A full gap above the Fermi level: the charge density wave of monolayer VS2
Camiel van Efferen, Jan Berges, Joshua Hall, Erik van Loon, Stefan Kraus, Arne Schobert, Tobias Wekking, Felix Huttmann, Eline Plaar, Nico Rothenbach, Katharina Ollefs, Lucas Machado Arruda, Nick Brookes, Gunnar Schoenhoff, Kurt Kummer, Heiko Wende, Tim Wehling, Thomas Michely
AA full gap above the Fermi level: the chargedensity wave of monolayer VS Camiel van Efferen, ∗ , † Jan Berges, ‡ , ¶ Joshua Hall, † Erik van Loon, ‡ , ¶ StefanKraus, † Arne Schobert, ‡ , ¶ Tobias Wekking, † Felix Huttmann, † Eline Plaar, † NicoRothenbach, § Katharina Ollefs, § Lucas Machado Arruda, (cid:107)
Nick Brookes, ⊥ Gunnar Sch¨onhoff, ‡ , ¶ Kurt Kummer, ⊥ Heiko Wende, § Tim Wehling, ‡ , ¶ andThomas Michely † † II. Physikalisches Institut, Universit¨at zu K¨oln, Z¨ulpicher Straße 77, 50937 K¨oln,Germany ‡ Institut f¨ur Theoretische Physik, Universit¨at Bremen, Otto-Hahn-Allee 1, D-28359Bremen, Germany ¶ Bremen Center for Computational Materials Science, Universit¨at Bremen, Am Fallturm1a, D-28359 Bremen, Germany § Fakult¨at f¨ur Physik und Center f¨ur Nanointegration Duisburg-Essen (CENIDE),Universit¨at Duisburg-Essen, Carl-Benz-Straße, 47057 Duisburg, Germany (cid:107)
Institut f¨ur Experimentalphysik, Freie Universit¨at Berlin, Arnimallee 14, 14195 Berlin,Germany ⊥ European Synchrotron Research Facility (ESRF), Avenue des Martyrs 71, CS 40220,38043 Grenoble Cedex 9, France
E-mail: eff[email protected] a r X i v : . [ c ond - m a t . m t r l - s c i ] J a n bstract In the weak-coupling Peierls’ view, charge density wave (CDW) transitions aremetal-insulator transitions, creating a gap at the Fermi level. However, with strongelectron–phonon coupling, theoretically the effects of the periodic lattice distortioncould be spread throughout the electronic structure and give rise to CDW gaps awayfrom the Fermi level. Here, using scanning tunneling microscopy and spectroscopy, wepresent experimental evidence of a full CDW gap residing in the unoccupied states ofmonolayer VS . Our ab initio calculations show anharmonic coupling of transverse andlongitudinal phonons to be essential for the formation of the CDW and the full gapabove the Fermi level. The CDW induces a Lifshitz transition, i.e., a topological metal–metal instead of a Peierls metal–insulator transition. Additionally, x-ray magneticcircular dichroism reveals the absence of net magnetization in this phase, pointing toa coupled CDW–antiferromagnetic ground state. The many-body ground states of two-dimensional (2D) materials, wherein the reduceddimensionality leads to the enhancement of correlation effects, have been extensively re-searched in recent years. Of particular interest are the coexistence or competition betweena charge density wave (CDW) phase, as found in many 2D transition metal dichalcogenides(TMDCs), with superconducting and magnetic phases.
Since these phases can be stronglydependent on the substrate or the defect density, the intrinsic properties of 2D materialsare difficult to determine experimentally. Added to this, CDWs themselves are the subjectof an ongoing controversy regarding the driving force behind the CDW transition and theexact structure of the electronic system in the CDW phase.
CDWs in 2D materials do not necessarily show the Peierls type behavior expected fora one-dimensional chain of atoms, where lattice distortions open an electronic gap at thenesting wavevector. In many (quasi-)2D cases, CDWs form in the complete or partial absenceof Fermi surface nesting, suggesting that the driving mechanism behind their formation liesbeyond a simple electronic disturbance. It has been questioned whether the concept of2esting is essential for understanding CDW formation.
Instead, a strong and wavevector-dependent electron–phonon coupling is often predicted to be the driving force behind thetransition. For these CDWs, spectral reconstructions are not limited to a small energywindow around the Fermi energy, but can occur throughout the entire electronic structure.In this case, if the Fermi surface persists in the CDW phase with a different topology, thetransition is a metal–metal Lifshitz transition. However, even for the well-studied strong-coupling TMDCs 2H-NbSe and 1T-TaS , no experimental verification of a clearCDW gap located away from the Fermi energy has been provided to date.Metallic 1T-VS is not only a promising electrode material in lithium-ion batteries, but also a prototypical d system, expected to host strongly correlated physics. It is a CDWmaterial and a candidate for 2D magnetism with layer-dependent properties, mak-ing it a model system for investigating complex ground states. Although difficult to synthe-size, bulk 1T-VS has been well studied, with many authors finding a ( √ × √ ◦ CDWtransition at around 305 K when it was prepared via the de-intercalation of Li.
