The study of topological properties of materials and corresponding phase transitions has received tremendous interest in the condensed-matter community in recent years1,2. A particularly interesting class of materials are three-dimensional topological crystalline insulators (TCIs)3,4 where degeneracies in the surface electronic band structures are protected by point-group symmetries of the crystals. The first material systems to realize this state were found within the class of IV–VI narrow-gap semiconductors: SnTe5,6, (Pb,Sn)Te7,8 and (Pb,Sn)Se9,10. These materials, crystallizing in the rock-salt structure, host a set of four Dirac points on specific surfaces and a great advantage in particular of the solid solutions is the tunability of the bulk band gap—and hence their topological properties—by several independent parameters11, such as composition8,12, temperature9,10,12, pressure13 or strain14. Moreover, since the surface Dirac points are protected by crystalline symmetries, it has been predicted that selectively breaking these symmetries can lift the surface-state degeneracies and the carriers acquire masses5,15. Indeed, signatures of such massive surface Dirac fermions have been observed recently in Landau-level-spectroscopy experiments on (001) surfaces of (Pb,Sn)Se crystals16 and a surface distortion has been identified to be at the heart of the phenomenon17 (Fig. 1). However, since these scanning tunnelling microscopy/spectroscopy (STM/STS) studies only provide low-temperature data, the evolution of the surface-state mass/gap with temperature remains elusive. It is particularly interesting to ask whether the four (ungapped) Dirac points ever exist simultaneously on the (001) surface of (Pb,Sn)Se crystals in the TCI state, whether the partially gapped surface states can be observed directly by k-resolved methods, and whether one can switch between the massive and massless states by varying extrinsic parameters.

Figure 1: The (001) surface and TCI surface band structure of (Pb,Sn)Se.
figure 1

(a) Sketch of the undistorted (001) surface with bulk lattice constant a. The black and red circles represent the cations and anions, respectively. The mirror lines of the surface are indicated by solid lines; (b) Sketch of the surface band structure corresponding to a. The particle-hole-symmetric band structure consists of the well-known hybridized Dirac cones, featuring a Lifshitz transition indicated by the constant-energy contours. The given labels denote the high-symmetry points of the surface Brillouin zone (, , , ) and the positions of the Dirac points (, ); (c) Sketch of a distorted (001) surface (solid circles) originating from the surface shown in a (dotted circles) by shifting the cation and anion sublattices with respect to each other along one of the mirror lines as suggested by STM measurements17. The dashed line indicates the breaking of the second mirror symmetry by this distortion; (d) Sketch of the surface band structure corresponding to c. The broken mirror symmetry induces a gap in one pair of the surface states. In this work, x and y (and kx and ky, correspondingly) are defined in the surface as shown. These coordinates are rotated by 45° about z with respect to the conventionally used bulk coordinates.

In this study, we address these open questions using high-resolution angle-resolved photoelectron spectroscopy (ARPES) measurements on (Pb,Sn)Se mixed crystals. Consistent with the previous STM/STS studies we directly observe robust gapped Dirac states on the (001) surface at low temperature. Most importantly, however, the evolution of the gap/mass is tracked by composition- and temperature-dependent experiments and a transition to entirely massless surface Dirac states is witnessed. We suggest that the surface distortion detected by STM vanishes at sufficiently high temperatures, thus leading to the possibility of tuning the surface states employing structural changes. Subsequently, this is summarized in a revised topological phase diagram for (Pb,Sn)Se (001).


While studying the detailed nature of the bulk band inversion in (Pb,Sn)Se by high-resolution ARPES12, low-temperature data of certain samples with comparably high SnSe content were found to contain a peculiar distribution of spectral weight around the Dirac points. An example of such spectra (not included in ref. 12) is depicted in Fig. 2a. The characteristic band structure of the hybridized parent surface Dirac cones schematically shown in Fig. 1 is clearly observed close to the point of the (001) surface Brillouin zone. However, also a distinct gap opening at the Dirac nodes on the - line is revealed. Following this initial finding and reports of similar observations using STM/STS techniques16, the gap formation has been studied systematically across the available temperature and composition ranges.

Figure 2: Surface-state Dirac gap in (Pb,Sn)Se (001).
figure 2

(a) ARPES spectra showing the surface-state dispersion of Pb0.63Sn0.37Se in the vicinity of at T=9 K. The reciprocal-space cuts are taken parallel to the borders of the surface Brillouin zone through the Dirac points at as labelled. The photoelectron intensity is encoded in a linear colour scale for each plot. The black circles represent the obtained peak positions from a two-Voigtian fit to the energy distribution curves. Vertical error bars correspond to the statistical uncertainties from the fit. A gap of about 25 meV is observed consistently across the spectra; (b) Surface-state gap as a function of temperature for various cleaved (001) surfaces of Pb1−xSnxSe. Only gap values, which can be reliably determined by our analysis are shown. At higher temperatures vanishes within the resolution of our experiment. The vertical arrows coloured equivalently to the data series indicate the lowest temperatures for which the Dirac points have been found intact within the experimental resolution. The dotted arrow corresponds to the series x=0.23 III represented by open pentagons. Solid arrows belong to series marked by filled symbols. For data sets without corresponding arrows no gapless data in the TCI phase have been acquired. Since the density of data points in temperature varies substantially from one data set to another, the positions of the arrows cannot be simply interpreted as gap-closing temperatures; (c) as a function of SnSe content x for selected temperatures. The dash-dotted lines indicate the rough trend in the observed gap size. The vertical error bars in b,c represent the statistical uncertainties from the energy-distribution-curve analysis scaled by a factor of 2 to take into account possible systematic shortcomings in the model used.

