Symmetry dictated grain boundary state in a two-dimensional topological insulator
Hyo Won Kim, Seoung-Hun Kang, Hyun-Jung Kim, Kisung Chae, Suyeon Cho, Wonhee Ko, Se Hwang Kang, Heejun Yang, Sung Wng Kim, Seongjun Park, Sung Woo Hwang, Young-Kyun Kwon, Young-Woo Son
SSymmetry dictated grain boundary state in a two-dimensional topological insulator
Hyo Won Kim, ∗ Seoung-Hun Kang, Hyun-Jung Kim, Kisung Chae, SuyeonCho, Wonhee Ko,
1, 4
Se Hwang Kang, Heejun Yang, Sung Wng Kim, SeongjunPark, Sung Woo Hwang, Young-Kyun Kwon,
2, 6 and Young-Woo Son † Samsung Advanced Institute of Technology, Suwon 13595, Korea Korea Institute for Advanced Study, Seoul 02455, Korea Division of Chemical Engineering and Materials Science, Ewha Womans University, Seoul, 03760, Korea Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA Department of Energy Science, Sungkyunkwan University, Suwon, 440-746, Korea Department of Physics and Research Institute for Basic Sciences, Kyung Hee University, Seoul, 02447, Korea
Structural imperfections such as grain boundaries (GBs) and dislocations are ubiquitous in solidsand have been of central importance in understanding nature of polycrystals. In addition to theirclassical roles, advent of topological insulators (TIs) offers a chance to realize distinct topologi-cal states bound to them. Although dislocation inside three-dimensional TIs is one of the primecandidates to look for, its direct detection and characterization are challenging. Instead, in two-dimensional (2D) TIs, their creations and measurements are easier and, moreover, topological statesat the GBs or dislocations intimately connect to their lattice symmetry. However, such roles of crys-talline symmetries of GBs in 2D TIs have not been clearly measured yet. Here, we present the firstdirect evidence of a symmetry enforced Dirac type metallic state along a GB in 1T’-MoTe , a proto-typical 2D TI. Using scanning tunneling microscope, we show a metallic state along a grain boundarywith non-symmorphic lattice symmetry and its absence along the other boundary with symmorphicone. Our large scale atomistic simulations demonstrate hourglass like nodal-line semimetallic in-gapstates for the former while the gap-opening for the latter, explaining our observation very well. Theprotected metallic state tightly linked to its crystal symmetry demonstrated here can be used tocreate stable metallic nanowire inside an insulator. Grain boundary (GB) or dislocations are importantstructural imperfections in studying nature of polycrys-tals [1]. Besides their classical properties, nowadays,various interesting topological states along the GBs ordislocations can be realized inside topological insulators(TIs) [2–7], although their direct detection and charac-terization are challenging [5, 7]. In two-dimensional (2D)TIs, interesting topological properties of GBs [7, 8] canbe detected more easily using local probes. Among manytwo-dimensional (2D) materials, various grain boundarieswith distinct spatial symmetries can be realized in a2D transition metal dichalcogenide, 1T’-MoTe thanksto its special crystal structure. Because the 1T’ phaseof MoTe can be understood as a static lattice distortionfrom its more symmetric but unstable 1T phase [9], thereare chances to create disparate structural phase bound-aries between crystalline domains with different orienta-tions while growing the 1T’ structure (hereafter we willdrop 1T’ for convenience). Our study of one-dimensional(1D) topological metallic states along GBs of MoTe was based on scanning tunneling microscopy (STM) andspectroscopy (STS). In as-grown MoTe sample surfaces,we observed an unprecedented GB structure satisfyinga non-symmorphic glide-reflection symmetry and topo-logical metallic state associated with it. Quite contraryto this, we could distinguish a semiconducting bound-ary state along the other boundary with a symmorphicmirror symmetry. Our measurement of existence of topo-logical states depending on the symmetries of GBs wasfurther confirmed by large scale atomistic simulations. We first identify the topological nature of the surfacelayer of an as-grown MoTe bulk sample. Although wemeasured tunneling spectra on the surface of the bulksample, our