How-ever, recent chemical vapour deposition grown samples and powder samples prepared underhigh pressure show no CDW transition.
Based on their finding of a phonon instability ata wavevector corresponding to the ( √ × √ ◦ CDW, Gauzzi et al. point out that bulk“VS is at the verge of CDW transition” but not a CDW material. Due to a similar diffi-culty in synthesis, the properties of monolayer (ML) 1T-VS have proven equally elusive. Theoretical calculations had predicted ferromagnetism and a CDW with a wavevector of2 / When it was first synthesized however, scanning tunneling microscope (STM)measurements did not reveal a CDW, presumably due to strong hybridization with theAu(111) substrate, similar to the case of 2H-TaS on Au(111). Here we report the growth of stoichiometric VS monolayers on the inert substrategraphene (Gr) on Ir(111) via a two-step molecular beam epitaxy (MBE) synthesis developedfor sulfur-based TMDCs. Using a combination of STM, scanning tunneling spectroscopy(STS), and ab initio density functional theory (DFT) calculations, we determine the spatial3 igure 1:
Structure of VS on Gr/Ir(111) at 7 K and 300 K. (a) Large scale STM topographof monolayer VS islands with small bilayers present. A height profile along the horizontalblack line is shown below the image. (b, c) Atomically resolved STM images of ML VS at7 K (b) and 300 K (c). The Fourier transform of each image is shown as an inset, with the1 × structure in red and the superstructure spots indicated in gray. In both images, anatomic model for the superstructure is included as an overlay, highlighting a 9 × √ ◦ (b)and 7 × √ ◦ unit cell (c). The model depicts the top sulfur atoms, with their apparentheight in STM coded in orange (low) and yellow (high).Measurement parameters: (a) 80 ×
50 nm , I t = 0 . V t = −
800 meV, (b) 6 × , I t = 0 . V t = 400 meV, (c) 6 × , I t = 1 . V t = − . We observe a (cid:126)q ≈ / islands. X-ray magnetic circular dichroism (XMCD) mea-surements at 7 K and 9 T robustly show vanishing total magnetization. The coupling of theCDW to an antiferromagnetic spin state, energetically favored in DFT calculations, couldexplain this observation, providing interesting prospects for future research on the interplayof CDW and magnetism.The typical morphology of the MBE-grown ML VS islands on Gr/Ir(111) is shownin the large-scale STM image in Fig. 1(a). The monolayer islands are fully covered by a4triped superstructure which is present regardless of island size or defect density and occursin domains, typically separated by grain boundaries. In the topograph of Fig. 1(b), takenat 7 K, the VS lattice is resolved. We find that ML VS has an average lattice constantof a VS = (3 . ± . (cid:6) A, in good agreement with the bulk lattice parameter of 3 . (cid:6) A of1T-VS . The similarity of the lattice parameters indicates the absence of epitaxial strain,consistent with the random orientation of the VS with respect to the Gr. The stripes ofthe superstructure have an average periodicity of (2 . ± . a VS . Close analogues to thisstructure have previously been observed in ML VSe . There, a superstructure of identicalsymmetry is attributed to a CDW [compare Fig. S1 in the Supplementary Information(SI)].The superstructure is found to persist up to room temperature, as can be concluded fromthe STM topograph in Fig. 1(c), taken at 300 K. At this temperature, the superstructureappears spontaneously only on larger islands, suggesting that the transition temperaturebetween the superstructure and the undistorted phase is not far above room temperature.Indeed, on smaller islands the STM tip can be used to reversibly switch between undistortedVS and the superstructure, shown in the SI (Fig. S2). This directly excludes the possibilitythat the superstructure is another stoichiometry of ML VS , for instance a S-depleted phase.We conclude that the superstructure is most likely of CDW origin.For the DFT calculations below, the experimental wave pattern must be approximated bya commensurate supercell. Two close approximations are overlaid on the atomic resolutionimages in Fig. 1(b, c). The gray boxes in Fig. 1(b) and (c) indicate 9 × √ ◦ and7 × √ ◦ structure with periodicities (2 . ± . a VS and (2 . ± . a VS , respectively.The Fourier transformed topographs shown as insets display the wavevectors of the CDW(gray arrows). They are temperature independent within the margins of error and found tobe (cid:126)q CDW(7 K) = (0 . ± . (cid:126)q CDW(300 K) = (0 . ± .
