Topological phase diagram and saddle point singularity in a tunable topological crystalline insulator
Madhab Neupane, Su-Yang Xu, R. Sankar, Q. Gibson, Y. J. Wang, I. Belopolski, N. Alidoust, G. Bian, P. P. Shibayev, D. S. Sanchez, Y. Ohtsubo, A. Taleb-Ibrahimi, S. Basak, W.-F. Tsai, H. Lin, Tomasz Durakiewicz, R. J. Cava, A. Bansil, F. C. Chou, M. Zahid Hasan
TTopological phase diagram and saddle point singularity in a tunable topologicalcrystalline insulator
Madhab Neupane,
1, 2
Su-Yang Xu, R. Sankar, Q. Gibson, Y. J. Wang,
5, 6
I. Belopolski, N. Alidoust, G.Bian, P. P. Shibayev, D. S. Sanchez, Y. Ohtsubo, A. Taleb-Ibrahimi,
7, 8
S. Basak, W.-F. Tsai, H. Lin,
10, 11
Tomasz Durakiewicz, R. J. Cava, A. Bansil, F. C. Chou, and M. Z. Hasan
1, 12 Joseph Henry Laboratory, Department of Physics,Princeton University, Princeton, New Jersey 08544, USA Condensed Matter and Magnet Science Group, Los Alamos National Laboratory, Los Alamos, NM 87545, USA Center for Condensed Matter Sciences, National Taiwan University, Taipei 10617, Taiwan Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA Department of Physics, Northeastern University, Boston, Massachusetts 02115, USA Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley, California 94305, USA Synchrotron SOLEIL, Saint-Aubin-BP 48, F-91192 Gif sur Yvette, France UR1/CNRSSynchrotron SOLEIL, Saint-Aubin, F-91192 Gif sur Yvette, France Department of Physics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan Centre for Advanced 2D Materials and Graphene Research Centre,National University of Singapore, Singapore 117546 Department of Physics, National University of Singapore, Singapore 117542 Princeton Center for Complex Materials, Princeton University, Princeton, New Jersey 08544, USA (Dated: October 1, 2018)We report the evolution of the surface electronic structure and surface material properties of atopological crystalline insulator (TCI) Pb − x Sn x Se as a function of various material parametersincluding composition x , temperature T and crystal structure. Our spectroscopic data demonstratethe electronic groundstate condition for the saddle point singularity, the tunability of surface chem-ical potential, and the surface states’ response to circularly polarized light. Our results show thateach material parameter can tune the system between trivial and topological phase in a distinctway unlike as seen in Bi Se and related compounds, leading to a rich and unique topological phasediagram. Our systematic studies of the TCI Pb − x Sn x Se are valuable materials guide to realize newtopological phenomena.
PACS numbers:
I. INTRODUCTION
A topological insulator (TI) material differs from aconventional insulator in the sense that a TI featuresmetallic surface states that can only be removed by go-ing through a topological quantum phase transition .Understanding the key parameters of a material thatare relevant to its nontrivial topology is vital for uti-lizing the protected surface states in applications .A topological crystalline insulator (TCI) is a new sym-metry protected topological phase that is protected bycrystalline space group symmetries . The surfacestates of this novel topological phase are predicted tohost many uniquely-new quantum phenomena, includingsurface spin filtering , strain-induced crystalline symme-try protected Chern currents , correlation physics dueto surface electronic singularities , none of which arepossible in the much studied Z TI . The realizationof these proposals requires the ability to control a TCImaterial to be topological or non-topological as a func-tion of various material parameters, and to understandthe properties of the protected surface states under allthese parameter conditions.Recently, the TCI phase is experimentally realized inthe Pb − x Sn x Se(Te) material system . It is interest- ing to study the TCI materials at various sets of param-eters (composition x , temperature T , lattice constant a ,crystal structure, etc.). Studying the Pb − x Sn x Se(Te)system as a function of these material parameters is notjust a material detail, but it is in fact important becausevarying these parameters usually leads to a change oftopology between a nontrival TCI and a trivial state.Furthermore, the exploration of TCI surface state prop-erties and novel utilization are limited due to the lack ofunderstanding of their material phase diagram.In this report, we systematically study the surface elec-tronic groundstate of the Pb − x Sn x Se at various compo-sitions x , temperatures T , and crystal structures by us-ing angle resolved photoemission spectroscopy (ARPES),circular dichroism ARPES (CD-ARPES) and transportmeasurements. Our results reveal a remarkable tunablityin terms of the topological nature and the surface stateelectronic properties as a function of these parameters.Specifically, we show that by varying the composition x ,Pb − x Sn x Se undergoes an electronic band inversion at x ∼ .
