c)) are flat and shiny. Figures 2(d) and 2(e) illustrate surface atomic structures of the (001) and (100) surfaces, respectively. We carried out Fermi surface mapping measurements over a large momentum space on these two kinds of Na3Bi samples; the observed three-fold symmetry in Fig. 2(f) and the mirror symmetry in Fig. 2(g) are consistent with the cleaved surfaces being (001) and (100) surfaces, respectively.
3.2. ARPES on the (001) surface of Na3BiThe access of three-dimensional Brillouin zone in studying the Dirac nodes in Na3Bi asks for a precise determination of momentum for each ARPES measurement. While it is straightforward to determine the in-plane momentum because it is conserved during the photoemission process, the determination of the out-of-plane component is tricky. Since the photo-emitted electrons have to overcome a potential barrier (inner potential V0) before they escape from the sample surface, the determination of the out-of-plane momentum requires the value of V0 beforehand. The kz can then be determined from the incident photon energy (hν) and the emission angle θ by taking the formula[49]
![](cpb_25_7_077101/cpb161036eqn1.gif)
where
W is the work function,
Eb denotes the binding energy,
θ is the angle between the emission direction and the sample normal direction. The
kzs for a given photon energy form a curved surface in momentum space which is related to different emission angles
θ of the photoelectrons. For simplicity and to avoid ambiguity, the
kz we mention below refers to the value for the normal direction photoemission.
The inner potential V0 of the Na3Bi (001) surface can be determined from ARPES measurements on the (001) surface by varying the phonon energy (see Fig. 3). Figure 3(a) shows photoemission spectra (EDCs) taken along the sample normal direction at various photon energies between 40 eV and 116 eV. Figure 3(b) presents a two-dimensional image that shows the photoemission intensity as a function of the binding energy and the incident photon energy for the normal emission. From Figs. 3(a) and 3(b), focusing on the bands at Eb ∼ 2.0 eV and ∼1.0 eV, it is obvious that the repetition appears around 56 eV and 101 eV which correspond to two Γ points along the Γ-A direction. By using Eq. (1), taking c′ = 0.5c = 0.5 × 9.655 Å = 4.8275 Å as we will explain below, the inner potential V0 for the (001) surface of Na3Bi is estimated to be around 9.0 eV (sensitively depends on the determination of the repeated photon energies).
Figure 4 presents the Fermi surface and band structures taken on the (001) surface of Na3Bi measured by lab ARPES system with a photon energy of 21.2 eV. In Fig. 4(a), we sketch the two three-dimensional Dirac nodes located near the Γ point along the Γ–A line in the bulk Brillouin zone, as expected for Na3Bi from theoretical calculations.[16] The projection of the 3D bulk Dirac nodes to the surface Brillouin zones at the (001) and (100) surface are also shown. The red dashed line connecting the two Dirac cones for the (100) surface represents topological nontrivial surface state with arc structure.[16] The atomic structures of both (001) and (100) surfaces are also drawn in Fig. 4(a). The 3D Dirac nodes are predicted to be located at ± kzc (kzc = 0.085 Å−1) along A–Γ–A direction in Na3Bi (Fig. 4(a)). The kz value estimated for the photon energy of 21.2 eV, taking V0 ∼ 9.0 eV as determined before, is around 0.02 Å−1. This is quite close to the value of kzc, especially when considering the finite momentum resolution along the kz direction. Therefore, He Iα with 21.2 eV photon energy is suitable to probe the 3D Dirac nodes on the (001) surface of Na3Bi.
Figure 4(b) presents the Fermi surface of Na3Bi over a large momentum space that covers both the first and second hexagonal surface Brillouin zones of the (001) surface. The repetition of the strong signal near the three Γ points is consistent with the (001) surface. Figure 4(c) shows the band structure measured along the
direction, where six bands can be well resolved and labeled in sequence from the Fermi level to high binding energy by α, β, γ, δ, ε, and ζ. For comparison, the calculated result with the same energy scale and momentum direction with kz = 0.02 Å−1 is also presented in Fig. 4(d). Within experimental uncertainness, the measured band structures in Fig. 4(c) show a perfect match with the calculated result in Fig. 4(d). From Figs. 4(c) and 4(d), also from more results below, it is clear that the spot-like Fermi surface feature at the zone center in Fig. 4(b) originates from the bulk band α, which is linearly dispersive near the Fermi level and forms the three-dimensional Dirac cones.