03) ΓK, just in between thoseof the unit cells: (cid:126)q ×√ ◦ = 9 /
14 ΓK ≈ .
643 ΓK and (cid:126)q ×√ ◦ = 2 / ≈ .
667 ΓK.
Ab initio density functional perturbation theory (DFPT) calculations of the acoustic5 K K L TZ Γ M M K Γ i P hononene r g y ( m e V ) (a) (b) 9× √ ◦ (c) S up.VS low.ref.inv.7× √ ◦ Figure 2:
Lattice instabilities in ML 1T-VS from first principles. (a) Acoustic phonondispersion from DFPT. The labels L, T, and Z indicate dominant longitudinal, transverse,and out-of-plane mode character. The leading instabilities of the L and T mode are locatedat (cid:126)q = 1 / (cid:126)q ≈ / × √ ◦ and7 × √ ◦ supercells with lattice distortions derived from structural relaxation using DFT.The yellow outline in (b) encloses a primitive cell of the 9 × √ ◦ structure. Coloredstripes in the background indicate the wavelengths 2 . a and 2 . a corresponding to thewavevectors 2 / /
14 ΓK of the instability. Dashed lines and crosses mark reflectionplanes and inversion centers. Vanadium and sulfur atoms are represented by black and yellowdots, their undistorted positions by gray shadows. All displacements are drawn to scale.6honon dispersion of undistorted ML 1T-VS confirm that a structural instability and corre-sponding tendencies towards CDW formation exist for the experimental (cid:126)q vector. Figure 2(a)shows that the longitudinal– and transverse–acoustic modes feature imaginary frequenciesin several parts of the Brillouin zone. The wavevector (cid:126)q ≈ / × √ ◦ and 7 × √ ◦ commensurate supercells, which canapproximately host an integer multiple of the observed wavelength. The vanadium atomsare displaced from their symmetric positions by up to 8 % of the lattice parameter, while thepositions of the sulfur atoms remain almost unchanged. In the calculated configurations, thetotal free energy is reduced by about 24 meV per VS unit. The vanadium displacement hascomponents in both the transverse (vertical) and longitudinal (horizontal) direction, eventhough the instability of the phonons at q = 2 / × by a combination of STS experiments and simulated d I/ d V maps based on the abinitio calculations using the 7 × √ ◦ and 9 × √ ◦ unit cells. STS spectra wereused to locally probe the DOS of ML VS , at 7 K (black line) and 78 . igure 3: Spatially and electronically resolved CDW phase in ML VS . (a) d I/ d V spectrataken with a Au tip on ML VS at 78 . × √ ◦ (cyan) and 9 × √ ◦ (indigo)CDW phases of ML 1T-VS . (b) Atomically resolved STM topograph of ML VS takenat V t = 175 meV. (c, d) Fourier-filtered d I/ d V conductance maps of the same region as(b) taken at V t = 75 meV (c) and V t = 275 meV (d). A linear yellow (maximum) to blue(minimum) color scale is used to depict the d I/ d V intensity. The indigo box indicates thesame location in (b), (c), and (d) and corresponds to a single 9 × √ ◦ unit cell of theCDW. In the same color scale DFT-simulated d I/ d V maps below (c) and above (d) thegap of the CDW are overlaid as insets. The maps show the integrated DOS from 0 meV to137 meV (c) and from 137 meV to 275 meV (d). Additionally, the Fourier transforms of theconductance maps are shown in the upper left corners with the 1 × f = 777 . T = 78 . I t = 0 . V r.m.s. = 6 meV; T = 7 K, I t = 0 .
45 nA, V r.m.s. = 4 meV, (b–d) T = 7 K, 5 . × . , I t = 0 . V r.m.s. =10 meV. 8slands. The most prominent feature is the gap located at about 0 .
175 eV, which is absentin calculations of undistorted ML VS . At 78 . I/ d V signal vanishescompletely, corresponding to a full gap in the DOS. In most other characteristic features thespectra agree qualitatively.In the same figure, ab initio calculations for the DOS of VS , structurally relaxed in the7 × √ ◦ (cyan) or 9 × √ ◦ (indigo) unit cell, are shown. Both unit cells feature quitesimilar structures, as expected for close-lying (cid:126)q vectors. Most striking, for both cases a fullgap in the unoccupied states is predicted. They only differ in size: 0 .
13 eV and 0 .
