## Nailing down the topology of a semimetal

Topological insulators are exotic materials that have a conducting surface state that can withstand certain types of material imperfection. Theoreticians have predicted a different kind of surface state in related three-dimensional topological Dirac semimetals, which do not have an energy gap in the band structure of the bulk. Xu *et al.* used photoemission spectroscopy to map out the band structure of the material Na_{3}Bi and detected the predicted surface state. Their results may lead to further insights into the physics of topological matter.

*Science*, this issue p. 294

## Abstract

The topology of the electronic structure of a crystal is manifested in its surface states. Recently, a distinct topological state has been proposed in metals or semimetals whose spin-orbit band structure features three-dimensional Dirac quasiparticles. We used angle-resolved photoemission spectroscopy to experimentally observe a pair of spin-polarized Fermi arc surface states on the surface of the Dirac semimetal Na_{3}Bi at its native chemical potential. Our systematic results collectively identify a topological phase in a gapless material. The observed Fermi arc surface states open research frontiers in fundamental physics and possibly in spintronics.

The realization of topological states of matter beyond topological insulators has become an important goal in condensed-matter and materials physics (*1*–*15*). In the topological insulators Bi_{1-x}Sb_{x} and Bi_{2}Se_{3} or topological crystalline insulators such as Pb_{1-x}Sn_{x}Te(Se), the bulk has a full insulating energy gap, whereas the surface possesses an odd or even number of spin-polarized surface or edge states (*3*, *14*–*18*). These are symmetry-protected topological states (*19*). Very recently, the possibility of realizing new topological states in materials beyond insulators, such as metals or semimetals, has attracted much attention (*1*–*13*). Semimetals are materials whose bulk conduction and valence bands have small but finite overlap; the lack of a full bandgap implies that any topological states that might exist in a semimetal should be distinct from the topological states studied in insulating materials. Theory has proposed two kinds of topological semimetals: the topological Dirac and Weyl semimetals (*2*, *7*–*12*). Their low-energy bulk excitations are described by the Dirac and Weyl equations, respectively. For both types, the bulk conduction and valence bands are predicted to touch at multiple discrete points in the bulk Brillouin zone (BZ) and show linear behavior in the vicinity of those points (*2*, *7*–*12*). For topological Dirac semimetal material candidates (*9*, *10*), theoretical calculations (*9*–*11*) show that the surface states are a pair of Fermi arcs (Fermi arc surface states) that connect the bulk bands at the energy of the bulk Dirac nodes, which can be realized at the sample’s native chemical potential. These double Fermi arc surface states are disjoint on the surface, and they connect to each other through the bulk bands. Moreover, they are predicted to show a wide range of exotic properties, such as unusual polarization (*13*), Friedel oscillations in tunneling experiments (*6*), and novel quantum oscillations in transport (*12*). However, they have not yet been observed in real materials.

Experimentally, a number of compounds have been identified to be three-dimensional (3D) Dirac semimetals (or 3D analogs of graphene), such as BiTl(S_{0.5}Se_{0.5})_{2}, (Bi_{0.94}In_{0.06})_{2}Se_{3}, Bi_{1-x}Sb_{x}, Na_{3}Bi, and Cd_{3}As_{2} (*20*–*25*). Only the latter two are theoretically believed to be topologically nontrivial (*2*), but existing experimental data (*20*–*25*) are insufficient to prove their topological nature. We experimentally uncovered the nontrivial topological nature of the semimetal Na_{3}Bi. We achieved this by observing a pair of spin-polarized Fermi arc surface states at the native Fermi level on the surface of our samples using high-resolution angle-resolved photoemission spectroscopy (ARPES). Our observation of Fermi arc surface states lays the foundation for studying fundamentally new physics in nontrivial metals and may also have potential for device applications (*1*, *2*, *4*–*15*).