measurements clearly indicate that the top-most layer behaves as a single layer detached from therest of sample. Such a behavior has already been ob-served in a similar material [10] as well as other layeredcrystals, e.g., graphene-like behavior of topmost layer ofgraphite [11] and Bi surface layer [6]. In our samplewith GBs, the presence of GBs in the topmost layer pro-motes the lattice mismatch between the topmost layerand the others, further ensuring the single layer spec-trum in the topmost one. Our simulated STM imagesand STSs also support the single-layer-like behavior ofthe topmost layer of the bulk MoTe sample. The simu-lated images of the single layer MoTe with varying ap-plied voltage also agree with our observations very well(Supplementary Section I). Thanks to the effective isola-tion, we were able to measure the topologically protectedmetallic state along the truncated step edge (Supplemen-tary Section II) by its dI/dV spectra and STM imagesagree well with previous studies [10, 12, 13].In MoTe , the chalcogen atoms (Te atoms) form quasi-one-dimensional parallel chains where adjacent chains(denoted by Te1 and Te2 rows, respectively) have al-ternative height variation with respect to the transitionmetal plane as shown in Fig. 1a. The inequivalent rows ofTe atoms will play an important role in determining thelocal crystalline symmetry along GBs later. The heightalternation spots in our STM images with different con- a r X i v : . [ c ond - m a t . m e s - h a ll ] J u l FIG. 1. (Color online) Morphology and electronic structureof a 1T’-MoTe . a, Schematic illustration of 1T’-MoTe . Thegreen box represents a 1T’-MoTe unit cell containing Te1 andTe2 with a height difference relative to the transition metalplane. b, c, STM topograph of the 1T’-MoTe surface neara defect at Vs = 20 mV and Vs = 300 mV, respectively (It =0.5 nA). Scale bars are 1 nm. The contrast between Te1 andTe2 in the STM image depends on the applied voltages. d,e, experimentally obtained and DFT-calculated dI/dV spectraat Te rows, Te1 (black line) and Te2 (blue line), respectively. trast. We found that this contrast difference between thetwo chains also depends on the applied bias voltages; forexample, the contrast at 300 mV is reversed in Fig. 1ccomparing to that at 20 mV in Fig. 1b (A defect is usedas a marker to identify Te1 and Te2 rows in Figs. 1b andc.) (See details in Supplementary Section III). To figureout the effect of the electronic state, we measured thelocal dI/dV spectra of Te1 and Te2 rows. As is matchedvery well with images, the values of dI/dV at differentrows cross each other at 20 mV in Fig. 1d. The dif-ferent energy levels of Te1 and Te2 p-orbitals from ourfirst-principles simulation can explain alternating contri-butions from two chalcogen atoms in determining crys-tal structures of MoTe precisely (Supplementary SectionIII).Having established a single-layer-like behavior of thetopmost layer, now we turn to formation mechanism ofGBs. Suppose that a structural phase transition of a ma-trix from the high symmetry (1T) phase to the low one(1T’) through dimerization of adjacent Mo rows occursin multiple locations. There are six orientation variants,resulted from the six-fold improper rotational symmetry(see black and grey dotted arrows in Fig. 2a). Excludingtrivial and high energy structures, the interfaces betweentwo 1T’ phase crystals in different orientations can makefour inequivalent symmetric tilt GBs with the angles ofeither 60 ◦ or 120 ◦ , which can be categorized by the GBoperations that map the right side onto the left of crys- FIG. 2. (Color online) Possible grain boundaries from sixorientation variants of 1T’-MoTe . a, Three symmetry-equivalent directions (black arrows) of structural distortionin the 1T structure and their opposite three directions (greendotted arrows), which are distinguished by the directions to-wards top and bottom Te atoms, respectively. The rectangu-lar is a primitive unit cell of the 1T’ phase, which correspondsto the 1 ×√ b–e, Possible grainboundary structures determined by the direction of structuraldistortion with respect to the boundary. To help guide theeyes, dimerized Mo atoms are shaded in blue and black colors. tal