## Half-Massless

Certain materials, such as topological crystalline insulators (TCIs), host robust surface states that have a Dirac (graphene-like) dispersion associated with massless carriers; the breaking of protective symmetry within such materials should cause the carriers to acquire mass. **Okada et al.** (p. 1496, published online 29 August) used scanning tunneling microscopy to map out the energies of the electronic levels of the TCI Pb

_{1-x}Sn

_{x}Se as a function of the strength of an external magnetic field. The massless Dirac fermions coexisted with massive ones, presumably as a consequence of a distortion of the crystalline structure affecting only one of the two mirror symmetries.

## Abstract

In topological crystalline insulators (TCIs), topology and crystal symmetry intertwine to create surface states with distinct characteristics. The breaking of crystal symmetry in TCIs is predicted to impart mass to the massless Dirac fermions. Here, we report high-resolution scanning tunneling microscopy studies of a TCI, Pb_{1}-_{x}Sn_{x}Se that reveal the coexistence of zero-mass Dirac fermions protected by crystal symmetry with massive Dirac fermions consistent with crystal symmetry breaking. In addition, we show two distinct regimes of the Fermi surface topology separated by a Van-Hove singularity at the Lifshitz transition point. Our work paves the way for engineering the Dirac band gap and realizing interaction-driven topological quantum phenomena in TCIs.

Topological crystalline insulators (TCIs) (*1*–*5*) are a class of materials that possess a new type of electronic topology that arises from crystalline symmetry; this gives rise to surface states with an unusual band dispersion and spin texture. In particular, the (001)-oriented surfaces of IV-VI semiconductor TCIs (Fig. 1A) harbor two generations of Dirac cones at different energies (Fig. 1B). The band structure of (001) surface states can be visualized by starting from a pair of Dirac points located at the points at the edge of the surface Brillouin zone (BZ), with the Dirac point energies close to the conduction and valence band edge, respectively (*6*). The hole-branch of the upper Dirac cone and the electron-branch of the lower Dirac cone coexist inside the band gap. The hybridization between these two branches leads to an avoided crossing in all directions except along the line, where a band crossing is dictated by electronic topology of TCI and protected by the (110) mirror symmetry (*6*). Such a band reconstruction generates a new pair of low-energy Dirac nodes displaced symmetrically away from each point (Fig. 1B).

The distinctive band structure of the TCI surface states has two important consequences. First, the formation of low-energy Dirac nodes is accompanied by a Lifshitz transition (*7*): As the Fermi energy moves deep into the band gap, the Fermi surface changes from concentric pockets of opposite carrier types centered at into a pair of disconnected pockets across the BZ boundary (Fig. 1B). The change of Fermi surface topology occurs via a saddle point in the surface band structure, which has been predicted to lead to a divergence in the electronic density of states known as Van Hove singularity (VHS) at the saddle-point energy (*6*). Although a VHS generally enhances interaction effects and drives electronic instabilities (*8*–*11*), in TCIs interaction effects in combination with band topology may result in novel correlated states (*12*, *13*). Second, when mirror symmetry is broken—either spontaneously or by external perturbations—the TCI surface states acquire a gap, creating massive Dirac fermions and providing an exciting avenue to control the properties of Dirac materials through strain or doping.

We used a low-temperature scanning tunneling microscope (STM) to track the dispersion and the density of states of Pb_{1}-_{x}Sn_{x}Se with a nominal doping of *x* = 0.5 [determined to have an actual doping level of *x* = 0.34 (*14*)], which lies in the topological regime (*5*). Single-crystal Pb_{1}-_{x}Sn_{x}Se samples were cleaved in ultrahigh vacuum at ~80 K before being inserted into the STM, and all measurements were obtained at ~4 K. Although Pb_{1}-_{x}Sn_{x}Se has a three-dimensional (3D) crystal structure (Fig. 1A), it cleaves along the [001] plane, resulting in the Pb/Sn-Se surface shown in Fig. 2A. STM topography reveals a square lattice whose inter-atom spacing of 4.32 Å indicates a preferential imaging of either the Pb/Sn or the Se sublattice.

