Multivalley Free Energy Landscape and the Origin of Stripe and Quasi-Stripe CDW Structures in Monolayer MX2 Compounds

Ultrathin sheets of transition metal dichalcogenides (MX2) with charge density waves (CDWs) is increasingly gaining interest as a promising candidate for graphene-like devices. Although experimental data including stripe/quasi-stripe structure and hidden states have been reported, the ground state of ultrathin MX2 compounds and, in particular, the origin of anisotropic (stripe and quasi-stripe) CDW phases is a long-standing problem. Anisotropic CDW phases have been explained by Coulomb interaction between domain walls and inter-layer interaction. However, these models assume that anisotropic domain walls can exist in the first place. Here, we report that anisotropic CDW domain walls can appear naturally without assuming anisotropic interactions: We explain the origin of these phases by topological defect theory (line defects in a two-dimensional plane) and interference between harmonics of macroscopic CDW wave functions. We revisit the McMillan-Nakanishi-Shiba model for monolayer 1T-TaS2 and 2H-TaSe2 and show that CDWs with wave vectors that are separated by 120° (i.e. the three-fold rotation symmetry of the underlying lattice) contain a free-energy landscape with many local minima. Then, we remove this 120° constraint and show that free energy local minima corresponding to the stripe and quasi-stripe phases appear. Our results imply that Coulomb interaction between domain walls and inter-layer interaction may be secondary factors for the appearance of stripe and quasi-stripe CDW phases. Furthermore, this model explains our recent experimental result (appearance of the quasi-stripe structure in monolayer 1T-TaS2) and can predict new CDW phases, hence it may become the basis to study CDW further. We anticipate our results to be a starting point for further study in two-dimensional physics, such as explanation of “Hidden CDW states”, study the interplay between supersolid symmetry and lattice symmetry, and application to other van der Waals structures.

Recent progress in STM technique has deepened the understanding of the CDW states 22,25,29,30 . These materials exhibit various CDW phases which are characterized by domain walls (topological defects). Interesting CDW phases are the stripe phase with stripe domain walls and the triclinic (T) phase with quasi-stripe domain walls ( Fig. 1(d,e)). Surprisingly, the stripe phase shows up only on heating from the C phase. T domain appear in coexistence with NC domains on cooling, but the T phase without coexistence appears only on heating from the C phase.
Our method is to use a Ginzburg-Landau model for a CDW with general wave vectors Q i ( ) (i = 1, 2, 3) satisfying the triple-Q condition (see Methods for detail). Historically, this model was first introduced by McMillan 9,31,32 to explain the IC-C phase transition. Nakanishi and Shiba analyzed this model more carefully by including higher order harmonics in the order parameter configuration, finding the presence of the NC phase in 1T-TaS 2 33 ( Fig. 1(d)). Experimentally, the C, IC, NC, stripe and T phases all satisfy the triple-Q condition. It is noteworthy that the free energy only involves the terms compatible with the crystal symmetry and the only input from experimental data is the incommensurate wave vectors Q i IC ( ) for the IC phase. Nevertheless, as shown by Nakanishi and Shiba, the free energy has an unexpected minimum corresponding to the NC phase. Thus, it is natural to ask whether there are more hidden minima if one makes a thorough search which was not feasible at the time.
We revisit the McMillan-Nakanishi-Shiba models 9,33,34 for monolayer 1T-TaS 2 and 2H-TaSe 2 without interaction between domain walls. First, we show that CDWs with Q i ( ) s that are separated by 120° (i.e. with the three-fold rotation symmetry of the underlying lattice) contain a free-energy landscape with many local minima. Then, we remove this constraint and show that free energy local minima corresponding to the stripe and T phases appear. Finally, we explain the origin of stripe and T domain walls and discuss the implication of our results.

