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[Shunsuke Yoshizawa et al. - Phys. Rev. Lett. 132, 056401 (2024) Visualization of Alternating Triangular Domains of Charge Density Waves in 2H-NbSe2 by Scanning Tunneling Microscopy.pdf](https://mdr.nims.go.jp/filesets/ee5ebaba-d76e-451c-9fe6-93d0d6d81df9/download)

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[Shunsuke Yoshizawa](https://orcid.org/0000-0003-3380-5473), [Keisuke Sagisaka](https://orcid.org/0000-0002-5089-4271), [Hideaki Sakata](https://orcid.org/0009-0005-4860-6735)

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[Visualization of Alternating Triangular Domains of Charge Density Waves in 2H-NbSe2 by Scanning Tunneling Microscopy](https://mdr.nims.go.jp/datasets/76bfb36a-9b22-4a9c-bce1-86fdc1518e1d)

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Visualization of Alternating Triangular Domains of Charge Density Waves in 2H-NbSe2 by Scanning Tunneling MicroscopyVisualization of Alternating Triangular Domains of Charge Density Waves in 2H-NbSe2by Scanning Tunneling MicroscopyShunsuke Yoshizawa ,1,* Keisuke Sagisaka ,1 and Hideaki Sakata 21Center for Basic Research on Materials, National Institute for Materials Science, 1-2-1 Sengen, Tsukuba, Ibaraki 305-0047, Japan2Department of Physics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku, Tokyo 162-8601, Japan(Received 3 April 2023; revised 27 October 2023; accepted 21 December 2023; published 29 January 2024)The charge density wave (CDW) state of 2H-NbSe2 features commensurate domains separated bydomain boundaries accompanied by phase slips known as discommensurations. We have unambiguouslyvisualized the structure of CDW domains using a displacement-field measurement algorithm on a scanningtunneling microscopy image. Each CDW domain is delimited by three vertices and three edges ofdiscommensurations and is designated by a triplet of integers whose sum identifies the types ofcommensurate structure. The observed structure is consistent with the alternating triangular tiling patternpredicted by a phenomenological Landau theory. The domain shape is affected by crystal defects and alsoby topological defects in the CDW phase factor. Our results provide a foundation for a completeunderstanding of the CDW state and its relation to the superconducting state.DOI: 10.1103/PhysRevLett.132.056401The charge density wave (CDW) is a cooperative pheno-menon characteristic of low-dimensional metals [1,2].CDWs that coexist with other ordered states, exemplifiedby superconductivity, are of particular interest in view ofthe interplay between multiple orders [3]. Notable exam-ples are CDWs in cuprate superconductors [4,5], kagomesuperconductors [6,7], and transition metal dichalcoge-nides (TMDCs) [8]. 2H-NbSe2 is a prototypical TMDCthat exhibits both the CDW and superconducting states. Itundergoes a triple-Q CDW transition, where three equiv-alent CDW wave vectors are 120° apart, at a temperature ofTCDW ∼ 30 K. Although neutron and x-ray diffractionexperiments indicate an incommensurate periodicity [9,10],scanning tunneling microscopy (STM) studies haveunveiled locally 3 × 3 commensurate domains [11–16].Domain boundaries serve as discommensurations, wherethe phase of the CDW shifts over a short distance [17].Superconductivity occurs below the critical temperature ofTc ∼ 7 K. A recent STM study [18] reveals the existence ofthe Cooper-pair density wave state, which shares the sameperiodicity as the CDW with a constant phase difference.These two modulated states have a common domainboundary. The results of another study [19] imply thatthe superconductivity competes with the CDW dependingon the commensurate local structure. To better understandthe interplay between the CDW and superconductivity,precise knowledge of the domain structure of the CDW isindispensable.Despite extensive study, the domain structure of theCDWof 2H-NbSe2 remains elusive. As shown in Fig. 1(a),the crystal structure of 2H-NbSe2 consists of Nb atomsforming a triangular lattice and Se atoms protruding up anddown from the Nb plane. STM observations and densityfunctional theory calculations have identified two typesof commensurate domain with