Observation of Dirac Node Formation and Mass Acquisition in a Topological Crystalline Insulator

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Science  27 Sep 2013:
Vol. 341, Issue 6153, pp. 1496-1499
DOI: 10.1126/science.1239451


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 Pb1-xSnxSe 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.


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, Pb1-xSnxSe 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) (15) 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 Embedded Imagepoints 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 Embedded Image 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 Embedded Image point (Fig. 1B).

Fig. 1 Crystal structure and schematic band structure of TCIs.

(A) Crystal structure of Pb1-xSnxSe. The top view reflects the [001] plane, which is the surface seen in STM. The Sn and Pb atoms are expected to randomly occupy the blue sites. (B) Schematic band structure of the surface state showing the surface Brillouin zone as a blue plane and the four pairs of Dirac nodes, placed across the Embedded Image point in momentum space. (C) Typical dI/dV spectrum in zero field. (D) Cuts along two high-symmetry directions showing the calculated surface-state dispersion [calculated from our fit to the SM data (14)] of one of the four Dirac cones inside the BZ. The energy scales EDP1, EvHs+, EvHs–, EDP2+, and EDP2– represent the Dirac point associated with the Dirac node deep in the band gap, the two Van Hove singularities associated with the saddle points in the dispersion, and the two higher-energy Dirac points associated with the Dirac nodes at the Embedded Image point, close to bottom of the conduction band and the top of the valance band, respectively.

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 Embedded Image 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 (811), 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 Pb1-xSnxSe 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 Pb1-xSnxSe samples were cleaved in ultrahigh vacuum at ~80 K before being inserted into the STM, and all measurements were obtained at ~4 K. Although Pb1-xSnxSe 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.

Fig. 2 STM image- and position-dependent spectra on Pb1-xSnxSe.

(A) Typical 40-nm STM topographic image at –100-meV bias voltage with clearly resolved atoms as seen in the smaller-scale zoomed-in image in the inset. (B) Averaged STM spectrum at 7.5 T showing Landau levels (blue). The second derivative spectrum (purple) shows the LL peaks much more clearly. Second derivative of spectra were therefore used to identify the peak positions. (C) Intensity plot of the second derivative LL spectra at 7.5 T as a function of position along the line shown in (A). The homogeneous nature of the sample is reflected in the lack of variation in the peak energy with position.

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 EDP1 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 Pb1-xSnxSe 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, EDP1. In addition, we found other nondispersing peaks, which have been labeled E*, E+*, and EDP2– 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 (1517). 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 Embedded Image. By using this scaling behavior as a constraint, we obtained the peak index assignments for the dispersing LLs (Figs. 2B and 3B).

Fig. 3 Change in Fermi surface topology measured by Landau level spectroscopy.

(A) Linecut averaged STM spectra at various values of magnetic field strength. (B) Second derivative of spectra shown in (A) that were used to trace the peak positions. (C) Plot of LL peak positions with Embedded Image. The data here are from spectra at the 10 different magnetic fields shown in (B). The pink dotted lines in (A) and (B) indicate the energies of the VHS peaks, which persist in magnetic field. (D) Plot of theoretically calculated Fermi surface area with energy, overlaid with experimental LL peak positions [from spectra in (B)] as a function of nB [details are available in (14)]. The jump in area corresponds to the Lifschitz transition. (E) Schematic of evolution of the constant energy contours in momentum space with energy showing the Lifschitz transition and associated jump in Fermi surface area as we go from the bottom to the middle. The dotted circle at top indicates the inner surface-state band, which is expected to merge with the bulk conduction bands at higher energies.

The resulting plot of the LL energy with Embedded Image 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 EvHS+ 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 EDP1) 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 EDP1, whereas all the other LL peaks (except EDP2–, which we discuss later) can in principle be accounted for by the surface-state dispersion. If, however, the Dirac node (at EDP1) 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 EDP1, 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 (2123). Because only one of the two sublattices was visible in our STM images (Fig. 2A), we could not directly image this distortion.

Fig. 4 Coexistence of massless and massive Dirac Fermions.

(A) Schematic band structure around the massless and massive Embedded Image points, respectively. (B and C) Theoretical calculation of LL fan diagram (B) without and (C) with a symmetry-breaking term added to the Hamiltonian, respectively. (C) shows a comparison between the data points obtained from the experimental LL spectra and theory, with the theoretical parameters adjusted to match the data. The red and blue lines refer to the theoretical LLs from the massive and massless surface-state bands, respectively. (D and E) Schematic arrangement of surface atoms and band structure (D) without and (E) with a crystal distortion that leads to one broken mirror symmetry plane.

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, EDP1. 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 EDP2–, 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 EDP1, 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 EDP2–. Directly after the Dirac cones across the BZ merge, Dirac points appear at Embedded Image, as shown in Fig. 1D (EDP2+ and EDP2–) and fig. S3 (14), which also result in nondispersing features in the LL calculations. Although this is suggestive that the experimental EDP2– may originate from the Dirac point at Embedded Image, 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 EDP1 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 Pb1-xSnxSe constitutes a realistic, tunable platform for exploring previously unknown topological states emergent via coupling to symmetry-breaking interactions.

Supplementary Materials

Materials and Methods


Figs. S1 to S4

Reference (24)

References and Notes

  1. Materials and methods are available as supplementary materials on Science Online.
  2. 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.
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