In order to verify the Dirac cone structure formed by the α band, we take fine Fermi surface mapping of the (001) surface in the first surface Brilouin zone as shown in Fig. 4(e). The spot-like Fermi surface at the zone center is the Dirac point since the Fermi level of the freshly cleaved sample is just pinned at the Dirac point. Figure 4(f) shows the detailed band structure evolution around the
point. In the energy range between the Fermi level and a binding energy of 0.8 eV, only α and β bands are observed as shown in Fig. 4(f). When the momentum cut is right across the
point (cut 4 in Fig. 4(e)), the α band in Fig. 4(f4) just crosses the point and shows a linear dispersion over an energy range of ∼0.5 eV. When the momentum cuts move away from the
point, the top position of the α band sinks below the Fermi level and can be well described by the hyperbolic cone-shaped curves as shown by the dotted lines. Such a momentum evolution of the band structures confirms the Dirac cone-like structure of the α band. For a 3D Dirac cone with the Dirac nodes at Fermi level, it usually has the form
. Here, ax,y,z are associated with the cone shape (slope of the cut across the generatrix of the projected 2D gapless cone) and bx,y,z are the momentum loci of the 3D Dirac nodes in the Brillouin zone. The cone shape of the α band on the (001) surface of Na3Bi can be well reproduced by fitting parameters ax = ay = ∼1.5 eV·Å, bx = by = 0 Å−1, regardless of the very small contribution of the kz direction.
Our observation of a Dirac cone on the (001) surface of Na3Bi and the corresponding momentum dependence show some overall resemblance to the previous results.[17] However, we note that there appears to be a difference on the exact band that forms the Dirac cones. In our measurements, the α band is strong and is the origin of the Dirac cone. This is consistent with the theoretical calculations which indicate that it is the topmost band α below the Fermi level that forms the 3D Dirac cone.[16] However, we note that the intensity of α band in the previous measurements[17] is rather weak probably due to photoemission matrix element effect under different measurement conditions. The difference could be quantitatively determined and shown by the Dirac cone shape fitting parameters which have values of ∼1.5 eV·Å in our case, in contrast to those of ∼2.5 eV·Å in the previous measurement.[17] In this case, which band forms the purported Dirac cones needs to be further checked.[17]
In order to explore the two 3D Dirac nodes separated along the A–Γ–A direction as sketched in Figs. 4(a) and 5(a), we measured the band structures of the Na3Bi (001) surface at different out-of-plane kzs with different photon energies by using synchrotron radiation ARPES, as shown in Fig. 5(c), with the photon energies used and the corresponding kzs marked. In Fig. 5(c), the dashed red lines are fitted to the α band according to the Dirac cone shape. The linearly dispersive band touches the Fermi level at the photon energy of 53.3 eV and moves away from the Fermi level with further increase of the photon energy. It touches the Fermi level again at the photon energy of 58.4 eV and sinks below the Fermi level with the further increase of the photon energy.
The photon-energy-dependent band dispersions in Fig. 5(c) clearly indicate the 3D nature of the Dirac points in Na3Bi. With the estimated inner potential of ∼9 eV for the (001) surface of Na3Bi, we can calculate the value of the kz for each incident photon energy with Eq. (1).[49] Meanwhile, we pick up the binding energy of the top of the α band in Fig. 5(c) and plot them as a function of the related kz in Fig. 5(b), where the red circles represent the experimental data and the light gray dashed line is the guide to the eye. The kz value associated with the photon energy of 56 eV is very close to one of the unique momentum kzc located at + 0.085 Å−1 along A–Γ–A direction. Although there is a slight deviation of the experimental data from the theoretically expected values due to the inherently limited kz resolution of the ARPES measurements, one can see a nearly linear dispersion along the kz direction between kzc and point A. In between the two kzc momentum points, the dispersion deviates from the expected linear dispersion due to the hybridization of the two bulk Dirac cones, which are located very close to each other along the kz direction. The gray dashed line approximately tracks the α band dispersion along A–Γ–A direction which gives the rest 3D Dirac cone parameters in the kz direction by az = ∼ 0.5 eV·Å and bz = ∼ ±0.12 Å−1. Figure 5(d) shows the evolution of the calculated bands with kz. Note that the upper-most α band is responsible for the 3D Dirac cone formation, with a dispersion slope around 1.5 eV·Å at the Dirac point (Fig. 5(d3)), which is flatter than the β band with a slope near 2.7 eV·Å (Fig. 5(d3)). The slopes of the α and β bands provide us another way to determine the band that forms the 3D Dirac cone. The slopes of our observed α and β bands in Figs. 4(f) and 5(c) are consistent with the calculated results (Fig. 5(d3)). The linear dispersive of the α band along the out-of-plane kz direction, together with the results in the kx–ky plane in Fig. 4, constitute overall evidence of the linear dispersion of the Dirac fermions along all the three momentum directions that unambiguously establishes the existence of the 3D bulk Dirac cones in Na3Bi, with a set of parameters given by ax = ay = ∼ 1.5 eV·Å, az = ∼0.5 eV·Å and bx = by = 0 Å−1, bz = ∼±0.12 Å−1.