structure. We identified that those GB operationsare 60 ◦ mirror, 60 ◦ glide-reflection, 120 ◦ two-fold rota-tion and 120 ◦ screw, and the corresponding boundarystructures are shown in Figs. 2b-e. Those low energyGBs were found by a simple GB model considering pointgroup symmetry in addition to the coincidence site lat-tice theory (Supplementary Section IV). Furthermore, wefound that 60 ◦ glide-reflection and 120 ◦ two-fold rotationboundaries are energetically favorable ones because theirstrains are smaller than the others. We confirmed that FIG. 3. (Color online) STM topograph and dI/dV spectra of60 ◦ glide reflection and 120 ◦ two-fold rotation boundaries of1T’-MoTe . a, STM topograph of 60 ◦ glide reflection bound-ary (Vs = 0.3 V, I = 0.5 nA). b, Averaged dI/dV spectrataken at left area (black line), and right area (blue line) anddI/dV spectrum obtained at the position indicated by the reddot in a (red line). c, STM topograph. d, dI/dV maps ob-tained over the area shown in c for bias voltages V = 100, 0and −
70 mV from top to bottom. e, STM topograph of the120 ◦ two-fold rotation boundary (Vs = 0.3 V, I = 0.1 nA). f, Averaged dI/dV spectrum taken at left area (black line),and dI/dV spectrum obtained at the position indicated bythe white dot in e (green line). g, STM topograph, h, dI/dVmaps obtained over the area shown in g for bias voltages V= 200, 85, 0, − −
100 and −
200 mV from top to bottom.The dashed white lines are used as guides indicating 120 ◦ boundaries. those two structures match the experimentally observedones (Figs. 3a, e) and are approximately 110 meV and570 meV more stable than the other structures, respec-tively. We note that the 120 ◦ two-fold rotation boundarywas theoretically predicted as a twin boundary inducedby strain in single crystalline 1T’-MoTe [14], which wasalso observed experimentally [15, 16] while the 60 ◦ glide-reflection boundary has never been reported yet.At the 60 ◦ glide-reflection boundary, the STM imagein Fig. 3a shows its non-symmorphic symmetry natureclearly. The bright rows of chalcogen atoms in the rightside of the boundary show abrupt discontinuities to theleft side ones. The bright rows in the right side can maponto the left ones by simultaneous operations of mirrorand a half-unit translation with respect to the GB. Themeasured dI/dV spectrum at the boundary in Fig. 3b(red line) reveals a peak near −
70 mV whereas the localdI/dV spectrum far away from it (black and blue lines)shows no feature at −
70 mV similar with the local spec-trum of the pristine sample (Fig. 1d). We note that theobserved spectrum in Fig. 3b is very similar with recentlyreported topological edge states in 1T’-WTe [10, 12, 13]and our results on the step edge of 1T’-MoTe (Supple- FIG. 4. (Color online) Theoretical electronic structures of60 ◦ glide-reflection and 120 ◦ two-fold rotation boundaries. a, Simulated STM image of 60 ◦ glide reflection boundary onthe Vs = 0.3 V. b, Simulated dI/dV at left area (black line)and boundary (red line) denoted by black and red dots in a,respectively. c, Band structure for the 60 ◦ glide reflectionboundary with twin boundary structure. d, Simulated STMimage of 120 ◦ two-fold rotation boundary on the Vs = 0.3 V.e, Simulated dI/dV at left area (black line) and boundary(red line) denoted by black and red dots in d, respectively. f, Band structure for the 120 ◦ two-fold rotation boundary withtwin boundary structure. mentary Section II). The peak is spatially localized at theboundary within 2 nm width as clearly seen in the spa-tial dI/dV maps in Fig. 3d and Supplementary SectionII.The spatially and energetically localized one-dimensional metallic state along the GB with non-symmorphic symmetry is in sharp contrast to states inthe other boundary with symmorphic symmetry. Unlikethe case of the 60 ◦ glide-reflection boundary, the STMimage of the 120 ◦ two-fold rotation boundary shows thecontinuous bright rows across the boundary with kinksas shown in Fig. 3e. So, the right side STM image canmap onto the left one by two-fold rotation operation.Its dI/dV spectrum in Fig. 3f also shows the starkdifferences: two peaks at 85 and -30 mV (green line thatlocate away from the energetic position of the gap at −