21 eV forthe 9 × √ ◦ and 7 × √ ◦ cell, respectively. The location of the gap matches the STSdata. That the width of the gap in the spectrum is smaller than in DFT might stem fromthe ground-state calculation assumed in DFT, overestimating the V-atom displacement atrealistic temperatures.While many of the characteristic features of calculated and measured DOS (peaks, min-ima) seem to agree, the experimental spectra appear to be compressed with respect to theDFT calculated DOS. This is indicative of strong electron-electron correlations beyond theapproximations of DFT (compare Fig. S4).These results suggest that a large or full gap at E F , as proposed for 1T-VSe , isunlikely to intrinsically occur for a CDW with the observed wavevector. The occurrence of agap at E F would however be possible as a consequence of substrate- or defect-induced doping,or due to an additional symmetry breaking, e.g., in the spin channel, which would removeadditional spectral weight around the Fermi level. This is discussed in the SI (Fig. S6) forML 1T-VSe .With theory and experiment largely agreeing on the electronic structure, we turn to therelation between the gap and the CDW measured on the VS islands. For that purpose,d I/ d V conductance maps were taken on either side of the gap (both in the unoccupiedstates), in the location shown in Fig. 3(b). The maps help to distinguish structural from9 M K Γ − − . . E l e c t r onene r g y ( e V ) (a) 0 DOS (arb. unit) Γ MK(b) CDW sym.
Figure 4: (a) Electronic band structure, density of states, and (b) Fermi surface of ML 1T-VS from DFT. Data for the undistorted structure and the 7 ×√ ◦ CDW is shown in blackand blue, respectively. The transition between the two is visualized in the SupplementaryVideo 1. The CDW data has been unfolded to the Brillouin zone of the undistorted structure.Here, the linewidth corresponds to the overlap of CDW and undistorted wave functionsfor the same (cid:126)k point. Analogous results for 1T-VSe , the 9 × √ ◦ CDW, and atomicdisplacements according to the transverse–acoustic phonon instability alone are shown inFigs. S6 and S7.electronic contributions, providing a close approximation of the spatial distribution of theDOS at the selected energies. As shown in Fig. 3(c, d), we find two different DOS distribu-tions on either side of the gap (see Fig. S5 for in-gap DOS). Both distributions are lockedinto the distorted lattice periodicity. They are out-of-phase, as seen in the indigo unit celldrawn in the same location in Fig. 3(b, c, d), the DOS maxima below the gap correspondto DOS minima above the gap and vice versa. This behavior is perfectly analogous to thatfor a CDW with a symmetric gap around E F . Simulated d I/ d V maps derived from theDFT DOS for a 9 × √ ◦ CDW are shown as an overlay in Fig. 3(c, d). In Fig. 3(c), thesimulation reproduces both the alternating rows of single and zigzag atoms and the DOSminima between the rows. Its counterpart in Fig. 3(d) shows higher DOS contrast thanexperiment, but presents the same qualitative features. With the simulated maps based onthe anharmonic displacement patterns of Fig. 2(b, c), the close agreement with experimentemphasizes the need to look beyond the harmonic approximation to understand this type ofCDW.To deepen our understanding of the system, we calculated the spin-degenerate band10tructure, density of states, and Fermi surface of ML 1T-VS with DFT. The results areshown in Fig. 4 and the Supplementary Video 1. In the undistorted case, we find a singleelectronic band at the Fermi level, which strongly disperses between M and K and features aVan Hove singularity in the unoccupied states, as shown in Fig. 4(a). The Fermi surface, de-picted in Fig. 4(b), consists of cigar-shaped electron pockets around the M points. For smalldistortions, partial gaps open at the Fermi level (e.g., between M and K). With increasingamplitude of the distortion, the gaps become larger, especially above the Fermi level, andthe bands are heavily reconstructed also for high energies (Supplemental Video 1). However,since the downwards-dispersing bands along Γ–M are only slightly shifted downwards andremain above the Fermi level near Γ, there is no complete gap at the Fermi level, as seen inthe density of states. On the other hand, the originally flat portion of the band structure be-tween Γ and K now disperses downwards and crosses the Fermi level. Altogether, the Fermisurface is reconstructed and not completely destroyed by the lattice distortion. The CDWtransition is thus a Lifshitz metal–metal transition with a change in Fermi surface topology,instead of the usual Peierls metal–insulator transition. As shown in the SI (Figs. S7 and8), the harmonic approximation does not capture the Fermi surface reconstruction. Theunexpectedly large anharmonicity occurs since the anharmonic part of the lattice distortioncouples to segments of the Fermi surface that are not affected by the harmonic distortion.Together, the harmonic and anharmonic electron–phonon coupling change the Fermi surfacefrom M-centered electron pockets to the single Γ-centered hole pocket visible in Fig. 4(b).Prompted by the prediction of ferromagnetism for ML 1T-VS in its (cid:126)q = 2 / we also examined the magnetic properties of VS , by means of x-ray magneticcircular dichroism (XMCD). The ML VS samples were grown in situ and investigated withSTM beforehand to make sure that the right stoichiometry and coverage were obtained. ASTM topograph of the sample investigated by XMCD is shown in the SI (Fig. S9). Theblue curve in Fig. 5 represents the x-ray absorption spectrum averaged over both helicitiesand external field directions. The overall line shape is very similar to previous bulk crystal11
