**3. Results and discussions: PIPT in 2D MoTe2**

Firstly, we calculate the phonon spectra of high-symmetry hexagonal 2H phase and low-symmetry monoclinic 1 T<sup>0</sup> phase. The results are depicted in **Figure 1** and fit well with previous reports [34]. The color from light yellow to dark red indicates the Bose-Einstein distribution of phonon states calculated at 300 K. Since the 2H phase has *D*3*<sup>h</sup>* point group at Γ point, the vibration modes can be decomposed by irreducible representations as [35]

**Figure 1.**

*Phonon dispersion of (a) high-symmetry 2H and (b) low-symmetry 1 T*<sup>0</sup> *MoTe2 monolayer and irreducible representation for optical phonons at the* Γ *point. The occupation number (calculated by Bose-Einstein distribution at 300 K) is marked by the colormap.*

*Photo-Induced Displacive Phase Transition in Two-dimensional MoTe2 from First… DOI: http://dx.doi.org/10.5772/intechopen.108460*

$$
\Gamma\_{2H} = E'' + A\_1' + E' + A\_2'' \tag{3}
$$

We find that the *E*<sup>0</sup> , *E*00, *A*<sup>00</sup> <sup>2</sup> phonons display well correspondance to the character of 2H to 1 T<sup>0</sup> phase transition. The microscopic atomic displacements are shown in **Figure 2**(**d**-**f**) by arrows. Under photo-excitation by photons with energies of 1.58, 1.96, 2.34, and 2.63 eV, we calculate the one dimensional potential energy surface (PES) along these vibrations to figure out the responsible modes of phase transition. The results are summarized in **Figure 2**(**a**-**c**). The 0 eV photo-excitation represents the displacement in dark environment for reference. Without photo-excitation, there is an energy barrier of 80 meV between 2H phase and 1*T*<sup>0</sup> phase. As the optical excitation increases, the origin energy minimum of 2H phase bifurcates and the newly emergent energy minima correspond to 1T<sup>0</sup> phase [2, 29].

The upper and lower layers of Te atoms move in the monolayer plane along opposite directions in E" mode (**Figure 2**(**d**)). In the dark environment, the Te atoms do not experience any damping force that drives into another phase. With increasing portion of excited electron states, the potential energy surface becomes flattened and finally forms a lineshape of Mexican hat at critical photon energy of 1.96 eV. This indicates a frozen soft mode that leads to a displacive phase transition. For the A2" phonon, both Mo atoms layer in the middle and Te atom layer at the side move vertically in different directions (**Figure 2**(**f**)). Similar with the foregoing circumstance of E" phonon, a Mexican hat lineshape also forms if the photon energy is larger than the critical one. But the E' mode shows totally different behaviors under photoexcitation (**Figure 2**(**e**)). For E' mode, both Mo atom layer and Te atom layers move in the monolayer plane, but along opposite directions. The one-dimensional potential surface of E' phonon shows asymmetric feature where the right-hand-side energy maximum gradually lowers down and finally becomes an valley at 2.63 eV of photonexcitation, which indicates that a new metastable phase emerges. We notice that for the photons with energy between 1.96 and 2.63 eV, the energy maximum does not become an energy minimum, thus the E' does not anticipate the phase transition process and only E" and A2" phonons make a contribution. After 2.63 eV excitation where the energy minimum along E' mode emerges, the E' mode starts to make a contribution simultaneously.

Besides the total energy surface, we can justify the stability of a phase by inspecting the phonon spectra. Since new energy minima emerge from a parabolic

#### **Figure 2.**

*One-dimensional potential energy surface at different excitation energies in the eigenspaces of (a) E", (b) E', and (c) A2" phonons, and their corresponding structures (d–f) at the minima of the potential energy surface at 2.63 eV.*

potential, which reveal new metaphases, the phonon spectra are also expected to collap and become softened. The frozen phonon mode can lead to structural phase transition, which breaks the original high symmetry [31, 36]. The E" and A2" phonon frequencies are investigated using density functional perturbation theory under different photon excitations. The results are shown in **Figure 3**. The phonon frequencies of E" and A2" phonon become imaginary as expected, which indicates a finite damping force and energy decreasing along the phonon mode. The critical photon energy fits well with the one of previous potential energy surfaces.

In the general Landau framework, we can choose an order parameter, which becomes finite at the vicinity of phase transition region, and the Landau free energy is expanded as a fourth-order polynomial of it, as follows:

$$F(u) = F\_0 + au^2 + bu^4\tag{4}$$

The coefficients *a*,*b* are related with the external condition. The change of sign of *a* can lead to an emergence of new energy minimum at *u*∝ ffiffiffiffiffiffiffiffi � *<sup>a</sup>* 2*b* p . By choosing here the photon energy as an external condition indication and the phonon frequencies as the corresponding order parameter. We fit the frequency of E" and A2" modes against the photo-excitation energy and obtain good agreement given by *<sup>ω</sup>*∝ð Þ *EC* � *<sup>E</sup> <sup>γ</sup>* 2, where E*<sup>C</sup>* = 1.96 eV and *γ* ¼ 0*:*24, as shown in **Figure 3**. The result is similar as the wellknown Curie-Weiss law, except that the critical exponent is 0.12. We can call it as the modified Cuire-Weiss law for PIPT. The sudden softening of E" and A2" phonon modes further proves the PIPT.