70 mV (black line)). The spatial distributions of thepeaks at 85 and −
30 mV in Fig. 3h indicate that thestates associated with the peaks are broadly distributedalong the boundaries and that their intensities are quitelower than that of the 60 ◦ glide-reflection boundary(further dI/dV maps are in Supplementary Section V).The band crossing in our calculated band structuresin Fig. 4c. clearly explains the origin of the metal-lic states in the 60 ◦ glide-reflection boundary. Thenon-symmorphic symmetry along the 60 ◦ glide-reflectionboundary can realize the partner change along the in-variant line and forces the formation of one-dimensionalhourglass [17] type metallic GB states as shown in Fig. 4c.Since the non-symmorphic symmetry along the GB guar-antees the band crossing with a linear dispersion at thelow energy [18], the spectroscopic signature in Fig. 3b and4b looks quite similar with one shown in the edge of topo-logical insulating MoTe . Contrary to this case, however,the symmorphic symmetry along the 120 ◦ two-fold rota-tion boundary cannot form the band crossing (Fig. 4f).Instead, the topological edge states from the left side ofthe GB interact with those from the right side resultingin gapped boundary states as shown in Fig. 4f. So, theband edges in the boundary states make two peaks be-low and above the charge neutral energy shown in Figs. 3fand 4e (Supplementary Section VI).Our theoretical simulations of STM images and dI/dVspectra agree with experimental observations quite well.In Figs. 4a and d, the simulated STM images for the twodifferent GBs structures are displayed. The images of thefully relaxed atomic geometries are computed with theconstant current condition [19] and shows that the pat-terns of bright rows of chalcogen atoms clearly respectthe underlying crystal symmetries of GBs. We also sim-ulated the local dI/dV spectra right on top of GBs andaway from them as shown in Figs. 4b and e. The result-ing spectra also agree well with our observations showinga single peak at the energetic position of bulk gap forthe 60 ◦ glide-reflection boundary, while two split peaksfor the 120 ◦ two-fold rotation boundary. Our projected1D band structures along the momentum parallel to thegrain boundary direction (Figs. 4c and f) show sharp dif-ferences between two GBs. In Fig. 4c for the formerstructure, the entangled metallic bands fill the band gapof the bulk MoTe , while the split conduction and va-lence band states inside the gap are shown in Fig. 4f forthe latter. So, it is evident that these contrast featuresof the band structures are responsible for the differentdI/dV spectra for the two disparate GBs.Considering that the typical growth method for theTMDs [20], the GB between crystalline domains with dif-ferent orientations are inevitable. Thus, our experimen-tal demonstration of protected metallic states here notonly provides the first direct evidence of existence of thesymmetry enforced Dirac type metallic state along thedislocated atomic defects with the distinct crystal sym-metry but also shows a possible route to draw the robustmetallic nanowires inside the insulator once the growthof crystalline structure of TMDs can be controlled. Sample preparation
High quality 1T’-MoTe single crystals were synthe-sized using NaCl-Flux method. Stoichiometric mixtureof Mo and Te powders were sintered with sodium chloride(NaCl) at 1373 K for 30 hours in evacuated silica tubes. Then, the samples were cooled to 1223 K with a rateof 0.5 K/hr. Rapid cooling down to room temperaturewas achieved by quenching with water. 1T’-MoTe sin-gle crystals for this study have large magnetoresistanceand residual resistance ratio, exceeding 32,000 and 350,respectively, indicating that Te deficiency is less than1 % [21]. Scanning tunneling microscopy/spectroscopymeasurements
We performed the experiments in a commercial low-temperature STM (UNISOKU Co., Ltd., Osaka, Japan)at 2.8 K. 1T’-MoTe single crystal sample was cleavedin an ultrahigh vacuum chamber ( 10 − Torr) at roomtemperature and then transferred to the low-temperatureSTM sample stage, where the temperature was kept at2.8 K. The STS measurements were performed using astandard lock-in technique with a bias modulation of5 mV at 1 kHz.