10 515 520 525 530 535 540 I n t en s i t y ( a r b . un i t ) Photon energy (eV)L L (a) ( — + + — – )/2( — + – — – )×10 − . . M o m en t ( — B ) (b) Figure 5:
Magnetic properties of ML VS . (a) Plotted in blue is the x-ray absorptionsignal averaged over both helicities and directions of the B field. The corresponding XMCDis shown in red. All measurements have been conducted in B fields of ± − . unit. The magnetic moment on vanadium is ± . µ B , ± . µ B ,or zero; the largest magnetic moment on sulfur is ± . µ B .measurements and clearly fits to a 3 d configuration. The red signal in Fig. 5 is theXMCD magnified by a factor of 10, where no signal above the noise level is visible. Thisimplies that the total magnetization vanishes. Sum rule analysis would yield an upper boundof 0 . µ B per V-atom. Since it cannot be strictly applied to the case of the V2, 3 edges, this analysis yields only a zero order estimate of the upper bound, but we can safely concludethat neither ferromagnetic nor paramagnetic behavior is present in this system.We investigated magnetic order in ML VS using spin-polarized DFT. We were ableto stabilize both ferromagnetic and antiferromagnetic structures within the 7 × √ ◦ supercell. In fact, magnetically ordered phases are preferred over nonmagnetic phases with upto 1 . . unit cell for antiferromagnetic (ferromagnetic) order. Althougha full account of magnetism needs to go beyond the DFT level, our ab initio calculations pointto magnetic moment formation. Thus an antiferromagnetic groundstate is both consistentwith experiment and plausible from DFT calculations.12oth the electronic and magnetic results for VS shed some light on the properties ofthe isoelectronic compound VSe , which displays a CDW of the same periodicity. Ourcalculations strongly suggest that also for this system anharmonic effects are relevant andthat a full gap opens in the unoccupied states (compare Fig. S6 in the SI). It should beworthwhile to carefully analyze the DOS of the unoccupied states for this material. Similarto VS , antiferromagnetic ordering could explain the absence of net magnetization in XMCDexperiments. Spin-polarized STM might be able to detect the magnetic ground state forboth VS and VSe .In conclusion, 1T-VS defies the common phenomenology of CDW formation, as thecomplete CDW gap occurs above the Fermi level, there is giant anharmonic longitudinal–transverse mode–mode coupling, and the CDW formation is accompanied by a change ofthe Fermi surface topology. The unconventional CDW appears to host further electroniccorrelations as signalled by the quasiparticle renormalization and magnetic moment forma-tion. In this respect, it is reminiscent of correlated phases in superlattice structures suchas Star of David phases, moir´e superlattices, and doped cuprate superconductors. Inthe latter class, lattice anharmonicities are central to boosting superconductivity under THzoptical driving. The case of VS presents new terrain: electron–lattice effects are at strongcoupling and anharmonicities are giant, intertwining an electronic Lifshitz transition withelectron correlations. We note that the full gap in the DOS, situated within 0 . with the full gap in theunoccupied states provides a paradigmatic case study of strong-coupling CDWs in general.13 ethods The Ir(111) crystal is cleaned by grazing incidence 1 . + ion exposure and flashannealing to 1500 K. A closed monolayer of single-crystalline Gr on Ir(111) is grown byroom temperature exposure of Ir(111) to ethylene until saturation, subsequent annealing to1300 K, followed by exposure to 200 L ethylene at 1300 K. The synthesis of vanadium sulfides on Gr/Ir(111) is based on a two-step MBE approachintroduced in detail in Ref. for MoS . In the first step, the sample is held at room temper-ature and V is evaporated at a rate of F V = 2 . × atoms / (m s) into a sulfur backgroundpressure of P gS = 1 × − mbar built up by thermal decomposition of pyrite inside a Knud-sen cell. This results in dendritic TMDC islands of poor epitaxy. To make the islands largerand more compact, the sample is flashed in a sulfur background to 600 K.The VS layers were analyzed by STM, STS and low-energy electron diffraction (LEED)inside a variable temperature (30 K to 700 K) ultrahigh vacuum apparatus and a low-temperature STM operating at 7 K and 78 . WSxM was used for STMdata processing. XMCD measurements have been conducted at the beamline ID32 of theEuropean Synchrotron Radiation Facility (ESRF) in Grenoble, France. The VS sampleswere grown in situ inside the preparation chamber and checked with LEED and STM beforeX-ray absorption spectroscopy measurements. To be surface sensitive, the measurementswere conducted in the total electron yield mode under normal incidence. The measurementtemperature was 7 K and fields of 9 T were used. The spectra were recorded at the L , -edges,i.e., using the dipole allowed transition from 2 p states into the 3 d shell potentially generatingmagnetism.All DFT and DFPT calculations were performed using Quantum ESPRESSO . Weapply the PBE functional and norm-conserving pseudopotentials from the
PseudoDojo table.
