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Multivalley Free Energy Landscapeand the Origin of Stripe andQuasi-Stripe CDW Structures in
Monolayer MX2 Compounds
Keiji Nakatsugawa

1,2, Satoshi Tanda1,2 & Tatsuhiko N. Ikeda

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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 stateof 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 Coulombinteraction 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 thesephases 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 formonolayer 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 withmany local minima. Then, we remove this 120° constraint and show that free energy local minimacorresponding 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 ofstripe 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.
Usually, anisotropic structures such as stripe phases can be explained by anisotropic interaction between theconstituent atoms, electrons, or liquid crystal polymers1. Anisotropic structures in charge density waves (CDWs)
have been explained likewise. CDWs are periodic modulations of electric charge density in low-dimensional
conductors2–6. The stability of CDWs containing stripe domain walls7,8 was first studied by free energy theories9,10. But these theories assumed that stripe domain walls can exist in the first place, probably due to weak
computational facilities at that time. The appearance of stripe domain walls and quasi-stripe (triclinic) domain
walls8,11,12 has been explained by Coulomb interaction between domain walls13 and three-dimensional stacking14.
However, are these interactions indispensable? It would be ideal to have rich structures with the least amount ofinteractions.
In this article we report that stripe and quasi-stripe CDW domain walls can appear without anisotropic interactions and 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 consider the transition
metal dichalcogenide (MX2) compounds 1T-TaS2 and 2H-TaSe2 (Fig. 1(a,b)) which have recently experienced aresurgence of interest due to rich physical content and potential applicability to nanoscale electromechanics15–28.
1

Department of Applied Physics, Hokkaido University, Kita 13, Nishi 8, Kita-ku, Sapporo, 0608628, Hokkaido, Japan.
Center of Education and Research for Topological Science and Technology, Hokkaido University, Kita 13, Nishi 8, Kita-ku,
Sapporo, 060-8628, Hokkaido, Japan. 3Institute for Solid State Physics, University of Tokyo, Kashiwa, 277-8581, Chiba,
Japan. email: keiji_nakatsugawa@eis.hokudai.ac.jp; tanda@eng.hokudai.ac.jp; tikeda@issp.u-tokyo.ac.jp

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Figure 1. CDW structures of 1T-TaS2 and 2H-TaSe2 in equilibrium. (a) Structure of 1T-TaS2. (b) Structure of
2H-TaSe2. These materials have two-dimensional layered crystalline structures like graphene. Here, the red
spheres represent Ta atoms and the yellow spheres represent S or Se atoms. (c) The Ta atoms form a triangularlattice with superlattice vectors a and b. A and B are commensurate (C) CDW superlattice vectors(A = (μ + ν )a + ν b, B = − ν a + μ b, |A| = |B| = μ2 + μν + ν 2 |a|). The CCDW superlattice for 1T-TaS2(μ = 3, ν = 1) and 2H-TaSe2 (μ = 3, ν = 0) are shown with green spheres. The Ta atoms on the star-of-Davidor hexagonal lines are shifted toward the central green Ta atoms. (d) Temperature dependence of CDW phases
in bulk 1T-TaS2. (e) Temperature dependence of CDW phases in bulk 2H-TaSe2. The blue arrows represent thecooling cycle. The red arrows represent the heating cycle. C: Commensurate phase, T: Triclinic phase withquasi-stripe domain walls (stretched honeycomb lattice). NC: Nearly commensurate phase, IC:
Incommensurate phase, N: Normal metal phase. The domain walls of the NC, T, and stripe phases are depictedwith solid lines.

Recent progress in STM technique has deepened the understanding of the CDW states22,25,29,30. 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 Cphase.
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 Q (1) + Q (2) + Q (3) = 0 (see Methods for detail). Historically, this model was first
introduced by McMillan9,31,32 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 theNC phase in 1T-TaS233 (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 theonly input from experimental data is the incommensurate wave vectors Q (ICi ) 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 models9,33,34 for monolayer 1T-TaS2 and 2H-TaSe2 without interaction between domain walls. First, we show that CDWs with Q (i ) s that are separated by 120° (i.e. with thethree-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-TaS2 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 Q (1) + Q (2) + Q (3) = 0 implies
Q (1) = Q (2) = |Q (3)|. Numerical results of CDW free energy F[{Q (i )}; T ] at temperature T are shown in Fig. 2.
Each Q (i ) (i = 1, 2, 3) defines a two-dimensional reciprocal space. To characterize the set {Q (i )} we use two components of Q (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 Q (1) = Q(x ) (shown in
Fig. 2(b) with black lines). We have used the free-energy parameters in ref. 33. Important wave vectors are the

