A thermal conductivity switch via the reversible 2H-1T' phase transition in monolayer MoTe₂

The structures of 2H-MoTe₂ and 1T'-MoTe₂ exhibit ABA and ABC stackings, respectively. The Mo atom locates in a hexagonal and distorted octahedral coordinate in 2H-MoTe₂ and 1T'-MoTe₂, respectively, as shown in Figs. 1(a) and 1(b). Interestingly, recent research reported that the two phases could be converted into each other within a short time (∼ ps) via laser radiation.^[18] The lattice constants are a₁ = b₁ = 3.54 Å for 2H-MoTe₂, and a₂ = 3.45 Å, b₂ = 6.37 Å for 1T'-MoTe₂, agreeing with previous experimental results,^[8,17,27] as listed in Table 1. Then, the bond length was measured in the optimized unit cell. The 2H-MoTe₂ has a Mo–Te bonding length of 2.73 Å, while 1T'-MoTe₂ shows three Mo–Te bonding lengths of 2.75 Å, 2.79 Å, and 2.84 Å.

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To distinguish the differences in Mo–Te bonding between the two phases, we analyzed the overlap between Mo and Te atomic orbitals. It is known that the larger the atomic orbital overlap, the stronger the bonding.^[28] The overlap can be quantified by the covalency (C_covalency) between the atomic orbitals based on the electronic density of states (DOS).^[29] As shown in Figs. 2(a) and 2(b), the total electronic DOS of MoTe₂ is dominated by Mo-4d and Te-5p orbitals, suggesting that these two orbitals form the Mo–Te bonding. The covalency between Mo-4d and Te-5p is calculated using the following expression:^[30]

$$\begin{eqnarray*}&&{C}_{{\rm{covalency}}}=-|C{M}^{{\rm{Mo}}}-C{M}^{{\rm{Te}}}|,\end{eqnarray*}$$

in which CM^Mo is the band center of Mo-4d and CM^Te is the band center of Te-5p

$$\begin{eqnarray*}\begin{array}{lll} & & C{M}^{{\rm{Mo}}}=\displaystyle \frac{\displaystyle {\int }_{{\varepsilon }_{2}}^{{\varepsilon }_{1}}\varepsilon {g}_{{\rm{4d}}}^{{\rm{Mo}}}(\varepsilon){\rm{d}}\varepsilon }{\displaystyle {\int }_{{\varepsilon }_{2}}^{{\varepsilon }_{1}}{g}_{{\rm{4d}}}^{{\rm{Mo}}}(\varepsilon){\rm{d}}\varepsilon },C{M}^{{\rm{Te}}}=\displaystyle \frac{\displaystyle {\int }_{{\varepsilon }_{2}}^{{\varepsilon }_{1}}\varepsilon {g}_{{\rm{5p}}}^{{\rm{Te}}}(\varepsilon){\rm{d}}\varepsilon }{\displaystyle {\int }_{{\varepsilon }_{2}}^{{\varepsilon }_{1}}{g}_{{\rm{5p}}}^{{\rm{Te}}}(\varepsilon){\rm{d}}\varepsilon },\end{array}\end{eqnarray*}$$

where ${g}_{{\rm{4d}}}^{{\rm{Mo}}}(\varepsilon)$ and ${g}_{{\rm{5p}}}^{{\rm{Te}}}(\varepsilon)$ are the orbital projected DOS from Mo-4d and Te-5p atomic orbits, respectively.

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The covalencies of the 2H and 1T' phases are –2.15 and –3.36, respectively, indicating that the Mo–Te bonding in 2H-MoTe₂ is stiffer than that in 1T'-MoTe₂. This result is further verified by the electron localization function (ELF) as shown in Fig. 2(c). The 2H-MoTe₂ has stronger bonding, while 1T'-MoTe₂ shows three weaker types of bonds (bonding inhomogeneity). Consequently, the weaker bonding and bonding inhomogeneity in 1T'-MoTe₂ results in lower thermal conductivity, as shown in the following section.

The thermal conductivity is calculated based on rectangular cells, which are plotted in the gray wireframe in Fig. 1. The 1T'-MoTe₂ displays more significant anisotropy in thermal conductivity than 2H-MoTe₂ (see Fig. 3(a)). Impressively, 2H-MoTe₂ has a larger thermal conductivity along the Y direction than that along the X direction, whereas 1T'-MoTe₂ exhibits higher thermal conductivity along the X direction than that along the Y direction. As shown in Fig. 3(b), the thermal conductivities of the 2H (1T') phase along the X and Y directions are 43.26 (11.98) W/(mK) and 45.84 (4.95) W/(mK) at 300 K, respectively. For the 2H phase, these calculated values are consistent with previous reports.^[31] The thermal conductivity switching ratios along the X and Y directions are 3.61 and 9.26 in MoTe₂, respectively. Such switching ratios are comparable to, or even higher than, those of the previously reported thermal modulation ratios of MoS₂ (0.8),^[15] graphene (2.7),^[32] 2D ferromagnets (1.31–5.73),^[33] and polymers (8–10),^[34] etc.^[35,36] Recent research reported that the two phases could be converted into each other within a short time (∼ ps) via laser radiation,^[18] gate voltage,^[8] and strain.^[17] Moreover, the thermal conductivity of 2D materials was measured in many ways, such as via the time-domain heat reflectance method and the laser flash method.^[37] Monolayer MoTe₂ therefore can be used for high-performance thermal switching devices in practice.