20 and a structural transition at 0 . ≤ x ≤ . − x Sn x Se sys-tem in its unique way. Our studies show that the topo-logical nature and the surface state electronic structureof Pb − x Sn x Se are further distinctly sensitive to tem-perature. Moreover, we provide momentum-resolved evi- a r X i v : . [ c ond - m a t . m e s - h a ll ] A ug dence for the existence of a saddle point singularity in themirror-protected surface states of Pb − x Sn x Se. Thesesystematic studies of Pb − x Sn x Se in terms of its topolog-ical nature and the surface state electronic structure asa function of the material parameters provide a valuablematerial reference to study various topological phenom-ena.
II. METHODS
High-resolution ARPES measurements:
Singlecrystals of Pb − x Sn x Se used in these measurements weregrown by standard growth method (see ref. ). Lowphoton energy (15 eV-30 eV) and temperature depen-dent ARPES measurements for the low-lying electronicstructures were performed at the Synchrotron Radia-tion Center (SRC) in Wisconsin and the Stanford Syn-chrotron Radiation Centre (SSRL) in California withR4000 electron analyzers whereas high photon energy( ∼
60 eV) measurements were performed at the Beamlines12 and 10 at the Advanced Light Source (ALS) in Califor-nia, equipped with high efficiency VG-Scienta SES2002and R4000 electron analyzers. Samples were cleaved insitu and measured at 10-300 K in vacuum better than1 × − torr. They were found to be very stable andwithout degradation for the typical measurement periodof 20 hours. Energy and momentum resolution were bet-ter than 15 meV and 1% of the surface Brillouin zone(BZ), respectively. Surface deposition:
Sn deposition ARPES measure-ments were done in The CASSIOPEE beamline, Soleil,France from Sn source, which was thoroughly degassedbefore the experiment. Pressure in the experiment cham-ber stayed 1 × − torr. The deposition rate (˚A/min)was monitored using a commercial quartz thickness moni-tor (Leybold In con Inc., Model XTM/2). The depositionrate of Sn was 0.3 ˚A/min. CD-ARPES measurements:
CD-ARPES measure-ments for the low energy electronic structure were per-formed at the Synchrotron Radiation Center (SRC),Wisconsin, equipped with high efficiency VG-ScientaSES2002 electron analyzer. The polarization purity isbetter than 99% for horizontal polarization (HP) and bet-ter than 80% for RCP and LCP. Samples were cleaved in situ and measured at 20 K in a vacuum better than1 × − torr. Energy and momentum resolution werebetter than 15 meV and 1% of the surface Brillouin zone(BZ), respectively. Synchrotron-based XRD:
Synchrotron-based X-ray diffraction (XRD) measurements were performedwith X-ray beamline at the Spring-8 light source whichhas access to hard X-rays. The synchrotron X-ray diffrac-tion patterns were analyzed using the General Struc-ture Analysis System (GSAS) program following theRietveld profile refining method. The final refinementswere carried out assuming a cubic symmetry with a spacegroup of Fm-3m and taking the pseudo-Voigt function for the peak profiles.
