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

Various types of DWs are found in the quenched phase of Fig. 1a and 1b, which are created by applying a positive voltage pulse from a STM tip cho15 . 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 respected 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.

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) cho16 ; cho17 ; park19 . Three minor DWs of DW-3, DW-6 and DW-12 appear sparsely with shorter lengths, but other six possible $straight$ DWs cannot be found in the present STM images. Interestingly, the $zigzag$ type of DWs are 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 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 are almost degenerated 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) is 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.

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 a smaller imperfect triangles in the left and the right side of the DW, respectively (Fig. 3f). These STM images are well reproduced in the simulated STM images, especially the asymmetric double rows along the DW. The present double row structure is distinguishable from the previously suggested model with a single row cho17 . These two models are very close in their energies, only 9 meV difference per supercell, and their electronic structures are very similar. Thus, these structures may coexist (Supplementary 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 cho17 , and it is more stable ($\sim$ 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 ($\sim$ 29 meV/supercell) than the above DW-4A structure. However, its electronic structure is not consistent with previous STS spectra cho17 (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 park19 . Other minor DWs also have their characteristic atomic structure, which are determined by their own CDW phase shift vectors and the internal structural relaxation (Supplementary Fig. 5 and 6).

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 band gap (Fig. 3i), which is consistent with the previous STS study cho17 . There exist two in-gap states localized along 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) are almost degenerated 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 unitcells 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 a 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 unitcell.

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 band gap of 0.1 eV due to the charge transfer from the edge of 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 is shifted upward just above the Fermi level (indicated by purple arrow in Fig. 3j), which is consistent with the previous STS spectra cho17 . 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 become 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 band gap of 0.25 eV with spin-split bands (Supplementary Fig. 8), therefore there is a possibility of the 1D magnetism along this DW (Supplementary Fig. 10). However, this is closely related to the controversial issue of the magnetic ordering for the CCDW phase faze80 ; law17 ; klan17 ; kart17 ; pal20 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 a higher energy indicating the donation of electrons from domains to DW (Table I and Supplementary Fig. 7). This is consistent with the peak shift of Hubbard states at domain edges observed in the previous experiment cho17 .

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 though 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 has the $\times$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 su79 , while we do not establish a quantitative model. The notable difference between DW-3,4 and DW-1,4/DW-2,8 is that the domain edge states of DW-3,4 preserve the Mott insulating character with upper and lower Hubbard states largely while those on DW-2,8 and DW-1,4 show a large in-gap states at around 0.1 eV without strong spin polarization (Supplementary Fig. 11). This is related to how the DS clusters on domain edges are distorted due to the neighboring imperfect DS clusters within DWs.

The single crystal 1T-TaS₂ were grown by iodine vapour transport method in the evacuated quartz tube. Before growth of the sample, the powder 1T-TaS₂ was sintered for 48 h at 750 ^∘C. We repeated this process two times to get poly crystals. To get high-quality sample, the seeds were slowly transported by iodine at 900$\sim$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.

The STM measurements were carried out at $T$ = 4.3 K with a commercial STM (SPECS) and the mechanically sharpened Pt–Ir wires were used for STM tips. All of STM images were obtained with the constant current mode and the bias voltage $V_{s}$ was applied to the sample. The detailed method of the textured CDW domain formation and the preparation of single crystal 1T-TaS₂ are similar to previous study cho16 .

The DFT calculations were carried out by using the Vienna $ab$ $initio$ simulation package kres96 within the Perdew-Burke-Ernzerhof functional of the generalized gradient approximation perd96 and the projector augmented wave method bloc94 . We used the single-layer model of 1T-TaS₂ with a vacuum spacing of about 13.6 Å. We assumed the 2D-like Mott states at the surface limit based on most of DWs are not coupled between the first and the second layer (Supplementary Fig. 12) and the interlayer coupling of the DWs in a bilayer structure is not affecting substantially the electronic structure of the DWs (Supplementary Fig. 13). A plane-wave basis set of 259 eV and a 6$\times$6$\times$1 $k$-point mesh for the $\sqrt{13}\times\sqrt{13}$ unit cell are used. All atoms were relaxed until the residual force components became smaller than 0.02 eV/Å. An on-site Coulomb energy (U = 2.3 eV) was included for Ta 5$d$ orbitals to reproduce the Mott gap size in experiment cho15 . Similar calculation methods were successfully used in our previous study park19 .