Zoology of domain walls in quasi-2D correlated charge density wave of 1T-TaS2

Domain walls in correlated charge density wave compounds such as 1T-TaS2 can have distinct localized states which govern physical properties and functionalities of emerging quantum phases. However, detailed atomic and electronic structures of domain walls have largely been elusive. We identify using scanning tunneling microscope and density functional theory calculations the atomic and electronic structures for a plethora of discommensuration domain walls in 1T-TaS2 quenched metastably with nanoscale domain wall networks. The domain walls exhibit various in-gap states within the Mott gap but metallic states appear in only particular types of domain walls. A systematic understanding of the domain-wall electronic property requests not only the electron counting but also including various intertwined interactions such as structural relaxation, electron correlation, and charge transfer. This work guides the domain wall engineering of the functionality in correlated van der Waals materials.


INTRODUCTION
Metal-insulator transitions and the emergence of superconductivity and other exotic quantum phases from strongly interacting electronic systems are central issues in condensed matter physics and important for device applications. Such interacting electronic systems are recently attracting particular interest in twodimensional (2D) materials with the extraordinary tunability provided [1][2][3] . In particular, quasi 2D charge-density-wave (CDW) compounds and their monolayer versions with strong interactions are under the focus of huge research activity because of rich quantum phases emerging from the commensurate CDW (CCDW) ground state with exotic electronic properties such as a spin frustrated Mott insulator in 1T-TaS 2 and an excitonic insulator in 1T-TiSe 2 4,5 . Suppressing the CCDW order in these materials by heat, pressure, doping, ultrafast optical, and electrical pulses leads to various quasi-metallic metastable phases which consist of CCDW domains and discommensuration domain walls (DWs)  . The electronic properties of these 'frustrated' phases have been known to be governed by electronic states residing on DWs created [24][25][26][27][28] . Those topological in-gap states have been suggested to play crucial roles in the emerging superconductivity [26][27][28][29] and the memristic resistivity switching of DW phases 14 .
In this work, we focus on the DW phases excited from the Mott CDW ground state of 1T-TaS 2 30-32 with strong electron correlation 4 and strong spin frustration 33,34 . The atomic structure of CCDW in 1T-TaS 2 has a unit of the David-star (DS) cluster with a ffiffiffiffiffi 13 p ffiffiffiffiffi 13 p supercell 35 , which features one unpaired electron in the central Ta atom. This electron has been known to fall into a spin-frustrated Mott-insulator state by the onsite Coulomb repulsion 4 . A recent STS study observed two types of isolated DWs among the various types of DW configuration 18,19 , as line defects within this 'correlated' CCDW phase, which have insulating electronic states due to strong electron correlation and structural relaxation within them 36 . Above 180 K, the Mott insulating CCDW phase transits spontaneously into a metallic phase with a nanoscale honeycomb DW network, which is called a nearlycommensurate CDW (NCCDW) phase 29,[37][38][39][40][41][42] . Our recent work identified this DW with another type of structure and a metallic ingap state 29 , which is absolutely different from the two types of insulating DWs within the CCDW phase 36 . A theoretical model was suggested 29,43 to connect it to the emerging superconductivity from this phase under pressure and doping 26 . This series of recent research revealed the various electronic properties and huge structural degrees of freedom within DWs of 1T-TaS 2 CDW phases. However, only a small fraction of DWs are identified among those expected from the large (at least thirteen-fold) degeneracy of the CDW structure and a systematic understanding of their diversity and complexity has been lacking. This information would be essential for the controllability over the functionality and the emerging or hidden phases brought by DWs [10][11][12][13] .
In this paper, we explore the structural and electronic properties of the 1T-TaS 2 surface layer, which was manipulated by scanning tunneling microscope (STM) voltage pulses into nanoscale metastable DW networks. The metastable DW networks are endowed with an enhanced variety of DWs 18,19 . We identify more than ten different DW structures and their topological vertices. The atomic structures and electronic states of most of these DWs are determined by direct comparisons of STM data and density functional theory (DFT) calculations. A systematic understanding of the structural and electronic variety of DWs is obtained through extensive DFT calculations, which unravel the diverse interactions involved such as structural relaxations, electron correlation effect, and charge transfer from neighboring domains.