First-principles calculations:
The first-principlescalculations are carried out within the framework of thedensity functional theory (DFT) using projector aug-mented wave method as implemented in the VASPpackage . The generalized gradient approximation(GGA) is used to model exchange-correlation effects.The spin orbital coupling (SOC) is included in the self-consistent cycles. The cutoff energy 260 eV is used forboth PbSe and SnSe systems. To capture the physi-cal essence of the surface state electronic structure ofthe TCI phase in an alloy Pb − x Sn x Se system, we takeSnSe and assume that SnSe crystalizes in the FCC struc-ture and then calculate the (001) surface electronic struc-ture. We also calculate the electronic structure of PbSe(FCC) and SnSe (orthorhombic) using experimentallydetermined crystal structure and lattice constant. Forall calculations, the (001) surface is modeled by periodi-cally repeated slabs of 48-atomic-layer thickness with 24˚A vacuum regions and use a 12 × × k − point mesh over the (BZ). k · p model: We use an effective surface k · p Hamiltonian of topological crystalline insulator on(001) plane to obtain the surface bands dispersion. III. RESULTS AND DISCUSSIONA. Crystal structure and Brillouin zone symmetry
Since the nontrivial topology in a TCI state fundamen-tally relies on crystalline symmetries, we first characterizethe crystal structure of Pb − x Sn x Se. Fig. 1a shows thecrystal structure of PbSe ( x = 0 .
0) and SnSe ( x = 1).Based on X-ray diffraction (XRD) data (Fig. 1b), the x = 0 . x = 1sample has a primitive orthorhombic structure (Pnma, a , b or c are equal and thePb/Sn site is displaced from the center of the unit cell,see Fig. 1a right.We study core level spectra in a binding energy range10 eV - 56 eV, Fig. 1c. For the x = 0 composition, weobserve the Pb 5 d ( ∼ d ( ∼ x = 0 .
23, we observe the Pb 5 d ( ∼ d ( ∼ d ( ∼ x = 1, we observe the Sn 4 d ( ∼ d ( ∼ x values where the crystal has rocksalt structure, the elec-tronic structure is known to have a direct band gap atthe L points . These are projected to the ¯ X points onthe (001) surface, see Fig. 1d. As x varies within therocksalt range, the system undergoes a band inversion ateach L point, see Fig. 1e. The inverted Pb − x Sn x Se(Te)
Crystal structure and Brillouin zone symmetry FIG. 1: (a) Crystal structure of PbSe with ideal NaCl type and SnSe with orthorhombic distortion. (b) X-ray power diffractionpatterns for Pb . Sn . Se and SnSe systems. (c) Core level spectroscopic measurements of PbSe, Pb . Sn . Se and SnSesystems. Various core level energy peaks are marked on the curves. (d) The first Brillouin of Pb − x Sn x Se. The mirror planesare projected onto the (001) crystal surface as the ¯ X -¯Γ- ¯ X momentum cut mirror lines (shown by red solid lines). (e) First-principles calculations of band dispersion along high-symmetry momentum space cuts. Red lines and blue areas represent thesurface and bulk bands, respectively. compositions are expected to show topological surfacestates protected by mirror symmetries. These are thesignatures of the TCI state . B. Tunable topological surface state and saddlepoint singularity.
