Intertwined chiral charge orders and topological stabilization of the light-induced state of a prototypical transition metal dichalcogenide
Yaroslav A. Gerasimenko, Peter Karpov, Igor Vaskivskyi, Serguei Brazovskii, Dragan Mihailovic
DDual vortex charge order in a metastable state created by anultrafast topological transition in 1T-TaS Yaroslav A. Gerasimenko, ∗ Igor Vaskivskyi, and Dragan Mihailovic † CENN Nanocenter, Jamova 39, SI-1000, Ljubljana, Slovenia andDepartment of Complex Matter, Jozef Stefan Institute,Jamova 39, SI-1000, Ljubljana, Slovenia (Dated: December 25, 2017)
Abstract
Many body systems in complex materials undergoing non-equilibrium phase tran-sitions may self-organise into ordered metastable emergent states with new and un-expected functionalities . Here, using large-area scanning tunneling microscopywe reveal an intricate chiral vortex structure and complex tiling of charged electrondomains in the metastable metallic state in 1T-TaS created by a non-equilibriumtopological transition initiated by a single femtosecond optical pulse. A Moir´e analy-sis shows that the interference of non-equilibrium nested Fermi surface (FS) electronsleads to the creation of charge vortices ~ D on a length scale of ∼
70 nm. On a muchsmaller scale of ∼ nm, domain configurational patterns appear, which show boundvortex-antivortex pairs, discommensurations, domain wall (DW) crossings and DWkinks, consistent with a rapidly quenched Berezinskii-Kosterlitz-Thouless (BKT) -transition. Revealing the detailed mechanism for the transition leads the way to de-sign of long-range charge-ordered metastable states with intricate emergent propertiesunder controlled non-equilibrium conditions. a r X i v : . [ c ond - m a t . s t r- e l ] D ec hotoexcited metastable states in crystals have lifetimes which usually range from pi-coseconds to microseconds , which makes it hard to investigate their structure in sufficientdetail to reveal transient mesoscopic or microscopic order . Consequently, the detailedmechanisms for photoinduced metastability have not been very well understood till now.Even though it is of fundamental importance to prove the principle of the existence of LROcreated of a transient emergent state, no direct experimental evidence has so far been pre-sented in any system exhibiting a photoinduced phase transition. Uniquely, as a modelsystem, 1T-TaS (TDS) has a metastable state with a temperature-tunable lifetime, whichis usefully long at low temperatures for detailed experimental investigations. This opensthe possibility of studying not only the fine structure, but also the origin and mechanismof the metastability with high resolution scanning tunnelling microscopy (STM). In thislayered di-chalcogenide system (Fig. 1a) the competition of lattice strain, a FS instabil-ity and Coulomb interactions lead to a number of different phases and a multidimensionalmulti-parameter phase diagram, where strain, chemical pressure, doping and laser photoex-citation fluence play different roles . In equilibrium, the high-temperature metallic stateof TDS is unstable towards a three-pronged quasi-two dimensional FS nesting instability causing a transition at T IC = 540 K from a single-band metal to a uniform three-directionalincommensurate (IC) charge density wave (CDW) state. On cooling this transforms into a’nearly-commensurate’ (NC) state at T NC = 350 K as a result of incommensuration strain,giving almost commensurate patches separated by smooth discommensurations (DCs) .Eventually the state becomes fully commensurate (C) and insulating below 190 ∼
220 K, asshown schematically in Fig. 1b. The localized electrons in the C state have a remark-able star-of-David (SD) ’polaron lattice’ structure, (Fig. 1c) where exactly one electron islocalised on the central Ta of the SD and the 12 surrounding Ta atoms are symmetricallydisplaced towards it. Photoexcitation of the material by a single ultrafast optical pulse wasshown to cause an insulator-to-metal transition (IMT) to a hidden metastable (H) statewithin a narrow range of photoexcitation fluences and pulse lengths. An indication of pos-sible long range order (LRO) came from the narrowness of the collective amplitude mode ofthe CDW whose frequency shifts by 3% at the transition . IMT switching caused by directcharge injection through electrodes was recently demonstrated in TDS while switchingby STM tip revealed that the uniform CCDW can be locally broken up into non-periodicdomains. No LRO was observed however, in either case . A hint of new states was found2n ultrafast electron diffraction (UED) experiments at significantly larger photoexcitationdensities, but the k -space resolution was insufficient to resolve the ordering vector or thedomain structure, so the fundamental question whether metastable LRO can form undernon-equilibrium conditions remains unanswered. Here, for the first time we reveal LROin an transient emergent state in a non-thermal transition using in-situ ultrafast opticalswitching in combination with a low-temperature high resolution STM. The studies reveala remarkable dual vortex structure of polaron ordering in the emergent state, quite unlikeany of the other states in this system, or in any other known material . Results
We photoexcite freshly in-situ cleaved single crystals of TDS with focused 50 fs singlepulses at 400 nm within a UHV STM chamber. A scanning electron microscope (SEM) isused to select a homogeneous region of the crystal without strain and for precise positioningthe tip and the beam on the sample (Fig. 1e). The pulse fluence was adjusted to justabove switching threshold at 0.9 mJ/cm , taking care to avoid heating the lattice to theNC state (see supplementary Fig. 5). To determine both LRO and local structure of chargemodulation, STM images of C, NC and H states were measured on different length scales(Fig. 2). The large area scan shows a regular polaron lattice in the C ground state at 4.2K (Fig. 2a). In the NC state this breaks up into a modulated structure of domains and”soft” DWs (Fig. 2c). The corresponding atomic-scale images are shown in Fig. 2b andd respectively. In the photoexcited H state at 4.2 K (Fig. 2e,f) the uniform C lattice istransformed into domains of diverse sizes and shapes, separated by sharp DWs. The actualpattern is different every time the experiment is performed, indicating that the domainstructure is determined by fluctuations rather than predetermined by sample defects andimperfections.Fourier transforms (FTs) of the large-area scans demonstrate a set of peaks shown inFig. 