Density functional theory and tight-bindingcalculations
To investigate the structural and electronic propertiesof 1T’-MoTe and their grain boundaries, we performed ab initio calculations based on density functional theory(DFT) [22, 23] as implemented in VASP code. [24, 25]Projector augmented wave potentials [26, 27] was em-ployed to describe the valence electrons, and the elec-tronic wave functions were expanded by a plane wave ba-sis set with the cutoff energy of 450 eV, and the atomicrelaxation was continued until the Hellmann-Feynmanforce acting on every atom became lower than 0.03 eV/˚A .For more precise calculations, we included the dipole cor-rection. The Perdew-Burke-Ernzerhof (PBE) form [28,29] was employed for the exchange-correlation functionalin the generalized gradient approximation (GGA). TheBrillouin zone (BZ) was sampled using a 10 × × k -grid for the primitive unit cell of 1T’ MoTe . The spin-orbit coupling (SOC) effect and on-site Coulomb repul-sion (U) are included in all calculations. These param-eters have been thoroughly tested to describe the exactcharacteristic in the 1T’ MoTe single layer [30]. Forthe optimization of GB structures of 1T’ MoTe , weadopted the basis consisting of pseudo-atomic orbitals(PAOs) with a single- ζ basis set, as implemented in theSIESTA code [31, 32]. The XC functional was treatedwith GGA as used for VASP. The behavior of valenceelectrons was described by a norm-conserving Troullier-Martins pseudopotential [33] with scalar-relativistic ef-fect in the Kleinman-Bylander factorized form [34] 4 d s p s p ≥
30 ˚A throughout the study. The BZwas sampled using a 1 × × × × k -grids for 120 ◦ and 60 ◦ GBs of 1T’-MoTe , respectively. To investigateelectronic properties of the GBs, we started by construct-ing Slater-Koster type tight-binding (TB) model whichsuccessfully reproduces the DFT band structure for themonolayer 1T-MoTe near the Fermi level. Here, we as-sumed five d orbitals on each Mo atom and s , and three p orbitals on each Te atom. Further details of the tight-binding Hamiltonian H in real space are in Supplemen-tary Section VII. Supplementary Section I. Dependence on thenumber of layers of 1T’-MoTe FIG. S1. Layer thickness dependence of 1T’-MoTe . a, Ex-perimental STM results. b, c, DFT-calculated results of a sin-gle and double layer, respectively. STM images were obtainedat 300, 200 and 20 mV for both experiments and calculations.Experimentally obtained dI/dV spectra are quite similar tothat of the single-layer rather than the double-layer. Result-ing different simulated dI/dV characteristics of the single anddouble layers of 1T’-MoTe , the heights changes of Te1 andTe2 rows are differently varied depending applied bias volt-ages. In the double layer, Te1 is higher than Te2 at 100, 200and 300 mV. Supplementary Section II. Topologically protectedmetallic state of 1T’-MoTe . FIG. S2. Topological edge states at the edge of 1T’-MoTe .a. STM topography of 1T’-MoTe (Vs = 0.3 V, I = 0.1 nA)and profile of the height along the dashed gray line. b. dI/dVspectra obtained at the positions indicated by black, red andblue dots in