We systematically study the electronic structure ofthe symmetry-protected topological surface states inPb . Sn . Se ( x = 0 . . Sn . Se ( x = 0 . . However, the saddle pointsingularity, the surface chemical potential tunability andthe spin-orbit polarization have not been experimentallycharacterized with momentum resolution. Here, we focuson these key properties of the symmetry-protected topo-logical surface states, which are crucial for the utilizationof the TCI phase in Pb − x Sn x Se.In Figs. 2a, b we show the dispersion of the topologi-cal surface states in Pb . Sn . Se along the mirror line, ¯Γ- ¯ X -¯Γ, as well as along the ¯ M - ¯ X - ¯ M direction, which isperpendicular to the mirror line. We observe two sur-face Dirac cones along the mirror line, on opposite sidesof the ¯ X point. We find that these two Dirac cones arenear each other in momentum space, so they hybridizewith each other at binding energies far from the energyof the Dirac point, E DP . In Fig. 2c we show constant en-ergy contour maps at different binding energies, E B . Wefind that the Dirac points are at E B = E DP = 70 meV.At energies E B slightly above or below E DP , the con-stant energy contour evolves from two points into twounconnected pockets. As we scan in E B , we find thatthe size of the pockets increases until they come togetherand undergo a Liftshitz transition. A Lifshitz transitionin the band structure is associated with a van Hove sin-gularity in the density of states (DOS), since the bandstructure is nearly flat around the saddle point wherethe bands touch . This observation is also predictedby our first-principles calculation of the surface and bulkDOS, see Fig. 2d. For still higher E B , the constant en-ergy contour changes into two concentric contours bothenclosing the ¯ X point. The inner contour shrinks with in- Tunable topological surface state and saddle point singularity. FIG. 2: Observation of thesaddle point singularity. (a)Schematics of surface banddispersion of the TCI phasealong the mirror line ¯Γ- ¯ X -¯Γ and the perpendicular tothe mirror line ¯ M - ¯ X - ¯ M mo-mentum space cut-directions.Five important features of thesurface state, including Diracpoint of the upper Dirac cones(UDP), van Hove singular-ity of the upper Dirac cones(VH1), two Dirac points alongthe ¯Γ- ¯ X -¯Γ mirror line (DP),van Hove singularity of thelower Dirac cones (VH2) andDirac point of the lower Diraccones (LDP) are marked. (b)ARPES measured dispersionsalong the ¯Γ- ¯ X -¯Γ and ¯ M -¯ X - ¯ M momentum space cut-directions, where five featuresare also marked. (c) Experi-mental observation of the Lif-shitz transition - the bindingenergies are noted on the con-stant energy contours. (d)Calculated density of state(DOS) for the surface statesand the bulk bands using the k · p model . ARPES datapresented here were measuredat low temperature (T ∼ creasing E B until it forms an upper Dirac point at large E B . In summary, there are five important energies inthe surface state band structure: the Dirac point (DP),the Lifshitz transition energy for the upper Dirac surfacestates ( E VH1 ) and lower Dirac surface states ( E VH2 ), aswell as the Dirac points for the upper and lower Diracsurface states (UDP and LDP), all of which are noted inFigs. 2a-c.In order to identify these key features, especially theinteresting van Hove singularities, we further study thesurface states’ electronic structure. We focus on the con-stant energy contour at the Lifshitz transition energy, E B = 40 meV (Fig. 3a). The two dots (green and blue)in Fig. 3a mark the momentum space locations where thetwo unconnected contours merge. Their energy and mo-mentum space coordinates are experimentally identifiedto be ( E B , k x , k y ) = (40 meV , , ± .
02 ˚A − ). To exper-imentally establish the saddle point band structure, wefocus on the blue dot in Fig. 3a and study the energy-momentum dispersion cuts along three important mo-mentum space cut-directions, labelled as cut 1, 2, and 3.Cut 1 and 2 (Figs. 3c and d) show the dispersion alongthe horizontal ( k x ) and vertical ( k y ) directions across the blue dot. Interestingly, the blue dot is found to be a lo-cal band structure minimum along cut 1 shown in Fig.3c, whereas it is a local maximum along cut 2 (Fig. 3d).Observation of a local minimum and local maximum atthe same momentum space location (the blue dot) un-ambiguously shows that it is a surface band structuresaddle point. The observation of a surface momentum-space saddle point implies that there exists certain inter-mediate cut-directions (between cut 1 and 2), where thesurface band structure is flat in the vicinity of the bluedot. Indeed, as shown in Fig. 3e, for cut 3, we found thatthe surface states are nearly flat near the saddle point.The observed flat band structure (along cut 3) is asso-ciated with a divergence of the surface density of states(DOS), a surface van Hove singularity (VHS). Indeed, apronounced peak is observed at the energy correspond-ing to the saddle point, namely E B = 40 meV, as labelledby “VH1” in Fig. 3f. Additionally, we observe a signif-icant dip of angle-integrated photoemission intensity atthe binding energy of E B = 70 meV, which correspondsto the energy of the Dirac points (DP). In order to betterhighlight the other three features (UDP, LDP, VH2), wepresent a second derivative of the angle-integrated pho- Tunable topological surface state and saddle point singularity. FIG. 3: Observation of the saddle point curvature. (a) ARPES constant energy contour in the vicinity of an ¯ X point inthe (001) surface BZ of Pb . Sn . Se at binding energy 40 meV, which corresponds to the saddle point energy of the upperDirac cones. The ¯ X points locate at ( k x , k y ) = ( ± . − ,