3 with high reciprocal space resolution, allowing us to discern fine details of the longrange charge modulations described by the vectors Q ( i ) NC , Q ( i ) C and Q ( i ) H , for the NC, C and Hstates respectively. Here i = 1 , , ◦ to each other.FTs of atomic resolution scans (Figs. 3a,d) allow us to calibrate the charge modulationvectors with respect to the underlying atomic lattice vectors a ∗ and b ∗ .3n the C state, CDW wavevectors ± Q ( i ) C appear as the six brightest peaks in the FTin Fig. 3a. Less intense atomic peaks coincide exactly with the linear combination of theCDW vectors, a ∗ = 3 Q (1) C − Q (2) C , defining the commensurability of the C CDW with theunderlying lattice. The NC state has additional domain structure, giving rise to peak split-ting. Fig. 3b shows one bright CDW superlattice peak surrounded by several less intensesatellite peaks. The bright peak corresponds to fundamental CDW wavevectors Q ( i ) NC , whilethe satellites characterise the domain periodicity, k ( i ) domain shown in Fig. 3c. The appear-ance of the satellites in the NC state is well understood in the plane wave picture of theCDW, where anharmonic interactions produce secondary distortions - described in terms ofharmonics of the fundamental wave Q ( i ) NC . Their wave vectors are related to the latter as Q (1) sat = n ( a ∗ + Q (2) NC ) − (3 n − Q (1) NC , and in the NC state the order is limited to n = 1 . Theinterference of fundamental and the first harmonic wave results in “beatings” - large-scale periodic modulations of the distortion amplitude. The interference pattern has hexagonaldomains with inverse period k ( i ) domain = Q ( i ) NC − Q ( i ) sat separated by smooth domain walls. TheCDW is locally commensurate with the lattice inside the domain, and its only distinctionfrom the C-state is a decreased CDW amplitude near the domain walls (see SI).In the photoexcited state, we first ascertain the relation between the atomic lattice peaksand the CDW lattice peaks from a FT of atomic resolution STM image of several domains (Fig. 3d). From the position of the second order CDW peaks (for higher accuracy) we obtain φ = 13 . ◦ ± . ◦ , which differs by 0 . ◦ to the C state, and 0 . ◦ from the nearest reportedNC state (see Supplementary Information for error analysis). This is the first indicationthat the H state has different LRO than the C or NC states.The periodicity of the CDW and the domains in the H state can be analyzed similarlyto NC state from FTs of large area scans over 200 ×
200 nm . We see six high intensitypeaks (Fig. 3e). They result from the largest amplitude modulation in the topographicimage, which comes from individual polarons, and are associated with the fundamentalCDW vectors, Q ( i ) H . Their FWHM angular width is 0 . ◦ (cf. Fig. 3g) which shows that theH state Q ( i ) H s are homogeneously rotated w.r.t. the C state over a macroscopically large area( ∼ ×
200 nm), which is a clear indication that the H state has distinct LRO.In the H state the single satellite characteristic of the NC state is converted into a diffusestreak spanning a range of angles 1 . ◦ < Θ < . ◦ (Fig. 3e,f) which reflects the distributionof the domain sizes and domain shapes. The cross-section AA along the streak (Fig. 3h)4eveals that FT intensity is not random in k -space, but rather there are approximatelyequally separated peaks with different intensities, showing that some domain sizes are morefavourable than others. If we zoom into aggregates of the small or large domains, individualpeaks can be discerned (see SI for more detailed analysis). The fundamental vector Q ( i ) H stays the same, independent of the domain size, as a consequence of the LRO.The existence of different domain sizes can be qualitatively described by extending theharmonic picture. The first peak position in the streak is at Θ ≈ . ◦ and fits rather well theexpected position of Q sat with n = 1, while peaks at larger Θ appear as higher harmonics.Their positions can be fit by varying angle φ and length Q H /a ∗ , which gives φ = 13 . ◦ and Q H /a ∗ = 0 .
278 for n = 1 . . . . There it was tentatively associated with the triclinic CDWappearing in the equilibrium phase diagram, but our higher resolution allows us to associateit with the emergent photoinduced state (see Supplementary Fig. 12 for comparison).We continue by examining the charge displacements within domains and the associatedDWs. To characterize the topological structure, we define a misfit displacement vector ~ D asthe charge displacement in the H state with respect to the C lattice (see Fig. 4a for definition).Locally, the structure is composed of irregular domains with different ~ D (Fig. 4b-d). Thereare 12 possible discrete displacement vectors and ~ D = 0 (Fig. 4a), which can be separatedby 6 different types of DWs. Commonly, three DWs meet at a Z vertex resulting in pairs ofvortices and antivortices (labeled Y and ¯ Y ) such as shown in Fig. 4b . For each individual Y defect the misfit displacement vector sum is nonzero, e.g. D + D + D = 0, but foreach pair it must be zero: D + D + D + D = 0. The latter condition sets the localtopological rules for the misfit displacement vectors. In addition, a significant number of X wall crossings (Fig. 4c) and kinks ( K s) (Fig. 4d) are observed, higher energy non-trivialdefects created by photoexcitation which are not expected in equilibrium hexagonal tiling(Fig. 4b). A contour encircling an X defect results in a non-zero sum of the misfit vectors, D + D + D = 0, which makes this a non-trivial topologically protected defect. The K defect corresponds to a change of domain wall type without a change of the misfit field. Itcontains one incomplete polaron hexagram, a nontrivial defect with zero Burgers vector (seeSuppl. Fig. 18).In addition to the local vortex structure, there exists a long wavelength structure which5s even more intriguing (Fig. 5). Plotting the magnitude | ~ D| and angle α of the vectorfield ~ D in Fig. 5b and c respectively as a Moir´e interference pattern between the C latticeand the H lattice (see Fig. 5a), we observe a clear hexagonal lattice of vortices with aperiod L H = 2 π/ ( | Q H − Q C | ) ∼
70 nm (Fig. 5b-d). Repeated experiments show that thewinding number of the large-scale vortices is either n w = − n w = +1, but neverboth simultaneously, implying that there is a macroscopically spontaneously broken mirrorsymmetry in the formation of the LRO ~ D vector field. We also observe an accompanyingweak, but unambiguous modulation of the domain size with a similar wavelength shown inFig. 5e. The domain size distribution explains the streaks in the Fourier transform shownin Fig. 3e,f.While the long-wavelength order in the H state apparently requires tessellation of differentdomain sizes, the NC state (above ∼
200 K) displays a regular domain structure with aperiod L NC = 2 π/ ( | Q N C − Q C | ) ∼
15 nm and uniform domain sizes (Fig. 2c), separated by“smooth” domain walls (or discommensurations). Comparing with other reported states,samples only a few layers thick deposited on substrates which are inherently strained showeither no CDW at low temperature, or the presence of a “supercooled” metallic state with randomly distributed domains as reported by Ma et al . On the other hand, thicksamples such as the one used here, or unstrained free-standing monolayer samples do notreadily show a supercooled phase, so we cannot make any empirical connection between anysupercooled state and the H state. On the other hand, the STM tip-induced state at lowtemperatures displays a local domain structure with four types of Z vortices and domainsof different size separated by sharp DWs, but there is no evidence of the intricate long rangeorder observed in the H state (see SI for analysis). Discussion
The mechanism for the transition is quite unlike previously studied non-equilibriumsecond-order symmetry-breaking FS nesting transitions such as the one in TbTe , andthe trajectory of the C → H transition cannot be discussed in terms of conventional non-equilibrium symmetry-breaking with a Landau order parameter that vanishes at some crit-ical time t → t c . It requires consideration of the competing effects of FS hot-electroninterference, Coulomb interactions and particularly the incommensurability strain which6orce the ordering of the system on different timescales. The state is created under veryspecific conditions of laser fluence 0 . < F < . , within a time and temperaturewindow defined by the pulse length range 30 fs < τ p < . During this time, the lat-tice remains below the transition temperature to the NC state, while the peak electronictemperature is estimated ∼ , thermalizing with the lattice on a timescale of ∼ ∼
185 K (see Fig. 5 in SI for lattice temperature estimates). Transientphotodoping is an essential component: photoemission experiments show that short pulsephotoexcitation causes melting and a rapid <