distinct local structures[15,19,20]. One is characterized by the CDW maxima atSe sites, referred to as the chalcogen-centered (CC)structure. The other is characterized by the CDW maximaat hollow sites and is referred to as the hollow-centered(HC) structure. Gye et al. have proposed a complexstructure of these CDW domains [15], where HC domainsare classified into nine types depending on the phase ofmodulation, forming isolated hexagonal patches, and CCdomains surround HC domains and form a honeycombnetwork. However, Gye et al. did not provide a nonvisualmethod to identify the domain boundary. Furthermore, theyfocused on the domain of only the HC structure and treatedthe CC structure as a type of discommensuration. Thisasymmetric treatment of the two commensurate structuresand the ambiguity in domain identification may haveprevented a simpler understanding of the domain structure.Other groups reported the detection of discommensurationsas steep changes in the phase factor of the CDW extractedfrom topographic images [18,21], but the nonvisual deter-mination of the shape and the type of domains remainedunresolved.Here, we present a systematic approach to visualize theCDW domain structure of 2H-NbSe2 using topographicimages obtained by STM. Through a simple simulation, wedemonstrate that the local CDW structure is characterizedby a triplet of real numbers, denoted as ðn1; n2; n3Þ. In eachcommensurate domain, the triplet takes integer valuesand their sum modulo 3 determines the type of commen-surate structure. We determined the spatial variation ofðn1; n2; n3Þ by measuring the displacement fields of theCDW modulations relative to the atomic lattice in aPHYSICAL REVIEW LETTERS 132, 056401 (2024)0031-9007=24=132(5)=056401(6) 056401-1 © 2024 American Physical Societyhttps://orcid.org/0000-0003-3380-5473https://orcid.org/0000-0002-5089-4271https://orcid.org/0009-0005-4860-6735https://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevLett.132.056401&domain=pdf&date_stamp=2024-01-29https://doi.org/10.1103/PhysRevLett.132.056401https://doi.org/10.1103/PhysRevLett.132.056401https://doi.org/10.1103/PhysRevLett.132.056401https://doi.org/10.1103/PhysRevLett.132.056401topographic image, thereby revealing the domain structurewithout relying on visual interpretation. Each CDWdomain is bounded by three vertices and three edges,consistent with the alternating triangular tiling predicted byan empirical Landau theory. The deformation of the domainshape from an equilateral triangle is attributed to thepinning of discommensurations by crystalline defects.We used an ultrahigh-vacuum (UHV) cryogenic STMsystem based on the model USM-1300 from Unisoku Co.Ltd. The STM tip was fabricated by focused ion beamfrom a mechanically sharpened Pt-Ir wire and conditionedon a clean Au(111) surface prior to the experiment. Thesingle-crystal 2H-NbSe2 was grown by the chemical vaportransport method using iodine as the transport agent. Thesample was cleaved in UHV at room temperature andimmediately transferred to the STM head kept at a lowtemperature. All the STM data presented in this Letter wererecorded in a constant-current mode at 4.5 K in zeromagnetic field. The resistivity measurement of a sample inthe same batch gives the residual resistivity ratio of 42,Tc ¼ 7.3 K, and TCDW ≃ 30 K.Figure 2(a) shows a high-resolution topographic imageof a 100 nm × 100 nm area that clearly resolves the CDWmodulations as well as the triangular lattice of Se atoms.The image also displays several types of atomic-scaledefect. The Fourier transform of this image [Fig. 2(b)]displays sharp peaks at the periodicity of the crystal lattice(b1, b2, and b3) and slightly broader peaks at theperiodicity of the CDW (Q1, Q2, and Q3). The broadnature of the latter peaks reflects the finite size of CDWdomains. All the other peaks are considered as harmonicsof these periodicities. The line profiles in Fig. 2(c) indicatethat the length of Qj is approximately one-third that of bj.Upon closer examination of the topographic image, weobserve a difference in local atomic-scale structure.Figures 2(d) and 2(e) depict enlarged images cropped fromthe regions labeled A and B in Fig. 2(a), respectively.The former displays a repeated star-shaped pattern that ischaracteristic of the CC structure, while the latter displays arepeated clover-shaped pattern of the HC structure [15,19].Figures 2(f) and 2(g) show the corresponding Fouriertransforms. The patterns are almost identical to each otherat the pixel level. This indicates that the CC and HCstructures share the same periodic components but differ intheir phase.