0) or (0 , ± . − ). The momentum space coordinate of one ¯ X point is redefined to be ( k x , k y ) = (0 ,
0) for simplicity of presentation. The blue and green dots denote the momentum spacelocations of the two surface saddle points. The blue dotted lines indicate the momentum space cut-directions for cuts1 (alongthe ¯ M - ¯ X - ¯ M momentum space cut), cut2 (parallel to the ¯Γ- ¯ X -¯Γ momentum space cut), and cut3 (in between cut1 and cut2),which are centered at the blue dot. (b) Calculated surface state constant energy contour at the saddle point singularity energy(top), where outer blue elliptical and inner red circular features represent the contour from outer cone and inner child conerespectively, and a three-dimensional schematic of a saddle point (bottom). (c-e) ARPES dispersion maps (left) and theirsecond derivative images along cuts 1, 2, and 3. The white and green arrows point the saddle points (blue and green dots inPanel (a). (f) Momentum ( k x and k y ) integrated ARPES intensity as a function of binding energy (left). 2 nd derivative of theARPES intensity with respect to binding energy is presented to further highlight the features (right panel). The upper Diracpoint (UDP), upper van Hove singularity (VH1), Dirac point (DP), lower van Hove singularity (VH2) and lower Dirac point(LDP) are marked. (g) ARPES dispersion maps upon in situ Sn deposition on the Pb . Sn . Se surface. The dosage (time)for Sn deposition is noted. A different batch of sample, which is p − type with the chemical potential located below the Diracpoints, is used for the Sn deposition data shown in this panel. ARPES data presented here were measured at low temperature(T ∼
10 K) with photon energy of 18 eV. toemission intensity (Fig. 3f right), where all five featuresare identified and show qualitative correspondence withthe theoretical calculated results in Fig. 2d.Here, we demonstrate the ability to control the sur- face chemical potential via surface chemical deposition.As shown in Fig. 3g, Sn atoms are deposited onto thesurface of a p -type Pb . Sn . Se sample, whose nativechemical potential is found to be below the Dirac point.
Tunable topological surface state and saddle point singularity. FIG. 4: (a) Fermi surface plot of Pb . Sn . Se. (b) Circu-lar dichroism measurements of Pb . Sn . Se. The ARPESspectra are measured using the light of circularly polarizedright (CPR) and left (CPL) along two different momentamarked as D1 and D2 in (a). The circular dichroism is the dif-ference between those two spectra (CD= CPR − CPL). Thesedata were measured at temperature of 20 K with photon en-ergy of 18 eV at SRC U9 VLS-PGM beamline. (c) Schematicview of the CD results.
The deposition rate of Sn on the surface of the sample isestimated to be about 0.3 ˚A per minute. Our Sn surfacedeposition shows that the surface chemical potential canbe shifted across the energies of the Dirac point and thevan Hove singularity.We further perform circular dichroism (CD) ARPESmeasurements in attempt to probe the spin-orbit polar-ization of the topological surface states. Previously, spin-resolved ARPES results were reported on TCI states .CD-ARPES not only brings insight to the spin-orbitpolarization of a topological surface state, as was al-ready shown in TI Bi Se , but also holds promiseto manipulate the spin texture via circular polarizedphoton excitations . However, there have beenno CD-ARPES measurements performed on the TCIPb − x Sn x Se(Te) systems. Fig. 4b shows ARPES disper-sion maps of surface states measured using right circu-larly polarized (RCP) light and left circularly polarized(LCP) light for a p -type Pb . Sn . Se. The directionsof the dispersion cuts are noted as D1 and D2 in Fig.4a. A clear surface state CD response on the photoelec-tron signal from the lower Dirac cone surface states isobserved where the + k Dirac branch shows stronger re-sponse for RCP light and the − k Dirac branch showsstronger response for LCP light (Fig. 4b) for both lowerDirac cones. The magnitude of the CD response signaldefined, as I CD =( I RCP − I LCP )/( I RCP + I LCP ) is ob-served to be about 20% for incident photons with energy 20 eV at binding energy 100 meV, well below the chem-ical potential. The CD signal observed in Fig. 4b isconsistent with the expected spin-orbital texture of thesurface states (shown schematically in Fig. 4c).