Upon careful examination, we find that there are Dirac-cone-like structures forming spot-like Fermi pockets around the K̅ points on the (001) surface of Na3Bi (Fig. 6(a)). Figure 6(b) shows the band structure measured along the
line as indicated by the orange line in Fig. 6(a). The intensity of the bands around the K̅ point is much weaker than that around the
point, but their presence is quite clear. Near the Fermi level, the slope of the bands around the K̅ points is similar to that of the β band around the
point, as highlighted as red dashed lines in Fig. 6(b). From the band structure calculations on Na3Bi, there is a band gap about 1 eV near K̅ point and there are no surface states there.[16] The origin of the feature near K̅ points needs to be further studied. One possibility is that the features near K̅ points are due to folding of the bands near
point. This implies that there is a surface reconstruction on the (001) surface. According to the symmetry of the Fermi surface features around the K̅ points, the (001) surface with a
reconstruction (inset of Fig. 6(a)) and the corresponding shrunk Brillouin zone (red thick hexagon around
in Fig. 6(a)) could be consistent with the observations.
3.3. ARPES on (100) surface of Na3BiBesides the 3D bulk Dirac nodes, the topological nontrivial surface-state Fermi arcs connecting the two 3D bulk Dirac nodes is another significant and peculiar feature of the Dirac semimetal Na3Bi.[16] As shown in Fig. 4(a), ARPES measurements taken on the (100) surface will be able to directly and simultaneously reveal a pair of the 3D bulk Dirac nodes as well as the nontrivial surface state Fermi arcs between them, thus capturing the main and most important physics of Dirac semimetal. We managed to in situ cleave the Na3Bi single crystal samples to obtain the (100) surface, as shown in Fig. 2. We also carefully tuned the incident photon energy to vary the out-of-plane ky momentum in order for ARPES to detect the 3D bulk Dirac nodes at the momentum position with ky = 0. One suitable incident photon energy we find is ∼56 eV.
Figure 7 shows the measured Fermi surface and the constant energy contours (CECs) at different binding energies on the (100) surface of Na3Bi taken with a photon energy of 56 eV. The coverage of large momentum area allows us to find the repetition unit in the reciprocal space. From the crystal structure shown in Fig. 1(a) with lattice constants a = b = 5.448 Å and c = 9.655 Å, the corresponding unit cell of the (100) surface Brillouin zone should be a rectangle (Fig. 1(b)) with a′ = 1.15 Å−1 and c′ = 0.65 Å−1, as shown by a blue dashed rectangle in Fig. 7(a). It is clear that such a unit cell cannot explain our measured data (Fig. 7), particularly along the kz direction. In order to explain the periodicity of the observed data, we have to expand the unit cell length along the kz direction twice so that a′ = 1.15 Å−1 and c″ = 2 × 0.65 = 1.3 Å−1, as shown by the solid grey rectangle in Fig. 7(a). This new unit cell can basically explain the periodicity of our observed results. In this case, the intensity in the first and third Brillouin zones are much stronger than that in the second zones. This may be due to the photoemission matrix element effect.
We further examine the issue of the surface Brillouin zone on the (100) surface of Na3Bi by checking on the band structures, as shown in Fig. 8. Figures 8(a) and 8(b) are Fermi surfaces taken with photon energies of 56 eV and 21.2 eV, respectively. Figures 8(c) and 8(d) are band structures taken along the
momentum cut marked by gray lines in Figs. 8(a) and 8(b), respectively. From the spectral weight distribution in the Fermi surface (Fig. 8(a)) and the band structure along the
momentum cut (Fig. 8(c)) measured at 56 eV photon energy, it is clear that there is a periodicity along kz direction that is twice of the original c′ = 0.65 Å−1 defined by real space unit cell in Fig. 1(a), i.e., c′ = 2 × 0.65 = 1.3 Å−1 as shown by the thick grey rectangle in Fig. 8(a). The overall band evolution from e1 to e5 cuts (Fig. 8(e)) and from f1 to f5 cuts (Fig. 8(f)) is consistent with this assignment. Similar band structures measured along e1 cut (Fig. 8(e1)) and g1 cut (Fig. 8(g1)), and along e2 cut (Fig. 8(e2)) and g2 cut (Fig. 8(g2)), also support this picture. This double-kz periodicity is also clear in the Fermi surface (Fig. 8(b)) and the band structure along the
momentum cut (Fig. 8(d)).