Na_{3}Bi is a semimetal that crystalizes in the hexagonal P6_{3}/mmc crystal structure with *α* = 5.448 Å and *c* = 9.655 Å (*26*). First-principles bulk band calculations (*9*) show that its lowest bulk conduction (Na ) and valence (Bi 6*p _{x,y,z}*) bands possess a band inversion of about 0.3 eV at the bulk BZ center Γ (

*9*). The strong spin-orbit coupling in the system can open up energy gaps between the inverted bulk bands, but because of the protection of an additional threefold rotational symmetry along the [001] crystalline direction, the bulk bands are predicted to touch in two locations (Dirac nodes), even after spin-orbit coupling is considered (Fig. 1F, blue crosses). At the (001) surface, because the two bulk Dirac nodes project onto the same point in the surface BZ (Fig. 1F), Fermi arc surface states are not possible at the (001) surface. On the other hand, at the (100) surface, the two bulk Dirac nodes are separated on the opposite sides of the (100) surface BZ center (Fig. 1F). Consequently, the Fermi arc surface states connecting the bulk Dirac nodes are found in the (100) surface electronic structure calculation (Fig. 1H). In order to experimentally search for the Fermi arc surface states and to probe the topological number for the Dirac semimetal state in Na

_{3}Bi, we systematically studied its electronic structure and spin polarization at the (100) side-surface.

Figure 1A shows the ARPES-measured Fermi surface of our Na_{3}Bi sample at its native Fermi level. The measured Fermi surface consists of two Fermi “points” along the *k _{y}*

_{[001]}direction and two arcs that connect the two Fermi points. We then studied the evolution of constant energy contour as a function of binding energy

*E*

_{B}(Fig. 1B). As we moved

*E*

_{B}below the Fermi level, the two bulk Dirac points were found to enlarge into contours (hole-like behavior), whereas the two surface Fermi arcs shrank (electron-like behavior). In Fig. 1C, we show the energy dispersion for important momentum space cut directions (Fig. 1B). Surface states with a surface Dirac crossing are clearly observed near the Fermi level in cut β. On the other hand, for cuts α and γ, no surface states are observed but the bulk linear band is seen to cross the Fermi level (Fig. 1E). All these ARPES results regarding the band behavior (Fermi surface, electron- or hole-like behavior for each band, and energy dispersion cuts) are in qualitative agreement with the theoretical prediction (

*9*). Therefore, our ARPES data demonstrate that the Fermi surface of Na

_{3}Bi is described by two Fermi arc surface states that connect two bulk Dirac nodes at our samples’ native chemical potential.

In order to further confirm the 2D surface nature for the double Fermi arc surface states and the 3D bulk nature for the two bulk Dirac bands, we present Fermi surface maps at different incident photon energies. The double Fermi arcs are observed at all photon energies, whereas the bulk Dirac nodes are pronounced only at 55 to 58 eV (Fig. 2, A to C). Furthermore, we show the dispersion along cut β, which also confirms that the surface state spectra do not depend on photon energy (Fig. 2D). In contrast, the bulk Dirac bands disperse strongly in photon energy (Fig. 2E). The systematic photon energy (*k _{z}*)–dependent data in Fig. 2 clearly demonstrate that the electronic states at the Fermi level are localized at the surface (2D surface state nature) everywhere around the Fermi surface contour, except at the locations of the two bulk Dirac nodes, where the states disperse strongly in all three dimensions (3D bulk band nature).

Next, we studied the surface spin polarization along cut β, where a surface Dirac crossing within the bulk bandgap is observed. Figure 3, B and C, show the in-plane spin-resolved intensity and net spin polarization. The direction of spin polarization is reversed upon going from S-cut 1 to S-cut 2 (defined in Fig. 3A), which shows the spin-momentum locking property and the singly degenerate nature of the Fermi arc surface states along the cut β direction.