RESULTS
Domain walls and their network in the nanoscale quenched phase Various types of DWs are found in the quenched phase of Fig. 1a, b, which are created by applying a positive voltage pulse from an STM tip 32 . It consists of domains of a few tens of nanometers in the CCDW structure as separated by various different DWs. These DWs form an irregular network extending laterally to a size of up to a few hundred nanometers depending on the pulse voltage.
Most DWs are formed straightly along the CCDW unit vector with a single-phase shift between two adjacent CCDW domains. Due to the thirteen Ta atoms in the DS cluster, there are twelve possible straight DW configurations and the notation of DWs are defined as the relative position of the center DS clusters between two adjacent domains as shown in Fig. 1c and Supplementary Fig. 1. We also found the zigzag type of DWs, which can be defined by two alternating phase-shift vectors, one of which is rotated by 60 ∘ with respect to the other (see Fig. 1f). Thus, the zigzag DWs are rotated by 30 ∘ with respect to the straight DW. The twelve possible zigzag DW configurations are summarized in Supplementary Fig.  1. The DWs also construct the complex DW junctions including the X, Y, and + shape ( Supplementary Fig. 2). The honeycomb-like network of DWs can be seen as the combination of Y-shaped junctions ( Supplementary Fig. 3). The detailed properties of DWs will be discussed further below.

Population and energetics of domain walls
Among the twelve possible straight DW configurations, we found six types of DWs in the quenched phase. The second-type DW (DW-2) is the most popular DW structure and DW-4 and DW-1 are also frequently found. These DWs were already observed in the CCDW phase at low temperature (DW-2 and DW-4) and in the NCCDW phase at room temperature (DW-1) 18,29,36 . Three minor DWs of DW-3, DW-6, and DW-12 appear sparsely with shorter lengths, but the other six possible straight DWs cannot be found in the present STM images. Interestingly, the zigzag type of DWs is rather popular (less than DW-4 but more than DW-1) with relatively long DW lengths. We found three types of zigzag DWs, namely, DW-2,8, DW-1,4, and DW-3,4. The two indices given here indicate the two-phase shift vectors defining a zigzag DW (Fig. 1f).
We optimize the atomic structures of all twelve configurations of straight DWs by DFT calculations. The resulting energetics and electronic properties are summarized in Table 1. The fully relaxed DFT calculations predict that DW-2 is the most stable structure and the second-lowest energy configurations of DW-1 and DW-4 almost degenerate in energy. This energetic hierarchy is largely consistent with the observed population of major DWs. However, the present calculations cannot explain all the population details of the minor DWs of the DW-3, DW-6, and DW-12 and the other absent DWs in the experiment. This may be due to their short lengths and various boundary conditions imposed by neighboring domains, DWs, and DW junctions. In addition, we also observe that the energetically unfavored DWs, such as DW-5, DW-8, and DW-6 break into two DWs of lower energies as shown in Fig. 2. This partly explains the absence of the unfavored DWs. The energy costs of zigzag DWs (0.025, 0.028, and 0.030 eV/DS for DW-2,8, DW-1,4, and DW-3,4, respectively) are comparable to those of major straight DWs. This seems roughly consistent with their popularity in the experiment, being popular than the minor straight DWs. The present results indicate the importance of detailed atomic structures in understanding the DW phases.

Atomic structures of straight domain walls
The atomic structure of DW-2 in Fig. 3b is composed of two imperfect DS clusters of a similar shape, each with 12 Ta atoms but with distinct numbers, five (left) and six (right), of S atoms. Since bright protrusions in STM images represent top-layer S atoms, the different numbers of S atoms are well reflected in the corresponding STM images; larger triangular clusters and smaller imperfect triangles in the left and the right side of the DW, respectively (Fig. 3f). These STM images are well reproduced in the   Fig. 4), while we find the double row model to reproduce better the STM images observed.
The STM image of DW-4 exhibits also an asymmetric double row (named as DW-4B) of imperfect DS clusters with 22 Ta atoms (the lower part of Fig. 3g). Very interestingly, we find that this double row structure changes abruptly to a single row structure (DW-4A) with the same phase shift between neighboring domains (Fig. 3g). This evidences that a DW can have a few competing structures, due to its own internal structural degree of freedom. Our DFT calculation tells that the single-row structure (DW-4A) is formed with a single imperfect DS cluster (with nine Ta atoms) and distorted DS clusters at the neighboring domain edge. This structure is consistent with the previous structural model 36 , and it is more stable (~56 meV/supercell) than the double row (DW-4B) structure. The optimized atomic structures of DW-4A and DW-4B reproduce the observed STM images well. We also find another type of a single-row DW structure without a strong distortion of the domain edge, which is slightly less stable (~29 meV/supercell) than the above DW-4A structure. However, its electronic structure is not consistent with previous STS spectra 36 ( Supplementary Fig.  9).
The DW-1 structure comprising 25 Ta atoms is similar to the DW-2 but features an additional Ta atom between two imperfect DS clusters. The corresponding STM image shows rows of three clusters within DW-1 in contrast to the double rows of DW-2 (Fig.  3e). This structure is consistent with our previous work 29 . Other minor DWs also have their characteristic atomic structure, which is determined by their own CDW phase shift vectors and the internal structural relaxation (Supplementary Figs. 5 and 6).