50 fs transient shift of the chemical potentialabove the switching threshold µ c , which implies that FS nesting is modified while thesystem is out of equilibrium. The hot nested electrons interfere to condense into a tran-sient incommensurate charge modulated structure with a wavevector Q ∗ IC , reflecting thenon-equilibrium FS. The accompanying change in low-energy electronic structure occurs ona timescale <
400 fs, as recently shown by coherent phonon spectroscopy . Thereafter, asthe incommensurate electronic CDW tries to adjust to the nearest commensurate lattice Q C ,individual domains spontaneously form (Fig. 4e). However, local topological defects such as K s, and Y vortices without associated nearby anti-defects also appear, presumably asa consequence of the rapid quench through the transition. On a longer timescales , a con-tinuous tiling emerges with a LRO Q H , and the K and Y defects become bound to form X defects and Y − ¯ Y vortex-anti-vortex pairs (see Fig. 4f) following local topological rules .The process is remarkably similar to superconductors where the binding of unbound vorticesleads to the formation of a vortex lattice in a non-equilibrium Berezinsky-Kosterlitz-Thoulesstransition . The long-wavelength chiral vortex lattice of the misfit displacement vector field ~ D and particularly the periodic modulation of domain size with a wavelength of ∼
70 nmmay be considered as an emergent phenomenon associated with the quench and interferenceof nonequilibrium nested electrons.Finally, we remark on the implications of the fact that strictly speaking, LRO cannotexist in two dimensions . Out-of-plane stacking of domains and orbital ordering along maycritically contribute to the stability of the H state . STM experiments show that domainwalls on subsequent layers avoid each other (see Supplementary Fig. 13), indicating thatinter-layer Coulomb interactions may lead to interlayer ordering of the Z vortices. At thesame time the very large out-of-plane versus in-plane resistivity persisting into the H state ,implies a non-trivial role for out-of-plane interactions in establishing the observed long range7ortex order.The validation of the concept of formation of emergent LRO through many-body in-teractions under non-equilibrium conditions, and the underlying topological transformationmechanism revealed by these experiments represents a large step towards understanding, andwhence designing new emergent metastable states in complex materials. The present system,with its associated M-I transition may lead to advances in novel non-volatile all-electronicmemory devices without the involvement of ions or magnetism, through controllable switch-ing between electronic topologically ordered states. Methods
The results presented here were measured in the custom-built Omicron Nanoprobe 4-probe UHV LT STM with base temperature of 4.2 K and optical access for laser photoexci-tation (Fig. 1e). Crystals of 1T-TaS were synthesised by the iodine vapor transport method.Samples were cleaved in situ at UHV conditions and slowly cooled down to 4.2 K.Photoexcitation of 1T-TaS single crystals was performed at 4.2 K (C-state) with a single50 fs optical pulse at ∼
400 nm (second-harmonic generation from 800 nm Ti-Sapphire laser)focused onto a 100 µ m diameter spot within STM UHV chamber. This ensures highlyspatially homogeneous excitation on the scale of the STM scans ( ∼
200 nm). It is importantthat photoexcitation energy density is carefully adjusted to be slightly above the thresholdvalue ( ∼ . ) for switching to the hidden state. Significantly higher excitationenergies result in heating of the lattice above the phase transition temperature T NC − C ,which we want to avoid.The surface was characterised each time before photoexcitation to confirm a defect-freeinitial C state. The beam was aligned at low power to hit the apex of the STM tip. Thenthe tip was retracted and a high-power single pulse was applied (Fig. 1e). After approachingthe tip, an area of the order of 500 ×
500 nm was checked for homogeneity.Contrast adjustment in Fourier transforms is routinely used for analysis of STM data in1T-TaS and is described thoroughly in literature . Here all FT images has max/min ratioof ≈ .
5, approximately an order of magnitude of absolute FT amplitude. Only parts of theimages used for FT are shown in Fig. 2. Full-scale versions of those for the hidden stateare shown in Supplementary Fig. 19. Raw images were corrected to match triangular lattice8sing standard procedure (see Supplementary Information).
Acknowledgments
This work was supported by European Research Council advanced grant “TRAJEC-TORY”. We are grateful for discussions with S. Brazovskii and H. W. Yeom.
Author contributions
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SEM 4.2KUHVchamberlens
Crystal1T-TaS H C TaS TaS e- FIG. 1:
The structure and charge ordering of equilibrium and non-equilibrium statesof 1T-TaS and a schematic diagram of the experiment. a The crystal structure of 1T-TaS . b The regions of stability for charge-density wave states under equilibrium conditions on coolingand heating, and charge density simulations in the incommensurate (IC), nearly commensurate(NC), triclinic (T) or commensurate (C) states respectively (see Supplementary Information). c The polaron structure in the C state, where twelve atoms are displaced towards the central Tawith the one extra electron, as indicated by the arrows. d The lifetime of the H state createdby an ultrafast optical pulse in the C state is indicated by the colour . e Schematic view of thecombined STM and SEM system with optical access for ultrafast laser pulse excitation. f TheSEM image shows a cleaved a - b surface of TDS single crystal in contact with the sharp tungstenSTM tip. After beam alignment the tip is retracted and the area underneath is illuminated witha focused single ultrafast laser pulse. NCC NCH Ha b c de f IG. 2:
Raw STM images of the hidden and equilibrium CDW states in 1T-TaS : a Large-area STM image ( V t = −
800 mV, I = 1 . b Atomic resolution STM image of the Cstate, revealing the hexagram structure shown in Fig. 1 b and c. (a) Bias and current were adjustedto highlight the top sulfur layer without any admixture of the tantalum orbitals ( V t = 100 mV, I = 200 pA). The CDW modulation is larger than the atomic size and has a peak in the group ofthree S atoms, positioned above the central Ta atom of the SD hexagram (red) (the correspondingmodel is shown in the inset). Each star has the same atomic configuration, illustrating thatCDW order is commensurate with the atomic lattice. c Large-area STM image of the nearlycommensurate CDW state at 300 K ( V t = −
250 mV, I = 150 pA), showing a periodic array ofbrighter areas – domains – separated by darker, soft domain walls. d Atomic resolution STM imageof the NC state ( V t = −
200 mV, I = 1 . e An extensive-area STM image ( V t = −
200 mV, I = 100 pA) of theH state after switching with a single ultrafast optical pulse showing a complex sharp network ofdomain walls. f Atomic resolution STM image of several domains in the H state ( V t = −
800 mV, I = 1 . a cbed C NCH NC g / f A A’AA’ n = 1
H H h IG. 3:
Reciprocal space analysis of the metastable H state in relation to equilibriumstates: a
A Fourier transform (FT) of the atomically resolved STM image of C state (Fig. 2b)shows commensurability in reciprocal space: peaks of atomic (black) and CDW (red) latticescoincide at certain positions. The two lattices are rotated by φ = 13 . ◦ . a ∗ and b ∗ are unit vectorsof the atomic reciprocal lattice and ( Q ( i ) C , i = 1 ...