In the undistorted case, uniform meshes (including Γ) of 12 × (cid:126)q and 24 × (cid:126)k points are combined with a Fermi–Dirac smearing of 300 K. For a fixed cell height of 15 (cid:6) A,minimizing forces and in-plane pressure to below 1 µ Ry / Bohr and 0 . . (cid:6) A and a layer height of 2 . (cid:6) A. For the supercell calculations, appropriate (cid:126)k -point meshes of similar densities are chosen. Subsequent Fourier interpolation to higher (cid:126)k resolutions and the unfolding of electronic states is based on localized representationsgenerated with
Wannier90 . Acknowledgments
This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Researchfoundation) through CRC 1238 (project no. 277146847, subprojects A01 and B06). J.B.and A.S. acknowledge financial support by the DFG through GRK 2247 and computationalresources of the North-German Supercomputing Alliance (HLRN). E.v.L. is supported bythe Central Research Development Fund of the University of Bremen. L.M.A. acknowledgesfinancial support from CAPES (project no. 9469/13-3).
Note on authorship
T.Weh. and T.M conceived this work and designed the research strategy. J.H. and discoveredthe CDW and developed, with the assistance of T.Wek., the growth method. C.v.E. andE.P. conducted and analyzed the STS experiments. J.B., A.S., and G.S., with support fromE.v.L. and T.Weh., performed ab initio calculations. F.H. and S.K., with support from N.R.,K.O., L.A., N.B., K.K. W.K., and H.W., performed and analyzed the XMCD experiment.The results were discussed by all authors. C.v.E., J.B., J.H., E.v.L., S.K., T.Weh., and T.M.wrote the manuscript with input from all authors. The first three authors have comparablecontributions to this work.
Competing interests
The authors declare no competing interests. 15 upplementary Information
Figure S1: Unit cells
Figure 6:
Correspondence between unit cells in ML VSe literature and the unit cells usedin this paper. In (a) and (b), models of the 1T-VS atomic lattice are depicted with Vatoms in blue and bottom-S atoms in faint yellow. 7 × √ ◦ (a) and 9 × √ ◦ (b)superstructures are visible in the top-S atoms, which are drawn in two colors to mimic theexperimental apparent height in yellow (bright) and orange (dark). The dark gray rectanglesindicate the 7 ×√ ◦ (a) and 9 ×√ ◦ (b) unit cells. The blue rectangle is a 2 ×√ ◦ unit, the red rhombus a √ ◦ × √ ◦ .In the isotypic material VSe , a superstructure of same symmetry as in VS has been identi-fied and attributed to a CDW. In these studies, the superstructure was described by acombination of 2 × √ ◦ and √ ◦ × √ ◦ units, which we mark in our model inFig. 6(a, b) in blue and red, respectively. By the combination of a single 2 × √ ◦ and two √ ◦ × √ ◦ units, the 7 × √ ◦ CDW lattice can be described; two 2 × √ ◦ units and two √ ◦ × √ ◦ units make up the 9 × √ ◦ lattice.16 igure S2: Tip-induced switching between distorted and undis-torted phase (a) (b) (c) Figure 7:
Influence of STM tip on ML VS : the two consecutive STM scans in panel (a) and(b) document the STM tip induced switch from the undistorted to the superstructure CDWphase. Scan (c) is taken 15 minutes later and shows no more signs of the superstructure.Images taken at 300 K. Measurement parameters: (a–c) 7 × , I t = 0 . V t = −
90 meV.The presence of the superstructure at room temperature can also be influenced by the STMtip. Figure 7(a) and (b) show two consecutive STM scans, taken at the same position,tunnelling current, and bias. In panel (a), the STM reveals only hexagonal atomic orderinginside the small VS structure. In the successive STM scan in (b), the wave superstructure isobserved in the same region, with the phase transition apparently triggered by the interactionwith the STM. A subsequent STM scan taken about 15 min later, displayed in Fig. 7(c),again shows the absence of the superstructure. Figure S3: Nesting conditions and electron–phonon coupling
In Fig. 2(a) of the manuscript, we can observe three main instabilities in the acoustic phonondispersion of ML 1T-VS : two in the longitudinal branch at (cid:126)q ≈ / (cid:126)q ≈ / (cid:126)q ≈ / (cid:126)q points are compatible with the formation of a 4 × (a) LA, q = 1/2 Γ M q (b) TA, q = 2/3 Γ K q (c) TA, q = 9/14 Γ K q (d) LA, q = 3/8 Γ K q (e) TA, q = 4/3 Γ K (mod BZ) q (f) TA, q = 9/7 Γ K (mod BZ) max0
Figure 8:
Nesting conditions for different phonon wavevectors (cid:126)q . We show the relevantelectron–phonon coupling g ∗ (cid:101) g from (c)DFPT as a function of the ingoing electron momen-tum (cid:126)k (colour scale) together with the original Fermi surface (straight lines) and the Fermisurface shifted by − (cid:126)q (dashed lines). Nesting parts of the Fermi surface can only have a strongeffect on the phonons if they occur in (cid:126)k -space regions with significant electron–phonon cou-pling (dark/brown spots). While the (cid:101) g from DFPT is fully screened, the partially screened g from cDFPT excludes low-energy electronic screening. Together with the bare electronicsusceptibility χ , they determine phonon self-energy Π = g ∗ χ (cid:101) g responsible for the instabil-ities in the phonon dispersion. This analysis is equivalent to the fluctuation diagnostics inRef. 12. Coupling data obtained via the EPW code. . Interestingly, despite these favorable conditions, this is not the preferred groundstate of ML VS . Instead, a CDW with a wavevector near (cid:126)q = 2 / (cid:126)q = 9 /