Electronic structures of straight domain walls
The rich variations in DW structures result in a variety of localized electronic states. All twelve straight DWs exhibit in-gap states localized one-dimensionally along the DWs (Supplementary Fig.  7). The Mott gap of the CCDW phase is, thus, substantially reduced or closed at DWs. However, details of in-gap states depend on various different parameters such as the number of Ta atoms (d z electrons) in the DW region, the structural relaxation, the correlation effect, and the charge transfer from neighboring domains. For example, the DW-2 structure exhibits a rather large bandgap (Fig. 3i), which is consistent with the previous STS study 36 . There exist two in-gap states localized along with this DW at about 0.1 and 0.2 eV above the Fermi energy. The majority and minority spin bands for these in-gap states (Fig. 3i) almost degenerate and their band dispersions do not change substantially with the inclusion of the Coulomb energy U ( Supplementary   Fig. 7). That is, the two in-gap states are split trivially by the bonding-antibonding interaction due to the structural relaxation. The insulating property of DW-2 is basically due to the even number of electrons from the two identical imperfect DS unit cells within the DW.
In stark contrast, DW-1 with a similar atomic structure exhibits a metallic behavior with two in-gap bands crossing the Fermi level. These bands are due to an unpaired electron on the additional link Ta atom (marked by the black circle in Fig. 3a), which makes the total electron number within DW-1 odd. As shown in Fig. 3h, the band splitting is due to the local spin and the electron correlation. Except for this Ta atom with partially filled spin-polarized states, the atomic configuration with two identical imperfect DSs, each with twelve Ta atoms, in the DW-1 structure is similar to the DW-2 structure and the band dispersions of corresponding in-gap states above the Fermi level are comparable to each other. This comparison manifests the crucial importance of the number of electrons within a DW unit cell.
The atomic and electronic property of a DW can substantially be affected by the charge transfer in addition to the electron correlation. The single-row structure of DW-4A is endowed with an odd number of Ta atoms and the partially filled orbital is expected on the imperfect DS cluster within the DW. The DW-4A, however, exhibits an insulating character with a bandgap of 0.1 eV due to the charge transfer from the edge of the neighboring domain to the DW: We find that the fully filled state between −0.2 to −0.1 eV (the blue band) is spin-split and localized on one edge atom of the broken DS cluster of the DW (Supplementary Fig. 8). On the other hand, the lower and upper Hubbard states of the domain-edge DS cluster are shifted upward just above the Fermi level (indicated by the purple arrow in Fig. 3j), which is consistent with the previous STS spectra 36 . These behaviors indicate that electrons are transferred from the edge of the neighboring domains to the DW clusters. We further notice that the amount of charge transfer depends on the structural distortion of the DS clusters on domain edges (Supplementary Fig. 9). In the double row structure (DW-4B) of the same domain wall, the DW width becomes large with two broken DS clusters. A band localized at those imperfect clusters becomes partially filled due to the charge transfer from the domain region (indicated by the green arrow in Fig. 3k). The other states on the imperfect DS clusters within the DW open a relatively large bandgap of 0.25 eV with spin-split bands ( Supplementary  Fig. 8), therefore there is a possibility of the 1D magnetism along with this DW (Supplementary Fig. 10). However, this is closely related to the controversial issue of the magnetic ordering for the CCDW phase 4,33,34,44,45 thus the magnetic ordering predicted in the present calculations for the DWs has to be investigated further by experiments and more sophisticated calculations. The above comparison of two different DW structures for one DW type indicates the importance of the charge transfer between DWs and neighboring domains, as well as the structural degree of freedom. In most of DWs except the DW-11, the Hubbard bands of neighboring domains are shifted to higher energy indicating the donation of electrons from domains to DW (Table 1 and Supplementary Fig. 7). This is consistent with the peak shift of Hubbard states at domain edges observed in the previous experiment 36 .
Zigzag domain walls Figure 4 shows the atomic structures and STM images of zigzag DWs (DW-2,8, DW-1,4, and DW-3,4). The DW-2,8 and DW-1,4 consist of four types of Ta clusters and the clusters are connected to each other through distortions of outer Ta atoms. The DW-3,4 consists of three types of Ta clusters so that the width is narrower. These STM topographies, which are well reproduced in our DFT simulations, reveal that the zigzag DWs have an even larger structural degree of freedom. There exist two types of in-gap states related to these zigzag DWs, which are localized at DWs (the red lines in Fig. 4d obtained on the broken DW clusters mentioned above, which is indicated by red clusters in Fig. 4a, b, and c) and edges of neighboring domains (blue lines of Fig. 4d and blue clusters in Fig. 4a, b, and c), respectively. These states cover most of the Mott gap but the local density of states (LDOS) of the whole systems commonly touch zero at the Fermi level, which is close to a pseudogap. Note that the zigzag DWs have the × 2-ordered Ta clusters along the DW direction. We attribute the common insulating nature of zigzag DWs to the alternating hopping amplitudes similar to the insulating 1D zigzag dimer chain of the Su-Schrieffer-Heeger model 46 Fig. 11). This is related to how the DS clusters on  domain edges are distorted due to the neighboring imperfect DS clusters within DWs.