The overall density of states in this TCI can be obtained by measuring *dI*/*dV* spectra. A typical *dI*/*dV*(*eV*) spectrum in this material (Fig. 1C) shows a V-shaped density of states, with a minimum at ~–80 meV. By comparing this with the schematic band structure, we tentatively assigned the minimum at –80 meV to the Dirac point deep in the band gap labeled E_{DP1} in Fig. 1D and the approximately symmetric peaks on either side of this Dirac point (at ~–40 meV and ~–120 meV) to the VHSs at the saddle point energy, which as we show later in this paper is consistent with our data. To establish the surface-state dispersion, we developed a framework, which combines our magnetic field–dependent STM data with a theoretical model.

It is necessary to first understand the level of inhomogeneity in these samples. To do this, we compared spectra obtained at various spatial locations (Fig. 2C) and found a notable degree of spectral homogeneity over at least 300 Å, despite the presence of randomly distributed Sn atoms within the Pb lattice; we henceforth used linecut-averaged spectra along the line shown in Fig. 2A for our analysis. This homogeneity should be contrasted with the highly inhomogeneous nature of graphene as well as doped topological insulators. This important feature makes Pb_{1}-_{x}Sn_{x}Se a much more stable host for topological surface states and allows true access to physics at the Dirac point with a variety of experimental probes.

The line-cut averaged spectra at various magnetic fields (Fig. 3A) show clear Landau level peaks. Comparing the zero-field spectrum with the spectra at higher fields, we found that for nonzero magnetic fields, a peak located precisely at the density of states minimum (at ~–80 meV) emerges. It is nondispersive; its position does not change with magnetic field, which confirms its origin as the 0th LL located at the Dirac point, E_{DP1}. In addition, we found other nondispersing peaks, which have been labeled E_{–}*, E_{+}*, and E_{DP2–} in Fig. 3A. To understand the electronic structure, as a first step we analyzed the LL data within a semiclassical picture. For normal 2D bands with linear or quadratic dispersion, the semiclassical approximation is applicable, and a plot of the LL peak position as a function of *nB* (where *n* is the LL index and *B* is the magnetic field strength) can be used to obtain information on the dispersion (*15*–*17*). However, this requires us to index the dispersing LL peaks, which is a nontrivial task in this material. At energies away from the saddle points, the band structure is characteristic of Dirac fermions, with approximately linear dispersion (*6*); the peak energies can therefore be expected to collapse to one curve as a function of . By using this scaling behavior as a constraint, we obtained the peak index assignments for the dispersing LLs (Figs. 2B and 3B).

The resulting plot of the LL energy with is shown in Fig. 3C. The sharp discontinuity in the LL plot in Fig. 3C and the jump in the LL index from *n* = 2 to 6 occur at the same energy as the *dI*/*dV* peak labeled E_{vHS+} in Fig. 3, A and B, which suggests that they have a similar origin. Consulting the schematic band structure in Fig. 1D suggests that these features are a result of the Lifshitz transition; the flat dispersion at the saddle point creating the VHS as well as the missing peaks (*n* = 3,4,5). The jump in index corresponds exactly to the doubling of the Fermi surface area expected from the Lifshitz transition (Fig. 3E). The observation of VHSs in a topologically nontrivial material close to the Fermi energy opens the exciting possibility of achieving correlated states in a Dirac material. VHSs have previously been observed in graphene (*18*); however, unlike in graphene, symmetry-breaking interactions modifying existing topologically protected electronic states in TCIs have the potential of generating helical domain wall states (*2*) or Majorana fermion modes (*19*).

We now discuss the appearance of the peaks labeled E_{–}* and E_{+}* (~±10 meV from E_{DP1}) in Fig. 3A. The immediate observation is that the peak energies do not change with magnetic field strength. This rules out *g*-factor effects (or a Zeeman term). Furthermore, the appearance of the additional peaks is restricted to the vicinity of the Dirac point E_{DP1}, whereas all the other LL peaks (except E_{DP2–}, which we discuss later) can in principle be accounted for by the surface-state dispersion. If, however, the Dirac node (at E_{DP1}) is gapped out by the acquisition of a small mass term at zero-field, the resulting massive two-dimensional Dirac fermions will cause the appearance of a nondispersing *n* = 0 LL pinned at the energy of the Dirac mass (*20*). Mass acquisition could therefore potentially explain the existence of the E_{–}* and E_{+}* peaks. Their coexistence with the zero–mode LL peak at the gapless Dirac point E_{DP1}, however, places strong constraints on the origin of the Dirac mass. In TCIs, such a scenario can only be realized when mirror symmetry is broken in one direction. Because there are two mirror planes within the surface BZ, it is possible to selectively break mirror symmetry reflection with respect to one mirror plane such as (110), leaving the other mirror plane unaffected. This can be achieved, for example, by a rhombohedral or orthorhombic distortion of the crystal structure with a displacement of atoms at the surface (Fig. 4), which is a common distortion in this class of materials (*21*–*23*). Because only one of the two sublattices was visible in our STM images (Fig. 2A), we could not directly image this distortion.

