Strain engineering and the hidden role of magnetism in monolayer VTe₂

To understand the ground state property of VTe₂ monolayer and to elucidate the origin of multiple CDW phases, we performed the total energy calculations. As a first step, we assumed the nonmagnetic phase because there has been no experimental report yet on any magnetic order in this material. Fig. 1c presents a summary of our results. Here we considered two different lattice parameter choices; namely, the bulk experimental (open green symbols) and the theoretically optimized value (filled blue) while the internal atomic positions are fully relaxed in both cases. Monolayer VTe₂ is typically grown on a substrate of graphene/SiC or highly oriented pyrolytic graphite (HOPG) whose lattice parameters of 3.6–3.7 Å^{19, 21, 22, 24} are therefore not much different from bulk VTe₂ (3.644 Å) ¹⁸.

At the experimental lattice parameter, the bulk-like ^{29, 18} $3\times 1$ structure is found to be most stable and the $3\times 3$ is the next having the higher energy by $\sim$25 meV/f.u.. Note that 1T and $2\times 1$ phases are energetically unstable having much higher energy. Our result is therefore in good agreement with previous theoretical studies of Ref. 24, 23, but not with the experimental reports ^{28, 19, 24, 23, 21, 20, 22}.

On the other hand, the result from the optimized lattice parameter gives rise to the 4-formula-unit CDW ground state as reported in some experiments ^{19, 24, 23, 21, 20, 22}. As shown in Fig. 1c, the $4\times 1$ and $4\times 4$ phases have the lowest energy, and their energy difference is less than 1.9 meV/f.u.. The $2\sqrt{3}\times 2\sqrt{3}$ phase reported by Ref. 20 has quite high energy ²⁰. It is also important to note that the $2\times 1$ CDW, which was reported in the bilayer and a few layers of VTe₂ ^{25, 26, 23}, is not quite favorable energetically for the monolayer.

Our result demonstrates that good care must be taken in the simulation of experimental situations. It is well known that conventional approximations adopted in DFT typically produce systematic deviations in reproducing the experimental lattice property. For example, GGA-type approximation often overestimates the lattice parameter while LDA underestimates. It can therefore be important to note that adopting the experimental value can correspond to the situation that effectively imposes the contraction or enlargement of lattice rather than the optimized value for the given material. For such sensitive cases as VTe₂ and other transition-metal chalcogenides, in which multiple CDW phases and other competing orders are intertwined, this kind of computation details should be taken into account more seriously.

We now examine the relative stabilization of $4\times 1$ and $4\times 4$ CDWs in detail. Fig. 1d presents the calculated total energy difference as a function of the lattice parameter. The $4\times 1$ phase is always more stable in the entire range. Interestingly, both structures have their minimum energies at $a=3.50$ Å  and the difference becomes minimized at that point. The smallness of this energy difference, corresponding to $\sim$26.7 K/f.u., likely indicates the possible control of the ground state by strain for example.

Important is to check the dynamical stability of each phase. The calculation results of phonon dispersion are presented in Fig. 1e and f. Note that imaginary phonon mode is clearly identified not only for 1T (Fig. 1e) but also for the most stable $4\times 1$ phase (Fig. 1f). While the unstable 1T phonon observed in the middle of $\Gamma$–$\textrm{M}([1/2,0,0])$ line indicates the ‘$\times$4’ modulations such as $4\times 1$ and $4\times 4$ ^{19, 20}, the imaginary mode found in $4\times 1$ (at the middle of $\textrm{X}[1/2,0,0]$–$\Gamma$) indicates the $4\times 4$ modulations ²³. Given that the energetically most stable $4\times 1$ CDW phase is dynamically unstable, the result implies that the ground state is not well described within the nonmagnetic picture, and requires further investigation.