(a)(d) (e)(b)(c)(f) (g)FIG. 2. (a) Topographic image of a 100 nm × 100 nm area of acleaved surface of 2H-NbSe2. The data were recorded at aresolution of 2048 × 2048 but were resampled to 1024 × 1024 toreduce the file size. The image was flattened by subtracting athird-order polynomial fit from each line. The feedback conditionwas 500 pA at 50 mV. (b) Fourier transform of the topographicimage in (a). The dotted and solid circles indicate the spots of theatomic lattice (b1, b2, and b3) and those of the CDW (Q1, Q2,and Q3), respectively. (c) Line profiles of the Fourier transformimage along b1, b2, and b3 directions. (d) and (e) Croppedimages of the CC and HC domains indicated by the boxes labeledA and B in (a). (f) and (g) Fourier transform images of croppeddata of (d) and (e).3j =12SeSeNb(0, 0, 0) (1, 0, 0)(1, 1, 0) (1, 1, 1)(c)(d) (e)(b)(a)FIG. 1. (a) Top view (upper panel) and front view (lower panel)of the crystal structure of 2H-NbSe2 [22]. The unit cell isdepicted by a dotted rhombus (top view) and a dotted rectangle(front view). The solid rhombus indicates the 3 × 3 unit cell of thecommensurate CDW. (b)–(e) Simulated CDW images withseveral combinations of indices ðn1; n2; n3Þ. The dotted linesindicate the maxima of CDW components. The solid rhombusindicates the 3 × 3 unit cell of the CDWand is fixed to the atomiclattice to show the relative position of the CDW maxima.PHYSICAL REVIEW LETTERS 132, 056401 (2024)056401-2To investigate how the relative phase alters the topo-graphic appearance, we present a simple simulation. Weapproximate the charge density within a commensuratedomain by the sum of two functions of position r ¼ ðx; yÞ,ρðn1;n2;n3ÞðrÞ ¼ ρSeðrÞ þ cρðn1;n2;n3ÞCDW ðrÞ: ð1ÞHere, ρSeðrÞ ¼ ReP3j¼1 expðibj · rÞ represents the latticemodulations with maxima at Se sites, and ρðn1;n2;n3ÞCDW ðrÞ ¼ReP3j¼1 exp½iðQj · r − 2πnj=3Þ� provides the CDW mod-ulations with commensurate wave vectors Qj ¼ ð1=3Þbj.The indices j ¼ 1, 2, 3 represent the directions. Theconstant c controls the amplitude of the CDW modulationsrelative to the lattice modulations, and we set it to be ∼2.5.The CDWmodulations have phase offsets controlled by nj.Increasing nj by 1 translates the jth component of theCDW by a lattice constant, and increasing it by 3 restoresthe CDW to its initial position. The simulated topographicimages ρðn1;n2;n3ÞðrÞ for several combinations of ðn1; n2; n3Þare shown in Figs. 1(b)–1(e). In the case of (0,0,0)[Fig. 1(b)], the CDW and the atomic lattice are in phase,and CDWmaxima are located at Se sites at the corner of the3 × 3 cell. This condition yields a topographic image ofthe CC structure. In the case of (1,0,0) [Fig. 1(c)], CDWmaxima are located at hollow sites, and the image showsthe clover-shaped pattern of the HC structure. In the case of(1,1,0) [Fig. 1(d)], CDW maxima are located at Nb sites.The pattern resembles that of the HC structure, but theorientation of the clover is the opposite. In the case of(1,1,1) [Fig. 1(e)], the image again displays the CCstructure, but CDW maxima are at Se sites different fromthose for (0,0,0). By examining the structure of variouscombinations of ðn1; n2; n3Þ in this manner, we observe thatthe type of domain is determined by λ≡ ðP3j¼1 njÞ mod 3,with λ ¼ 0 corresponding to the CC structure and λ ¼ 1 tothe HC structure. All the possible structures are presentedin Sec. S1 of Supplemental Material (SM) [23].The precise determination of the triplet ðn1; n2; n3Þfrom the topographic image is essential for visualizingthe domain structure. To achieve this, we utilize a dis-placement detection algorithm proposed by Lawler andFujita et al. [26]. Assuming that the topographic image zðrÞcontains a CDW modulation with periodicity Qj that isaltered by a slowly varying displacement field ujðrÞ, weobtain