In order to confirm this scenario, we calculated the Landau level spectra theoretically by adding a new symmetry-breaking mass term to the recently developed ** k.p** Hamiltonian for TCIs (

*6*). The resulting LL spectrum as a function of magnetic field is presented as a fan diagram in Fig. 4. Comparing the LL fan-diagram without (Fig. 4B) and with (Fig. 4C) the symmetry-breaking term, the mass term (with an appropriately chosen mass of ~11 meV) partially splits the fourfold degenerate 0th LL, whereas the LL spectra at higher energies are not substantially affected.

Comparing the theoretical model for coexisting massive and massless Dirac fermions to our experimental data, we found good agreement between the two. The symmetry-breaking term gaps out two of the four nodes, but the sign of the mass term for the two gapped nodes is necessarily opposite, as dictated by time-reversal symmetry (*2*). As a consequence, the *n* = 0 LLs for the two massive Dirac bands appear on the upper and lower branch of the spectrum, respectively, each of which is nondegenerate. This explains our observation of two symmetric peaks near the Dirac point, E_{DP1}. All the other Landau levels are only weakly affected by the symmetry breaking.

By fitting the theoretical LLs to our STM data shown as green dots in Fig. 4C, excluding the peaks labeled E_{DP2–}, we can calculate the surface-state dispersion for the massive and massless cones (Fig. 1D and fig. S3). The theoretical saddle point energy is ~40 meV from the Dirac point E_{DP1}, which is once again consistent with our previous identification of the zero-field *dI*/*dV* peaks with VHSs. This analysis also provides a possible identification of the nondispersing feature labeled E_{DP2–}. Directly after the Dirac cones across the BZ merge, Dirac points appear at , as shown in Fig. 1D (E_{DP2+} and E_{DP2–}) and fig. S3 (*14*), which also result in nondispersing features in the LL calculations. Although this is suggestive that the experimental E_{DP2–} may originate from the Dirac point at , the theory and experimental energies are not identical. This could potentially be attributed to particle-hole asymmetry in the band structure, which is not included in the current model. The possibility of particle-hole asymmetry is also consistent with the observation that the LL data below E_{DP1} show deviations from the theoretical LL spectrum (Fig. 4C).

Our data show an enhanced, nearly singular density of states inside in the bulk band gap and tied to the surface-state spectrum near the Dirac point. The band structure enabled by the coexistence of both massive and massless Dirac fermions in the same surface spectrum—as well as the enhanced density of states in close proximity to the Dirac point—demonstrates that Pb_{1}-_{x}Sn_{x}Se constitutes a realistic, tunable platform for exploring previously unknown topological states emergent via coupling to symmetry-breaking interactions.

## Supplementary Materials

www.sciencemag.org/content/341/6153/1496/suppl/DC1

Materials and Methods

SupplementaryText

Figs. S1 to S4

Reference (*24*)

## References and Notes

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**Acknowledgments:**V.M. gratefully acknowledges funding from the U.S. Department of Energy (DOE), Scanned Probe Division under award DE-FG02-12ER46880 for the support of Y.O. and D.W. and NSF-ECCS-1232105 for the partial support of W.Z. S.D.W. acknowledges NSFDMR-1056625 for support of C.D. The work at Northeastern University is supported by the DOE Office of Science, Basic Energy Sciences contract DE-FG0207ER46352 and benefited from Northeastern University’s Advanced Scientific Computation Center. L.F. is partly supported by the DOE Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under award DE-SC0010526. M.S. was supported by P. A. Lee via grant NSF DMR 1104498. The work at Princeton and Princeton-led synchrotron*x*-ray–based measurements (angle-resolved photoemission spectroscopy) are supported by the DOE Office of Basic Energy Sciences, grant DE-FG-02-05ER46200. M.Z.H. acknowledges visiting-scientist support from the Lawrence Berkeley National Laboratory and support from the A. P. Sloan Foundation. We thank Y. Ran and I. Zeljkovic for useful discussions.