Our analysis (see Methods section) shows that most of the low-temperature spectra are gapped. Yet, at higher temperatures the Dirac nodes are—within the resolution of the experiment—overall intact. Figure 2b,c summarizes the ascertained gap sizes for the samples investigated during this study. While one has to note that the values of vary by about 3–4 meV for nominally similar sample compositions, nevertheless, a general trend to larger values for higher SnSe contents and lower temperatures is apparent.

Having directly established the low-temperature mass acquisition of the Dirac fermions at it remains to be investigated whether the Dirac points at do exist also on the distorted surface as suggested by the STM/STS studies16,17. Unlike STM/STS which is a local-probe technique, ARPES averages over a lateral sample region with a diameter of several tens of micrometres. Hence, a priori the observation of both massive surface states at and massless states at requires a long-range textured surface. If rather a multidomain structure is formed with distortions along x and y, respectively for different domains, then, a superposition of gapped and gapless states is expected to be seen. A further complication arises from the low photon energy and the experimental geometry of the ARPES set-up which necessitate the samples under study to be rotated by ±90° to enable the collection of electrons originating from . Since most likely laterally slightly different parts of the surface are probed before and after such a manipulation, the long-range-texture precondition is further intensified. Our experiments show that this requirement is not always met (and hence, overall massive Dirac states are observed). Nevertheless, indeed we do find Dirac nodes, and thus massless states, at coexisting with gapped states at at low temperatures. An example of the latter is shown in Fig. 3a through 3c, . It illustrates the evolution of the (001) surface states of Pb0.70Sn0.30Se across the different phases. Close to room temperature the sample is in a topologically trivial state characterized by the positive bulk gap at (projection of the bulk high symmetry point L)12. On cooling to T=150 K, the bulk bands invert and the sample enters the TCI phase with massless surface states. Lowering the temperature further leads to the gap formation at the positions of the former Dirac points , while the degeneracies at prevail, thus confirming the overall situation depicted in Fig. 1.

Figure 3: Topological transitions in (Pb,Sn)Se (001).
figure 3

(ac) ARPES data of Pb0.70Sn0.30Se (001) at different temperatures in the vicinity of : (b) depicts the absolute photoelectron intensity along the high-symmetry lines of the surface Brillouin zone encoded in a linear colour scale; (c) The second derivative of the data in b with respect to the energy encoded in a linear colour scale; (a) EDCs along the red dotted lines in the spectra in b. The arrows in b,c indicate the fundamental bulk band gap inverting as a function of temperature. The black dotted lines mark the minimally separated EDC peaks and thus indicate the presence or absence of the surface Dirac gap ; (d) Summary of the evolutions of the bulk gap (, interpolated colour map, see also ref. 12) and the largest observed surface Dirac gap (, contour lines, spacing 2.5 meV) in Pb1−xSnxSe (001). The open grey circles indicate the (x, T) positions of the data shown in ac; (e) Topological phase diagram for (Pb,Sn)Se (001) based on the observations of the spectral gaps shown in d. The symbols and corresponding error bars specify the temperature ranges in which the spectral changes shown throughout (ac) occur. The red symbols estimate a lower temperature limit for the surface Dirac gap formation. The dotted line marks the proposed transition line for the occurrence of the surface distortion in the topologically trivial state. All lines are guides to the eye.


To understand the full implications of this study, it is necessary and helpful to compare our results with the observations made in STM/STS experiments before. Zeljkovic et al.17 have demonstrated a low-temperature (T=4 K) surface Dirac gap in (Pb,Sn)Se reaching a value of about 25 meV at a SnSe content of about 38% in the solid solution. With decreasing x the gap is reported to decrease and it supposedly vanishes at a critical composition of about 17%. At the same time, the observed lattice distortion at the surface appears to have roughly the same magnitude for all studied samples. To reconcile the diminishing surface gap, it has been argued that the penetration depth of the surface state into the bulk increases when the size of the bulk gap decreases and thus the weight of the surface state in the outermost atomic layer probed by STM/STS decreases continuously. So, is our observation of the (x, T)-dependent gap merely the result of the varying surface-state penetration depth ? The answer to that question is found in Fig. 2b,c. At T=9 K, indeed decreases with decreasing x, but in total only by a few millielectronvolts from x=0.37 to x=0.19. A similar behaviour is found at x=0.23: On increasing the temperature from 9 K to 100 K (spanning essentially the entire band-inverted region) the observed gap is reduced only minutely. Hence, observed here is—if at all—only to a small extent influenced by the changing penetration depth of the surface states. Thus, possibly due to the increased probing depth of low-energy ARPES as compared with STM/STS (nanometres instead of ångströms), the gap can rather be regarded as a direct measure of the size of the lattice distortion at the surface15. This conclusion is seemingly at variance with the claims of ref. 17, yet, it should be noted that in the STM/STS work the reliably determined gap sizes span a region between about 15 meV and 30 meV, in full agreement with our data. The nonobservation of distinct features in the STS spectra due to the diminishing spectral weight in the top-most atomic layer does not imply an overall vanishing surface-state gap for samples close to the transition.