3) – of the CDW reciprocal lattice. b FT of theSTM image of NC state (Fig. 2c) shows the CDW reciprocal unit cell. c Expanded image of panel(b) shows the highest-intensity Q (2) NC peak (red), surrounded by 3 neighbours (blue). The domainmodulation vector is given by ~k (2) domain = ~Q (2) NC − ~Q (2) sat . (see Supplementary Fig. 3 for more details).Note that the FT amplitude is zero between the satellite peaks. d FT of the atomic resolutionSTM image in the H state (Fig. 2f). The black (red) hexagon shows the reciprocal unit cell of theatomic (CDW) lattice. The CDW rotation angle is φ = 13 . ± . ◦ . e FT of the large-scale STMimage of the hidden state (Fig. 2e) reveals a CDW unit cell with six bright peaks – fundamentalvectors (red) – and three-fold streaks protracting over Θ ≈ . ◦ . f Detailed structure of the Hstate streaks within the black square of panel (e). g , Azimuthal cross-section data of the Q (1) H fundamental CDW peak in the FT of the hidden state (e), compared with the Q C ( T = 4 . K ),and the NC state, Q NC ( T = 205 K , third-order peak, right axis). The lines show gaussian fits.FWHM = 0 . ◦ for the H state. h A cross-section of the H state FT in the panel (f) along theAA line with a 5-pixel lateral averaging. The arrow shows the n = 1 CDW harmonic (see text). a bc d ! " " ef IG. 4:
Local topological structure in the H state, and a schematic sequence of eventsin its formation. a , The twelve possible ~ D vectors connecting 0 and 1 −
12 atomic sites of aDavid star. The thirteenth is the identity operation ~ D = 0. b , Z vortex-antivortex pair separate4 domains. The paths defining the winding number are indicated by arrows. Colour shading showsthe misfit vector in the adjacent domains according to the scheme in the panel (a). b , X crossingof two domain walls separates four domains. A contour encircling it results in a non-zero sum ofthe misfit vectors, D + D + D = 0. d , A kink K is a defect site where a change of the domainwall type occurs without a change of ~ D . e-f A proposed sequence of events in the formation ofthe H state. Initially, disconnected domains with 6 kinks ( K s) are formed within the intermediateIC ∗ CDW state. Continuous tessellation emerges when K s bind into X crossings and Y − ¯ Y pairs.During this process the transient IC* state is converted into the H state, following both local rulesand long-range vortex ordering shown in Fig. 5. b ca e d50 nm IG. 5:
Long range topological order in the H and NC states. a
The definition of | ~ D| and its direction α w.r.t. the C lattice. b,c , Large scale real space images (from Fig. 2e) of the Hstate showing | ~ D| and α for each polaron with respect to its position in the initial C lattice. Theimage exhibits long range hexagonal vortex lattice on a scale of ∼
70 nm. The vortices (windingnumber -1) are indicated by white arrows. d , The Moir´e pattern between Q H and Q C latticescalculated on the same grid. The vortices of the misfit vector field are shown by white arrows. Acartoon in the bottom right corner schematically shows the actual observed hexagonal tessellationof domains: instead of continuous change, ~ D (shown by the black arrows) is fixed inside a domainand changes on crossing a domain wall. e , The modulation of the domain size in the H state on ascale of ∼
70 nm, emphasized by shading. upplementary Information for “Dual vortex charge order in ametastable state created by an ultrafast topological transition in1T-TaS ” Yaroslav A. Gerasimenko, ∗ Igor Vaskivskyi, and Dragan Mihailovic † CENN Nanocenter, Jamova 39, SI-1000, Ljubljana, Slovenia andDepartment of Complex Matter, Jozef Stefan Institute,Jamova 39, SI-1000, Ljubljana, Slovenia (Dated: December 25, 2017) a r X i v : . [ c ond - m a t . s t r- e l ] D ec upplementary Note 1: simulation of CDW states using Nakanishi-Shiba model Below we summarize how various CDW states in 1T-TaS are built in real and reciprocalspace. The fundamental CDW vectors ( Q (1) , Q (2) , Q (3) ) form the triangular basis, whichcreates star of David distortion of the Ta atomic lattice. They appear in the IC state alignedwith atomic lattice vectors, whereas in NC and C states they are shrunk and rotated. Onecan thus describe them as the following product: (cid:16) Q (1) Q (2) Q (3) (cid:17) = γ · − − · a ∗ b ∗ · cos( φ ) − sin( φ )sin( φ ) cos( φ ) , where a ∗ , b ∗ are reciprocal vectors of Ta triangular atomic lattice, γ = Q ( i ) /a ∗ is star ofDavid size and φ is the rotation angle between CDW and atomic lattices.In CCDW state the fundamental Q ( i ) C is determined by the commensurability condition: Q (1) C = 113 (3 a ∗ + b ∗ )or, equivalently, 3 Q (1) C − Q (2) C = a ∗ . Eq. 1
The above commensurability condition allows to calculate the ideal length of CDW vector, Q ( i ) C /a ∗ ≈ . φ ≈ . ◦ .Real space images of IC and C state can be readily built as the following sum: Z = < "X i exp(i Q ( i ) R ) + exp(i a ∗ R ) + exp(i b ∗ R ) , where R is in-plane real-space vector. The sum term gives CDW modulation, the last two– atomic modulation. Suppl. Fig. 1 a-d shows the real and reciprocal space models of ICand C states. In the commensurate case each star of David is centered at a certain Ta atomposition, resulting in the same atomic structure of all CDW units across the sample (
Suppl.Fig. 1 c). This feature can be checked experimentally in atomic resolution STM images andis indeed observed in C, NC and H states (main text, Fig. 2b,d,g). In contrast, violationof commensurability will result in irregular atomic configuration for each star of David, asillustrated in
Suppl. Fig. 1 a. 2 cb d
Supplementary Figure 1 : Model of IC and C states: a, c , Real space models. Star of Daviddistortion is shown by red polygon. In the commensurate state DS centers coincide with certainTa atoms and, in contrast, have random atomic surrounding in the incommensurate state. b, d ,Reciprocal space (FT) models, illustrating the relation between atomic (black) and CDW (red)lattices.
In NCCDW state the periodic domain structure appears, a model example of which isshown in
Suppl. Fig. 2 a. Domains appear as brighter areas, which are separated bydarker domain walls. CDW inside the domains is commensurate, but has different phase inneighboring domains. The phase change occurs inside the domain walls. To describe suchstate it is necessary to add the vectors describing a domain network to Q ( i ) NC basis of CDW.It can be modelled with simple approach used to interpret the early X-ray results , whichconsiders domain structure as a result of the interference of two triple CDWs. One of themis given by the three fundamental NCCDW vectors, whereas the other has smaller amplitudeand is given by three additional vectors chosen in such way, that their linear combinationwith fundamental ones will be equal to some vector of reciprocal atomic lattice. This givesthe new commensurability condition: Q (1) sat = − Q (1) NC + Q (2) NC + a ∗ . Eq. 2 b,c. This equation gives only one possiblecommensurability condition for reciprocal lattice vector G = a ∗ , but it can be extended tothe case of so-called higher harmonics G = na ∗ , n = 1 , , . . . , as shown in the main text.It is worth noting, that the domain wall is always two stars of David thick, independentof Q ( i ) NC , as long as ( Eq. 2 ) is fulfilled. This model also gives slowly varying modulation ofheight, peaking at the domain centre (cf.