DISCUSSION
This work provides a totally different approach and understanding from a recent theoretical work, which explained the insulating properties of straight DWs of 1T-TaS 2 in terms of the frustrated hopping within the triangular Mott insulator model 47 . In fact, the particular type of DWs mentioned in the previous model is metallic in our experiment and calculation. This discrepancy indicates clearly the limitation of this simple model, which ignores all the internal degrees of freedom of DWs.
This work uses the large tunneling current and bias to induce DW networks. While the mechanism is not fully clear yet, the network may be induced by the local heating or the electric field. The DW phases were also induced globally by different methods of heating 29 and optical pulses 10 . Since each method may have different levels of excitation and thermodynamics, the lateral and population distribution of different types of DWs induced can be different. For example, a recent STM study showed that the optical pulse produces irregular networks of the zigzag DWs and DW-2 configurations 48 . This is partly consistent with our work. On the other hand, as we reported recently, the thermally excited DW phase (the NCCDW phase above 180 K) has only the DW-1 type uniformly 29 . Nevertheless, what is very important here is that we tabulate most of the DW structures possible and provide a unified understanding of their properties.
We investigate systematically structural and electronic properties of DW structures of the 1T-TaS 2 surface layer, which was manipulated by scanning tunneling microscope (STM) voltage pulses to generate DW networks with a plethora of metastable DWs 18,19 . We identify more than ten different DW structures including the straight DWs discussed partly in the previous works, the newly observed zigzag DWs, and the breakdown of unstable DWs into multiple DWs of lower energy. The atomic structures and electronic states of most of these DWs are determined by direct comparisons of STM data and density functional theory (DFT) calculations including the on-site Coulomb interaction. Various types of DW configurations lead to rich electronic states in a narrow energy scale within the bandgap of 1T-TaS 2 . A systematic understanding of the structural and electronic variety of these DWs is obtained through extensive DFT calculations. The electronic property of these discommensurate DWs exhibits both metallic and insulating characters depending on their atomic structure, correlation, and charge transfer from neighboring domains. Our finding of the DW properties would help to understand the microscopic origins of various metastable phases, hidden phases, and emerging superconductivity, which are all attributed to the electronic states localized on DWs.

Sample preparations
The single crystal 1T-TaS 2 was grown by iodine vapor transport method in the evacuated quartz tube. Before the growth of the sample, the powder 1T-TaS 2 was sintered for 48 h at 750 ∘ C. We repeated this process two times to get polycrystals. To get a high-quality sample, the seeds were slowly transported by iodine at 900-970 ∘ C for 2 weeks. The tube was rapidly cooled down to room temperature in the air due to the metastability of the 1T phase.