Results
1T-taS 2 CDW states with 120° constraint. First, we consider the special case that the CDW wave vectors Q i ( ) (i = 1, 2, 3) are separated by 120°. In this case, the triple-Q condition  (1) since they uniquely determine Q (2) and Q (3) by the triple-Q condition and the 120° constraint. The CDW free energy can be visualized by varying these components. Figure 2(a) reproduces the results by Nakanishi-Shiba for the special wave vector path = x Q Q( ) (1) (shown in Fig. 2(b) with black lines). We have used the free-energy parameters in ref. 33 . Important wave vectors are the  , respectively, and are tilted from the primitive reciprocal lattice vectors G i by angles of 13.9° and 0°, respectively. N = 0 represents the charge density modulation by the fundamental wave Q i ( ) . The important idea by Nakanishi-Shiba is to include higher order harmonics with = … N 1, 2, 3, (see Methods section). In fact, the NC phase appears for N > 0.
In our extended analysis we have used two types of free energies: the first one (type-1) uses phenomenological parameters which reproduce a ring-like diffuse scattering obtained by an electron diffraction experiment 35 (also in ref. 25 for optically excited hidden CDW states), while the second one (type-2) uses McMillan's original free energy 9 analyzed with the method of Nakanishi-Shiba (see the Method section for more detail). The type-2 free energy is important because ring-like diffuse scattering were not observed for ultra-thin sheet of 1T-TaS 2 including monolayer 22 . Figure 2(b-d) show the free energy in the Q (1) space. Note that the same free energy is obtained for Q (2) and Q (3) because of the three-fold rotational symmetry.
Surprisingly, the free energy has a "multivalley landscape": there are many local minima besides those corresponding to the well-known IC, NC, and C CDWs. Each of these local minima corresponds to new CDW states with three-fold rotational symmetry. Despite of different global texture, type-1 and type-2 free energies show almost identical local minima. If the 120° constraint is removed, then these local minima are expected to move to new minima corresponding to the T phase.
2H-taSe 2 CDW states with 120° constraint. The multivalley free energy structure also exists in 2H-TaSe 2 as shown in Fig. 3. The commensurate wave vectors Q i C ( ) and the incommensurate wave vectors Q i , respectively, and they are parallel to the primitive reciprocal lattice vectors G i . Figure 3(a) reproduces the free energy along the special line Q x = 0 obtained by Nakanishi-Shiba 34 which explains the IC-C phase transition. We extend their analysis to the two-dimensional plane by varying Q x and find new local free energy minima like those of 1T-TaS 2 . Here, we have used the free-energy parameters in ref. 34 These local minima correspond to CDW states with three-fold rotational symmetry. Despite different global texture, type-1 and type-2 free energies show almost identical local minima. If the 120° constraint is removed, then these local minima are expected to move to new minima corresponding to the stripe phase.
The "multivalley landscape" of the type-1 free energy with N = 0, N = 1, N = 2, and N = 3 are given in (b). (c) show the type-1 free energy for larger Q Q , x y values. Note that the number of branches and local minima increase as N increases. New local minima appear near Q C . (d) The type-2 free energy with N = 0, 1, 2, and 3 show almost identical local minima, but it is easier to see how the CCDW state is obtained as → ∞ N .
Scientific RepoRtS | (2020) 10:1239 | https://doi.org/10.1038/s41598-020-58013-7 www.nature.com/scientificreports www.nature.com/scientificreports/ Anisotropic cDW phases. Here, we consider CDW states without the 120° constraint. Let ± R denote the matrices which rotate a wave vector = Q Q Q ( , ) x y by ±120°. If we set the 120° constraint, then the vectors (3) are identical. If the 120° constraint is removed, then the triple-Q condition implies that the vertices of (3) form a regular triangle (see Methods section). Consequently, stable or metastable CDW states with anisotropic domain walls are obtained if local minima from the free energy shown with Q (1) , Q (2) , Q (3) form a regular triangle.