However, by examining the band structures measured at 56 eV and the Fermi surface measured at 21.2 eV (Fig. 8(b)), there seems to be another period that agrees with the original unit cell defined by the real space crystal structure in Fig. 1(a). Looking at the measured band structures carefully along e1 cut (Fig. 8(e1)) and e2 cut (Fig. 8(e2)), there are clear bands that are marked by α’ (red dashed lines in Fig. 8(e)) and γ’ (blue dashed lines in Fig. 8(e)) although their intensity is strongly suppressed. Considering the presence of these α’ and γ’ bands, one may see the similarity of band structures between (e1-e5) cuts and (f1-f5) cuts. This is consistent with a smaller weak period along kz direction with a normal value of π/c (dashed blue rectangular). The spectral weight distribution in the Fermi surface mapping measured at 21.2 eV (Fig. 8(b)) is also consistent with the presence of this weak period that has a normal unit cell expected from real space crystal structure in Fig. 1(a).
We note that there is a controversy in the literatures regarding the selection of the basic Brillouin zone in the kx–kz (100) plane.[17,22] From an indirect photon energy dependent measurement on the (001) surface, the size of (100) surface Brillouin zone is used, which is derived directly from the real space crystal structure with a′ = 1.15 Å−1 and c′ = 0.65 Å−1.[17] In another direct measurement on the (100) surface of Na3Bi, a much large Brillouin zone unit cell is used, which expands the length twice along both kx and kz directions, i.e., a′ = 1.15 × 2 = 2.3 Å−1 and c′ = 0.65 × 2 = 1.3 Å−1.[22] In our case of direct ARPES measurements on (100) surface of Na3Bi over a large momentum space, we find there is a strong period and a weak period along kz direction. For the strong period, we need to double the length along kz direction, while keep the length along the kx direction intact, i.e., a′ = 1.15 Å−1 and c′ = 0.65 × 2 = 1.3 Å−1. Such a Brillouin zone doubling along kz direction is not present in the theoretical calculations; it is also different from the previous results.[17,22] The expansion of the Brillouin zone is unusual because in most cases one would see the shrinkage of the Brillouin zone due to real space structural reconstruction that leads to expansion of unit cell size in the real space. The origin of the reciprocal space unit cell expansion in Na3Bi needs further investigation. One possibility may be related with the presence of an inversion center in Na3Bi which makes the two Na3Bi layers in one unit cell equivalent, as seen in Fig. 1(a). On the other hand, we also show the signature of a weak period that is consistent with the normal unit cell expected from the real space crystal structure of Na3Bi (Fig. 1(a)).
Figure 9(a) presents the Fermi surface near the zone center measured with a photon energy of 56 eV on the (100) surface of Na3Bi. Two strong spots located around ±0.15 Å−1 (close to kzc = ±0.085 Å−1) along
(the same as A–Γ–A when ky = 0) direction are observed with a mirror symmetry with respect to the
point, which are very close to the position of the 3D bulk Dirac nodes. This is similar to that reported before on the (100) surface of Na3Bi.[22] These indicate that the observed two spots are associated with the two 3D bulk Dirac nodes on the kx–kz plane with ky = 0, as schematically shown in Fig. 4(a). In order to further demonstrate that these two spot-like features are from 3D bulk Dirac nodes, we examine the band structures carefully by making dense momentum cuts cover the two points, as shown in Fig. 9(c). In the energy range between the Fermi level and a binding energy of 1 eV, two bands of α band and β can be observed. The α bands are fitted with the Dirac cone shape indicated by the dashed red lines. It exhibits a clear linear dispersion when the momentum cut approaches the Dirac nodes (panel 3 in Fig. 9(c) and panel 6 in 9(c)). The slope parameter is found to be ax = ∼1.5 eV·Å. From a symmetrical view, it is natural to derive ax = ay = ∼1.5 eV·Å. The momentum evolution of the band structures along the kz direction confirms the simultaneous observation of the two 3D bulk Dirac nodes that are located along the A–Γ–A direction with a mirror symmetry with respect to the Γ point of the bulk BZ. Figure 9(b) plots the energy position of the top α band from panels 1–8 in Fig. 9(c) (red circles) as a function of kz, with the dashed line as the guide to eyes. The linear dispersion of the Dirac bands along kz and the overall M-shaped curve in Fig. 9(b) are similar to that shown in Fig. 5(b) which have the same quantities measured by two different approaches, i.e., az = ∼ 0.5 eV·Å, bz = ±0.15 Å−1 (small deviation from ±0.12 Å−1). Their consistency again confirms the observation of a pair of 3D bulk Dirac nodes along the A–Γ–A direction, with a set of parameters by ax = ay = ∼1.5 eV·Å, az = ∼0.5; eV·Å, and bx = by = 0 Å−1, bz = ∼±0.15 Å−1 (small deviation from that of (001) surface with bz = ∼±0.12 Å−1).