the displacement field using the relationexp½−iQj · ujðrÞ� ∝Xr0zðr0Þ expð−iQj · r0Þwðr − r0Þ; ð2Þwhere wðrÞ ¼ ð2πσ2Þ−1 exp½−jrj2=ð2σ2Þ� is a two-dimensional Gaussian with a spatial extent of σ. We setσ ¼ 1 nm in the present analysis. From the phase compo-nent of the right-hand side, we obtain the (apparent)displacement field ujðrÞ≡ ujðrÞ ·Qj=jQjj for each j.However, this quantity includes an extrinsic image defor-mation vðrÞ, which originates from the creep effect of thepiezoelectric scanner or the drift of the sample. To eliminatethis extrinsic effect, we extract vðrÞ by applying the Lawler-Fujita algorithm to the periodicities bj of the crystal lattice.We then obtain the intrinsic displacement fields of theCDW modulations by subtracting vjðrÞ≡ vðrÞ · bj=jbjjfrom ujðrÞ. These intrinsic displacement fields are thendivided by the interplanar spacing to obtain the spatialdependences of ðn1; n2; n3Þ. A complete description of thisprocedure is presented in Sec. S2 of SM [23].Figures 3(a)–3(c) show the images of n1ðrÞ, n2ðrÞ, andn3ðrÞ determined from the topographic image in Fig. 2(a).Each image displays a step-terrace structure, and thehistograms in Figs. 3(d)–3(f) show sharp peaks at integervalues, highlighting the successful determination of njvalues. The results also show that the region is mostlycovered by commensurate domains. The domain structurewas visualized in Fig. 3(g) by plotting λðrÞ≡fPj nint½njðrÞ�g mod 3, where nint denotes the nearestinteger. The region is composed of alternating domains ofλ ¼ 0 (CC) and λ ¼ 1 (HC), along with small areas withλ ¼ 2 at the intersections of domain boundaries. Eachdomain is enclosed by three vertices and three edges and isidentified by a unique triplet of integers ðn1; n2; n3Þ. Therealization of this domain structure indicates that thediscommensurations of the Q1, Q2, and Q3 componentsare not independent but are forced to intersect at a singlevertex. A direct comparison of the domain structure and thetopographic image is shown in Fig. S7 of SM [23]. We haveconfirmed the reproducibility of the present findings onanother 2H-NbSe2 sample (see Sec. S4 of SM [23]).We also evaluated the width of the discommensurations.For each j, we defined the discommensurate region as theregion where njðrÞ is closer to a half-integer than to aninteger. We then estimated the width by dividing the areaof the discommensurate region by the total length of thediscommensuration. Details are provided in Sec. S3 ofSM [23]. The estimated width was about 3 nm for all CDWcomponents. This is not an artifact but the real width of thediscommensuration, because the value is larger than thebroadening induced by the σ of the Gaussian function wðrÞ.This finding implies why Gye et al. [15] proposed a picturedifferent from ours, where the HC domains were suggestedto form isolated patches separated by continuous CCregions. This interpretation arises when the discommensu-rate region in our definition is misassigned to the CCregion.The domain structure revealed here is consistent withthe results of the studies based on the phenomenologicalLandau theory in the 1970s–1980s [17,27–29]. In particu-lar, McMillan’s theory of discommensuration [17] wasextended to describe the CDW states of TMDCs witha focus on 2H-TaSe2 [27–29]. These studies classifiedthree possible commensurate structures, depending on thePHYSICAL REVIEW LETTERS 132, 056401 (2024)056401-3location of the threefold rotation axis, and showed thatthese structures were labeled by three integers, as we havefound in our analysis. Nakanishi and Shiba [28,29] pro-posed a generic form of the free energy with manyparameters and performed a numerical minimization fora parameter set that gives the situation where two typesof commensurate structure become equally stable. Theypredicted an alternating triangular tiling pattern of the twotypes of commensurate domain [29]. Figure 4(a) shows theCDW image simulated by minimizing Nakanishi-Shiba’sfree energy. Details of the numerical simulation are shownin Sec. S5 of SM [23]. Although the parameter choice wasnot optimized to describe 2H-NbSe2, the predicted patternis essentially equivalent to the domain structure obtainedin our experiment. Note