First, we consider the T phase. From the triple-Q condition there are four independent degrees of freedom. Here, we fix two degrees of freedom, namely, the angles between domain walls, to visualize the free energy. The angles between domain walls are known from experiments 12 . Here, we consider the angles Figure 4(a) shows the free energy with N = 1. We have used the free energy parameters from Nakanishi-Shiba 33 . Then, we find a triplet of local minima which forms a regular triangle (black). This result is surprising because the T phase was previously explained by inter-layer interaction 14 , but our calculation is done for a monolayer crystal. Experimental result of the T phase in bulk 1T-TaS 2 (green triangle) 12 is in good agreement with our calculation. The domain size in our calculation is smaller than bulk crystal but may become closer to experiment if we further consider Coulomb interaction between domain walls or inter-layer interaction. Aside from this quantitative detail, we conclude that anisotropic domain walls for the T phase can be formed without inter-layer interaction and, hence, in a monolayer 1T-TaS 2 .
Next, we consider the stripe phase. According to experimental results 7 , the stripe phase is obtained by minimizing the free energy with the constraint = Q Q (1) C . Figure 4(b) is the free energy ( N = 1) visualized with Q (2) and Q (3) . Clearly, these local minima form a regular triangle with Q C (1) which corresponds to the stripe phase. Again, the domain size is smaller than the stripe domain of bulk 2H-TaSe 2 (green triangle) 7 . The domain size may become closer to experiment if we further consider Coulomb interaction between domain walls. Therefore, we conclude that anisotropic domain walls for the stripe phase can be formed naturally even in a monolayer 2H-TaSe 2 with the least amount of interaction.

Discussion and conclusion
First, we summarize the results presented in the previous sections. We revisited the McMillan-Nakanishi-Shiba models for 1T-TaS 2 and 2H-TaSe 2 with general CDW wave vectors Q i ( ) (i = 1, 2, 3) separated by 120°. A "bird's-eye view" of CDW free energy landscape with multivalley structure reveals the presence of multiple local minima which correspond to different CDW states. Then, we removed the 120° constraint and observed local minima for www.nature.com/scientificreports www.nature.com/scientificreports/ the stripe and T phases. These results are surprising because we resorted to neither Coulomb interaction between domain walls nor inter-layer interaction, which had been considered as the origins of the stripe and phases.
To explain these results, we consider the following mechanism. CDW phases between the IC and C phases are accompanied with domain walls, where each domain contains a CCDW with a different phase: such domain walls minimize the commensurability energy between the CDW and the underlying lattice. The average domain size is inversely proportional to | − | Q Q i i ( ) C ( ) . As temperature decreases, Q i ( ) approaches Q i C ( ) and the domain size diverges: The ground state is the C phase without domain walls. As temperature increases from the ground state, energy increase due to thermal fluctuation induces domain walls. Since the formation of domain walls requires energy, these domain walls are created one by one. These domain walls are expected to be anisotropic rather than hexagonal, since the formation of hexagonal domain walls require a global arrangement. This model is analogous to the formation of the Abrikosov vortex lattice in a type-2 superconductor, where vortices are formed one by one and finally form a lattice structure as the number of vortices increase.
Moreover, for 1T-TaS 2 , the C phase is closer (in the reciprocal lattice space) to the T phase than the NC phase, i.e. the T phase has larger domains which is energetically favored. Therefore, as temperature increases, 1T-TaS 2 will first make a phase transition from the C phase to the T phase; phase transition to the NC phase occurs when the NC domain size and the T domain size become comparable (Fig. 5) 12 .
We note that this induced anisotropy model has experimental support. The low temperature CDW phase of a free-standing monolayer 1T-TaS 2 with multiple T domain walls has been observed with scanning transmission electron microscopy 22 . In addition, 1T-TaS 2 with a single domain wall has been observed using scanning tunneling microscopy 36 .