C. Topological phase transitions and topologicalphase diagram for Pb − x Sn x Se.
We now study the topological nature of Pb − x Sn x Seas a function of the composition ( x ), temperature ( T ),and crystal structure. We note that the temperatureand composition dependent studies of Pb − x Sn x Se (with0 ≤ x ≤ . Our goalhere is to map out an entire topological phase diagram forPb − x Sn x Se (0 ≤ x ≤ . Sn . Se atdifferent temperatures. At a low temperature, T = 20K, we observe two Dirac cones near the ¯ X point alongthe ¯Γ- ¯ X -¯Γ cut, which are associated with the nontrivialTCI phase. We repeat this measurement at 100 K, 180K and 200 K, and we find that as the temperature risesthe two Dirac points move closer to each other. At ∼ X point. Themomentum space distance between the two Dirac pointsare measured to be 0 .
05 ˚A − , 0 .
03 ˚A − , 0 .
01 ˚A − , and0 ˚A − at 20 K, 100 K, 180 K and 200 K, respectively.Finally, at 300 K we observe that a gap opens in the sur-face states, reflecting the transition into a topologicallytrivial phase (rightmost panel, Fig. 5a-b).To further study the thermodynamic evolution ofthe surface state electronic structure, we performsynchrotron-based X-ray diffraction measurements onthe sample as a function of temperature. As shown inFig. 5c, the peak of the diffraction 2 θ angle is clearlyfound to move towards smaller values with rising tem-perature showing that the lattice constants increase withincreasing temperature (Fig. 5d). The variation of themomentum distance between two Dirac points with tem-perature (Fig. 5d, right axis) shows an evolution of thetrivial state from TCI. This is qualitatively consistentwith thermodynamic expansion of the lattice constant.We study the TCI phase in Pb − x Sn x Se as a functionof composition x . Fig. 6 shows ARPES measurementsof the low energy states of various compositions togetherwith the two end compounds, namely PbSe and SnSe.For PbSe, we observe low-lying bulk conduction and va-lence bands with a clear band-gap of ∼ .
15 eV. Thissuggests that the system is topologically trivial for x = 0.As x is increased (see Fig. 6 ), the low-lying bulk bandsare observed to approach each other, and eventually theirenergy levels invert. We find that the critical composi-tion for band inversion is at x ∼ .