Suppl. Fig. 2 a). Real space image is then givenby: Z = < "X i (cid:16) exp(i Q ( i ) NC R ) + exp(i Q ( i ) sat R ) (cid:17) + exp(i a ∗ R ) + exp(i b ∗ R ) . Full details of the nearly commensurate state are captured with Nakanishi-Shiba theory ,which considers NC state as a product of CCDW and periodic domain structure overlayedon top of it: Z = X i exp( Q ( i ) C R )Ψ ( i ) ( R ) , where Q ( i ) C are fundamental CCDW wavevectors andΨ ( i ) ( R ) = X l,m,n ≥ l · m · n =0 ∆ lmn exp(i q ( i ) lmn R )is the modulation responsible for the domain structure.Modulation periods are given by: q ( i ) lmn = lk (1) domain + mk (2) domain + nk (3) domain + q ( i ) . Eq. 3
Here the last term is the discommensuration vector: q ( i ) = Q ( i ) C − Q ( i ) NC Eq. 4 with Q ( i ) NC being the fundamental NCCDW wave vector (see Suppl. Fig. 2 d). q ( i ) there-fore represent the deviation from commensurability. The correct domain periodicity shouldresult in the commensurate CDW inside the domains and thus can be calculated usingcommensurability condition similar to that for fundamental vectors ( Eq. 1 ): k (1) domain = 3 q (1) − q (2) Eq. 5
The above equations are illustrated in Figure d.These two approaches are equivalent and connected by the relation ( Suppl. Fig. 2 c): k (1) domain = Q (1) NC − Q (1) sat . Eq. 6 Q ( i ) NC and Q ( j ) sat where i = j . This is equivalent to several of ( l, m, n ) coefficients being non-zero in NS model ( Eq.3 ). For example, the satellite − k (3) domain will be seen near Q (2) NC as a result of the followingvector equation (see Suppl. Fig. 2 e): − k (3) domain = − Q (1) NC − Q (3) sat . Eq. 7
Below we will refer to these satellites as − k ( i ± domain , where i determines the fundamental peak Q ( i ) NC , near which the satellite ( i ±
1) is observed. They are often seen in FT of STM imagesobtained in NC state, though their intensity is much smaller compared to that of k ( i ) domain (cf. Fig. 3c in the main text). The relation between these satellites and domain packing isillustrated in Suppl. Fig. 3 .One can readily see that David stars have different spacing inside the domain walls,depending on the combination of satellites used in simulation. The interference patternproduced by multiple satellites slightly deviates from the pure commensurability, so it ishard to determine which atomic sites David stars occupy inside the domain walls. To get aqualitative understanding, we have calculated pair distribution functions (PDF) for Davidstar centers (cf.
Suppl. Fig. 4 b) obtained in simulations and compared them to the one forTa atoms (cf.
Suppl. Fig. 4 b). The results are shown in
Suppl. Fig. 4 a, where the firstthree panels correspond to David stars PDF and the last one is atomic. In the latter, peaksare marked by the respective atomic configurations, i.e. 2 + 1 is the sum 2 ~a + ~b . In the casewhere just one satellite, k ( i ) domain , the largest DS peak coincides with the 3 + 1 configuration.Adding more satellites, − k ( i ± domain , shifts it slightly, as one can expect from the change of thevector sum length. More importantly, each configuration produces additional peaks locatedclose to atomic other than the 3 + 1, going through the possible 2, 2 + 1 and 3 shifts, in nicecorrespondence to the theoretical analysis by Ma et al. .5 cbd e Supplementary Figure 2 : Real and reciprocal (FT) space models of NCCDW: a , Realspace model of NCCDW state with overlaid tantalum lattice. Angle between CDW and atomiclattices used in the simulation is φ = 11 . (i)NC / a ∗ = 0 . b ,Reciprocal space model showing relation between atomic and CDW lattices. Atomic vectors areshown in black, Q ( i ) NC – in red, Q ( i ) sat – in blue. This model uses commensurability condition ( Eq.2 ) in the NC state to calculate satellites, Q ( i ) sat , position. c , Zoom-in of the reciprocal space model,showing in details the CDW unit cell. This panel demonstrates the relation between satellitesand domain structure vectors, k ( i ) domain (green). d , Zoom-in of the Q (2) NC area of the reciprocalspace model, demonstrating the relation between Nakanishi-Shiba model and satellite structure.Discommensuration vectors of Nakanishi-Shiba model, q ( i ) = Q ( i ) C − Q ( i ) NC , ( Eq. 4 ) are shown inblack. Their vector sum determines the domain period, k ( i ) domain according to the equation ( Eq.5 ). e , Possible satellite positions near the fundamental peak for the case of Q (2) NC . One of themcorresponds to k (2) domain , other two are described by − k ( j ) domain , where j = 1 , Q ( i ) NC and Q ( j ) P (see ( Eq. 7 )). b d e c f Supplementary Figure 3 : Relation between satellites and packing of hexagonal do-mains in NCCDW: a , Simulation with Q ( i ) NC and k ( i ) domain satellites showing the hexagonaldomains with clearly visible domain walls in between. This image is characteristic to NCCDWstate. Top inset shows the FT. White square emphasizes the domain wall. b, c , Simulation with Q ( i ) NC and − k ( i − domain (in (b)) or − k ( i +1) domain (in (c)) satellites. Whereas domains stay almost in place,the domain walls structure is different. None of (b) or (c) alone are realized experimentally inNCCDW state. d, e , Simulation with Q ( i ) NC , k ( i ) domain and k ( i − domain (in (d)) and k ( i ) and − k ( i +1) i (in(e)). These images are also characteristic to NCCDW state, though compared to (a) they havedifferent phase shift inside the domain wall, resulting in different star of David arrangements (seeareas inside white squares). f , Simulation with Q ( i ) NC , k ( i ) and both − k ( i +1) domain and − k ( i − domain isequivalent to the (a) case. c b r – r r – r Supplementary Figure 4 : Pair distribution functions for interference patterns pro-duced by multiple satellites: a , Pair distribution functions for simulations with Q ( i ) NC and k ( i ) domain (panel 1), plus k ( i − (panel 2) or − k ( i +1) i (panel 3). Panel 4 contains atomic pair distri-bution function. b, c , Point patterns with numerically determined centers of David stars (b) andTa atoms (c) used for PDF calculations. upplementary note 2: fluence dependence of amplitude mode temperature The peak lattice temperature is determined from the frequency of the amplitude mode(AM) (
Suppl. Fig. 5 a), whose temperature dependence is determined from independentlow-fluence measurements (
Suppl. Fig. 5 b). A plot of T AM vs. fluence ( Suppl. Fig. 5 c)shows that at threshold, the temperature of the AM reaches 150 ±
10 K. The AM is themode which is most strongly coupled to the electrons, so its temperature is inevitably byfar the highest of all the lattice modes. a b c d
Supplementary Figure 5 : The determination of the lattice temperature after pho-toexcitation: a , The time-dependence of the AM frequency measured after photoexcitationby time-resolved coherent phonon reflectivity response at different excitation fluences and b , itstemperature dependence with low fluence. c , The temperature of the AM as a function of fluence.At threshold fluence for switching (0.85 mJ/cm ) T AM is 150 ±
10 K (arrow). d , Electron andlattice temperatures calculated within the two-temperature model for 1T-TaS2 for the opticalpulse fluence of 1 mJ/cm applied at T = 4 . upplementary note 3: error estimation in HCDW φ angle measurement To determine the Q H vector length and angle from FT of the atomic-resolution image(main text, Fig. 3d) we used peak positions for atomic and second order CDW reflexes. Thelatter have higher pixel resolution, compared to the first order ones. The values obtainedthis way were averaged among the three independent directions, Q ( i ) H , i = 1 , ,
3. With theFT resolution available in atomically-resolved scans the error is limited by the pixel size.The error bars are thus set as an error introduced by the single pixel shift of the reflexposition on the average value of either angle or length.