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Figure 2. Numerical results of 1T-TaS2 free energy in the Q(1) = (Q x , Qy ) space (in units with |Gi | = 1).
(a) Reproduction of Nakanishi-Shiba’s result33. x parameterizes the wave vector Q(x ) with Q(0) = Q IC,
Q(x∗) = Q NC (nearly commensurate), and Q(1) = Q C. Q(x ) is shown with solid black lines in (b). 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 x , Qy 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 → ∞.

commensurate wave vectors Q (Ci ) and the incommensurate wave vectors Q (ICi ) which have the norms|Q (Ci )| /|Gi | = 1/ 13 ≈ 0.277 and |Q (ICi )| /|Gi | = 0.283, respectively, and are tilted from the primitive reciprocallattice vectors Gi by angles of 13.9° and 0°, respectively. N = 0 represents the charge density modulation by thefundamental 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 phenomenologicalparameters which reproduce a ring-like diffuse scattering obtained by an electron diffraction experiment35 (alsoin ref. 25 for optically excited hidden CDW states), while the second one (type-2) uses McMillan’s original freeenergy9 analyzed with the method of Nakanishi-Shiba (see the Method section for more detail). The type-2 freeenergy is important because ring-like diffuse scattering were not observed for ultra-thin sheet of 1T-TaS2 including monolayer22. Figure 2(b–d) show the free energy in the Q (1) space. Note that the same free energy is obtainedfor 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 stateswith 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-TaSe2 CDW states with 120° constraint. The multivalley free energy structure also exists in
2H-TaSe2 as shown in Fig. 3. The commensurate wave vectors Q (Ci ) and the incommensurate wave vectors Q (ICi )have the norms |Q (Ci )| /|Gi | = 1/ 9 ≈ 0.333 and |Q (ICi )| /|Gi | = 0.325, respectively, and they are parallel to theprimitive reciprocal lattice vectors Gi. Figure 3(a) reproduces the free energy along the special line Qx = 0
obtained by Nakanishi-Shiba34 which explains the IC-C phase transition. We extend their analysis to thetwo-dimensional plane by varying Qx and find new local free energy minima like those of 1T-TaS2. Here, we haveused 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.

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Figure 3. Numerical results of 2H-TaSe2 free energy in the Q(1) = (Q x , Qy ) space (in units with |Gi | = 1). (a)Reproduction of the result by Nakanishi-Shiba34. x parameterizes the wave vector Q(x ) with Q(0) = Q IC andQ(1) = Q C. Q(x ) is shown with a black line in (b) The “multivalley landscape” of the free energy with N = 0, 1,
2, and 3 are given for the type-1 (b) and type-2 (c) free energies.

Anisotropic CDW phases. Here, we consider CDW states without the 120° constraint. Let R ± denote thematrices which rotate a wave vector Q = (Q x , Qy ) by ±120°. If we set the 120° constraint, then the vectors{Q (1), R−Q (2), R+Q (3)} are identical. If the 120° constraint is removed, then the triple-Q condition implies that thevertices of {Q (1), R−Q (2), R+Q (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 freeenergy. The angles between domain walls are known from experiments12. Here, we consider the angles
δφ1 = 360° − δφ2 − δφ3, δφ2 = 180° − φC, δφ3 = 150°, where φC ≈ 13.9° is the angle between Q (Ci ) and Q (ICi ).
Figure 4(a) shows the free energy with N = 1. We have used the free energy parameters from Nakanishi-Shiba33.
Then, we find a triplet of local minima which forms a regular triangle (black). This result is surprising because theT phase was previously explained by inter-layer interaction14, but our calculation is done for a monolayer crystal.
Experimental result of the T phase in bulk 1T-TaS2 (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-TaS2.
Next, we consider the stripe phase. According to experimental results7, the stripe phase is obtained by mini(2)mizing the free energy with the constraint Q (1) = Q (1)C . Figure 4(b) is the free energy ( N = 1) visualized with Q(3)(1)
and Q . Clearly, these local minima form a regular triangle with Q C which corresponds to the stripe phase.
Again, the domain size is smaller than the stripe domain of bulk 2H-TaSe2 (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-TaSe2 with the least amount of interaction.