a. c. dI/dV spectra taken across the step edgeof a 1T’-MoTe . Supplementary Section III. Bias-dependence ofSTM topographs and dI/dV maps of 1T’-MoTe . FIG. S3. Bias-dependence of STM topographs and dI/dVmaps of 1T’-MoTe . a, STM topographs and b, dI/dV mapsfor sample bias voltages V = 20, 50, 100, 200 and 300 fromleft to right. Scale bar is 1 nm. A defect is in Te2, which isused for a marker. In the STM images obtained at 20, 50 and100 mV Te1 is brighter than Te2. At 200 mV the contrast ofTe1 and Te2 is similar, and at 300 mV Te2 is brighter thanTe1. In the dI/dV maps at 50, 100, 200 and 300 mV Te2is brighter than Te1. c, Band structure of monolayer 1T’-MoTe , d, Contributions of Te1 and Te2 p orbitals plotted onthe band structure. Supplementary Section IV. Symmetry andformation energy of grain boundaries
FIG. S4. Structure of the MoTe . Atomic structures of theMoTe crystal in a, T (P-3m1) and b, T’ (P2 /m) phases.The large white balls represent Mo atoms, and Te atoms aremarked as the smaller balls. A phase transition from T toT’ phase is indicated by the arrows along ± y directions nextto the Te atoms. The primitive unitcell for the T phase isshown in dashed lines in a. c, A point group element of − Here, we demonstrate a grain boundary (GB) modelto provide atomic models of the observed GBs. Themodel provides possible atomic rearrangements at theGB based on coincidence site lattice (CSL) theory andpoint group analysis, especially when the given crystalis stabilized by lowering its symmetry, e.g., Jahn-Tellerdistortion. Instead of constructing a GB directly fromthe original crystal which has lower symmetry, we gen-erate GB models in two-step process. Firstly, we cre-ate a bi-crystal separated by a symmetric tilt GB fromhigh-symmetry intermediate crystal by using CSL theory.The intermediate crystal can be found by searching forthe best-matched-fit molecular geometry from the origi-nal crystal with the lower symmetry [35]. After that, theintermediate crystals in each domain are distorted backto the original crystal.
FIG. S5. Structural units near the two-fold symmetric grainboundaries with a, symmorphic and b, non-symmorphic sym-metries.FIG. S6. Structural units near the mirror-symmetric grainboundaries with a, symmorphic and b, non-symmorphic sym-metries.
In the case of a monolayer MoTe , a distorted octa-hedral motif in a T’ phase can be best-matched to theregular octahedron with a point group symmetry of − by Schoenflies nota-tion), and the space group of the crystal is changed fromP2 /m (no. 11) phase to P-3m1 (no. 164) (T phase) asshown in Figs. S4a and b. In addition to the improper ro-tation, there are three additional two-fold rotation axesperpendicular to the improper rotation axis and threevertical mirror planes (Fig. S4c). This indicates that aGB, of which relative angle of rotation (2 θ ) is 60 ◦ , willcreate a C y -symmetric interface with the GB angle of π − θ , i.e., 120 ◦ (Fig. S5). Similarly, 2 θ of 120 ◦ resultsin M y -symmetric interface with the GB angle of 60 ◦ (Fig.S6). The above also hold when fractional translationis followed, and both of the “symmorphic” and “non-symmorphic” GBs are energetically degenerated prior tothe distortion. Once the distortion is taken into accountas indicated by arrows in Figs. S5 and S6, the energies ofsymmorphic and non-symmorphic GBs become different,and the four GB models as in Figs. 2b-e are constructed.Since each of the models is tractable by GB operationsby which one side of the bi-crystal can be mapped ontothe other, we label each of the GB models as follows: 60 ◦ mirror { M y | } , 60 ◦ glide-reflection { M y | t } , 120 ◦ two-foldrotation { C y | } and 120 ◦ screw { C y | t } , respectively. Supplementary Section V. Differential conductancemaps
FIG. S7. Differential conductance maps. a. STM topographand dI/dV maps of 60 ◦ glide-reflection boundary for bias volt-ages V = 200, 100, 0, − − − −
70 and −
200 mV fromtop to bottom. b. STM topograph and dI/dV maps of 120 ◦ mirror boundary for bias voltages V = 200, 100, 85, 0, − − −
100 and −
200 mV from top to bottom
Supplementary Section VI. DFT-calculated DOS ofTe atoms in 120 ◦ two-fold rotation boundary FIG. S8. DFT-calculated DOS projected on s , p x , p y and p z orbitals of Te atoms in 120 ◦ two-fold rotation boundary. Supplementary Section VII. Tight-binding model for1T’-MoTe To investigate electronic properties of the grain bound-aries, we start by constructing Slater-Koster type tight-binding (TB) model which successfully reproduces theDFT band structure for the monolayer 1T’-MoTe nearthe Fermi level. Here, we assume five d -orbitals on eachMo atom and s − , and three p -orbitals on each Te atom.The tight-binding Hamiltonian H in real space is givenas follows: H = (cid:88) (cid:104) i,j (cid:105) (cid:88) σαα (cid:48) (cid:104) t αα (cid:48) i,j c † i c j + c.c. (cid:105) + H soc , where i ( j ) labels the atomic sites, σ spin and α ( α (cid:48) ) or-bitals, and t αα (cid:48) i,j is a transfer matrix, which can be param-eterized depending on the direction and distance betweenpair of orbitals through the Slater-Koster formula [36].The H soc represents the effect of on-site spin-orbit cou-pling (SOC), H soc = − λ Mo ˆ S · ˆ L Mo − λ Te ˆ S · ˆ L Te where λ Mo(Te) are SOC parameters for Mo (Te) atom,ˆ S is the spin 1/2 operator, and ˆ L Mo(Te) is the angularmomentum operator of Mo (Te) atom, respectively [37].Then, TB parameters are fitted by minimizing fitnessfunction F , F = (cid:88) n, k ω n k ( E TB n k − E DFT n k ) where ω n k is weight at the k-point and n -th eigen-value, and E TB(DFT) n k is eigenvalue obtained by TB (DFT)for monolayer 1T’-MoTe . We use a nonlinear least-squares method bases on the Levenberg-Marquardt al-gorithm [38, 39]. With this procedure, we successfully fitthe DFT band structure for 1T’-MoTe , its topologicalproperties and the orbital character compared with thosefrom the DFT calculations. ∗ E-mail: [email protected] † E-mail: [email protected][1] A. P. Sutton and R. W. Balluffi,
Interfaces in crystallinematerials