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Meanwhile, the contributions of acoustic and optical phonon branches to thermal conductivity are presented in Fig. 3(b). More than 75% of the contribution to thermal conductivity originates from acoustic branches. More convincing evidence can be seen from the cumulative thermal conductivity with frequency, in which phonons at frequencies below 3 THz, i.e., acoustic phonons, mainly contribute to the accumulation of thermal conductivity (see Fig. S3). To further explore the origin of different thermal conductivity between the 2H and 1T' phases, we next explore the effect of phase transition on the harmonic and anharmonic properties of phonons.

Figures 4(a) and 4(b) display phonon dispersions of the 2H and 1T' phases, respectively. The unit cell of the 2H (1T') phase has 3 (6) atoms and, accordingly, 9 (18) phonon branches. There are no imaginary frequencies, suggesting dynamical stability. Compared with 2H-MoTe₂, 1T'-MoTe₂ has a negligible phononic gap between acoustic and optical branches, which enhances the phonon scattering process. Based on the phonon eigenvectors, the phonon vibrational modes of the 2H and 1T' phases at the G-point are illustrated in Figs. S4(a) and S4(b), respectively. The phonon vibrational modes of 1T'-MoTe₂ exhibit more directions and weaker strength than those of 2H-MoTe₂, showing that the 2H to 1T' phase transition induces more scattering and overall softening of vibration modes. Consequently, 1T'-MoTe₂ has reduced phonon propagation speed and lower thermal conductivity compared with 2H-MoTe₂.

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Figures 4(e) and 4(f) shows that compared with 1T'-MoTe₂, 2H-MoTe₂ has higher group velocities of ZA and LA phonons, and lower group velocities of TA phonons. The optical phonon group velocity of 1T'-MoTe₂ is larger than that of 2H-MoTe₂. Overall, the acoustic phonons of 2H-MoTe₂ bear larger group velocities compared with those of 1T'-MoTe₂. Therefore, the reduced group velocity with the 2H to 1T' phase transition is responsible for the lower thermal conductivity in MoTe₂.

Phonon relaxation time is an important factor which governs thermal conductivity.^[5,38] Figure 5(a) shows that 1T'-MoTe₂ shares smaller phonon relaxation time than 2H-MoTe₂ over the whole frequency range, which implies that the phonon scattering rate can be reduced by the phase transition from the 2H to the 1T' phase. This result agrees with our aforementioned analysis of the phonon dispersion. Phonon anharmonicity is important to phonon propagation,^[39,40] which is evaluated using the Grüneisen parameter (γ) in this study. The γ is calculated using $\gamma (q)=-\displaystyle \frac{V}{\omega (q)}\displaystyle \frac{\partial \omega (q)}{\partial V}$.^[41] As presented in Fig. 5(b), the values of γ for ZA phonons are negative, and those for TA, LA and optical phonons are positive. The main difference in γ between the two phases is the γ of ZA phonons, and the 1T' phase has a larger γ of ZA than the 2H phase. Therefore, there is a stronger anharmonicity in 1T'-MoTe₂, further confirming the lower thermal conductivity.

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Finally, we found that the anisotropy in thermal conductivity is attributed to unevenly distributed Mo and Te atoms in the geometric configurations of the rectangular cell. To clearly describe the difference in anisotropic behavior between 2H-MoTe₂ and 1T'-MoTe₂, we calculated the Grüneisen parameter γ_q,s for the s th mode using ${\gamma }_{q,s}=-\displaystyle \frac{a}{2{\omega }_{s}({\boldsymbol{q}})}\cdot \displaystyle \frac{{\rm{d}}{\omega }_{s}({\boldsymbol{q}})}{{\rm{d}}a}$ along the G–Y (X) and G–X (Y) directions (a is the lattice parameter). As shown in Fig. S5, the absolute values of γ_q,s for the ZA, TA, and LA phonon modes of 1T'-MoTe₂ along the Y direction are larger than that of the X direction, as that of 2H-MoTe₂ along the X direction is slightly greater than that of the Y direction, which supports our predicted thermal conductivity anisotropy.