We remark that it is very natural that the stripe and T phases correspond to local minima of the free energy rather than global ones. Experimentally, these phases (without coexistence) appear only upon heating from the commensurate state. The intralayer mechanism of the incommensurate-striped phase boundary was first studied by Littlewood and Rice 10 where the stripe free energy minimum can be a global one due to competition between the elastic and commensurability energies (see Supplementary Information for further discussions). However, if the free energy minima for these phases were the global minima at some temperature, then these phases would be observed upon cooling as well. In addition, the coexistence of NC and T has been observed for non-equilibrium condition upon rapid cooling, where the fraction of T-domain depends on cooling rate 12 . This coexistence is (1) and local minima for Q (2) and Q (3) form a regular triangle. Therefore, these local minima correspond to the stripe phase. (2020) 10:1239 | https://doi.org/10.1038/s41598-020-58013-7 www.nature.com/scientificreports www.nature.com/scientificreports/ rather a "proof " of the multivalley free-energy structure. For rapid cooling, temperature gradient leads to energy fluctuation and a portion of the sample can stay in the local minimum for the T phase.
Next, we explain what condition distinguishes stripe and T domain walls. While 2H-TaSe 2 has a parity-conserving commensurate structure (Fig. 1(c)), 1T-TaS 2 breaks parity in the first place, and hence they cannot have stripe domain walls. So, from our results we predict that MX 2 compounds with triple-CDW can have domain walls with an angle of φ°− 180 C , where φ C is the angle between Q i C ( ) and Q i IC ( ) . From the commensurability condition μ ν − = where G i are reciprocal lattices of the crystal, we obtain φ = . Then, we see that µ ≠ 0 and ν ≠ 0 give T domain walls (such as 1T-TaS 2 with μ = 3 and v = 1), while μ ≠ 0 and v = 0 give stripe domain walls (such as 2H-TaSe 2 with μ = 3 and v = 0). In particular, we predict that the ultrathin TaSe 2 with a new × 7 7 commensurate structure (that is, μ = 2 and v = 1) 37 can also have T domain walls.
The appearance of the stripe and T phases with the least amount of interaction can also be explained in the context of entropy. The Kosterlitz-Thouless transition 38,39 , for instance, is stabilized by the production of vortices. Creation of vortices requires energy, but vortices also increase entropy and minimize the free energy of the system. Similarly, stabilization of the stripe and T phases is related to entropy release/increase which is asymmetric with heating/cooling. The stripe and T phases break the 3-fold rotational symmetry of the CCDW state. The T phase further breaks parity. Lower symmetry due to production of symmetry-breaking domain walls implies larger entropy, i.e. lower free energy.
From these discussions, we conclude that the T and stripe phases can appear naturally even in monolayer MX 2 . Our results imply that Coulomb interaction between domain walls or inter-layer interaction may be secondary factors for the appearance of these phases. Note that our result does not contradict with Nakanishi-Shiba's model of stripe and T phases based on inter-layer stacking 14 : their work still applies for MX 2 with higher number of layers. Our result also explains our recent experimental result, namely the appearance of the T phase in monolayer 1T-TaS 2 .
As future applications of this work, the appearance of various local minima may predict new CDW phases. Addition of Coulomb interaction between domain walls and local charge modulation may explain metal-to-insulator transition with domain wall network [28][29][30] and mysterious CDW states such as the hidden CDW states 24,[40][41][42] , hence our model may become the basis to study CDW further. Our analysis may be extended to study the interplay between supersolid symmetry and lattice symmetry, and application to other van der Waals structures 43 .  12 . The NC domain size (cooling) are from ref. 44 . We note that the same NC domain sizes were verified but not reported in ref. 12 . (c) Similarly, the ground state of 2H-TaSe 2 is the CCDW state without domain walls. As temperature increases, we speculate that domain walls are formed one by one. The temperature dependence is shown here for bulk 2H-TaSe 2 7 . r r r r r Here the complex order parameters ψ r ( ) 2 . Type-1 and Type-2 free energies explained in the Result section correspond to different forms of − ∇ e i ( ). In their study of 1T-and 2H-polytypes, Nakanishi and Shiba considered µ ν = ( , ) (3, 1) and µ ν = ( , ) (3, 0), respectively, but their method can be generalized and applied to other CDW materials with different µ ν ( , ) values. The complex order parameters of a CCDW state is ψ ( 1) and = q q . The free-energy is obtained by varying general Q i ( ) s. In this case, we assume that each ∆ lmn i ( ) are real. These wave vectors are depicted in Fig. 6.