Discussion and Conclusion

First, we summarize the results presented in the previous sections. We revisited the McMillan-Nakanishi-Shibamodels for 1T-TaS2 and 2H-TaSe2 with general CDW wave vectors Q (i ) (i = 1, 2, 3) separated by 120°. A “bird’s-eyeview” 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

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Figure 4. Free energy of T and stripe phases. (a) Free energy of 1T-TaS2 without the 120° constraint(at temperature T = 225K ). Local minima for Q(1), Q(2) and Q(3) form a regular triangle. These local minimacorrespond to the T phase with quasi-stripe domain walls. (b) Free energy of 2H-TaSe2 without the 120°(2)
(3)
constraint (at temperature T = 100K ). The point Q(1) = Q (1)C and local minima for Q and Q form a regulartriangle. Therefore, these local minima correspond to the stripe phase.

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 areaccompanied 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 isinversely proportional to |Q (i ) − Q (Ci )|. As temperature decreases, Q (i ) approaches Q (Ci ) 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 thanhexagonal, 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-TaS2, 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-TaS2
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 ofa free-standing monolayer 1T-TaS2 with multiple T domain walls has been observed with scanning transmission
electron microscopy22. In addition, 1T-TaS2 with a single domain wall has been observed using scanning tunneling microscopy36.
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 thecommensurate state. The intralayer mechanism of the incommensurate-striped phase boundary was first studied
by Littlewood and Rice10 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 beobserved 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 rate12. This coexistence is

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Figure 5. A model for the formation of anisotropic domain walls. (a) The ground state of 1T-TaS2 is the Cphase without domain walls. As temperature increases, domain walls are formed one by one. As the number ofdomain walls increase, we obtain the T phase. The domain size decreases as temperature increase further. Then,
T-NC phase transition occur when the NC domain size and the T domain size become comparable (280K forbulk 1T-TaS2). The temperature dependence is shown here for bulk 1T-TaS212. (b) Bulk data of 1T-TaS2 domain
size. The T domain size (heating) are from ref. 12. The NC domain size (cooling) are from ref. 44. We note that thesame NC domain sizes were verified but not reported in ref. 12. (c) Similarly, the ground state of 2H-TaSe2 is the
CCDW state without domain walls. As temperature increases, we speculate that domain walls are formed one byone. The temperature dependence is shown here for bulk 2H-TaSe27.

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 aparity-conserving commensurate structure (Fig. 1(c)), 1T-TaS2 breaks parity in the first place, and hence
they cannot have stripe domain walls. So, from our results we predict that MX2 compounds with triple-CDW
can have domain walls with an angle of 180° − φC , where φC is the angle between Q (Ci ) and Q (ICi ) . From thecommensurability condition μQ (Ci ) − ν Q (Ci +1) = Gi where Gi are reciprocal lattices of the crystal, we obtain
μ+ν/2. Then, we see that µ ≠ 0 and ν ≠ 0 give T domain walls (such as 1T-TaS2 with μ = 3 and
φC = cos−1
μ 2 + μν + ν 2

v = 1), while μ ≠ 0 and v = 0 give stripe domain walls (such as 2H-TaSe2 with μ = 3 and v = 0). In particular, wepredict that the ultrathin TaSe2 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 thecontext of entropy. The Kosterlitz-Thouless transition38,39, 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 asymmetricwith 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
MX2. 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 stacking14: their work still applies for MX2 with higher numberof layers. Our result also explains our recent experimental result, namely the appearance of the T phase in monolayer 1T-TaS2.
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 network28–30 and mysterious CDW states such as the hidden CDW
states24,40–42, 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
structures43.