C Hidden
Supplementary Figure 6 : Parametric diagram of CDW states: the behavior of thefundamental CDW vector with temperature (equilibrium: NC and C) or after optical excitation(H) is shown in terms of the length (Q (i) / a ∗ ) and the angle between atomic and CDW lattices ( φ ).Hidden states parameters we measured are shown by two red points: one from atomic resolution(larger error), another – from satellite positions and real space analysis (error is equal to experi-mental width of the fundamental peak). Blue dot marks the Q NC position in NC state measuredwith STM on the same sample at T = 205 K. Literature data for the lowest-temperature valuesof Q NC measured with STM is shown by black ( T = 215 K) and gray ( T = 240 K) points. Forthe latter, the length is deduced from the approximate period of 1 . . For reference, yellowline and diamonds show NC states observed with X-ray diffraction down to T = 195 K. At lowertemperatures, CDW vector changes discontinuously to C state (blue diamond). Yellow dotted lineshows the continuation of the NC trend. Red dashed line connects NC and C state. No stablestates were observed in this region of parameters before. Large discrepancy of STM and X-ray datalikely originates from the distortion of NC structure as the transition to C-state is approached. ca II I I II
Supplementary Figure 7 : Single Q H in the optically switched hidden state: a Part ofSTM image (main text, Fig. 2e) of the optically switched hidden state. The areas with large andsmall domains are marked with squares I and II. b, c , FT of the areas I and II: the fundamentalvector (red) Q (1) H remains the same, while the satellite positions (blue) changes in accord with thereal-space domain sizes. Therefore, in the FT of the full image multiple satellites correspond tothe same fundamental vector. Supplementary Method 1: Fitting FT peaks position in hidden state withNakanishi-Shiba model
In order to get high value of satellite spread angle Θ and small deviation of the CDWrotation angle φ from its C-state value within the NS model, we use k ( i ) domain harmonics (seealso Suppl. Fig. 16 d). The fitting procedure of the real data with NS model is given below.In the first step we estimate the required number of harmonics. To this end, we findthe smallest angular distance between the fundamental, Q (1) H , peak and the satellite peaksaround, which in this case is Θ ≈ . ◦ . This value appears the same, independent of thefundamental peak (i.e. i = 2 ,
3) It gives a rough estimate of k (1) domain length and the numberof groups as Θ max / Θ = 5. Indeed, one can see from the cross-section that the peaks along k (1) domain direction can be separated into five groups ( Suppl. Fig. 8 ).In the second step, we determine the averaged peak positions for each group. Peaksalong the k ( i ) domain direction in the experimental data ( Suppl. Fig. 9 a,b) are additionallysplit in the radial direction by 2-3 pixels. This gives the characteristic scale of the additionalmodulation equal to 1 / ∼
70 nm. The only periodicity that can be11 upplementary Figure 8 : Q (1) H : the highest peak corresponds to the fundamental vector Q (1) H , whereas smaller peaks show thatsatellites are grouped near five positions. found on such scale in real space image corresponds to the distance between groups of smalldomains, separated by groups of larger domains and vice versa. This kind of modulationcannot be described by NS model. We collapse them into ’average’ single peak using thesimple centroid procedure described below.To this end we estimate seed positions, that should be located at multiples of k (1) domain andhence lie on a straight line. Then we take rectangular area centered at seed centers and 1 . ◦ wide along k (1) domain and 10 pixels in the radial direction and calculate the weighted centroidfor each of these areas. All the real experimental FT points included in the averaging,not just detected peaks. The resulting centroid FT coordinates are used as average peakpositions to be fitted with NS model.Fit is done using the least squares method. The quantity minimized is the deviation ofall five NS harmonics of k (1) domain from the average experimental points. In the NS model wevary the Q (1) H parameters φ and Q (1) H /a ∗ and thus change direction and length of k (1) domain .The result of the fit is shown in Suppl. Fig. 9 b. The comparison of the STM data withthe real-space simulation of the individual harmonics is presented in the
Suppl. Fig. 9 c-fand shows nice correspondence of the domain sizes.The emergence of higher harmonics underlines the differences between hidden and equi-librium states. The distinction that takes place in the reciprocal space can be illustrated onthe three-dimensional phase diagram, which shows simultaneously the fundamental CDWvector position, ( φ, Q ( i ) /a ∗ ), and corresponding harmonics position, Θ ( Suppl. Fig. 10 ).It appears, that with temperature and photodoping 1T-TaS explores different axes of this12 c d e f b Supplementary Figure 9 : Fitting FT peaks position in the hidden state with NSmodel: a , Part of the Fourier transform (Fig. 3e) of the large-scale STM image of hidden stateshowing Q (1) H peak (red), its satellites (blue) and their position with respect to the origin. b , Zoom-in of the Q (1) H peak area in (a) revealing the details of the Nakanishi-Shiba model: fundamentalpeak is shown in red, satellite peaks – with blue circles, fitted k (1) domain harmonics – in green. c, d ,Close-up of the two regions in Fig. 2e (main text), demonstrating the domains most similar to thereal-space model of the third, e , and of the fifth, f , harmonics of k ( i ) domain . Contrast was adjustedin the model images to emphasize the domains. phase diagram. Indeed, within the phenomenological model presented before , the fulldomain wall length is linked to the difference between the photodoping concentration ofelectrons and holes, n d = n e − n h , and their chemical potentials being equal at the sametime, µ d = µ e = µ h . Given the domain walls are arranged into the regular hexagonal struc-ture, one could extract the CDW vector Q H ∼ Q C − πn d . For high values of n d , which is thecase in the experimentally observed H state, deviation of Q H from Q C should be large, incontrast to the measured values. The system thus chooses to rotate Q H slightly, but excitemultiple harmonics instead. We expect such changes to emerge from different nature of thetransition, as discussed in the main text. 13 upplementary Figure 10 : Parametric ( φ, Q ( i ) /a ∗ , Θ) diagram of CDW states illustrat-ing the difference between the equilibrium and hidden states in 1T-TaS : Blue surfaceshows the first harmonic, Q sat , satellite positions Θ within the Nakanishi-Shiba model (respectiveangles and vectors are shown on the sketch). Solid lines show experimentally observed equilibriumtrajectory of NCCDW (blue) state upon cooling, which then changes discontinuously to CCDW(green) state as temperature is lowered. Hidden state (fitted φ and its value from atomic resolu-tion scan are close) is located in between C and NC states, and does not lie on the continuationof NC trajectory (dotted line). Most importantly, hidden state satellites (domain structure) followdifferent axis: instead of rotating CDW vector, higher harmonics are excited. upplementary Discussion 1: breakdown of Nakanishi-Shiba model at low temper-atures Nakanishi-Shiba model used to describe the NC state in 1T-TaS is based on the mod-ulation of CDW amplitude by the interference of two waves. The first one is fundamentalwith the period Q NC and the second is n = 1 harmonic with the period Q sat (see ( Eq. 2 )).Modulation of the amplitude can be clearly seen at high temperatures, where David stardistortion is the largest in the center of a domain and decreases towards its edge (main text,Fig. 2c)The model predicts that higher harmonics n >
Suppl. Fig. 11 a.The height profile (
Suppl. Fig. 11 c) confirms that David star distortion is the same inde-pendent of the distance to the domain wall ( x = 7 nm). Next, we apply band pass filter tothe Fourier transform of the image: the band is a ring including only the first CDW Brillouinzone and all the Nakanishi-Shiba harmonics in it. Suppl. Fig. 11 b shows the same twodomains as before but in the bandpassed inverse Fourier transform. The respective heightprofile (
Suppl. Fig. 11 c) clearly shows, that David star distortion becomes smaller towardsthe domain wall. Thus, the harmonics observed in the H state are not related to the domainflatnees. This result also shows that the latter comes from the increased intensity in thehigher Brillouin zones of CDW. 15 c b
Supplementary Figure 11 : Absence of relation between domain flatness and har-monics in the H state: a , Zoom of a domain boundary in the STM image of the opticallyswitched state. b , Zoom of the same area as in (a) but in the inverse of the bandpassed FT of theSTM image. The band includes the first CDW Brillouin zone only. c , Height profiles along thecross-section shown by the dashed red line in (a) and (b). Black and red lines are guides for theeye. b d NCT T c T Supplementary Figure 12 : Triclinic CDW state: a , STM image of a triclinic CDW statemeasured on heating at 240 K ( V t = −