Recently, the intriguing interplay of charge, spin, and lattice is found to play a key role in determining the polymorphic bulk ground state of VTe₂ ¹⁸. For monolayers, on the other hand, the role of magnetism has not been explored. Considering that the direct detection of magnetic order is experimentally challenging for ultra-thin systems, particularly the AFM order, the intriguing possibility of magnetism and its ‘hidden’ role interplaying with other degrees of freedom are certainly worthy of being investigated.

Here we examine the 1T, $2\times 1$, and $4\times 1$ structure together with FM and AFM spin order. We take both single-stripe AFM (sAFM) and dAFM order into account. The former is found to be a well converged solution only for 1T, and dAFM order is the ground state for 1T, $2\times 1$, and $4\times 1$; see Fig. 2a–c. The total energy results are summarized in Fig. 2d. First of all, let us take note that magnetic orders significantly reduce the energy of the system. In 1T phase, FM and AFM orders have almost the same energy whereas the AFM order is clearly more stable in $2\times 1$ and $4\times 1$ CDW phases. Interestingly, the $2\times 1$ phase is less favorable energetically than 1T unless magnetism is considered. Combined with AFM order, however, the $2\times 1$ CDW phase becomes stabler than 1T by about 30 meV/f.u.. In the case of $4\times 1$, we found, the initial FM order converges into a ‘ferrimagnetic’ phase having the ‘up-down-down-up ($\uparrow$-$\downarrow$-$\downarrow$-$\uparrow$)’-type order carrying a non-zero net moment. The most stable configuration is dAFM also for $4\times 1$ whose energy is almost the same with that of dAFM $2\times 1$ CDW ($\Delta E_{2\times 1-4\times 1}^{\rm dAFM}$ < 0.5 meV/VTe₂) which can be regarded as being consistent with previous experiment ^{23, 24, 20}.

A remarkable difference between $2\times 1$ and $4\times 1$ phase is in their dynamic stability. Fig. 2e and f show the calculated phonon dispersion. In $2\times 1$ CDW phase, clearly noted is the imaginary phonon mode centered at X point ([1/2,0,0]), which indicates the structural modulation toward $4\times 1$ type as schematically represented by arrows in Fig. 2b. In fact, the phonon modes of $4\times 1$ phase are all quite stable as shown in Fig. 2f. It is important to note that the dynamical stability is achieved only when we take the spin order into account (compare Fig. 2f to Fig. 1f), demonstrating the key role of magnetism in stabilizing CDW ground state. Our result thus provides an intriguing new example of ‘hidden’ interplay of magnetism with charge and lattice in 2D materials.

As the conventional experimental techniques of determining spin order (e.g., neutron scattering) can be severely limited for 2D materials, Raman spectroscopy has been a useful alternative, albeit indirect, tool ^{30, 31, 32}. To facilitate the possible future experimental study, we calculated Raman-active modes shown in Fig. 2f; see the red lines in the left panel which can be directly compared with experiments. Due to the structural similarity, the $4\times 1$ modes are not much different from those of $2\times 1$ phase (Fig. 2e), and many states are located near 100 cm^-1. However, the states near $\leq$50 cm^-1 are found only in $4\times 1$ CDW phase. In fact, these modes are the vibrations that break the $2\times 1$ structural symmetry and lead to the $4\times 1$ CDW. As depicted in Fig. 2g, these three have the opposite direction atomic movements from the $2\times 1$ chain. Thus, their experimental verification can be the key signature of $4\times 1$ phase stabilized by ‘hidden’ dAFM spin order.

The active interplay of spin, charge, and lattice immediately indicates the exciting possibility of controlling material properties. As a concrete example, here we examined the strain effect. Fig. 3a shows the calculated total energies with respect to the most stable magnetic and CDW configuration at a given lattice parameter. The blue square, red circle, and green diamond symbols represent the lowest energy spin order of 1T, $2\times 1$, and $4\times 1$ phase, respectively. It is clearly noted that the most stable CDW pattern changes from $4\times 1$ to $2\times 1$, and this phase transition occurs in between $a=$3.6 and 3.644 Å.