20. It is interesting tonote that the spectrum of the x = 0 . ∼
20 K) shows the presence of weak spec-tral weight in the bulk band gap and at the boundary
FIG. 5: Thermal evolution ofthe topological surface states(TCI → Trivial). (a) Disper-sion maps along the mirrorline ¯Γ- ¯ X -¯Γ at different tem-peratures. These data aremeasured with photon energyof 18 eV at SSRL BL 5-4.(b) Energy dispersion curves(EDCs) of the selected spec-tra shown in (a). The mea-sured temperatures are notedon the plots. The blue dashlines are guide to the eye. (c)Synchrotron-based tempera-ture dependent X-ray diffrac-tion (SXRD) measurementsfor Pb . Sn . Se. The peakis observed to shifts towardsthe lower angles with in-creasing temperature, whichconfirms the picture of thethermal expansion of lattice.(d) Experimentally measuredvalues of the lattice con-stant at different tempera-tures (left axis). The mo-mentum distance between twoDirac points as a functionof temperature is plotted inthe right axis. The errorbar represents an uncertaintyof estimating the momentumdistance between two Diracpoints.FIG. 6: Topological phase transition in Pb − x Sn x Se. ARPES measured dispersion map of PbSe, Pb . Sn . Se, Pb . Sn . Se,Pb . Sn . Se, Pb . Sn . Se, Pb . Sn . Se and SnSe. Note that PbSe, Pb . Sn . Se and SnSe are observed to be insulators. TheDirac-like surface state dispersion is observed in Pb . Sn . Se, Pb . Sn . Se, Pb . Sn . Se while no surface state is observedin Pb . Sn . Se. PbSe, Pb . Sn . Se, Pb . Sn . Se, Pb . Sn . Se and Pb . Sn . Se are measured at low temperature ( ∼ . Sn . Se sample is p − typewhere the Fermi level is below the Dirac point. Energy gap is observed in Pb . Sn . Se and SnSe measured at temperature of100K. All these spectra were measured parallel to the ¯ M - ¯ X - ¯ M momentum cut. Topological phase transitions and topological phase diagram for Pb − x Sn x Se. FIG. 7: Topological phase diagram of the Pb − x Sn x Se system. For composition range of 0 < x < .
45, the system is inthe single crystalline FCC phase. The bulk band of Pb − x Sn x Se undergoes a band inversion with Pb/Sn substitution. Thecritical composition x c is ∼ .
20 depending on the temperature. The conduction and valence band states representing oddand even parity eigenvalues are marked as L − and L +6 , respectively. For composition range of 0 . < x < .
75, the systemshows multi-structural-phase (cubic and orthorhombic phases coexist. See the XRD data in the inset for x = 0 . . X point. The inset in the bottom right conner shows the resistivitymeasurements on SnSe, which proves its insulator nature. BS FS and SS FS denote the bulk state Fermi surface and surfacestate Fermi surface, respectively. The arrows at bottom note the compositions where our ARPES studies have been performed.For composition range of 0 . < x <
1, the system is in a single crystalline orthorhombic phase. of the bulk valence and conduction bands. These maybe preformed surface states in the vicinity of topologicalphase transition. Analogous preformed states have re-cently been observed experimentally . Next, we con-sider the other end compound, SnSe, as shown in theright panel in Fig. 6. At the Fermi level, no electronicstates are observed. Instead, a fully gapped electronicstructure with the chemical potential inside the band gapis found. This is because SnSe is in primitive orthorhmo-bic structure . This crystal structure breaks both in-version and mirror symmetry, removing the fundamentalsymmetries which support the TCI phase. Our obser-vation of an insulating state in SnSe is consistent withprevious studies where the band gap is reported to be aslarge as ∼ . Our systematic ARPES studies andtransport measurements show a rich topological phase di-agram in Pb − x Sn x Se. A summary is presented in Fig. 7.The blue and red lines represent the energy levels of the lowest lying bulk conduction and valence bands. Start-ing from PbSe ( x = 0), the system has a non-invertedband-gap of ∼ .
15 eV. As x increases, band inversiontakes place and the system enters the TCI phase. Theinverted band-gap increases until the system enters themulti-(crystal structure)-phase regime at x (cid:38) .
45, wherecubic and orthorhombic structures coexist. Finally, for x (cid:38) .
75, the system becomes a large band gap trivial in-sulator with primitive orthorhombic structure. Distinctphase transitions are observed, labelled x c1 , x mp (where‘mp’ stands for multi-phase) and x c2 . The first transition( x c1 (cid:38) .
2) is due a decrease in the lattice constant, whichincreases the effective spin-orbit strength, while the sec-ond transition ( x mp (cid:38) .
45) corresponds to the coexis-tence of cubic and orthorhombic structures and the thirdtransition ( x c2 (cid:38) .
75) is the result of a drastic change incrystal structure to orthorhombic phase. Therefore, ourexperimental data reveal a delicate relationship among
Topological phase transitions and topological phase diagram for Pb − x Sn x Se. − x Sn x Se.
IV. CONCLUSION