Altogether, our observations are summarized in Fig. 3d. On the basis of the distinction of the massive and massless surface states in the bulk-band-inverted region we are able to determine a transition line below which the lattice distortion at least in the (001) surface occurs (Fig. 3e). Given the finite resolution of our experiment (in terms of energy as well as temperature), it should be regarded as a lower limit and hence the indicated TCI phase with completely massless surface states might occupy a somewhat smaller, yet still finite, area in the (x, T) phase diagram. Since STM also provided evidence for a lattice distortion in the topologically trivial phase at low temperature17, although inaccessible by our spectroscopic method, we suspect the transition to occur at finite temperatures in this phase of the solid solutions as well.

The (Pb,Sn)Se mixed crystals with a SnSe content in the studied range are found to crystallize in the rock-salt structure without bulk structural phase transitions. Nevertheless, the occurrence of a surface distortion further corroborated by this study should not entirely catch us by surprise. SnSe itself features an orthorhombic crystal structure18 and various closely related IV–VI solid solutions undergo ferroelectric phase transitions19. Also, in (Pb,Sn)Se the transverse-optical phonon mode at Γ softens as the temperature is lowered and the SnSe content is increased20. Occasionally, for Pb0.59Sn0.41Se even bulk structural changes were reported and found to be suppressed only with increasing carrier densities21, suggesting displacive transitions driven by electron–phonon coupling22,23.

It is too early to conclude on the exact mechanism which leads to the observed surface structure. Specifically, given the small but still clearly detectable variations between different measured samples with comparable SnSe content, it appears that the local nature of this lattice distortion somewhat depends on the details of each surface. However, its overall tunability by independent parameters inferred from the (x, T) phase diagram (Fig. 3e) lays the foundation for achieving a controlled mass generation in the topological surface Dirac states. It will be interesting to see in the future how other TCI surfaces behave24,25, both of single crystals and of thin films26,27, where controlled tuning of the crystal structure, for example, by strain, promises to be a viable avenue to open gaps in topological surface states in addition to breaking symmetries by electric28 or magnetic fields15.

Eventually, it is worth noting that by specifically breaking both mirror symmetries as well as the rotational symmetries of the surface, all Dirac points are expected to be gapped in the material studied here15. If a domain structure is formed, then a topologically protected 1D conducting state percolating through the surface along the domain walls would be the result5. Such states can potentially be engineered in very thin films grown on specific vicinal surfaces or by employing TCIs naturally featuring suitable lattice distortions.

In summary, the evolution of the surface-state gap/mass in (Pb,Sn)Se (001) has been determined systematically by means of high-resolution ARPES. Our results are consistent with the proposed underlying surface distortion at low temperatures. The observation of only massless Dirac surface states at higher temperatures indicates a structural transition of the surface. While its detailed nature is yet to be investigated by systematic structural and/or vibrational studies, we have shown experimentally the possibility to tune between massive and gapless topological states using this very transition.


Crystal growth and characterization

The n-type (Pb,Sn)Se single crystals with typical doping levels of a few 1018 cm−3 studied in this work have been grown using the self-selecting vapour-growth method and characterized by X-ray diffraction and energy-dispersive X-ray spectroscopy29,30. High-resolution ARPES studies were carried out on samples cleaved in a (001) plane in ultrahigh vacuum at room temperature. Temperature-dependent measurements of solid solutions with SnSe/PbSe ratios between 0 and 0.6 were carried out at the BALTAZAR laser-ARPES facility using linearly polarized light with an excitation energy 10.5 eV and a THEMIS-1000 time-of-flight analyzer collecting the photoelectrons31. The overall crystal-momentum and energy resolution is better than 0.008 Å−1 and about 5 meV, respectively.

Analysis of the photoelectron spectra

The decision whether the surface states in the TCI phase are massive/gapped is based on an analysis of the energy-distribution curves (EDCs) in the close vicinity of the Dirac nodes . In a first step, the EDCs are modelled by sums of two Voigtian lines on a small strictly monotonic linear to cubic background. The resulting line widths are compared with a fit of a single Voigtian (on the same background) to the EDC at . In case the single line (full width at half maximum) at is found to be broader by >5 meV than the broadest two-line constituent, we call the states gapped and the two-line peak separation determines the gap size . This criterion is based on the scattering of the determined line width at high temperatures and the energy resolution of the experiment, which are both on the order of 5 meV.

Additional information

How to cite this article: Wojek, B. M. et al. Direct observation and temperature control of the surface Dirac gap in a topological crystalline insulator. Nat. Commun. 6:8463 doi: 10.1038/ncomms9463 (2015).