Direct observation and temperature control of the surface Dirac gap in a topological crystalline insulator

Since the advent of topological insulators hosting Dirac surface states, efforts have been made to gap these states in a controllable way. A new route to accomplish this was opened up by the discovery of topological crystalline insulators where the topological states are protected by crystal symmetries and thus prone to gap formation by structural changes of the lattice. Here we show a temperature-driven gap opening in Dirac surface states within the topological crystalline insulator phase in (Pb,Sn)Se. By using angle-resolved photoelectron spectroscopy, the gap formation and mass acquisition is studied as a function of composition and temperature. The resulting observations lead to the addition of a temperature- and composition-dependent boundary between massless and massive Dirac states in the topological phase diagram for (Pb,Sn)Se (001). Overall, our results experimentally establish the possibility to tune between massless and massive topological states on the surface of a topological system.

T he study of topological properties of materials and corresponding phase transitions has received tremendous interest in the condensed-matter community in recent years 1,2 . A particularly interesting class of materials are threedimensional 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: SnTe 5,6 , (Pb,Sn)Te 7,8 and (Pb,Sn)Se 9,10 . 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 parameters 11 , such as composition 8,12 , temperature 9,10,12 , pressure 13 or strain 14 . 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 masses 5,15 . Indeed, signatures of such massive surface Dirac fermions have been observed recently in Landau-level-spectroscopy experiments on (001) surfaces of (Pb,Sn)Se crystals 16 and a surface distortion has been identified to be at the heart of the phenomenon 17 (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.
In this study, we address these open questions using highresolution 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).

Results
While studying the detailed nature of the bulk band inversion in (Pb,Sn)Se by high-resolution ARPES 12 , 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 X point of the (001) surface Brillouin zone. However, also a distinct gap opening D L X at the Dirac nodes L X on the G-X line is revealed. Following this initial finding and reports of similar observations using STM/STS techniques 16 , the gap formation has been studied systematically across the available temperature and composition ranges. Our analysis (see Methods section) shows that most of the lowtemperature spectra are gapped. Yet, at higher temperatures the Dirac nodes are-within the resolution of the experimentoverall 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 D L X 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 L X it remains to be investigated whether the Dirac points at L Y do exist also on the distorted surface as suggested by the STM/STS studies 16,17 . 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 L X and massless states at L Y requires a longrange 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 Y. Since most likely laterally slightly different parts of the surface are probed before and after such a manipulation, the longrange-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 L Y coexisting with gapped states at L X 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 Pb 0.70 Sn 0.30 Se across the different phases. Close to room temperature the sample is in a topologically trivial state characterized by the positive bulk gap at X (projection of the bulk high symmetry point L) 12 . On cooling to T ¼ 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 L X , while the degeneracies at L Y prevail, thus confirming the overall situation depicted in Fig. 1.

Discussion
To understand the full implications of this study, it is necessary and helpful to compare our results with the observations The answer to that question is found in Fig. 2b,c. At T ¼ 9 K, D L X indeed decreases with decreasing x, but in total only by a few millielectronvolts from x ¼ 0.37 to x ¼ 0.19. A similar behaviour is found at x ¼ 0.23: On increasing the temperature from 9 K to  ARTICLE 100 K (spanning essentially the entire band-inverted region) the observed gap is reduced only minutely. Hence, D L X 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 surface 15 . 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 temperature 17 , 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 structure 18 and various closely related IV-VI solid solutions undergo ferroelectric phase transitions 19 . Also, in (Pb,Sn)Se the transverse-optical phonon mode at G softens as the temperature is lowered and the SnSe content is increased 20 . Occasionally, for Pb 0.59 Sn 0.41 Se even bulk structural changes were reported and found to be suppressed only with increasing carrier densities 21 , suggesting displacive transitions driven by electron-phonon coupling 22,23 .
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 behave 24,25 , both of single crystals and of thin films 26,27 , 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 electric 28 or magnetic fields 15 . 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 here 15 . If a domain structure is formed, then a topologically protected 1D conducting state percolating through the surface along the domain walls would be the result 5 . 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.

Methods
Crystal growth and characterization. The n-type (Pb,Sn)Se single crystals with typical doping levels of a few 10 18 cm À 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 spectroscopy 29,30 . 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 photoelectrons 31 . The overall crystal-momentum and energy resolution is better than 0.008 Å À 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 L À Á . 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 L. In case the single line (full width at half maximum) at L is found to be broader by 45 meV than the broadest two-line constituent, we call the states gapped and the two-line peak separation determines the gap size D L . 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.