14 ΓKdevelops, which features only approximate nesting and a slightly reduced coupling strength,as seen in Fig. 8(b, c). As discussed in the main text, the formation of the CDW canonly be understood considering anharmonic effects. Phonon modes that appear stable in theharmonic approximation contribute significantly to the final atomic displacements, especiallythe transverse–acoustic modes for twice the momenta of the unstable modes, i.e., (cid:126)q = 9 / (cid:126)q = 4 / (cid:126)k -space regions of considerable coupling, except that differentpairs of pockets are involved. Figure S4: Compression of electronic spectrum
In Fig. 3(a) of the main text, the DOS from DFT appears to be wider than the experimentalspectrum. Dynamic electronic correlation effects beyond DFT are a possible source of thisdiscrepancy, since they can lead to band renormalization. More precisely, they effect a massenhancement of the electrons, i.e., the quasi-particle dispersions become flatter than whatis expected from theories like DFT. In the case of purely local correlations, this effect isdescribed by a single renormalization factor Z or the corresponding mass enhancement factor1 /Z . Figure 9 shows that we obtain a good match between experimental and theoreticalspectra by setting Z = 0 .
8. This is indicative of moderate electronic correlations. Forcomparison, examples range from diverging mass enhancement at Mott–Hubbard transitions,via mass enhancement factors of about 10 to 1000 in Kondo or heavy fermion systems, toenhancement factors between 1 and 10 in transition metal compounds like metallic Cr orFe-based superconductors. The mass enhancement factor of 1 /Z ≈ .
25 puts VS at similarelectronic correlation strengths as, e.g., metallic Cr. The rise in the normalized d I/ d V beyond − . igure 9: Compression of experimental spectrum relative to DFT-calculated DOS. The 7Kspectrum from the main manuscript is compared to the calculated DOS for the 7 × √ ◦ and 9 × √ ◦ unit cells in (a). In (b), the calculated DOS is compressed to about 80 %of its original width. The red arrows in (a, b) indicate three major features in the spectrumand DOS that can be harmonized between them when the DOS is compressed.from the Gr/Ir(111) substrate, which can come to dominate the signal for large V when theVS has a small DOS. In this case, the Gr spectrum (not shown) diverges beyond the Ir(111)surface state at −
190 meV.
Figure S5: Suppression of CDW Fourier intensity within the gap
Apart from the different charge distributions on either side of gap discussed in the mainmanuscript, d I/ d V maps taken within the gap show a clear suppression of the CDW. For aquantitative analysis, we have Fourier analyzed the d I/ d V maps and normalized the CDW20 igure 10: Suppression of CDW within the gap. (a) Logarithmic plot of the CDW intensityin the Fourier transform of d I/ d V conductance maps, normalized to the 1 × I/ d V spectrum is plotted in order to indicate the location and width of thegap. (b–d) d I/ d V conductance maps taken at the voltages indicated in (a). Measurementsettings: (maps) 9 . × . , I t = 0 . I t = 0 . I/ d V spectrum) f = 777 . I t = 0 . V r.m.s. = 6 meV. All data takenat T = 7 K.peak in the Fourier spectrum with respect to the 1 × R = I CDW /I × is observed to fall by an order of magnitude within the gap. Since agap of other than CDW origin would have the same value of R in- and outside of the gapregion, this is another clear indication of the relation between gap and CDW. Figure S6: Comparison to VSe and different supercells In Fig. 11(a), we show additional DFT results for ML 1T-VSe , both in its undistorted phaseand in a 9 ×√ ◦ configuration, for comparison with the literature on this material, wheresuch a CDW has been reported repeatedly. As points of reference, corresponding resultsfor ML 1T-VS in the undistorted, 9 ×√ ◦ , and 7 ×√ ◦ configuration are displayed inFig. 11(b, c). Our calculations suggest that VSe and VS are very similar in their electronic21 a) × √ R ◦ −0.5 0 0.5 1 D O S ( a r b . un i t ) Electron energy (eV) (b) × √ R ◦ −0.5 0 0.5 1Electron energy (eV) (c) × √ R ◦ −0.5 0 0.5 1Electron energy (eV) Figure 11:
Fermi surface and density of states of ML (a) 1T-VSe and (b, c) 1T-VS in theundistorted phase as well as for the (a, b) 9 × √ ◦ and (c) 7 × √ ◦ CDW as obtainedfrom DFT. The CDW data has been unfolded to the Brillouin zone of the undistortedstructure. Here, the linewidth corresponds to the overlap of CDW and undistorted wavefunctions for the same (cid:126)k point.structure. In the distorted phase, VSe will also have a full gap in the unoccupied states,which however only spans 0 .