that the theoretical modelwas proposed before STM began to be used to observeCDWs [30], and it took nearly 40 years for its directverification in actual material.The observed domain structure is heavily deformedcompared with the predicted equilateral triangles. Thisdeformation is attributed to the pinning of discommensu-rations by defects in the crystal, as discussed in theoreticalstudies [31,32]. Figures 4(b) and 4(c) show the topographyof the same region at 4.5 K before and after a thermal cycleacross TCDW. The thermal cycling has significantly alteredthe domain structure. Before the thermal cycle, a protrudingdefect P1 was located on a discommensuration. After thethermal cycle, P1 and P2 were on discommensurations andP3 was also close to a discommensuration. This observa-tion indicates that these protruding defects potentially actas pinning centers. It is reasonable that only some of thedefects pin discommensurations, because the present sam-ple surface has a large defect density compared with thespacing between discommensurations. Avoidlike defect V1also appears to pin discommensurations. The exact roles ofthe different types of defect remain unclear in the presentstudy. In addition, it is still uncertain why defects locallyinduce CDW above TCDW [16,33], but attract discommen-surations below TCDW. Although elucidating these points isbeyond the scope of this study, our domain visualizationmethod should be crucial for such investigations.(a) (b) (c)(g)(d) (e) (f)FIG. 3. (a)–(c) Images of n1ðrÞ, n2ðrÞ, and n3ðrÞ. The arrowsindicate the locations of topological defects. (d)–(f) Histogramsof n1ðrÞ, n2ðrÞ, and n3ðrÞ. (g) Image of λðrÞ. The CC and HCdomains are depicted as white and blue areas, respectively. Atriplet ðn1; n2; n3Þ is written in each domain. The circles anddotted curves are the vertices and edges of the ð0; 1;−1Þ CCdomain, respectively.(b)(a)(c)FIG. 4. (a) Simulated topographic image of a 160a × 90aregion (a is the lattice constant). (b) Topographic image of a25 nm × 25 nm region recorded after the initial cooling fromroom temperature to 4.5 K. (c) Topographic image of the sameregion recorded after thermal cycles across TCDW. The whitecurves represent the discommensurations. The feedback condi-tion was 500 pA at 50 mV. The CC and HC domains aredisplayed in different colors for clarity.PHYSICAL REVIEW LETTERS 132, 056401 (2024)056401-4The changes in the domain structure induced by thermalcycling suggest that the observed domain structures cor-respond to metastable states even at 4.5 K. Smoothrelaxation to the ground state was probably prevented bythe presence of topological defects, as the density of topo-logical defects depends on the cooling rate [34], a param-eter not controlled in this study. Topological defects in theCDW state of 2H-NbSe2 are known to be excited as vortex-antivortex pairs in the phase factor of the CDW [21,35]. Inour case, vortices and antivortices exist at the terminationsof the discontinuities in the image of n1ðrÞ [Fig. 3(a)].Interestingly, we find that topological defects also exist inn2ðrÞ and n3ðrÞ at the same locations as those in n1ðrÞ[Figs. 3(b) and 3(c)]. This highlights the fact that thetopological defects in this system cannot exist independ-ently in a single CDW component, but always appear asvortex-antivortex pairs in any two of the three components.This can preventPj nj from changing by more than onearound the vortex, thereby avoiding the formation of aregion with λ ¼ 2. Owing to this intertwined nature of thethree CDW components, vortex-antivortex pairs of eachCDW component cannot annihilate independently butmust do so simultaneously. This restriction may hinderthe relaxation to the ground-state domain structure free oftopological defects.In conclusion, our low-temperature STM observations,combined with systematic data analysis, have provided anaccurate depiction of the domain structure of the CDW statein 2H-NbSe2. Our findings reveal that each CDW domainis characterized by a triplet of integers, and the domainsform an alternating triangular tiling that is deformed by thepresence of crystalline and topological defects. 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