In order to resolve the topological nontrivial surface states that connect the two 3D bulk Dirac nodes, we examine the band structures along
direction measured on the (100) surface of Na3Bi. The band structure obtained by the VUV laser-based ARPES[46] is presented in Figs. 10(a1) and 10(b1) with their second derivative images with respect to the energy shown in Figs. 10(a2) and 10(b2), respectively. As seen more clearly from Fig. 10(b2), there are three main band features where the weaker one crosses the Fermi level, and a stronger one (labeled as α band) and a weaker one (labeled as β band) are below the Fermi level. A close examination on the band near the Fermi level (Figs. 10(a1) and 10(a2)) indicates two branches of a sharp band dispersing linearly from a binding energy of ∼150 meV and crosses the Fermi level. These observations are rather consistent with the band structure calculations (Fig. 10(g)). In this case, the band that crosses the Fermi level is assigned to be the surface state that is predicted for the (100) surface and forms an arc-shaped Fermi surface, as labeled by SS with gray dash guide lines in Figs. 10(a1), 10(a2), 10(b1) and 10(b2). To further verify the surface state nature of the SS band, we performed photon-energy-dependent ARPES measurements with synchrotron radiation light source, as shown in Figs. 10(c)–10(f). The SS band, highlighted by the gray dashed lines, shows little change with the photon energy in terms of the band slope and the Fermi momentum. On the other hand, the other bulk bands, marked by red arrows and black dashed lines, exhibit clear photon energy dependence. These photon-energy-dependent measurements again demonstrate the surface state nature of the SS band on the (100) surface.
We note that there is a sample aging effect during the ARPES measurements of Na3Bi, on both (100) surface and (001) surface. Figure 11 shows the effect of the sample aging on the measured Fermi surface and band structures of the (100) surface. For the fresh (100) surface of Na3Bi, a Fermi surface with two clear nodes is clearly observed (Fig. 11(a)). The band along
across the
point (Fig. 11(c)) is below the Fermi level. However, after about 2 hours in vacuum, the sample gets aged and measured Fermi surface shows a single hole-like piece (Fig. 11(b)). The corresponding band across the
point crosses the Fermi level (Fig. 11(d)). It turns out that there is a band shift upwards by about 80 meV during the sample aging of about two hours which causes hole doping into the sample. This indicates that the aging control of the sample during the ARPES measurements of Na3Bi is crucial for the revelation of the bulk 3D dirac nodes and the nontrivial arc Fermi surface.
Having identified the 3D bulk Dirac nodes and the surface state, we take further investigations to resolve the unique Fermi surface arcs connecting the two 3D Dirac nodes on the (100) surface of the Na3Bi. Figures 12(a)–12(f) show the constant energy contours at different binding energies measured on the (100) surface of Na3Bi with a photon energy of 56 eV. The red and gray dashed lines are drawn for the bulk Dirac nodes and surface bands, respectively. For comparison, the calculated results are also presented in Figs. 12(g)–12(l). In Figs. 12(a)–12(f), the spectral weight that connects the two bulk Dirac nodes and takes a unique arc shape originates from the surface state, which can be more clearly observed when the binding energy is above 60 meV. The calculated results (Figs. 12(g)–12(l)) also show that the nontrivial surface state can penetrate deep below the Fermi level, forming rugby-like arc contours. The surface state we observe here matches well with the theoretical calculations.[25] The overall evolution of the two bulk Dirac nodes and the arc surface state with the binding energy are well reproduced from the theoretical calculations, except for the issue of the doubling length along kz as mentioned above. Our ARPES results on the (100) surface are therefore consistent with the theoretical predictions of the 3D Dirac nodes and the Fermi arc structure on the Na3Bi surface.