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Figure 6. The elements of the McMillan-Nakanishi-Shiba model. Note that in this article we focus on|q(1)| = |q(2)| = |q(3)| and each |q(i )| are separated by a 120°. (a) CCDW wave vectors Q (Ci ) define the CCDW
3charge density ρ(r) = ρ [1 + ∆C ∑ cos(Q (Ci ) ⋅ r + θCi)]. The free energy is calculated as a function of a general
0

i=1

(i )

wave vector Q with the triple-Q condition Q(1) + Q(2) + Q(3) = 0. (b) The deviation vectors q(i ) = Q(i ) − Q (Ci )projected on the Q(1) space form a regular triangle. Here, R ± are rotation matrix of ±120°. If Q(i ) are separated
by 120°, then this triangle reduces to a point. (c) The harmonics form a triangular lattice. k (i ) are wave vectors
responsible for the formation of domain walls (the image shows an example for (μ , ν ) = (3, 1)). (d) A general3i)i)
CDW state is given by ρ(r) = ρ [1 + ∑ ∑ l , m , n ≥0 ∆(lmncos(Q (lmn⋅ r)] with 3N (N + 1) higher harmonics.
0i=1l⋅ m⋅ n = 0

Methods

McMillan’s phenomenological free energy for triple CDW is9,31,32
F =

2

2

3

4

3

∫ d r{a(r)α (r) − b(r)α (r) + c(r)α (r) + Σi=1ψi (r)e( − i∇)ψi(r)∗+d(r)[|ψ1(r)ψ2(r)|2 + |ψ2(r)ψ3(r)|2 + |ψ3(r)ψ1(r)|2 ]}Here the complex order parameters ψi(r ) at position r are related to the electron charge density
ρ(r) = ρ0(r){1 + α(r)}, α(r) = ∑ i3=1Re[ψi(r)] . ρ0(r) is the charge density in the normal state. The index i (= 1,2, 3) specifies the three components of the triple CDW. a(r), b(r), c(r), d(r) are position-dependent coefficients
which have the periodicity of the crystal lattice and can be written in the form a0 + a1∑ i6=1e iGi ⋅r , etc., where Gi
denote the six shortest reciprocal lattice vectors. The commensurate wave vectors Q (Ci ) = |Q (Ci )|(cos φCi , sin φCi )are defined by the conditions μQ (Ci ) − ν Q (Ci +1) = Gi (i = 1, 2, 3), |Q (Ci )| = |Gi | / μ2 + μν + ν 2 . Type-1 andType-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(i )parameters of a CCDW state is ψi(r) = ∆ Ce i Q C ⋅r +iθCi where Q (Ci ) are the wave vectors of the CCDW state and θCi
are constant phases which minimize the free energy. For other CDW states, ψi(r) can be expanded as a functioni)·r where(i ) iQ (lmnof the deviation vectors q(i ) = Q(i ) − Q (Ci ). Then we may expand ψi(r) as ψi(r) = ∑ l , m , n ≥0 ∆lmne
l·m·n = 0(i )l , m, n are integers and Q lmn= l k (i ) + mk (i +1) + nk (i +2) + Q(i ), k (i ) = μq(i ) − ν q(i +1) and q(i ) = Q(i ) − Q (Ci ),and we have introduced the cyclic notations such as q(4) = q(1) and q(5) = q(2). The free-energy is obtained by(i )are real. These wave vectors are depicted in Fig. 6.
varying general Q(i ) s. In this case, we assume that each ∆lmn(i )aredeterminedbynumericallycalculating
the
set
of
couplednon-linear differential equations ∂F(i) = 0.
∆lmn

Received: 26 September 2019; Accepted: 8 January 2020;
Published: xx xx xxxx∂∆lmn

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Acknowledgements

We thank D. Mihailovic, M. Kosterlitz, D. Nelson, and K. Inagaki for stimulating discussions. T.N.I. acknowledge
support by JSPS KAKENHI Grant No. JP18K13495.

Author contributions

T.N.I. and S.T. conceived the project. K.N. and T.N.I. performed theoretical calculations. T.N.I. developed thesimulation code and discovered the multivalley structure. K.N. analyzed numerical data and discovered theorigin of stripe/quasi-stripe structures. S.T. made interpretations based on actual experimental evidence. K.N.
prepared figures and wrote the manuscript with feedback from S.T. and T.N.I. All authors discussed the results
and approved the final manuscript.

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Competing interests

The authors declare no competing interests.

Additional information

Supplementary information is available for this paper at https://doi.org/10.1038/s41598-020-58013-7.
Correspondence and requests for materials should be addressed to K.N., S.T. or T.N.I.
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