Moreover, this charge-lattice pattern transition is followed by a magnetic transition as the lattice parameter further increases, and the system becomes eventually FM at $a\approx$ 3.7 Å. Fig. 3c and d shows the relative stabilization energy of the two most stable spin orders within the $2\times 1$ and $4\times 1$ CDW phase, respectively. While dAFM remains most stable for $4\times 1$ in the entire range of strain values, the ground state for $2\times 1$ changes from dAFM to FM. The calculated total energies for 1T structure are presented in Fig. 3b.

Fig. 3f summarizes the evolution of the ground state properties as a function of strain. The first transition is in between two CDW phases of $4\times 1$ and $2\times 1$ while both maintain the dAFM spin order. The second is a spin reordering transition from dAFM to FM within the $2\times 1$ CDW.

Fig. 3e presents the phonon dispersion of the predicted ground state of FM $2\times 1$ phase under tensile strain. It confirms that this FM $2\times 1$ phase is dynamically stable. We emphasize that this interesting possibility of controlling the ground state becomes feasible through the spin-lattice-charge coupling. Also, the ‘hidden’ spin order (in the sense that an AFM order is hardly identified in the monolayer via conventional methods) would unveil itself once the system becomes FM (which is much easier by means of magnetic circular dichroism (XMCD) and/or magneto-optical Kerr effect (MOKE) for example). To facilitate the experimental verification, we once again calculated the Raman spectra shown in the left panel of Fig. 3e. It can serve as a useful guideline or prediction for future experiments.

As presented in Fig. 2, the spin structure of 1T, $2\times 1$, and $4\times 1$ are basically similar although the CDW makes the inter-atomic or inter-‘stripe’ spacing different. To understand the magnetic interactions in greater detail, we performed the magnetic force linear response calculation ^{33, 34, 35, 36, 37, 38, 39}. Here we fix the lattice symmetry to the most symmetric 1T because it provides the most useful insight for the underlying origin of the spin ground state evolution. The convention of our spin Hamiltonian is $\mathcal{H}=-\sum J_{i,j}\bf{e_{i}}\cdot\bf{e_{j}}$ where $\bf{e_{i,j}}$ refers to the normalized unit spin vector and $i,j$ represents the atomic sites. The results are summarized in Fig.4b. At the lattice parameter $a=3.5$ (blue squares) and $3.6$ Å  (red circles), the first and the third neighboring $J$’s are AFM (negative sign) while the second is FM (positive sign). Note that this exchange interaction profile makes the dAFM order stable. As $a$ further increases to 3.7 Å  (green diamonds), $J_{1,2,3}$ become all FM. This result is therefore in good agreement with the predicted ground state evolution shown in Fig. 3f. Our results show that the sign changes of $J_{1}$ and $J_{3}$ are the key for the AFM to FM transition induced by tensile strain.

We also estimate the ‘inter-stripe’ coupling $\mathcal{J}_{n}^{\text{stripe}}$ which represents the magnetic interaction of a given V site with its $n$-th neighboring ‘stripe’ as depicted in Fig. 4a. Note that a single index $n$ is enough because $\mathcal{J}_{n}^{\text{stripe}}$ should be the same for any choice of V site due to the high symmetric 1T structure. From the lattice structure, this atom-‘stripe’ interaction can be expressed as follow:

The results are summarized in Fig. 4c. At $a\leqq 3.6$ Å, $\mathcal{J}_{1}^{\text{stripe}}$ has a small positive value, indicating the FM coupling while $\mathcal{J}_{2}^{\text{stripe}}$ is AFM. As the lattice parameter increases up to $a=3.7$ Å, $\mathcal{J}_{1}^{\text{stripe}}$ and $\mathcal{J}_{2}^{\text{stripe}}$ change to become positive, stabilizing the FM ground state, which is again in support of our prediction of magnetic phase transition. The rough estimation of FM critical temperature simply by summing up all couplings is $T_{\textrm{c}}\simeq\sum J_{ij}\bf{e_{i}}\cdot\bf{e_{j}}\simeq\textrm{216}$ K.