We used the observed electronic and spin structure for the (100) Fermi arc surface states to obtain the topological number for the Dirac semimetal state in Na_{3}Bi. Because of the existence of the two bulk nodes (slices 2 and 6), our ARPES data (Fig. 3, D and E) show that the bulk bandgap closes and reopens as one goes across each bulk node (slice 2 or 6). We carefully examined the ARPES electronic structure for all 2D-*k* slices, where the bulk is gapped (slices 1, 3 to 5, and 7). It can be seen from Fig. 3, E and F, that slices 1 and 7 are gapped without any surface states; slices 3 and 5 do have surface states, but the surface states are gapped and do not connect across the bulk bandgap; however, for slice 4 (*k*_{[001]} = 0), our data show gapless Dirac surface states that span across the bulk gap. Therefore, our data indicate that slice 4 has a nontrivial 2D topological number. Because slice 4 (the *k*_{[001]} = 0 plane) is invariant under the time-reversal operation, it is possible for slice 4 to have a nonzero 2D number ν_{2D} = 1 (*2*, *14*, *15*). If that is the case, then slice 4 can be viewed as a 2D quantum spin Hall-like system, and one would expect a Dirac surface state with a helical spin texture connecting the bulk conduction and valence bands. Indeed, this is what we observe in our ARPES (Fig. 3E) and spin-ARPES (Fig. 3, A to C) data. Therefore, our data demonstrate that Na_{3}Bi is a topological Dirac semimetal that features a 2D topological number ν_{2D} = 1 for the *k*_{[001]} = 0 plane, which is consistent with the prediction in (*2*). Further elaborations regarding the topological properties of Na_{3}Bi are presented in the supplementary materials (figs. S1 and S2). Figure 4, A to D, shows schematically the difference between a trivial and topological Dirac semimetal.

Our observations of topological surface states, the way in which they connect to the bulk Dirac cones, and their spin momentum locking demonstrate the topology of a Dirac semimetal in line with the theoretical prediction (*2*). The observed Fermi arc surface states represent a type of 2D electron gas that is distinct from that of the surface states in a topological insulator (*27*). In a typical topological insulator such as Bi_{2}Se_{3}, the surface states’ Fermi surface is a closed contour. In sharp contrast, the Fermi surface in Na_{3}Bi consists of two arcs, which are bridged by the two bulk nodes (Fig. 4, E and F). Therefore, as one goes along the surface Fermi arc and reaches a bulk node, the wave function of the surface state gradually loses its surface nature and becomes a bulk band. Such exotic behavior does not exist in surface states of Bi_{2}Se_{3} or any known topological insulator. In surface electrical transport experiments at high magnetic field, it would be interesting to study how the surface electrons wind along the arc and enter the bulk singularity as considered by a recent theoretical work (*5*), if the chemical potential can be placed precisely at the bulk Dirac node so that the bulk carrier density is vanishingly small (*5*). Finally, it would be interesting to introduce superconductivity or magnetism in these new topological metals (*9*, *10*, *28*). In particular, breaking time-reversal symmetry via magnetism can split the Dirac nodes into Weyl nodes (Fig. 4E).

## Supplementary Materials

## References and Notes

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**Acknowledgments:**The work at Princeton and Princeton-led synchrotron-based ARPES measurements are supported by U.S. Department of Energy grant DE-FG-02-05ER46200. H.L. acknowledges the Singapore National Research Foundation (NRF) for support under NRF award no. NRF-NRFF2013-03. C.-Y.H. and W.-F.T. are supported by the Ministry of Science and Technology (MOST) in Taiwan under grant no.103-2112-M-110-008-MY3. T.-R.C. and H.-T.J. are supported by the National Science Council, Taiwan. H.-T.J. also thanks the National Center for High-Performance Computing, Taiwan; the Computer and Information Network Center–National Taiwan University, Taiwan; and the National Center for Theoretical Sciences, Taiwan, for technical support. Crystal growth was supported by the Army Research Office Multidisciplinary University Research Initiative on topological insulators, grant no. W911NF-12-1-0461. F.-C.C acknowledges the support provided by MOST-Taiwan under project number 103-2119-M-002-020-MY3. G.B. is supported by grant NSF-DMR-1006492. We gratefully acknowledge M. Leandersson, C. M. Polley, J. Adell, T. Balasubramanian, K. Miyamoto, T. Okuda, and M. Hashimoto for their beamline assistance at the Maxlab, the Hiroshima Synchrotron Radiation Center (Hisor), and the Stanford Synchrotron Radiation Lightsource (SSRL). We also thank C. Fang, B. A. Bernevig, and A. Vishwanath for discussions. ARPES data were collected at the beamline 5-4 at the SSRL at the Stanford Linear Accelerator Center in California, USA; the beamlines I3 and I4 at the MAX-lab in Lund, Sweden; and the beamline-9B of the Hisor in Hiroshima, Japan.