14 eV. At the Fermi energy, only a partial gap is expected.Though this has been observed in experiment, most studies on ML VSe agree on alarge gap located at E F . From our understanding, such a gap would require a filling ofthe downwards dispersing bands near Γ, which are not gapped in the CDW configuration.According to our DFT calculations for both VS and VSe on a 9 × √ ◦ (7 × √ ◦ )supercell, 1 / ≈ .
11 (1 / ≈ .
14) additional electrons would shift the gap to the Fermienergy. This charge could be provided by, e.g., the substrate or defects. The subtlety ofthe interactions responsible for the shift of the gap is highlighted by considering two recentARPES investigations of ML VSe , both grown on bilayer Gr. In one case, a full gap at E F was measured, whereas in the other only a partial Fermi surface gap between M and Kwas observed. igure S7: Harmonic component of CDW displacement Γ MK(a) CDW sym. Γ M K Γ − − . . E l e c t r onene r g y ( e V ) (b) 0 DOS (arb. unit) (c) (d) Figure 12: (a) Fermi surface, (b) band structure and density of states, and (c) top and(d) side view of the crystal structure of ML 1T-VS for an atomic displacement in thedirection of the unstable transverse–acoustic phonon modes alone. For comparison to thefull CDW structure, see Figs. 2 and 4 of the manuscript. For an animated version of thisfigure, see the Supplementary Video 2.While the relaxed crystal structure of the CDW from DFT features atomic displacements inboth transverse and longitudinal directions, the phonon instability is mainly of transversecharacter (compare Fig. 2 of the main text). In Fig. 12, we show results for the CDWdisplacements with all anharmonic contributions removed. As seen in Fig. 12(a), about onethird of the cigar-shaped Fermi pockets withstand the distortion. At the same time, thereis no full gap in the electronic low-energy spectrum, see Fig. 12(b). As evident from thetop and side view of the crystal structure in Fig. 12(c, d), the atoms are almost exclusivelydisplaced in the transverse in-plane direction. Figure S8: Bands along extended Brillouin-zone path
In Fig. 4(b) of the manuscript and the above Fig. 12(b), we show the electronic band struc-ture of ML 1T-VS in the 7 × √ ◦ phase along a selected high-symmetry path Γ–M–K–Γ23 . . E l e c t r onene r g y ( e V ) (a) Γ MKM K M K Γ M K Γ M K − . . E l e c t r onene r g y ( e V ) (b) Figure 13:
Electronic band structure of ML 1T-VS along an extended Brillouin-zone path(a) for the full 7 × √ ◦ CDW displacement and (b) without anharmonic displacementcomponents.of the undistorted phase only. Once the distortion breaks the hexagonal symmetry, this pathis not representative of the full Brillouin zone anymore. For completeness, in Fig. 13, wethus reproduce the respective data along an extended path. This emphasizes the differencesbetween the bands for the full 7 × √ ◦ CDW structure [Fig. 13(a)] and without anhar-monic displacement components [Fig. 13(b)]. Most prominently, there is no gap between M (cid:48) and K in the latter case.
Figure S9: XMCD sample morphology
In the main manuscript, we describe the magnetic properties of VS as measured by XASand XMCD. Figure 14 displays the sample morphology of the investigated sample. Like thesamples shown in the main text, the island shape is dendritic. By comparison to substrate24 igure 14: Magnetic properties of VS : STM topograph illustrating the sample morphologyof the XMCD measured sample. Image size: 100 ×
50 nm .step edges, the monolayer height is measured to be 7 (cid:6) A. The sample has a ML coverage ofabout 40 %. Distinct height levels indicate three layers.
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