800 mV, I t = 100 pA) demonstrating domains (stripes)elongated in one direction (orange arrow) and b , its Fourier transform. CDW unit cell is shown inred. Only 4 of 6 CDW peaks are clearly split, whereas two other ( ± Q (1) T ) has no satellites. Thistype of loss of hexagonal symmetry of the domain structure (not CDW itself) is the signature ofthe triclinic state . Amplitude modulation peaks are marked green. c , zoom-in of the central partof FT (b) showing that hexagonal symmetry of amplitude modulation is also lost in the triclinicstate. (b)-(c) allows to tell unambiguously triclinic state from hidden. d , the same zoom-in for FTof NCCDW state at 300 K (from Fig. 3c in main text), showing the hexagonal symmetry of theamplitude modulation. upplementary Figure 13 : Domain wall behavior in adjacent layers:
STM image of thehidden state measured with large setpoint current ( V t = −
800 mV, I t = 1 . layer beneath. Three domain walls in the top layer (dark groovesapprox. 2 DS wide) are marked with dashed yellow lines. Domain walls beneath can be identifiedby the darker areas within the domains and are marked with dashed red lines . Crossing of domainwalls in the adjacent layers appears as a dip (see arrow). It is clearly seen, that the domain walls inthe adjacent layers tend to avoid each other. This observation suggests that the role of interlayerinteractions cannot be ruled out. upplementary Discussion 2: analysis of hidden state obtained by electrical switch-ing with STM tip Switching from CCDW to metastable metallic hidden state can be done also with elec-trical pulse applied between two contacts on top of 1T-TaS crystal. Alternatively, thepulse can be applied between the crystal and STM tip , and the resulting state can beimaged in situ. Below we perform the same analysis of electrically switched state as the onepresented in the main text for the optically switched state.Here the best switching results were obtained for small tip-sample separation with a lowpulse voltage ( V p ∼ . V ). Suppl. Fig. 14 a shows that the homogeneous CCDW statebreaks down into an irregular array of domains separated by sharply defined walls (theyappear bright in the scan due to different sign of tip bias, compared to the figures in themain text), apparently similar to the optically switched hidden state (o-HCDW). Fouriertransform (
Suppl. Fig. 14 b) reveals the CDW unit cell with the six peaks and theirsatellites. The spread of the latter, Θ max ≈ . ◦ , is almost twice smaller compared to thatof o-HCDW, which reflects smaller domain walls density in case of electrical switching. Thezoom-in of the area around one of the peaks shows that satellite structure is smeared andintensity along − k ( i ± domain directions is vanishingly small, in contrast to more well definedstructure in the o-HCDW state. Even though FT resolution is somewhat worse due tosmaller size of the scan (80 nm vs
200 nm for o-HCDW), the fundamental peak is stronglysmeared, FWHM = 0 . ◦ , as can be seen from angular cross-section in Suppl. Fig. 14 c.The above results present strong evidence towards less homogeneous transition to hiddenstate and absence of the true LRO in case of electrical switching.Smaller Θ max in case of electrical switching suggests that optical pulse excites the elec-tronic system more stongly. The different kind of domain walls (absence of − k ( i ± domain satel-lites) implies that the nature of excitation also plays role in determining the resulting domainstructure. The above arguments allow us to discuss optical switching separately from theelectrical one. 19 b c Supplementary Figure 14 : Electric switching to hidden state with STM tip: a , Large-scale STM image of the domain state obtained after in situ switching with STM tip at 4.2 K( V t = 800 mV, I t = 500 pA); b , Fourier transform of the STM image in (a), showing the CDW unitcell (red). The length of satellite tail is Θ max ≈ . ◦ , which is smaller compared to 9 . ◦ observedin the optically switched hidden state. Inset shows the zoom-in of the Q (2) electric peak area, whichappears quite smeared with structure barely seen. The satellites corresponding to − k ( i ± domain areabsent. c , Angular cross-section of the Q (2) electric peak along q x (red) and q y (blue) FT axes. Solidlines are Gaussian fits. To illustrate the real space details of the domain structure, we have performed the map-ping of David stars misfit vectors. The results are shown in
Suppl. Fig. 15 . Misfit vec-tor field shows no distinct structure. Its amplitude changes randomly in space (
Suppl.Fig. 15 b), and its direction jumps between very different, sometimes opposite, values(
Suppl. Fig. 15 b). Such a behavior is qualitatively different from the Moir´e-like vor-tex structure observed for the long-range ordered H or NC states (see main text): gradualrotation of the direction (dark-blue to dark-red) and gradual increase of the amplitude from0 (green) to 1 (cyan) and then 1 + 1 (blue). This allows us to conclude that the H stateobtained by electrical switching does not have long range order.It should be noted, that domain walls in this state also show X and K crossings inaddition to the standard Y − ¯ Y ones characteristic of the pure hexagonal structure. Thisresemblance could suggest similar mechanism of transition.20 b c d Supplementary Figure 15 : Misfit vector map in the hidden state obtained by electricalswitching with STM tip: a , large-scale STM image of the H state obtained after in situswitching with STM tip at 4.2 K ( V t = 800 mV, I t = 500 pA); b , the map of the misfit amplitudeon the (0 , , c , the map of the misfit directionfor the image in panel (a); d , the legend to the panels (b) and (c), showing the definition of themisfit vector ~ D , its amplitude and direction. upplementary discussion 3: Nakanishi-Shiba model of supercooled NCCDW Since NS theory describes well the domain structure of NCCDW state, we can use it tomodel various outcomes that could be found for supercooled NCCDW state. Here we con-sider two relevant scenarios: (i) “frozen” domains with single Q NC and (ii) sum of “frozen”domains corresponding to different Q NC .The first scenario is equivalent to the trajectory, where the system is heated to sometemperature above C–NC transition and then is quenched to low temperatures. The modelFT corresponding to this scenario is shown in Suppl. Fig. 16 a, where the parameters arechosen to obtain large Θ angle (length of the streak) similar to that in experiment. Thiscorresponds to the temperatures as high as T = 300 K, since at lower temperatures Θ isknown to be smaller. In FT picture we should observe single fundamental Q ( i ) NC set of peaksand with one k ( i ) domain satellite, determined by the commensurability condition ( Eq. 2 ). Twodomains are unlikely to merge into larger one, given Q NC is fixed. Indeed, the mismatchbetween commensurate and real NCCDW lattices increases with size, resulting in energyloss, not gain. Hence, only k ( i ) domain corresponding to l + m + n = 1 in ( Eq. 3 ) are allowed.The only path of merging two domains into larger one can occur with simultaneous rotationof Q NC inside the domain, i.e. another Q NC will appear in FT (see Suppl. Fig. 16 b).This situation is discussed below.The second scenario corresponds to the trajectory, where high-temperature state isquenched at intermediate rates and different Q NC are present in real-space picture. An-other option is that single- Q rapidly quenched state has relaxed at low temperatures, asdescribed above. In both trajectories the FT of such image will result in a number of Q ( i ) NC and k ( i ) domain peaks (see Suppl. Fig. 16 c). This scenario has two distinct features in the FTpicture. First, fundamental peak will be smeared and satellites will no longer be equidistant(due to random choice of local fundamental vectors). Second, satellites will not lie on thesame line with fundamental peak. Both these features are in contrast to experimentallyobserved FT picture.For comparison the model CDW state with 5 harmonics used to fit experimental data forthe hidden state is shown in
Suppl. Fig. 16 d. The features of model FT of supercooled NCstate are absent here and the pictures are qualitatively differen. Thus hidden state cannotbe considered as a supercooled NCCDW state.22 bc d
NC (300K) NC relaxationSupercooled NC Hidden
Supplementary Figure 16 : Comparison of FT pictures for five-harmonic CDW (hid-den) and for various scenarios of supercooled NCCDW: a , First scenario: quenchedhigh-temperature NCCDW state with φ = 11 . (i)NC / a ∗ = 0 . k domain areallowed. b , HT NC state relaxes partially to a configuration closer to CCDW. Additional Q ( i ) NC (red dot) appears with e.g. φ = 13 . (i)NC1 / a ∗ = 0 .