The phonon vibrational modes of a material are closely related to its mechanical properties, and the strength of a mechanical property is measured by Young's modulus.^[43] For example, compared with Y₂Zr₂O₇, Y₃NbO₇ has a lower Young's modulus, which indicates a softer structure and softening of the vibration modes.^[44] Therefore, to provide more insights into the deferent phonon vibrational modes between the two phases, we calculated the Young's modulus based on the elastic modulus tensor.

The stiffness tensor (C_ij , i,j = 1, 2, 6) is the second partial derivative of the strain energy (E_s) as a function of strain (ε):^[45,46]

$$\begin{eqnarray*}&&{E}_{{\rm{s}}}=\displaystyle \frac{1}{2}{C}_{11}{\varepsilon }_{xx}^{2}+\displaystyle \frac{1}{2}{C}_{22}{\varepsilon }_{yy}^{2}+{C}_{12}{\varepsilon }_{xx}{\varepsilon }_{yy}+2{C}_{66}{\varepsilon }_{xy}^{2},\end{eqnarray*}$$

with ε = (a – a₀)/a₀, where a (a₀) is the strained (unstrained) lattice constant. As listed in Table 2, the elastic constants of 2H-MoTe₂ are C₁₁ = 83.99 N/m, C₂₂ = 84.02 N/m, C₁₂ = 21.43 N/m, and C₆₆ = 31.15 N/m, and those of 1T'-MoTe₂ are C₁₁ = 61.57 N/m, C₂₂ = 87.77 N/m, C₁₂ = 39.66 N/m, and C₆₆ = 31.42 N/m, satisfying the Born criteria ${C}_{11}{C}_{22}-{C}_{12}^{2}\gt 0$ and C₆₆ > 0.^[47,48] The structures of both phases therefore display mechanical stability.

Then, the Young's modulus (E) and orientation-dependent Young's modulus (Y) are derived as^[49]

$$\begin{eqnarray*}\begin{array}{lll} & & {E}_{x}=\displaystyle \frac{{C}_{11}{C}_{22}-{C}_{12}{C}_{21}}{{C}_{22}},{E}_{y}=\displaystyle \frac{{C}_{11}{C}_{22}-{C}_{12}{C}_{21}}{{C}_{11}},\\ & & Y(\theta)=\displaystyle \frac{{C}_{11}{C}_{22}-{C}_{12}^{2}}{{C}_{11}{s}^{4}+{C}_{22}{c}^{4}+\left(\displaystyle \frac{({C}_{11}{C}_{22}-{C}_{12}^{2})}{{C}_{66}}-2{C}_{12}\right){s}^{2}{c}^{2}},\end{array}\end{eqnarray*}$$

where s = sin θ and c = cos θ. The Young's modulus value of 2H-MoTe₂ in the X (Y) direction is 88.52 N/m (88.56 N/m), higher than that of 1T'-MoTe₂ (62.22 N/m for the X direction and 43.65 N/m for the Y direction). We find that the stiffness of 1T'-MoTe₂ is smaller, confirming the softening of the vibration modes. As shown in Fig. 6, 1T'-MoTe₂ exhibits an obvious anisotropic Young's modulus, in which the value in the X direction is much larger than that in the Y direction. On the other hand, 2H-MoTe₂ shows little anisotropic Young's modulus, where the value of the Young's modulus in the Y direction is slightly smaller than the value in the X direction. The anisotropy in the Young's modulus complies with the anisotropy of thermal conductivity calculated above.

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According to the stiffness tensor, the Poisson's ratio (v) and orientation-dependent Poisson's ratio are computed as follows:^[49,50]

$$\begin{eqnarray*}\begin{array}{lll} & & {v}_{xy}=\displaystyle \frac{{C}_{21}}{{C}_{22}},{v}_{yx}=\displaystyle \frac{{C}_{12}}{{C}_{11}},\\ & & v(\theta)=\displaystyle \frac{{C}_{12}({s}^{4}+{c}^{4})-\left({C}_{11}+{C}_{22}-\displaystyle \frac{({C}_{11}{C}_{22}-{C}_{12}^{2})}{{C}_{66}}\right){s}^{2}{c}^{2}}{{C}_{11}{s}^{4}+{C}_{22}{c}^{4}+\left(\displaystyle \frac{({C}_{11}{C}_{22}-{C}_{12}^{2})}{{C}_{66}}-2{C}_{12}\right){s}^{2}{c}^{2}}.\end{array}\end{eqnarray*}$$

Both the Poisson's ratios of 2H-MoTe₂ along the X and Y directions are 0.26, less than those of 1T'-MoTe₂ (0.45/0.64 for the X/Y direction). These values are greater than that of graphene (0.18), implying that 2H-MoTe₂ and 1T'-MoTe₂ are more flexible than graphene, and ensuring that the two phases can be transformed into each other. It is obvious from Fig. 6 that the values of v vary together with the change of direction, which also demonstrates the anisotropic properties.