The high-dimensional potential energy surface (PES) in the configuration space as a direct product of E" and A2" modes under different photo-excitation is depicted in **Figure 4(a–d)**. The photo-excitation is chosen as 1.0, 2.0, 3.0, 4.0% per unit cell. These results suggest that at � 3.0% excitation, the energy minimum bifurcates and two new asymmetric PES minima emerge. Besides, in comparison to the E" mode, the A2" mode apparently contributes more, which indicates that the phase transition is mostly triggered by A2" mode.

**Figure 3.** *(a) Phonon frequency of the E" and A2" modes as a function of photo-excitation energy.*

*Photo-Induced Displacive Phase Transition in Two-dimensional MoTe2 from First… DOI: http://dx.doi.org/10.5772/intechopen.108460*

**Figure 4.**

*The two-dimensional potential energy surface spanned by E" and A2" modes under (a) 1.0%, (b) 2.0%, (c) 3.0%, (d)4.0% photo-excitation per unit cell. Energies are in unit of eV.*

The excited electron states that contribute to the electron localization function (ELF) are shown in **Figure 5**. The PIPT is caused because the charge distribution exerts forces to the ions [37]. In **Figure 5**(**a**) we plot ELF of high-symmetry MoTe2 under different photo-excitation. In the dark environment, the electron gas connects the Mo atoms to form hexagons. As the electron gradually excited from the valance states, the electron gas becomes more localized into the center of Mo atoms triangles. The more electrons excited, the more localized into the triangle vertices. The localization in the triangle vertices is viewed as the microscopic reason that exerts forces on Mo and Te atoms to soften the E" and A2" phonon modes and induces displacive phase transition.

The second law of thermodynamics shows that free energy is the criterion of spontaneous process, and here we calculate the Helmholz free energy to describe the thermodynamic stability of MoTe2 at 2H and 1T<sup>0</sup> phases [38]. As shown in **Figure 6**, at 0 K, the calculated free energy difference between the two phases of MoTe2 is only 33 meV per rectangular unit cell at 0 K, indicating that 2H phase and 1T<sup>0</sup> phase of MoTe2 have similar stability at this environment. Furthermore, with the temperature rising, 1T<sup>0</sup> MoTe2 becomes more thermodynamically stable than 2H MoTe2 at temperatures higher than 190 K as shown in green line in **Figure 6**. When taking thermal strain into account as shown in orange line in **Figure 6**, the temperature of phase transition decreases to 110 K.

#### **Figure 5.**

*(a) 3D and (b) top view of the electron localization function of 2H MoTe2 at different excitation energies. The isoenergy surface for the electron localization function in a is 0.2.*

**Figure 6.** *Helmholtz free energy difference between 2H and 1 T*<sup>0</sup> *MoTe2 monolayers as a function of temperature.*

### **4. Conclusions and perspective**

In this chapter, it is demonstrated that a PIPT from 2H to 1T<sup>0</sup> phase occurs in monolayer MoTe2 as the excitation energy of photon is higher than the critical value of 1.96 eV by using first-principles calculations. Such structural phase transition is intermediated by the charge distribution transition after photo-excitation, contributing to phonon mode softening and newly emergent potential energy minima. The structural phase transition can be induced under the photo-excitation of � 3.0% electrons per unit cell. Further, this work expresses the phonon frequencies as order parameter and fit it with the external condition indication, namely the photonexcitation energy, and the result fits excellent with the modified Cuire-Weiss law. This research reveals the microscopic nature of the 2H-1T<sup>0</sup> PIPT and facilitates the fundamental research of non-equilibrium transient phase transition between normal semiconductor and the topological phase. Also, this theoretical finding is expected to trigger the experimental investigations of PIPT materials in the 2D materials.

By now, stimulated by the advanced experimental methods such as fourdimensional ultrafast electron diffraction (UED), and the recently developed realtime time-dependent functional theory (rt-TDDFT) method, the studies on the PIPT in newly emergent systems such as three-dimensional VO2 and two-dimensional IrTe have been able to manifest the atomic motions induced by light at femtosecond scale. Furthermore, it has been shown by rt-TDDFT simulations that, by laser pulse excitation, a large part of electrons are excited from valence bands to conduction bands, which not only enlarges the electron occupation in antibonding states, leading to the enhancement of intrinsic Coulumb repulsion and thus enlarged interatomic bond length, but also forms the so-called electron-hole plasma. The dense plasma softens and stabilizes acoustic phonon modes, which drives the distortion of crystal lattice. Recent studies also reveal the competition between coherent light-induced nonthermal collective motions and thermally induced disorded motions, when PIPT takes place. However, there are two important problems still existing. One problem is the nature of PIPT in different kinds of materials. In this work, the microscopic theory based on the Landau phase-transition theory has been developed to explain the PIPT in two-dimensional materials, but whether it is applicable for one- or three*Photo-Induced Displacive Phase Transition in Two-dimensional MoTe2 from First… DOI: http://dx.doi.org/10.5772/intechopen.108460*

dimensional materials remains unclear. Moreover, the relation between different PIPT-based phases is still needed to clarify. Therefore, it is believed that, it is still challenging to develop a globally microscopic theory for PIPT in this field. The other problem is related to the intentional control over the PIPT-based phases. If the different hidden phases of materials can be induced by light, is it feasible to control by laser pulse to induce a special phase? More experimental and theoretical works are still needed to be conducted.