279 closer to that of CCDW (13.9 ◦ , 0.277).The corresponding satellites are shown with blue dots. Green dotted lines show the direction ofall the k domain for the original Q ( i ) NC vector. c , Second scenario: domains with different Q NC (reddots) are frozen during quench ( φ = (11 . × . (i) / a ∗ = (0 . × . Q NC has itsown set of satellites with different k domain , shown with blue dots. The satellites no longer lie on thegreen dotted line, which connect the highest-temperature Q NC and its satellites. d , Five-harmonicCDW used to fit experimental data. Red vector shows the experimentally determined Q ( i ) H with φ = 13 . (i)H / a ∗ = 0 . k ( i ) domain . All of themcorrespond to either of three lines, which originate from fundamental peak (compare to panel (c)). upplementary Method 2: mapping misfit vectors in the H state David stars positions in the H state can be determined with high accuracy using imageprocessing methods. Each David star appears as a well-defined peak in STM images. Theaccuracy of finding the peak could be compromised mostly by the noise. The noise spatialscale can be divided into two categories: (i) slowly varying background on the scale of severalDS and (ii) spikes smaller than DS. Type-1 noise makes detection of DS inside domain wallshard. This problem is solved by using gradient-based algorithms, e.g. Laplacian of Gaussian.Type-2 noise can be overcome using the known separation of neighboring DS. Still algorithmperformance can be improved by preprocessing the image, based on the known global andlocal structure.Both noise sources can be most substantially reduced in the bandpassed inverse Fouriertransform of an experimental STM image. To this end, the band is selected to include onlythe fundamental and harmonic peaks. It is important that such procedure preserves thewhole domain wall structure (see
Suppl. Fig. 17 a,b for comparison). Another approach isto build cross-correlated image using the simulated David star shape and template matchingalgorithm. We have checked that different preprocessing methods result only in quantitativedifferences of several pixels in peak position, whereas qualitative behavior is the same. Belowwe use the maximum filter based on the dilation algorithm applied to bandpassed inverseFT image ( Suppl. Fig. 17 c).To map the misfit vectors of David stars in the H state with respect to their originalpositions in the C state, we have to know the latter. Here we note, that C order is locallypresent in the single domain. Therefore, we create the simulated C lattice which exactlycoincides with that in the largest domain we can find in the experimental image (
Suppl.Fig. 17 d). In this way two arrays containing H and C state centers are obtained. Misfitvectors ~ D are then calculated by finding the nearest C neighbor for each H state David starand measuring length and angle of the shift between them ( Suppl. Fig. 17 d,e). Angle isthen corrected by the difference between atomic axis and experimental x axis. Each H stateDavid star is then given two discrete indices: one shows the shift on (0 , , Suppl. Fig. 17 f,g). To assign theshift index we assume that detected position can have the error up to ± ±
30 degrees. a b c d f g e
Supplementary Figure 17 : Mapping misfit vectors ~ D in the H state: a , zoom of theoriginal image of the H state; b , zoom of the bandpassed inverse Fourier transform of the originalimage (a) with: c , David stars marked with the red dots; d , overlayed David star centers of thesimulated C lattice (black dots); e , overlayed displacement vector field ~ D ; f , overlayed discrete shiftindex – the length of ~ D on (0 , , g , overlayed interpolated angle index – thedirection of ~ D ; b c Supplementary Figure 18 : Inner structure of the X and K defects: a, b , Schematicdrawing of David star positions on a Ta lattice in (a) the X crossing and (b) the K kink (seemain text). Black arrows indicate the misfit vectors. In the case of the X crossing their sum alongthe closed contour is nonzero. In the case of the K kink there is no change in the misfit vector,rather the domain wall changes type (number of atoms shared by David stars touching each other).However, the edge David star (yellow background) is sharing atoms with three neighbors, ratherthan two, as every other ones. c , Twelve possible misfit vectors D connecting 0 and 1 −
12 atomicsites of a David star. The thirteenth is the identity operation ~ D = 0. Supplementary note 4: Correction of STM images.
STM fine X-Y piezoscanners were calibrated at low temperatures within ten percentsbefore measurements. Later each picture was corrected with FFT peaks to match thetriangular lattice of stars of David, i.e. using 2 fundamental CDW vectors. Correctionfor scanner calibration, drift and X-Y crosstalk was done either by calculating the affinetransform between the observed and ideal lattice. In all the cases 1.174nm was taken forCDW period, but this choice affects neither ratio between CDW and atomic periods northe angles. The ratios were checked for consistency with positions of atomic peaks in FTwhenever possible. The ratios were checked for all – H, C, NC – states.26 b Supplementary Figure 19 : Full images of the optically switched hidden state used foranalysis: a – large-scale and b – atomic resolution. For clarity, only parts of these images areshow in the main text in Fig. 2e,f. Fourier transform is done on the full images. ∗ Electronic address: [email protected] † Electronic address: [email protected] Nakanishi, K., Takatera, H., Yamada, Y., Shiba, H. The Nearly Commensurate Phase and Effectof Harmonics on the Successive Phase Transition in 1T-TaS . J. Phys. Soc. Jap. , 1509-1517(1977). Yamada, Y. and Takatera, H. Origin of the Stabilization of the Nearly Commensurate Phase in1T-TaS . Sol. Stat. Comm. , 41-44 (1977). Scruby, C. B., Williams, P. M., Parry, G. S. The role of charge density waves in structuraltransformations of 1T-TaS . Phil. Mag. , 255-274 (1975). Ma, L. et al . A metallic mosaic phase and the origin of Mott-insulating state in 1T-TaS . Nat.Commun. , 10956 (2016). Thomson, R. E., Burk, B., Zettl, A., Clarke, J. Scanning tunneling microscopy of the charge-density-wave structure in 1T-TaS . Phys. Rev. B , 16899 (1994). Wu, X. L. and Lieber, C. M. Direct Observation of Growth and Melting of the Hexagonal- omain Charge-Density-Wave Phase in 1T-TaS by Scanning Tunneling Microscopy. Phys. Rev.Lett. , 1150 (1990). Stojchevska, L. et al . Ultrafast Switching to a Stable Hidden Quantum State in an ElectronicCrystal.
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