Clarendon Press (1995).[2] C. L. Kane and E. J. Mele, Phys. Rev. Lett. , 226801(2005).[3] B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Science , 1757 (2006).[4] M. K¨onig, S. Wiedmann, C. Br¨une, A. Roth, H. Buh-mann, L. W. Molenkamp, X.-L. Qi, and S.-C. Zhang,Science , 766 (2007).[5] Y. Ran, Y. Zhang, and A. Vishwanath, Nat. Phys. (2009).[6] I. K. Drozdov, A. Alexandradinata, S. Jeon, S. Nadj-Perge, H. Ji, R. J. Cava, B. Andrei Bernevig, and A. Yaz-dani, Nat. Phys. , 664 (2014).[7] R. J. Slager, A. Mesaros, V. Juricic, and J. Zaanen, Phys.Rev. B , 241403 (2014).[8] V. Juricic, A. Mesaros, R. J. Slager, and J. Zaanen, Phys.Rev. Lett , 106403 (2012).[9] D. H. Keum, S. Cho, J. H. Kim, D.-H. Choe, H.-J. Sung,M. Kan, H. Kang, J.-Y. Hwang, S. W. Kim, H. Yang,et al., Nat. Phys. , 482 (2015).[10] L. Peng, Y. Yuan, G. Li, X. Yang, J.-J. Xian, C.-J. Yi,Y.-G. Shi, and Y.-S. Fu, Nat. Commun. (2017).[11] G. Li and E. Y. Andrei, Nat. Phys. , 623 (2007).[12] Z.-Y. Jia, Y.-H. Song, X.-B. Li, K. Ran, P. Lu, H.-J.Zheng, X.-Y. Zhu, Z.-Q. Shi, J. Sun, J. Wen, et al., Phys.Rev. B , 041108 (2017).[13] S. Tang, D. Zhang, Chaofan anf Wong, Z. Pedramrazi,H.-Z. Tsai, C. Jia, B. Moritz, M. Claassen, H. Ryu,S. Kahn, J. Jiang, et al., Nat. Phys. , 683 (2017). [14] W. Li and J. Li, Nat. Commun. , 10843 (2016).[15] C. H. Naylor, W. M. Parkin, J. Ping, Z. Gao, Y. R.Zhou, Y. Kim, F. Streller, R. W. Carpick, A. M. Rappe,M. Drndic, et al., Nano Lett. , 4297 (2016).[16] G.-Y. Wang, W. Xie, D. Xu, H.-Y. Ma, H. Yang, H. Lu,H.-H. Sun, Y.-Y. Li, S. Jia, L. Fu, et al., Nano Res. ,569 (2019).[17] Z. Wang, A. Alexandradinata, R. J. Cava, and B. A.Bernevig, , 189 (2016).[18] S. M. Young and C. L. Kane, Phys. Rev. Lett. ,126803 (2015).[19] J. Tersoff and D. R. Hamann, Phys. Rev. B , 805(1985).[20] M. Chhowalla, H. S. Shin, G. Eda, L.-J. Li, K. P. Loh,and H. Zhang, Nat. Chem. (2013).[21] S. Cho, S. H. Kang, H. S. H. S. Yu, H. W. Kim, W. Ko,S. W. Hwang, W. H. Han, D.-H. Choe, Y. H. Jung, K. J.Chang, et al., 2D Mater. , 021030 (2017).[22] P. Hohenberg and W. Kohn, Phys. Rev. (1964).[23] W. Kohn and L. J. Sham, Phys. Rev. (1965).[24] G. Kresse and J. Furthm¨uller, Phys. Rev. B , 11169(1996).[25] G. Kresse, Phys. Rev. B (1993).[26] P. E. Bl¨ochl., Phys. Rev. B , 17953 (1994).[27] G. Kresse and D. Joubert, Phys. Rev. B , 1758 (1999).[28] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.Lett. , 3865 (1996).[29] E. Artacho, D. S¨anchez-Portal, P. Ordej´on, A. Garc´ıa,and J. M. Soler, Phys. Status Solidi B , 809 (1999).[30] H. J. Kim, S. H. Kang, I. Hamada, and Y. W. Son, Phys.Rev. B , 180101 (2017).[31] E. Artacho, E. Anglada, O. Di´eguez, J. D. Gale,A. Garc´ıa, J. Junquera, R. M. Martin, P. Ordej´on, J. M.Pruneda, D. S´anchez-Portal, et al., J. Phys: Condens.Matter , 064208 (2008).[32] D. S¨anchez-Portal, P. Ordej´on, E. Artacho, and J. M.Soler, Int. J. Quantum Chem. , 453 (1997).[33] N. Troullier and J. L. Martins, Phys. Rev. B (1993).[34] L. Kleinman and D. M. Bylander, Phys. Rev. Lett. ,1425 (1982).[35] A. D. Banadaki and S. A. Patala, npj Comput. Mater. , 13 (2017).[36] J. C. Slater and C. F. Koster, G. F. Phys. Rev. , 1498-1524 (1954).[37] A. J. Pearce, E. Mariani and G. Burkard, Phys. Rev. B , 155416 (2016).[38] K. Levenberg, Quart. Appl. Math. , 164 (1944).[39] D. W. Marquardt, SIAM J. Appl. Math.11